Carnegie Mellon University
Research Showcase @ CMU
Computer Science Department
School of Computer Science
1983
The theory of straight homogeneous generalized
cylinders and A taxonomy of generalized cylinders
Steven A. Shafer
Carnegie Mellon University
Takeo Kanade
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The Theory of Straight Homogeneous
Generalized Cylinders
and
A Taxonomy of Generalized Cylinders
18 January 1983
Steven A. Shafer
Takeo K a n a d e
Computer
Science
Carnegie-Mellon
Department
University
This research was s p o n s o r e d by the Defense A d v a n c e d Research Projects Agency (DOD), ARPA
Order No. 3597, monitored by t h e Air Force Avionics Laboratory Under Contract F33615-81 -K-1539.
T h e views and c o n c l u s i o n s c o n t a i n e d in this d o c u m e n t are those of the authors and s h o u l d not be
interpreted as representing the official policies, either expressed or implied, of t h e Defense Advanced
Research Projects A g e n c y or the US Government.
The Theory of Straight
Homogeneous
Generalized Cylinders
15 January 1983
Steven A. Shafer
Takeo Kanade
Computer
Science
Department
Carnegie-Mellon
University
Abstract
In recent years, Binford's
scheme
in computer
by a lack of analytical
generalized cylinders have become
vision.
However,
is presented
Generalized
Cylinders,
coordinates
are not curved.
Generalized
Cylinders,
with
concerning
Then, projections,
some subclasses
name
Right
a linear
axis,
The bulk
are
formulae
contour
SHGCs),
generalized
used shape
cylinders
a lack of a precise
Cylinders
important
of the paper
because
is concerned
have constant
with
Straight
for which
results
several
of these
shapes.
Straight
object-centered
Homogeneous
for these shapes.
lines, and surface
are obtained
closed-form
hampered
shape but vary in size.
descriptions
The strongest
definition
natural
under which multiple
shadow
has been
the
normals
generators,
representation
and for several subclasses.
for points and surface
the conditions
of shapes.
Circular
for Generalized
in which the cross-sections
The results begin with deriving
shape.
involving
results at all levels, even including
In this paper, a definition
are presented
research
a commonly
can exist for a single
normals
are analyzed
for solids of revolution
methods
Theorems
for analyzing
(which
images
solid
for
we
are
presented.
This research was sponsored by t h e Defense Advanced Research Projects Agency (DOD), ARPA
Order No. 3597, monitored by the Air Force Avionics Laboratory Under Contract F33615-81 -K-1539.
The views and conclusions contained in this d o c u m e n t are those of the authors and should not be
interpreted as representing the official policies, either expressed or implied, of the Defense Advanced
Research Projects Agency or the US Government.
T a b l e of Contents
1 . Introduction
1.1 Importance of Generalized Cylinders
1.2 This Presentation
2. C l a s s e s of S h a p e s
2.1 Generalized Cylinders
2.2 Straight Generalized Cylinders
2.3 Straight H o m o g e n e o u s Generalized Cylinders
2.4 Subclasses of SHGC
3 . Fundamental T h e o r e m s and P r o b l e m s
3.1 Coordinates for SHGCs
3.1.1 Points on the Surface
3.2 Descriptions and Shapes
3.3 T h e Equivalent Right SHGC Problem
3.3.1 The Slant Theorem
3.4 T h e Alternate Axis Problem
3.4.1 The Pivot Theorem
3.5 Surface Normals of an SHGC
3.6 The Corresponding Normal Theorem
4 . P r o j e c t i o n s of SHGCs
4.1 Projected Contour Generators
4.1.1 Planarity of Contour Generators
4.2 Images of Right SHGCs
4.2.1 Contour Generators Under Perspective Projection
4.3 Projection and Imaging for Right Circular SHGCs
4.3.1 Occlusions and Singularities in Image Contours
4.3.2 Cross-Sections Described by Fourier Coefficients
4.4 Contour and Silhouette Analysis for Right Circular SHGCs
4.4.1 Contour Analysis
4.4.2 O c c l u d e d Contours and Silhouettes
4.4.3 Aligning the Image
4.5 Contour and Silhouette Analysis for Other RSHGCs
4.5.1 Contours of Right Linear SHGCs
4.5.2 Contours of Right Polygonal SHGCs
4.6 Contour Generators in Two Views
4.7 Analysis of Surface Normals
4.7.1 Individual Surface Normals of an RCSHGC
4.7.2 Aligning the Image for Analysis of Surface Normals
4.7.3 Cross-Sections For a Given Surface Normal
4.7.4 S h a d o w Geometry for RCSHGCs
4.7.5 Range Data Analysis for RCSHGCs
4.7.5.1 Method of Minimizing Derivatives
4.7.5.2 M e t h o d of Interpolating Normals
5. Conclusions
5.1 Future Work
5.2 A c k n o w l e d g e m e n t s
6 . Bibliography
I. S y m b o l s
II. P r o p e r t i e s of SHGCs
11.1 Straight H o m o g e n e o u s GC: SHGC
11.2 Right SHGC: RSHGC
11.3 Linear SHGC: LSHGC
11.4 Circular SHGC: CSHGC
11.5 Polygonal SHGC: PSHGC
iii
List of Figures
F i g u i re 1 :
Concept of a Generalized Cylinder
1
F i g u i re 2 :
Examples of Generalized Cylinders
2
F i g u i re 3 :
Imaging and S h a d o w Volumes are Generalized Cylinders
2
F i g u i re 4 :
Generalized Cylinder
4
F i g u i re 5 :
Straight Generalized Cylinder
6
F i g u i re 6 :
Straight H o m o g e n e o u s Generalized Cylinder
6
F i g u re 7 :
Linear SHGC
7
F i g u re 8 :
Circular SHGC
7
F i g u i re 9 :
Polygonal SHGC
8
9
F i g u re 1 0 :
Coordinate Axes for SHGCs
F i g u re 1 1 :
Coordinates of a Point on an SHGC
10
F i g u re 1 2 :
T h e Equivalent Right SHGC Problem
11
F i g u re 1 3 :
T h e Slant T h e o r e m
14
F i g u re 1 4 :
T h e Alternate Axis Problem
16
F i g u re 1 5 :
T h e Pivot T h e o r e m
18
F i g u re 1 6 :
Surface Normal of an S H G C
20
F i g u re 1 7 :
The C o r r e s p o n d i n g Normal T h e o r e m
21
F i g u re 1 8 :
Surface Normals of the S h a d o w Volume
23
F i g u re 1 9 :
C o n t o u r s and Contour Generators
24
F i g u re 2 0 :
Viewing Direction and Angle
25
F i g u re 2 1 :
End, Side, and O b l i q u e Views
26
F i g u re 2 2 :
Object and W o r l d Coordinate Systems
28
F i g u re 2 3 :
Image of a Point on a Right SHGC
29
F i g u re 2 4 :
Imaging Under Perspective Projection
29
F i g u re 2 5 :
C o n t o u r Generator on a Right Circular SHGC
31
F i g u re 2 6 :
C o n t o u r Generator in Side View
33
F i g u re 2 7 :
C o n t o u r Generator in Near-Side View
34
F i g u re 2 8 :
C o n t o u r Generator Singular W h e n Tangent to Line of Sight
35
F i g u re 2 9 :
C o n t o u r Generator Split at Near-End View
36
F i g u re 3 0 :
C o n t o u r Pieces Correspond to Disjoint Intervals of s
36
F i g u re 3 1 :
Aligned Image of an SHGC
38
F i g u re 3 2 :
C o n t o u r Analysis Results in Shape Description
39
F i g u re 3 3 :
RCSHGC With Near End Flat
41
F i g u re 3 4 :
Tangents at C o r r e s p o n d i n g Points Intersect on the Axis
43
F i g u re 3 5 :
C o n t o u r Generators Need Not Be At Vertical Extrema
44
F i g u re 3 6 :
Crease Contours for a Right Polygonal SHGC
44
F i g u re 3 7 :
Tangents of Crease C o n t o u r s Intersect on the Axis
45
F i g u re 3 8 :
C o n t o u r Generators From T w o Points of View
47
IV
Figure 39:
S h a d o w Line is a C o n t o u r Generator for t h e Light S o u r c e
47
Figure 40:
Two Angles Define t h e Illumination Direction
48
Figure 4 1 :
S h a d o w Line on a Right Circular SHGC
49
Figure 42:
K n o w l e d g e of a Single Surface Normal
50
Figure 43:
Analyzing a Single S u r f a c e Normal
52
Figure 44:
Complete and Partial Alignment of the Image
53
Figure 45:
S o u r c e s of K n o w l e d g e A b o u t S u r f a c e Normals
53
Figure 46:
Cross-Section of a Right Circular SHGC T h r o u g h a Point
54
Figure 47:
S h a d o w Geometry Provides Surface Normals Along S h a d o w Line
56
Figure 48:
C o r r e s p o n d i n g Point on T a n g e n c y C o n t o u r Generator
57
Figure 49:
At Most O n e C o r r e s p o n d i n g Point Exists
57
Figure 50:
Range Data Yields an Image of Surface Normals
58
Figure 5 1 :
T w o S u r f a c e Normals on the Same Cross-Section
59
Figure 52:
T w o Surface Normals on Different Cross-Sections
59
Figure 53:
Only C o m p a r e S u r f a c e Normals with Similar Values of x
61
Figure 54:
Fitting Ellipses to Interpolated Surface Normals
62
1
1 . Introduction
In recent years, t h e generalized
cylinders
proposed by Binford [2] have become an increasingly
important tool for shape description in image understanding. However, generalized cylinders have in
the past been defined primarily in informal terms, leading to several different definitions for
"generalized cylinder" (or "generalized c o n e " ) and resulting in a lack of rigorous geometric analysis
of these shapes.
In this paper, w e present a formal definition of generalized cylinders and several
interesting subclasses. A number of theorems and results are then derived, including formulas for the
coordinates of the constituent points and surface normals, and several results relevant to projections
(i.e. images) of generalized cylinders.
1.1 Importance of Generalized Cylinders
F i g u re 1 : C o n c e p t of a Generalized Cylinder
A generalized
cylinder
is loosely characterized by having an axis (a space curve w h i c h forms a
spine for the shape), a cross-section
(a 2-D c o n t o u r w h i c h sweeps along the axis), and a
sweeping
rule <a rule for transforming the cross-section as it is swept along the axis) (figure 1) [2].
Generalized cylinders are an important class of shapes for several reasons.
They are primarily
important because of the wide variety of man-made and natural objects w h i c h can be represented as
generalized cylinders. For example, many objects w h i c h stand up have a vertical axis (counteracting
gravity) and a horizontal cross-section w h i c h c h a n g e s only slightly (if at all) as it follows the axis: a
vase, a lamp, a flower stem, a skyscraper, a c u p , a blade of grass, a box (figure 2). For many of these
shapes, representation as a polyhedron requires defining many identical facets, w h i c h fails to take
into a c c o u n t the simplicity of the basic structure involved.
In addition, machining processes may
F i g u re 2 :
Examples of Generalized Cylinders
p r o d u c e shapes w h i c h are r o u n d e d generalized cylinders, for w h i c h polyhedral or surface patch
approximations introduce undesirable edges and corners. Generalized cylinders have been used for
describing pottery types for anthropology [6] and for modelling biological cell shapes [18].
eye
F i g u re 3 :
Imaging and S h a d o w Volumes are Generalized Cylinders
Generalized cylinders are important in c o m p u t e r image processing because of the fact that a
region of an image c o r r e s p o n d s to s o m e volume in space, defined by the imaging projection
(orthography or perspective). The volume so defined is a generalized cylinder, with axis along the
optical axis of the eye or camera (figure 3).
Similarly, w h e n an object casts a shadow, there is a
volume of space behind the object w h i c h will be shaded from the light s o u r c e .
generalized cylinder.
This volume is a
Thus, the problem of determining the shape of a s h a d o w is equivalent to the
geometric problem of determining the intersection of a generalized cylinder with another object (such
as a planar surface).
3
Generalized cylinders also have several properties w h i c h render them appropriate for geometric
modeling tasks, as pointed out by Marr and Nishihara [12]:
generalized cylinders are
volumetric,
w h i c h means they are characterized as o c c u p y i n g a volume of space rather than being a collection of
flat or curved patches; they have object-centered
coordinates,
w h i c h allow for rotations etc. for model
instantiation or to represent motion of limbs; they have a principal
axis w h i c h itself can be used as a
gross approximation to the volume represented; and they can be organized
hierarchically
to
represent coarse and fine details.
1.2 This Presentation
In this paper, w e begin by defining several classes of solid shapes.
Various definitions of
"generalized cylinder" or "generalized c o n e " in the literature are identified as corresponding to
certain of these classes.
A particular class, called Straight
Generalized
Cylinders
(SGC),
is
fundamental since the natural object-centered coordinate system has no curvature in the axes, and
hence is a linear transformation from world space coordinates.
Homogeneous
Generalized
Cylinders
A subset of SGC,
Straight
(SHGC), contains those SGCs w h o s e cross-sections have the
same shape but may vary in size; this allows a d e c o m p o s i t i o n of the shape description into size and
shape functions, u p o n w h i c h this work is based.
Several subclasses of SHGC are defined, with
particular properties w h i c h allow stronger statements to be made about them in various situations.
The remainder
of the paper consists of the elaboration of
important
properties of
Straight
Homogeneous Generalized Cylinders and of these subclasses.
The next section presents the most important formulae: the coordinates of points on an SHGC, and
the surface normal vector at each point.
Two important problems are identified: the existence of
different representations for the same solid shape, and the constancy of surface normals in certain
locations.
Theorems are proven in each case, showing that t h e Linear
SHGCs
(LSHGC)
(SHGCs
w h i c h expand the c o n t o u r by a scaling factor proportional to distance along the axis) are a very
important class by themselves. Since the imaging volumes and shadow volumes (discussed above)
are LSHGCs, these results may be especially useful.
The c o n s e q u e n c e s of projection of SHGCs (i.e. imaging) are then explored.
This includes a
general formulation of the imaging geometry, followed by an analysis of the problem of mapping
image points to points on the SHGC. The analysis of silhouettes of SHGCs is explored, including the
difficult problem of determing the contour
generators
-- points on the SHGC w h i c h are imaged on the
outline of the silhouette [11]. Next, the problem is addressed of interpreting the contour generators
from one point of view as seen from another: this arises in stereo and shadow geometry analysis. The
use of range-finder data for description of SHGCs is also discussed.
4
2. Classes of Shapes
W e will begin by defining some classes of solid shapes.
We begin with Generalized
Cylinders,
w h i c h introduce the most basic ideas in the definition, then specialize this to Straight
Cylinders,
Straight
Homogeneous
Generalized
Cylinders,
and several subclasses
Homogeneous Generalized Cylinders w h i c h w e call Linear, Right,
Circular,
Generalized
of
and Polygonal
Straight
SHGCs.
T h e names of these classes will be capitalized w h e n used to refer to their formal definitions.
This
terminology c o r r e s p o n d s to that of the authors in related work [16]. A summary of the symbols used
will be found in an Appendix.
2.1 Generalized Cylinders
V
Figure 4:
A Generalized
Cylinder
Generalized Cylinder
(GO, as s h o w n in figure 4, is a function w h i c h maps two parameters onto a
set of points in x-y-z space (i.e. the world). T h e two parameters are s, w h i c h measures distance along
the axis, and t, w h i c h indirectly measures distance along the cross-section contour; both s and t have
as domain the unit interval [0,1]. This development is similar to that of Ballard and Brown [1].
A Generalized Cylinder is specified by a three-tuple (A, E, a). A is the axis, w h i c h is a curve in
space defined in parametric form by A{s) = ( x , y , z ) (s).
A
A
A
The remainder of this discussion will desribe features of the shape relative to the axis itself rather
than in absolute x-y-z coordinates.
At each point A(s) on the axis, let the cross-section be described on a u-v plane, with A(s) at the
5
origin, and defined by the (constant) angle a. T h e ty-axis will be the direction of steepest descent of
the u-v plane from the tangent to the axis, a, the angle of inclination,
is the angle from the a-axis to
the tangent to the axis at A(s); a = 0 means that the a-axis is pointing t o w a r d s A(1), and a = <n
means that the u-axis is pointing t o w a r d s A(0).
On each u-v plane, the cross-section c o n t o u r is defined by the envelope
function
E(s,t) = (u ,
E
V)
E
(s,f). On t h e u-v cross-section plane for each value of s, the cross-section c o n t o u r is the set of points
E(s,t) for values of t from 0 to 1, inclusive. T h e c o n t o u r is normally expected to be closed,
i.e. E{s,0)
= E(s,1). The union of the c o n t o u r s is the Generalized Cylinder.
A c c o r d i n g to this strict definition, s o m e very peculiar shapes are admitted in GC, including those
with intersecting c o n t o u r s on different cross sections, those with singular points or arcs, and those
with cross-section c o n t o u r s w h i c h are open arcs, points, or even space-filling curves.
Since our
ultimate goal is the analysis of the shapes of c o m m o n objects, w e will generally e x c l u d e s u c h bizarre
cases from further consideration. However, since shapes with degenerate cross-sections (open arcs
or points) d o have several important properties, w e will note them w h e n appropriate.
The class GC includes very many shapes: all for w h i c h there exists a space curve (axis) s u c h that
each cross-section of the object on s o m e set of planes can be defined by a single c o n n e c t e d
component.
Binford [2] defines "generalized cylinder" to be a superset of GC, allowing rotational
sweeps and non-planar cross-sections. Since this definition includes s u c h strange shapes as those
described above, it is likely that some assumptions of normalcy were intended, t h o u g h not explicitly
stated, by Binford. Nevatia [13] has defined "generalized c o n e s " to be GC, with the condition that a
be equal to TT/2, i.e. the cross-section planes are orthogonal to t h e axis.
2 . 2 Straight Generalized Cylinders
Interesting subclasses of GC arise from imposing suitable restrictions on the various c o m p o n e n t
functions. The most important of these is the set of Straight
Generalized
Cylinders
(SGC), in w h i c h
the axis A is linear (figure 5). The axis is t h u s a line segment, and all u-v planes are parallel.
This class is important because all tangents to A(s) are parallel, as are all u-axes and all v-axes. W e
can therefore define vectors S, U, and V pointing in these directions, and assign a local (objectcentered) coordinate system using u-v-s coordinates. S u c h coordinates can of c o u r s e be defined for
all Generalized Cylinders; however, there will be no curvature in the coordinate axes for SGCs. The
local coordinates of an SGC are a linear transformation of world (x-y-z) coordinates.
6
F i g u re 5 :
Straight Generalized Cylinder
2.3 Straight Homogeneous Generalized Cylinders
F i g u r e 6 : Straight H o m o g e n e o u s Generalized Cylinder
W e define a Straight
Homogeneous
Generalized
Cylinder
(SHGC) (figure 6) to be an SGC in w h i c h
the envelope E c a n be d e c o m p o s e d into two functions by E(s,f) = r(s)C(0. T h e contour
= (u ,v )
c
Q
function
(t) describes the shape of the cross-section; the radius function r(s) describes its size.
C(t)
So,
the cross-section has a constant shape but may vary in size. An SHGC is specified by a four-tuple ( A
C, r, a).
We impose the restriction that the functions A and r be c o n t i n u o u s and differentiable everywhere,
and that the c o n t o u r C be c o n t i n u o u s and differentiable almost everywhere.
required, that the u-v origin be in the interior of the contour.
s c a l i n g " of s and f, i.e. | | d / \ / d s | | and ||dC/df|| are constants.
It is usual, but not
In addition, we will presume " u n i f o r m
7
This definition of generalized cylinder is essentially the same as Marr's "generalized c o n e " [11].
The bulk of this paper describes the properties of SHGCs.
2.4 Subclasses of SHGC
Additional restrictions on t h e various functions give rise to several subclasses of SHGC with
particular interesting properties:
Figure 8:
Circular SHGC
Linear SHGC (LSHGC) - SHGC with r linear (figure 7)
T h e size of the c o n t o u r varies linearly with distance along the axis. r(s) can be
written as r(s) = m ( s - s ) for s o m e values of m and s .
Q
LSHGCs: r(s )
Q
Q
Important relations for
= O a n d d r / d s = m. In the special case that r is constant, r(s) = r
for some value r .
Q
In this case, d r / d s = 0. LSHGCs are ruled surfaces
being Generalized Cylinders [5].
Q
as well as
8
Figure 9:
Right SHGC (RSHGC)
Polygonal SHGC
SHGC with a = TT/2
The u-v planes are normal t o the axis. There is no " d i r e c t i o n of steepest d e s c e n t "
relative to the axis, so the u-axis may be c h o s e n in any direction on the crosssection planes.
Circular
SHGC (CSHGC)
SHGC with C a circle centered at the origin (figure 8)
Without loss of generality, let C be a unit circle, C(f) = (u
Qi
2irt).
v ) (t) = (cos 2mt, sin
Q
All surfaces of solids of revolution are Right Circular SHGCs (but with o p e n
ends unless r(0) = 0 or r(1) = 0 ) .
Polygonal
SHGC (PSHGC) - SHGC with C polygonal (piecewise linear) (figure 9)
If C{t ) is a vertex for some t
Q
then the set of points P{sJ )
Qi
Q
is a crease
(ridge if C
convex there, valley if concave). Otherwise, P(s,f) is on a face; note that faces are
not necessarily planar in this definition.
On a face, C(t) is linear, s o it c a n b e
expressed as:
C(f) = ( u ( 0 , v (t))
c
Q
= (mj
for s o m e m , £> , m , and b .
u
u
vertices C(tJ
m
v
and C(t )
v
u
=
0
L
f -f
= (u ,v ),
2
u -u.
and
2
b
u
v
u
v
bj
f
v
then
2
\ M
=
with similar definitions for n? and 5 .
(/77 ,m ).
+
u
If the c o r r e s p o n d i n g s e g m e n t of C is b o u n d e d by
y
= (u vj
+ b , mj
0
- t
t u.
~
0
a
L
In addition, d C / d f
= (dt; /df,d\/ /dO
c
c
=
By t h e uniform scaling assumptions, t measures distance a r o u n d t h e
polygon.
In various situations, the c o n s e q u e n c e s of these properties will be s h o w n to be of special interest.
9
3. Fundamental Theorems and Problems
In this section, the formulae for the coordinates of a point and the direction of a surface normal for
an SHGC are presented. These give rise to some important problems w h i c h are explored, and several
relevant theorems are presented. These formulae and theorems provide the basis for the imaging
discussion in the next section.
3.1 Coordinates forSHGCs
For any SHGC, there is a natural u-v-s
preceding definitions.
object-centered coordinate system imposed by the
We will adopt t h e convention that the v-axis is c h o s e n to provide a right-
handed u-v-s c o o r d i n a t e system. T h e unit vectors in the axis directions will be denoted U, V, and S,
as s h o w n in figure 10.
F i g u re 1 0 : Coordinate Axes for SHGCs
However, l / _ L S o n l y if a = t t / 2 , i.e. in a Right SHGC. Therefore, it will be convenient to define an
orthogonal w-v-s coordinate system using W perpendicular to V a n d S. W lies in the US plane, with
an angle of m/2 - a between M/and U. For any point (u, v, s )
u v s
(where
u v s
denotes coordinates in
the u-v-s system), the corresponding coordinates in w-v-s are (u sin a, v, s + u cos
CL)^ S
The w-v-s
coordinates are important since the axes are independent of the c o m p o n e n t s of any particular
Generalized Cylinder. In an RSHGC, u-v-s and w-v-s coordinates are identical.
Except w h e r e otherwise stated, all coordinates in this paper will be given in the w-v-s system.
10
3.1.1 Points on the S u r f a c e
F i g u re 1 1 : Coordinates of a Point on an SHGC
For any values s and f, the point P(s, t) on the surface of the SHGC has u-v-s coordinates (figure 11)
P(s,f) = ( u ( 0 r(s), v (t) r(s), s )
c
c
y v s
and hence w-v-s coordinates
P(s,f) = (u (t)
Q
r(s) sin a, v (t) r(s), s + u (t)
Q
Q
r{s) cos a)
(3-1)
3.2 Descriptions and Shapes
A subtle problem arises from our definition of SHGCs (and other s h a p e classes):
defined t h e m , an SHGC is actually a description
itself.
as w e have
of a shape rather than being a specific solid s h a p e
Of course, each such description describes a unique shape; however, w e must attempt t o
decide when a single shape may have several different descriptions. Since a solid shape c o r r e s p o n d s
to an equivalence class of descriptions (i.e. SHGCs), w e will call two descriptions equivalent
when
they describe the same solid shape, as d i d Marr and Nishihara in [12].
There are four trivial changes possible in the s and t coordinates themselves while preserving
equivalence:
• the axis can be flipped end-over-end to yield a new SHGC (reversing the sense of the s
coordinate)
• the sense of t can be likewise reversed, a n d , if the c o n t o u r C(t) is closed, the point at
w h i c h t = 0 can be shifted to a n y w h e r e on the c u r v e
• the radius function r(s) can be multiplied by any constant scale factor, while the cross
section contour C(t) is divided by the same factor
11
• an RSHGC can have the u-v axes rotated about t h e origin arbitrarily (shifting the t
coordinate).
These transformations are sufficiently simple that no deeper discussion is needed.
There are, however, more significant variations in the possible descriptions of a specific shape as
an SHGC. We will investigate two of the principal types of variation: altering the orientation of the
cross-section planes, and altering the direction of the axis.
3.3 The Equivalent Right SHGC Problem
F i g u r e 1 2 : T h e Equivalent Right SHGC Problem
In figure 12, w e see a shape described as two different SHGCs, with cross-section planes at
different orientations.
What properties of the shape make this possible?
Since this question is so
general, w e will limit our attention to a more restricted (but still difficult) question: For what SHGCs
are there equivalent Right SHGCs?
This is interesting since the RSHGC seems to be a natural
" c a n o n i c a l " form of representation for a shape. We will ignore the effect of " b e v e l e d " ends resulting
from values of a not equal to 7 r / 2 .
12
To make this problem somewhat more tractable, w e will presume that the same axis A and radius
function r are to be used for the SHGC and RSHGC. (We conjecture, but have not proven, that this
presumption implies no loss of generality.) T h e problem can then be stated this way: Given an SHGC
G
1
= (A, C
G
2
= (A, C , r, 7r/2) contains the same points as G^>
v
r, a), with C , = (u vj
v
can some function C
%
2
= {u ,v )
2
2
be f o u n d such that the RSHGC
2
Now for each s and 1 , w e must have
1
1
PfiytJ
= {u^tjrisjsin
for s o m e s , with P on G
2
1
v
a, v^(tjr{sj,
s
1
+ u^tj
r(sj
cos a)
P on G , and with C ( f ) c o r r e s p o n d i n g to C , ^ ) for all t
2
2
2
1
r
Equating s-coordinates,
s
2
=
S
1
+
u (tJ ($Jcos
a
r
A
Equating ^-coordinates,
=
v,(tjr(sj
v {tjr[sj
2
so
Now, since
v {tj
9s
2
w e must have
_
9
r(s
+
1
u^Jffc^cosa)
9s
r^)
i.e. the ratio between r(s ) and /-(s^ is independent of s. So,
2
0 =
2
V^J —
r ( s
i
+
w J f J r t s ) c o s a ) - r(s
W e can differentiate this by f, using the fact that
v t / ( f ) r ( s j cos
a ) — J
13
3
3
dr .
dr
I
"97"d7' . Ht "d7'
s2 =
du
.
|
A
= r(s,)cosa—
s1+u1(t1)r(s1)cosa
d r •
-^2-|
du
2
t 1
s 1 + u 1 ( t 1 ) r ( s 1 ) c o s a
|
1
d r •
2
= ^ o s a - ^ ^ - l ^
to obtain the c o n d i t i o n :
du. , d r ,
_ |
_ |
2
2
0 , ^ ,
c
o
s
a
t
i
s
dt/ ,
- r ( s j cos a — H
df
M
, t 1
du
2
,
r(Si)
cos a
_|
1
I
t i
+
r
(
S
i
)
c
o
s
a
dr ,
_ |
s
, s 1
_
3
(
f
t
7
i
(
f
i
)
dr
_
|
s
2
, s 2
d r ,
2
_
i
dr ,
dr ,
Li
Lo
ds
ds
1
1
2
) s 2
2
+ r(Si)
,
do. | d r ,
.
= r i s / c o s a - ^ - ^ — ? ] ^
_|
Ui(fi)cos a
2
u,(t,)—
+
dr •
|
s 1
du. ,
_|
t i
dr ,
_
s i
d r
2
l s 2
.
)
So, one of the above terms must be 0. If r(s) = 0, G is degenerate (cross-section is a single point).
1
If cos a = 0, G is already a Right SHGC. If d ^ / d f = 0, G is degenerate (C(f) is a line of constant u).
1
1
If 0 = d r / d s , G is a Linear SHGC. A n d finally, if 0 = 1 + u^{t) d r / d s , w e can differentiate by s to
2
2
1
yield 0 = u^(t) d r / d s and G is either degenerate (0 = u^{t)) or Linear (0 = d r / d s ) .
2
2
2
2
1
So, if an SHGC G has an equivalent representation as an RSHGC, then either (1) G is degenerate,
1
1
(2) G is an LSHGC, or (3) G is already an RSHGC.
1
1
3.3.1 The Slant Theorem
T h e Equivalent RSHGC Problem suggests that t h e only nondegenerate SHGCs with equivalent
representation as RSHGCs are LSHGCs.
T h e converse is also true:
Every Linear SHGC has an
equivalent representation that is also an RSHGC. In other w o r d s , the set of all LSHGCs is a subset of
all RSHGCs.
This statement is called the Slant Theorem,
w h i c h says in effect that it doesn't matter what direction
the cross-section planes are taken relative to the axis of an LSHGC: for any direction, some c o n t o u r
function C can be found to describe t h e shape (ignoring the possible beveling of the ends) (figure 13).
To prove the theorem, let G
1
= (A, C
r, a) be an LSHGC with r = m (s - s ) and C , = {u
v
Q
(We will deal with the special case r{s) = r below.) T h e constituent points are
Q
P (s ,? ) = ( a ( f ) m ( s
1
1
1
1
1
1
- s )sina,
Q
^ (f )m(s
1
1
1
- s ), s
Q
1
+ u (f )/r?(s
1
1
1
-
s )cosa)
Q
v
vj.
14
F i g u r e 1 3 : The Slant Theorem
Now consider G
=
' 2^
v
L
e
t
2
w
2
a
n
2
= {A, C , r,
d
v
2
b
e
w h i c h is an LSHGC and RSHGC with A and r as above, and C
1 7 / 2 ) ,
2
d e f i n e c l
2
by-'
u At) sin a
uJf)
*
=
vJf)
=
3
1 +
m cos a
^ ( 0
1 + u^(t) m cos a
(This definition violates the uniform scaling assumption for f; however, the c o n t o u r can be d e f o r m e d
along its length t o eliminate this problem, and in any event this condition is not central to t h e
derivation being presented.) Each point of G has coordinates
2
P ($ >t )
2
2
= (u (t )
2
2
2
r(s ), v (f ) r(s ), s )
2
^ u {t )m{s
1
2
1
'
=
2
f
2
2
- s )sina
2
Now, for each s and f
2
2
v,(t )m(s
Q
2
- s )
2
Q
^ ^
define s and t by:
v
2
2
i
Then
,
, t / ^ ) rn ( s
1
+ u ^ ) m (s
*
1 + u^tj
v^(t ) m ( s
A
1
1
- s ) cos a - s ) s/n a
0
m cos a
+ ufi J m {s^ - s ) cos a - s )
Q
1 + t / ( ^ ) m cos a
r
s
1
+ u (f )m(s
1
1
Q
l
- s )cosa )
Q
Q
15
= ( u ^ ) m (s
- s ) sin a, v^(tj
1
m (s - s ), s
Q
1
Thus, for e a c h s and f , there exists a point P {s ,t )
1
G
v
2
2
2
Q
1
+ uftj
m (s
1
- s ) cos a)
Q
on G w h i c h is t h e same as the point P ( s , f ) on
2
1
i.e. G and G c o n t a i n the same points. T h e beveling of the ends of G., but not G
1
2
2
1
1
is reflected in
the restriction t h a t s must lie in t h e interval [0,1].
2
In t h e special case that r(s) is constant, w e have r(s) = r .
Q
PfeytJ
= (u (f )r s/no,v (f )r ,s
1
1
0
1
1
0
+
1
Then
u^/^^cosa)
So let u a n d v be defined by:
2
2
u {t)
= u.,(f)s/n a
2
T h e n , for each s and t
n
<2
v
define s and f by:
2
2
- N
Then w e have:
P ( s , f ) = ( u ^ f j ) r s/n a, v ^ f g ) f , s )
2
2
2
=
Q
Q
2
P^Vt)
and the above c o n c l u s i o n s hold true for this case.
So, for each LSHGC, there exists another description of the same shape w h i c h is both an LSHGC
and RSHGC, c o n t a i n i n g all the same points (but without beveled ends).
In this sense, the set of
LSHGCs is a subset of the set of RSHGCs.
3.4 The Alternate Axis Problem
Having explored t h e issue of c h a n g i n g t h e cross-section planes, w e can ask about moving the axis:
For what SHGCs are there equivalent representations with different axes, using the same crosssection planes (figure 14)? (This is k n o w n to involve a loss of generality with respect to the question:
For what SHGCs are there equivalent representations with different axes?
For example, a sphere
satisfies t h e latter c o n d i t i o n , but not t h e c o n d i t i o n w e are addressing here. We c o n j e c t u r e that only
shapes resembling certain regular polyhedra, of w h i c h the sphere is the limiting case, are excluded
from our analysis herein by the restriction to use the same cross-section planes.) W e will begin by
restricting the problem so that the t w o axes intersect s o m e w h e r e , and so that both axes intersect the
cross-section planes (i.e. the axes of t h e SHGCs are not parallel to the cross-section planes).
16
F i g u r e 1 4 : T h e Alternate Axis Problem
Suppose G
1
= ( A C
V
r
V
a J and G
r
with the same u-v planes, and with
oriented identically for G
= ( A , C , r , a ) are SHGCs representing the same s h a p e ,
2
2
AFI )
2
2
A {s )
=
0
2
2
for s o m e s .
Q
(We will presume t h e t/-axis is
Q
a n d G , w i t h o u t loss of generality since t h e u-v axes c o u l d be rotated
1
2
w i t h o u t affecting the following line of reasoning. Similarly, the exact definition of a
is not of interest
2
here -- it is sufficient to note that it is in fact constant since A is a line s e g m e n t and all u-v planes a r e
2
parallel.)
Let
2
+ 1) = ( u > A 2 ) u 1 v 1
V
A (S
Q
5 L e
A2
' ( A2*
V
of t h e u-v plane for G , for t h e value s = s
2
A (s)
= (0, 0 )
2
a
A2^
U
Q
r
e
t
h
e
c
o
o
r
d
i
n
a
t
e
s
i n
t
h
e
U
'
V
P
l a n e
f o r
G
1
o f
t
h
e
o r i
9
i n
+ 1. Then for any s,
u 2 v 2
since t h e axes A and A are linear.
1
2
For any point P(s,f) on
P(s,f) = {ufi)
=
G
rfi),
v
vfi)
- u
IUFI)RJS)
A 2
since the origin is translated by ( u
^(s), s )
( s - s ) , vfi)r (s)
Q
A 2
A
(s - s ) , v
Q
P(s,f) = (u (t) r ( s ) , v (f) r(s), s )
2
2
u 1 v 1 s
2
A 2
u 2 v 2 s
Equating u-coordinates,
ufi)r^(s)
- ^
A 2
(s-s ) =
u (t)r {s)
Q
2
Differentiating by s,
dr
1
dr
p
2
- % (s-s ), s)
2
0
u 2 v 2 s
(s - s )). But since P(s,f) must also be a point on G ,
Q
2
17
and differentiating by s again,
dV
d %
_u (0__d r _ _ d
2
1
^
2
L
u {t)
ds
2
"
2
ds
2
Now differentiating by f,
d \
d
uM)
df
uJt)
*— = o
1
ds
2
So,
d r
2
1
ds
If d r / d s
2
2
n
f-
= 0
or
2
d
uJt)
— = 0
uJt)
3
df
= 0, G is an LSHGC.
1
Suppose d ^ / d s
2
* 0. T h e n d / d f i/.,(f)/u (f) = 0, and u ( f ) = /(u.,(f) for s o m e constant k. But, w e
2
2
already k n o w that
u (t)r (s)
2
2
= ^(0^(8) -
u (s~s )
M
Q
so
uAt)
1
u (s-s )
= —^
—
^(s) kr (s)
A 2
Q
2
2
Differentiating by f, w e have
df
so u ^ f ) = u for s o m e constant u .
Q
Q
By similarly equating iz-coordinates, w e have v^{t) = v for some
Q
constant i/ , i.e. the cross-section is a single point, and G is thus a degenerate Generalized Cylinder.
Q
1
Thus, a nondegenerate shape can be represented as SHGCs with different (intersecting) axes only
if the shape is an LSHGC.
18
3.4.1 The Pivot T h e o r e m
Figure 15:
The Pivot Theorem
The alternate axis problem suggests that the only SHGCs with equivalent representations with
different axes are LSHGCs. T h e converse is also true: each LSHGC can be represented with the axis
pointing in any d i r e c t i o n , so long as it c o n t a i n s the apex point of the s h a p e (the point at w h i c h its
radius is 0) (figure 15). (We will observe a minor restriction on the direction of t h e axis later in this
section). This statement is called the Pivot Theorem
since it states that the axis of an LSHGC c a n
effectively be pivoted about the apex into any d i r e c t i o n .
To prove the t h e o r e m , let G
= (A
1
V
C
r,
V
and G
2
= (A ,
2
C , r, a ) be LGCs describing t h e
2
2
same shape, with r(s) = m ( s - s ) , with the same u-v planes (possibly rotated as in the previous
Q
section), and with
A^Sq)
=
A {s ).
2
Let the u-v coordinates of A (s
2
A (s)
2
= (u
A 2
(s-s ),
0
since the axes are b o t h linear.
Q
is the apex of the shape, since r(s )
{A^SQ)
Q
+ 1) on the u-v plane for G be ( t / , v
1
^ (s-s ))
2
= 0.)
Q
0
u
1
v
1
A 2
A 2
)
u 1 v 1
-
T
h
e
n
f o r
a
"
S f
19
Now, let C
A
= (u
u (t)
and C
v
= u (t)
2
2
and
For any point P ( s , 0 on G
=
2
-
A
P(s,f)
= ( u , v ) where:
2
^ ( O r f s ) ,
- u
A
2
A 2
( [ ^ ( f )
- - ^ - ] m ( s - s
r(s),
w h i c h is a point on G .
and G
1
v (f)
2
v
1
a
"
A
2
( S - S
r(s),
s )
) ,
0
u
0
2
v
2
) ,
^(f)/77
[ ^ ( 0
2
- - ^
A 2
(s-s ),
( S - S
2
s)
Q
Q
)
-
u 2 v 2 s
^ ( S - S q ) ,
- ] a t 7 ( s - s
0
m
) ,
s )
u
S )
2
v
2
u
2
v
2
s
s
s
So, each point of G
2
reasoning). Thus, G
-
m
1
1
T
=
Q
u
Q
( S - S
2
^ 2 .
( s - s ) , v (f)r(s) - v
(¿^(0/77
(i/ (f)
)
s )
=
=
-
A
1 f
( i / ^ O ^ s ) ,
= (u (t)r{s)
v (f) = v (t)
1
is also a point of G
2
(and vice versa, by similar
describe the same shape.
By two applications of the Slant T h e o r e m , w e can eliminate the requirement that A
the u-v planes of G
2
pass through
So, for any LSHGC, the axis may be defined in any direction, but must pass
y
t h r o u g h the apex.
T w o additional restrictions must be noted:
the shape may be beveled differently in the two
representations just as in the Equivalent RSHGC Problem; and, the u-v planes must pass completely
t h r o u g h t h e shape, so the axis is actually prevented from pointing directly away from t h e shape. This
latter restriction excludes the axis from belonging to the projection of the shape t h r o u g h its own apex
(figure 15).
3.5 Surface Normals of an SHGC
The surface normals for an. SHGC can be defined wherever the c o n t o u r function
C{i)
is
d i f f e r e n t i a t e (figure 16).
Equation (3-1) gave the coordinates of a point
P(s,f)
on an SHGC. From this, w e can calculate the
tangent vector to the surface in the direction of increasing s as
dP
3s
t
= ( u {t)
n
c
sin a
ór
ds
, v {i)
c
n
ór
ds
, 1 + u {t)
and the tangent in the direction of increasing t as
c
n
cos a
ór
ds
x
)
20
dP/dt
F i g u re 1 6 :
3P
,
du
= ( r{s) sin. a — r { s )
dt
dv
— r { s )
dt
r
dt
r
du
dv
= r(s) ( sin a — — c o
dt
dt
f
r
Surface Normal of an SHGC
r
cos a
du
a—
dt
r
s
x
So, the outward-pointing surface normal vector N'{s,t)
DP
du
—
dt
r
x
)
j
at P(s,0 is the cross p r o d u c t of these:
DP
DS
DT
,
du
d r
r
^ -
n
+
—
dt
d s
c
dv
n
= r{s) ( - v (t) cos a
^
-
+
M
dt
O
0
dr
cos a —
ds
dv
r
f * -
dt
d U ç
-sin
a
• u (t)
r
= rls)
(
dt
sin a
h(t) cos a
ds
v {t) sin a
Q
dt
+ — ~, u
ds
df
sin a —
df
dr
du
ds
dt
-
Q
)
h(t) sin a —— )
ds
w h e r e h(t), defined by
du
/7(f) =
is the Wronskian
u (t)
c
r
du /dt
df
dv /dt
Q
of the c o n t o u r f u n c t i o n s u
Q
Q
and v
Q
[14]; h{t) = 0 implies that the SHGC has a line
segment for a cross-section contour, i.e. is degenerate.
W e will use N{s,t) parallel to N'(s,t),
defined by
21
A/(s,f) =
N'{s,t)
.
dr
= ( /7(f) c o s a
ds
r(s)
dv
+ —
df
du
sin a —
dt
r
r
h(t) sin a
dr
ds
x
)
3.6 The Corresponding Normal Theorem
F i g u r e 1 7 : The Corresponding Normal Theorem
Figure 17 shows that the surface normals on a Linear SHGC are parallel along c o n t o u r s of constant
i, and for a Polygonal SHGC they are parallel along c o n t o u r s of constant s within a face.
stated in the Corresponding
Normal
Theorem:
SHGC is Polygonal iff for almost all t, dN/dt
An SHGC is Linear iff for all s, dN/ds
= (0,0,0).
For the first part of the t h e o r e m , we begin by noting that
dr
dr
= ( h(t) cos a — r - , 0 , - /7(f) sin a — 5 - )
ds*
ds*
2
= ( 0 , 0 , 0 ) ; an
This says that the surface normals for an
LSHGC d e p e n d only on f, and for a face of a PSHGC d e p e n d only on s.
dN
——
3s
This is
2
x
22
Now for an LSHGC, 6 r/ds
2
Conversely, if dN/ds
= 0, so dN/ds
2
= (0,0,0) for all s.
= (0,0,0) for all s, then either h(t) = 0, sin a and cos a are both 0, or d r / d s
2
2
= 0. If h(t) = 0, the shape is degenerate. It is impossible for sin a and cos a to both be 0. So, t h e
only interesting case is that in w h i c h d r / d s
2
2
= 0, i.e. the shape is an LSHGC.
For the s e c o n d part of the t h e o r e m , note that
dN
dt
,
( cos
dr
dh
Kill
VJI
a
A~
ds
d VV f >
d
V-i u1 4
drI I
V
- sin a — 5 - , - sin a
At*r
A*.
d
ds
2
VJ
+
At
df
At*r
d
dh
%
At
df
'
Kill
)
where
dh
d v
d u
2
2
r
=
c" W dt
- ^ - 'c ( ) —
" "
df
u
dt
c
v
v
2
Now for a PSHGC, d u / d f
2
_
2
c
O a n d dN/dt
f
c
v
w
r
S L
2
= 0 and d v / d f
2
2
c
= 0 for almost all t (except at t h e vertices), so dA?/df =
= (0,0,0).
Conversely, suppose dN/dt
= (0,0,0) almost everywhere. Using the v-coordinate, since sin a = 0
is impossible (it implies that a = 0, i.e. the axis is c o n t a i n e d in the cross-section planes), it must be
thatd u /df
2
2
c
dN
dt
= 0. T h e n wherever dN/dt
,
dr
d V
cos a — ) — f - ,
ds
dr
(1 + uJt)
c
So from the s-coordinate, either u (t)
Q
degenerate (planar).
If dr/ds
case is that in w h i c h d i v / d f
2
c
= (0,0,0), w e have
0, - uJt)
°
sin a
dr
ds
= 0 or d r / d s = 0 or d v /dt
2
= 0, using the u-coordinate, d v /dt
Q
2
= 0. If i / ( f ) = 0, the SHGC is
2
Q
2
= 0 and d v / d f
2
Q
2
dV.
j f - )
dr
c
2
= 0. So, the only interesting
= 0, i.e. the c o n t o u r C(t) is locally linear. W h e n this
is true for almost all t, C(t) is piecewise linear and the shape must be a Polygonal SHGC.
The Corresponding Normal Theorem is especially useful in s h a d o w geometry, since the " s h a d o w
v o l u m e " (the volume of space shaded by an object) is an LSHGC (figure 18) [17].
23
F i g u re 1 8 : Surface Normals of t h e S h a d o w Volume
24
4. Projections of SHGCs
In this section, w e will begin by exploring the projections of SHGCs onto images. T h e n , w e will
describe how images can be analyzed to determine the SHGCs d e p i c t e d . Although the formulations
will begin with the most general cases, most of our attention will be given to the case of Right Circular
SHGCs, since these are sufficiently constrained to allow interesting analysis from imagery without
additional knowledge sources.
4.1 Projected Coniour Generators
F i g u r e 1 9 : Contours and Contour Generators
Suppose w e have an SHGC, and w e project it along the direction of a vector F = ( f
w>
f , f ) , as in
v
s
figure 19. The c o n t o u r s along w h i c h the surface is tangent to the line of sight as seen from direction
F (i.e. o c c l u s i o n , or parallelism to F ) will be projected by the ends of the SHGC, or w h e r e N ± F , i.e. N
• F = 0. The points on the SHGC projected onto s u c h c o n t o u r s are called coniour
generators
[11].
(Of course, if the SHGC is opaque, some of the c o n t o u r s may be hidden from view.) From this point
on, we will use the term " c o n t o u r " to mean " i m a g e of a c o n t o u r generator" (in the above sense), a n d
"cross-section f u n c t i o n " to refer to C{t).
On the contour generators,
0 = N 'F = f
w
V
,
dr
( h(t) cos a
ds
dv
+ — )
df
n
x
du
- f sin a —^dt
r
v
where h(t) is the contour Wronskian, as previously d e s c r i b e d .
- f h(i) sin a
3
dr
ds
25
eye
Figuré 2 0 :
Viewing Direction and Angle
Unless otherwise stated, w e will presume in the following discussion of projection and imaging that
w e are dealing only with Right SHGCs, i.e. SHGCs with cross-sections perpendicular to thè axis. This
allows the simplification of rotating the w-v axes as desired; in particular, w e will presume that F i s in
the MAS plane, i.e. f
= 0. Without loss of generality, w e can then presume that F i s between - l,Vand
S; if the angle from F to S (the viewing angle) is <r, then F = ( - sin a, 0, cos a) (figure 20). Additional
simplification arises for an RSHGC since sin a = 1 and cos a = 0. T h e n , for an RSHGC, the contour
generator points satisfy
dv
0 = sin a — ^
dt
n
+ h(t) cos a
dr
ds
There are three interesting cases, illustrated in figure 2 1 : e n d , side, and oblique views.
(4-1)
If F || S
("end v i e w " ) , sin a = 0 and the c o n t o u r generator points satisfy
0
dr
= M o ds
dr
In this case, either h{t) = 0 o r — = 0. But, h{t) = 0 only if the RSHGC is degenerate (planar), so the
ds
only interesting case is dr/ds
= 0, i.e. the c o n t o u r generators are cross-sections at extrema (relative
maxima and minima) of r(s) (Marr's "radial extremities" [11]).
l f F | | W ("side v i e w " ) , cos a = 0 and the contour generator points satisfy
26
F i g u r e 2 1 : End, Side, and Oblique Views
So, the c o n t o u r generators are at the extrema of v ( f ) .
c
If F f t S a n d F}f M/("oblique v i e w " ) , there is no simplification from equation (4-1).
It is easy to imagine a dichotomy for oblique views between " e n d - l i k e " views and " s i d e - l i k e " views
as if there were an abrupt c h a n g e from one to the other as F swings from S t o W. However, there is
actually no abrupt c h a n g e , but rather a steady c h a n g e in the c o n t o u r generators.
4 . 1 . 1 P l a n a r i t y of C o n t o u r G e n e r a t o r s
A set of points { P } = { ( p , p , p ) } is planar iff there exist constants a, £>, c, and d s u c h that, for all
w
points in the set, ap
v
+ bp
W
s
+ cp
+ d = 0.
W e can use this to determine some conditions on
s
V
planarity of c o n t o u r generators.
In an end view, 0 = d r / d s along a c o n t o u r generator.
satisfy p
s
= s where d r / d s
= 0. So,
u
rt
°Pw
+
°Pv
+
P s -
Thus, the points on a c o n t o u r generator
S
0
=
0
and the contour generator must therefore be planar.
Further, the plane containing the c o n t o u r
generator is perpendicular to the viewing direction F = (0, 0,1) for an end view since F* P - s
Q
= 0.
27
In a side view, 0 = dv /dt.
But this is a function of f, so dv /dt
n
n
.
= 0 for some f .
n
Each s u c h
value of t defines a single contour generator. On this c o n t o u r generator, t = ? a n d
Q
Q
P(s,f ) = ( P . P > P ) = (u (t )r{sh
0
v
W
V
S
c( o)Pw- c( o)Pv
f
u
f
c
+
0
^s
so the contour generator is planar.
can be simplified to p
w
v ( f ) r ( s ) , s)
Q
+
0
=
c
Q
0
In addition, if for s u c h t w e have u (t )
Q
Q
Q
= 0, then the equation
= 0, i.e. the plane containing the c o n t o u r generator is perpendicular to F ( a n d
parallel to the image plane).
However, for oblique views, the c o n t o u r generators are generally not planar.
There is no easy
method for evaluating the planarity of the c o n t o u r generator defined by equation (4-1). Instead of the
above criterion, if specific functions t / ( 0 . Q(t)>
v
c
a n
d
r
( s ) are? k n o w n , the torsion
of the c o n t o u r
generator can be evaluated; it must be 0 for the c o n t o u r generator to be planar [4].
In the special case of LSHGCs, w e have r = m (s-s )
Q
and dr/ds
= m. So, the contour generator
satisfies:
dv
0 = sin a — +
dt
c
m h{t) cos a
(4-2)
The above expression is only a function of t. Thus, for a given c o n t o u r generator, the above equation
holds true for some value of t
Qi
and the c o n t o u r generator is planar by the same reasoning as used for
a side view. So, for an LSHGC, every c o n t o u r generator is planar.
4.2 Images of Right SHGCs
The world coordinate system is defined as s h o w n in figure 22, by aligning X and Y horizontally and
vertically (respectively) on the image plane, and letting Z point towards the eye (or camera) The
viewing direction F i s then the same as Z , i.e. F = ( 0 » 0 , 1 )
xyz
(where
x y z
denotes world coordinates).
The discussion of imaging in this paper will be primarily limited to orthographic
world point (x,y,z)
is mapped onto the image point (x,y)
xyz
projection,
in w h i c h a
.
xy
Suppose we are looking at an RSHGC. Without loss of generality, w e can presume that S is in the
X - Z p l a n e , between Z a n d X. Then the viewing angle o is measured from Z t o w a r d s S ( s i n c e F = Z ) .
Since WV-S
is right-handed, W is in the XZ
upward, V = Y. The important vectors are t h e n :
plane between X and - Z and 1/ points vertically
28
.Y=V
Z=F
\
eye P'
F i g u r e 2 2 : Object and W o r l d Coordinate Systems
F
= ( - sin O, 0, cos A)wvs
W
= ( 1 , 0, 0)
V
= (0, 1, 0)
S
= (0, 0,
wvs
wvs
1)
wvs
» <°' °'
V
= (cos a, 0, - sin A)
xyz
= (0, 1 , o
= (sin a, 0, cos A)
xyz
)xyz
For any point P,
p
=
s ) ^ = w IV + v V + s S
= (w cos
a
+ s s/n
a,
v, - w s/n
A point P(s,f) on an RSHGC is therefore
P(s,0 = (u (t)
Q
a
+ s cos
a)
xyz
r(s), v ( f ) r(s), s ) ^
c
= (u (t) r(s) cos A + s s/n a, v ( f ) r(s), - a ( f ) r(s) s/n A + s c o s a)
Q
c
Q
and its image under orthography is (figure 23)
/(s,0 = (x,y) ( s , f )
xy
= ( u ( f ) r(s) cos G + s s/n a, f ( f ) r f s ) ) ^
c
c
W e will presume that the image of t h e origin of the RSHGC is (0,0)
; otherwise, an additional
xy
translation of the image points will occur. In addition, w e are presuming here that there is no scaling
difference between w-v-s and x-y-z coordinates.
4.2.1 Contour Generators Under Perspective Projection
While the bulk of this discussion c o n c e r n s orthographic projection, in w h i c h the image of a point (x,
y, z)
is (x, y)
xyz
projection,
, it is also possible to analyze the c o n t o u r s of RSHGCs viewed under
xy
in w h i c h the image of (x, / , z)
is ( - x / z , - y/z)
xyz
perspective
if the unit of measure is t h e focal length
xy
29
Figure 23:
Image of a Point on a Right SHGC
of the lens (this is the same coordinate system as that of Shafer et al. [15]). Since translation in space
affects a perspective image, it is necessary to generalize the imaging model used above to allow for
the position of the object in the scene, as well as the possibility that the axis of the image of the object
does not pass t h r o u g h the origin of t h e image.
eye = Pe
Figure 24:
Imaging Under Perspective Projection
The contour generator analysis itself can be accomplished by considering the eye (center of the
lens) to be located at a point P
Q
= (w
&1
v
Qf
s ) in the object-centered coordinate system, as in figure
Q
24. Then, at each point P{s,t) on the surface of the object, the line of sight is the vector F(s,0 from the
eye to P(s,0, defined by:
30
F(s,f) = P(s,/) = (u {t)
P
e
r(s) - w , v {t) r(s) - v , s -
Q
e
Q
Q
s )
e
Along a c o n t o u r generator, w e still have N _L F , so 0 = A/ * F .
While both the imaging and c o n t o u r generator problems are therefore more difficult
under
perspective projection, they may still be solvable, particularly for relatively constrained cases s u c h as
the analysis of Right Circular SHGCs.
4.3 Projection and Imaging for Right Circular SHGCs
For Right SHGCs w h i c h are also Circular SHGCs (CSHGCs), there is considerable simplification in
the orthographic projection a n d imaging relationships. Recall that, for a Circular SHGC,
u (t)
= cos 2MT
Q
and
v {t)
sin 2MT
and
Q
= sin 2MT
So,
2TT
dt
cos 2MT
= 2M
and
dv
h(t) = u
=
{
0
n
dt
d i /
.
(
2IR cos
2
0
2MT + 2M sin
2
2MT
dt
2tt
The c o n t o u r generators for an RCSHGC therefore satisfy
dr
0 = 2M sin <j cos 2MT
+
2 7 r
cos a
ds
i.e. (figure 25)
1
t =
cos" (
1
-cota
2M
(4-3)
)
So along the c o n t o u r generator,
dr
u {t)
n
u
= COS2MT
= -cota
ds
where v ( f ) > 0 on the upper half of the shape, and v (t)<0
c
Q
on the lower half.
31
eye
F i g u r e 2 5 : Contour Generator on a Right Circular SHGC
Now, since t is a function of s along the c o n t o u r generator, the points P(s,f) along the contour
generator can be specified as P ( s ) , a function of s only:
C G
P
C G
( s ) = ( -cotar(s)—,
±r{s) yj 1 - cot
2
<x ( d r / d s ) , s )
2
ds
and the c o n t o u r generator includes points s u c h that v (t) is defined, i.e.
Q
1 - cot
dr
ds
a ( — )
ds
2
> 0
~
2
< \tan a\
The c o n t o u r generator is not generally planar in an oblique view.
On an RCSHGC, the image of a point P(s,f) is
/(s,f) = (r(s) cos 2mt cos a + s sin a, r{s) sin
2irt)
(4-4)
xy
and for a point on a c o n t o u r generator,
=
<
X
C G '
(
W( >xy
= V ~ (s)
S
cos
2
a
dr
r
sin a
ds
+ s sin a , r[s) V l
Further, the slope of the image contour, d y
the following derivation:
- cot
2
a (dr/ds)
2
)
w
xy
C Q
/dx
C G
, can be determined as a function of dr/ds
(4-5)
using
32
d
x
— 2 a .
n
cos
_
n
=
ds
a ,
2
(
sin
i
=
s / n
a
2
(
s
d r
_ _
)
+
(
_ ) 2 )
ds
G
o
1
(
dr
r
-
c
o
x
+
s
/
n
Q
ds
d
G
r
0
2
r
r(s) — 5 - - cor
d s "
2
d
G
2
r
(
\
)
d
1
j
s
dr d r
2
d
y
p
g
ds
=
V
l
-
c
o
f
2
a
(
d
r
/
d
)
s
2
4
^
"
ds
-
0
/
V
-
!
-
c
?
o
( d r / d s )
2
2
) c o f
d
=
(
1
/
V
l
-
c o f
a
2
( d r / d s )
2
)
(
1
-
c o f
a
2
r(s)
d y ^
dx
C Q
=
dy
c g
ds
ds
dx
=
(
1
/
^ ^
c
d
Q
$
2
^
(
d
r
/
d
g
)
2
}
r ( s )
ds ds
d
2
— 5 -
d s
<r
2
-
c o f
2
a
s
dr^
C Q
(
)
d
2
)
s
(
4
6
)
ds
4.3.1 Occlusions a n d Singularities in image C o n t o u r s
W h e r e | d r / d s j > ifan <x|, there will be n o points o n t h e c o n t o u r generator. This causes occlusion of
t h e c o n t o u r generator from view, with resulting discontinuities in t h e visible c o n t o u r s .
To study this p h e n o m e n o n , suppose w e begin with t h e object at a side view (<r = t t / 2 ) , a n d let us
study t h e c o n t o u r generator as w e rotate the object t o w a r d s an end view ( a = 0). At t h e start, tan G is
infinite a n d | d r / d s | < tan G for all s. T h e r e will b e a single c o n t i n u o u s c o n t o u r generator o n t h e
object, w h i c h will in fact be planar (running along t h e t o p a n d b o t t o m of t h e object) (figure 26).
As w e rotate t h e object slightly, decreasing G a n d hence tan a , as long as | d r / d s | < tan G t o w a r d s
that e n d . t h e c o n t o u r generator will still be c o n t i n u o u s (figure 27). However, it will n o longer b e
planar (unless the object is also a Linear SHGC, w h i c h w e will n o t consider further here).
From
equation (4-3), w e see that w h e r e d r / d s is 0, t = 1/4, i.e. t h e c o n t o u r generator is on " t o p " of t h e
object. (Of course, there is also an identical c o n t o u r generator on the bottom.) Where d r / d s < 0, t >
1/4 a n d the c o n t o u r generator is pushed away from us; w h e r e dr/ds
> 0, t h e c o n t o u r generator is
pulled towards us. T h e variation in t as w e travel along t h e shape is expressed by:
dt
.
I
I
I
T v d r
= ( 1 / 27r cos G ysin' G - cos* G (dr/ds)*
) — o "
ds
ds*
Since | d r / d s | < |fan G\ along t h e c o n t o u r generators, t h e sign of d r / d s d e p e n d s only on t h e sign of
d r/ds .
2
2
Note also that | d r / d s | > \tan G\ at t h e ends of t h e object, hence t h e contour generator no
longer includes the ends as t h e object is slightly rotated away from a side view.
33
image
top view
showing contour generator
t
^eye
dr/ds
Figure 2 6 :
Contour Generator in Side View
Let us presume for the moment that the object is thinner at the near e n d , i.e. d r / d s < 0 everywhere.
Eventually, we rotate the object so m u c h that d r / d s = -tan
is at a minimum (figure 28). At this value, since d r / d s |
C Q
generator is tangent to the line of sight at s .
/ds |
= - fan a and d r / d s
2
m
j
is a relative minimum for d r / d s ) , we have d x
a at some value of s, say s
m
C Q
/ds |
m
If w e rotate the object yet farther, there will be an interval (s , s.) around s
a
b
|
where d r / d s
= 0 (because s
m
m
I
= 0 and d y
2
t
= 0 . Thus, the contour
m
in w h i c h d r / d s < - tan
a, i.e. for w h i c h no c o n t o u r generator points.exist (figure 29). What has happened is that the former,
single contour generator has been split into two separate c o n t o u r generators, corresponding to
values of s s u c h that s > s
and s < s . Along the c o n t o u r for s > s , all points will be visible in the
b
a
b
image (i.e. none are o c c l u d e d by the object itself in this vicinity).
^cg^^cg
'
s
' ^ '
n
n
t e
a
s
w
e
approach s
b
Further, note that the limit of
from above, i.e. the c o n t o u r in the image becomes
asymptotically vertical as w e travel towards s : the c o n t o u r thus "flares o u t " towards the vertical in
b
this region.
34
image
top view
showing contour generator
Figure 2 7 :
Contour Generator in Near-Side View
Meanwhile, along the c o n t o u r for s < s . t h e object itself will o c c l u d e part of the c o n t o u r generator,
for values of s above some value s
a
c
(where s „ < s J (segment X in figure 30). S u p p o s e the c o n t o u r
c
a
generator is o c c l u d e d for s o m e s , w h e r e s < s < s . T h e n
Q
3 s > s s u c h that x ^ f s ^ = *
1
b
c
C G
Q
a
( s ) and l y ^ s ^ l > | y
0
C G
(s )|
n
This, unfortunately, c a n n o t be further simplified to a c o n d i t i o n on s without s o m e k n o w l e d g e about
the behavior of r(s) (the fact that d r / d s > - tan a above s
b
gives no useful constraint here) - thus,
segment X (in figure 30) is hard to characterize. In any event, r(s) can be determined for segments A,
e , and V.
What w e have seen is a single image c o n t o u r splitting into two parts c o n n e c t e d by a sort of " T "
j u n c t i o n ; the split o c c u r r e d at the point at w h i c h the c o n t o u r generator was tangent to the line of
sight. This illustrates an important kind of "special v i e w p o i n t " for curved surfaces: If a visible arc in
the scene is tangent to the line of sight, then a small variation in the viewpoint can cause a topological
c h a n g e in the image of the arc.
35
4 . 3 . 2 C r o s s - S e c t i o n s D e s c r i b e d by Fourier C o e f f i c i e n t s
T h e use of Fourier coefficients for t h e functions u {t) and v (t) allows more complex cross-sections
Q
c
than circular ones to be d e s c r i b e d , while retaining some degree of mathematical
tractability.
Suppose, for example, that the cross-section of an RSHGC is described by one set of Fourier
coefficients, w h i c h allows ellipses with arbitrary position, rotation, scale, and eccentricity. Using the
notation of Kuhl and Giardina [10], w e have:
C(t) = (u
Q1
v )(i)
Q
= (>A + a cos 2<n\ +
0
1
so
du
r
v
dt
= - 2773, sin 2mt + 2mb. cos 2<nt
1
1
sin 2*nt, C
Q
+ c cos 2it\ +
1
sin 2irt)
36
top view
image
showing contour generator
dr/ds
Figu re 2 9 :
Contour Generator Split at Near-End View
image
r(s)
Se
F i g u re 3 0 :
àv
dt
Sa
V J , M
Sb
Contour Pieces Correspond to Disjoint Intervals of s
= - 2 7 T C , sin 2*nt + 2md. cos 2irt
37
h(t) = 2TT ( {A d^
+ C bJ
Q
cos 2irt - ( A c
Q
Q
+ C ^ ) s/n 2?rf + afi
1
+
A
)
The points on the c o n t o u r generators must obey:
0 = Sin <7 (c^ COS 27Tf - C S/H 27Tf)
1
dr
,
- cos a
x
v (A d
Q
t = (1 / 2 7 r ) s / n -
1
1
+ C ^ ) cos 27rf - ( / \ c
Q
( (-eg + f\/e
2
+ ? - g )/(e
2
2
1
+ C ^ ) s/n 27rf +
+ b ^
)
+ f ) )
2
where
dr
e = -c+sinv
+ (>A c. + C L a J c o s a
n
1
0 1
0
V
d
s
dr
f =
s//7 a - ( A d . + C b J cos a
n
n
o 1
1
g = -(a cf
1
1
+ b^cjcoso
d
s
dr
—
Since f has been expressed here as a function of s along t h e c o n t o u r generators, w e can presumably
express points along the c o n t o u r generators as P
C G
( s ) and c o n t o u r points as /
C G
(s).
M u c h of the
following analysis of Right Circular SHGCs will therefore be applicable for RSHGCs with arbitrary
ellipses as cross-sections; while the closed-form expressions will be complex, actual numeric analysis
should not be difficult.
For cross-sections described by two sets of Fourier coefficients (i.e. A , a
Q
d ) , analysis is yet more difficult but still conceivable.
2
v
a , b
2
v
b , C , c
2
Q
v
c ,
2
d
v
For additional coefficients, however, s u c h
closed-form analysis seems beyond tractability.
4.4 Contour and Silhouette Analysis for Right Circular SHGCs
In image understanding, we are faced with the problem of analyzing image c o n t o u r s rather than
predicting them.
W e can accomplish this task using the above description of the properties of
c o n t o u r generators.
S u p p o s e we have a line drawing consisting of visible (i.e. unoccluded) c o n t o u r s , each of w h i c h is
the image of some c o n t o u r generator on a Right Circular SHGC. By analyzing the c o n t o u r s , we can
construct a description of the solid shape portrayed.
F i g u r e 3 1 : Aligned Image of an SHGC
First, w e need to determine the viewing angle a and to align the image as in figure 3 1 , so t h e
images of the endpoints of the axis A(0) and 4 ( 1 ) are at (0,0)
and (sin <x, 0)
xy
, respectively; this
xy
c o n f o r m s to the imaging model presented previously. T h e n , w e can analyze t h e c o n t o u r s to recover
the shape; for a Right Circular SHGC, w e need only determine r(s), t h e radius f u n c t i o n , to have a
complete description of the shape.
W e will begin by addressing the latter problem -- analysis of c o n t o u r s when the image is aligned
and a k n o w n . T h e n , w e will examine h o w to determine or and perform the alignment.
4.4.1 Contour Analysis
Along a c o n t o u r generator of a Right Circular SHGC, recall that equations (4-5) and (4-6) give
*
C G
(s), y
C G
( s ) , and d y
C Q
/dx
C G
as functions of s, r(s), and d r / d s . These allow us to solve for s, r(s),
and d r / d s , the shape description, as functions of *
(s), y ( )>
s
C G
C G
measured in the image. We obtain:
W > - W >
s
s
c o s 2
sin a
°
( ^r/
d
d x
CG)
CG
a
n
d
d
VcG
/ d x
CG'
w
h
i
c
h
c
a
n
^
e
39
image x,y
description r(s)
Figure 3 2 :
Contour Analysis Results in Shape Description
So, given any c o n t o u r point ( * ( s ) , / c G ^ ^ x y '
C G
a n c l
^
e
s
'°P
e
d
^cG
/ c l x
C G °*
t
h
e
c
o
n
t
o
u
r
a t
t h a t
point, we can determine s, r(s), and d r / d s at that point. By doing this for all c o n t o u r points, w e can
determine as much as possible about r(s) from the image (figure 32).
As stated earlier, this analysis presumes that a is k n o w n and that the image is appropriately
aligned.
4.4.2 Occluded Contours and Silhouettes
W e have seen how to analyze c o n t o u r s to determine values of r(s); now, w e will discuss h o w m u c h
of r(s) can be reconstructed in this manner (i.e. over what range of values of s). As w e already know,
there is a contour generator only where | d r / d s | < \tan a|; values of s for w h i c h | d r / d s | > \tan a\
therefore d o not c o r r e s p o n d to any points on a c o n t o u r generator, and r(s) c a n n o t be determined for
these values by examination of image c o n t o u r s . In addition, as described in section 4.3.1, the object
itself may o c c l u d e portions of the c o n t o u r generator from view.
For analyzing a silhouette, exactly the same methods and conditions apply, except that, using the
notation of section 4.3.1, the c o n t o u r for s < s will render invisible the c o n t o u r for s > s. w h i c h lies to
— c
— 0
the left of '
C G
( s ) ; thus, in figure 30, only segments A and B will be visible. This may be stated as
c
follows: The contour generator for s is o c c l u d e d or invisible if
Q
3s^s
Q
s u c h that x
C G
(
S l
) = *
C G
(s )and I W M
n
* IW o>l
s
The only difference between this case and the above discussion for s < s is that the requirement s
a
s
b
has been generalized to s
1
1
>
* s . Thus, silhouettes are simply images of c o n t o u r s in w h i c h certain
Q
portions of some c o n t o u r s are not visible.
If dr/ds
> 0, the situation is just the above viewed from the opposite direction (i.e. segments A, B,
40
and X in figure 30 will be visible, and segment V will be occluded).
In this case, when the c o n t o u r
generator splits, the arc for s < s is still o c c l u d e d , but t h e arc for s > s is a closed curve in the image
a
b
rather than flaring out as above.
Silhouette analysis will be identical; indeed, the silhouette of an
object is identical (to within a reflection) viewed from opposite directions.
In any of these cases, there will be an interval of s over w h i c h r(s) c a n n o t be c o m p u t e d , say (s., s.).
However, w e can c o m p u t e r(s.), r(s.), d r / d s | , and d r / d s | . For p r a c t i c a l image analysis, it is
'
i
i
possible to estimate r(s) over (s., Sj) by fitting a function w h i c h c o n f o r m s to these constraints. For
1
example, a cubic polynomial can be fit to the data:
r(s) = a s
3
+ bs
+ cs + d
2
so
dr
ds
= 3as
2
+ 25s + c
Then the following system of linear equations can easily be solved to determine the values of a, 5, c,
and d:
"
3
2
b
r(s,)
dr/ds |
J
J,
3s.
;
dr/ds *
•s. J
J
L
3s.
J
2
2
i
2s.
2 s
J
1
a
1
b
1
0
c
1
0
d
s.
1
4.4.3 Aligning the Image
The above analysis has presumed that w e k n o w t h e viewing angle a and have aligned the images of
the axis endpoints A(0) and 4 ( 1 ) onto (0,0)
and (sin a, 0)
xy
, respectively. W e will now address t h e
xy
problems of aligning the image and determining a.
Suppose w e are given an image of the c o n t o u r s of a Right Circular SHGC, arbitrarily scaled,
rotated, and translated, and viewed from an u n k n o w n angle.
W e can immediately determine the
image of the axis, since this will be an axis of symmetry in the image, and rotate the image so this is
horizontal. By translating the image, this axis can be made to line up with the x-axis. Brooks [3] and
Marr and Nishihara [12] discuss this issue of finding the image of the axis of a generalized cylinder.
W e must next determine w h i c h end of the object is nearer, so this can be placed on the right as in
our imaging model.
If the left end is closer, we will need to reflect the image about the / - a x i s , or
e q u i v a l e n t s rotate it 180° about the origin.
41
W e must also determine a and find the images of the axis endpoints.
Determining a is more
important, since it affects both the image alignment and the shape of the reconstructed function r(s).
The axis endpoints, on the other hand, affect only the scale and shifting (along s) of r(s).
Figure 3 3 :
RCSHGC With Near End Flat
If the closer end of the object is flat as in figure 33, i.e. r(1) > 0, then the edge of the cross-section at
that end will produce a contour in the image, w h i c h will be a vertically elongated ellipse. This is a very
useful configuration, since w e can easily determine w h i c h end of the object is closer. T h e n , w e know
that the center of the ellipse must be the image of the axis endpoint A(1); further, w e can c o m p u t e the
viewing angle a from the eccentricity of the ellipse, using cos a = b / a , w h e r e a and b are the
semimajor and semiminor axis lengths, respectively.
If the farther end of the object is flat and not o c c l u d e d , i.e. r(0) > 0 and d r / d s | < fan <x, then w e will
Q
se exactly half of an ellipse, w h i c h can be analyzed as above to determine w h i c h end of the object is
farther, the image of /1(0), and a.
If neither end can be analyzed as above, w e may be able to determine whether any " T " - s h a p e d
occlusions o c c u r along the contour; if so, the o c c l u d i n g c o n t o u r generator is on the nearer portion of
the object.
Failing this, w e c a n n o t from c o n t o u r s alone decide w h i c h end of the object is nearer.
Fortunately, there are many other potential sources for alignment information, s u c h as the object's
length or width, or surface normals as determined by photometry or range data.
It should be noted that, for a curved surface, scaling the figure along s and altering the viewing
angle a are not complementary; i.e. w e c a n n o t keep the image c o n t o u r s constant by altering the
viewing angle and compensating with elongation of the object.
The reason is that elongation does
not alter the position of the contour generators on the shape, but when the viewing angle is altered,
the c o n t o u r generators " c r e e p " along the surface to a different position. This o c c u r s because within
42
a local neighborhood of a point, the angle between the surface normal N and the viewing direction F
varies. This may be contrasted, for example, with polyhedra under orthography, in w h i c h this angle is
constant, causing c o n t o u r generators to remain in place, and allowing elongation and rotation of the
viewing direction to compensate to keep the image constant. The d é p e n d a n c e of c o n t o u r generator
position upon the viewing angle is the reason that s and r(s) d e p e n d u p o n a in a complicated manner
rather than being, say, proportional to sin a or cos a.
4.5 Contour and Silhouette Analysis for Other RSHGCs
There are several steps to be performed w h e n analyzing c o n t o u r s of any Right SHGC, as illustrated
in the above discussion of Right Circular SHGCs:
1. Finding the image of the axis line.
2. Finding the images of the axis endpoints.
3. Deciding w h i c h end of the axis is nearer.
4. Determining the viewing angle, a.
5. Determining t h e cross-section, C(f).
6. Determining the radius f u n c t i o n , r(s).
Steps 1-4 are required to align the image; t h e n , c o n t o u r analysis can proceed to perform steps 5-6.
If the near end of the RSHGC is flat and b o u n d e d by a sharp edge, i.e. r(1) > 0 and — L is finite, its
ds
edge produces a c o n t o u r w h i c h can be analyzed to assist in steps 1, 2, and 4, and to solve 3 and 5. If
the near end is not flat but the far e n d is flat and p r o d u c e s a visible contour, it can be used to assist in
steps 1, 2 , 4 , and 5, and to solve 3.
For solving step 1, w e can also use the fact that, for any two points with the same s-value but
different f-values, the tangents to the surface in the direction of increasing s will intersect at a point on
the axis (figure 34). This can be seen by noting that on an RSHGC, for given values of s and f,
P(s,f) = (u (t)
Q
r(s), v {t) r(s), s)
Q
dr
ds
So the vector from P in the direction dP/ds
intersects the axis at the point
43
Figu re 3 4 :
( 0, 0, s -
Tangents at Corresponding Points Intersect on the Axis
r(s)
dr/ds
Since this is independent of f, w e can take two points with the same s-coordinate, draw the respective
tangents dP/ds,
and the intersection of these latter will be some point on the axis. In the image, the
same relationship holds, using tangents to visible c o n t o u r s , providing the c o n t o u r generators are arcs
of constant i (so that their tangents are in fact dP/ds).
Of ocurse, to make use of this p h e n o m e n o n ,
some means w o u l d be required for finding points with the same s-coordinate. Surface markings or
knowledge of C{i) might make this possible.
4 . 5 . 1 C o n t o u r s of R i g h t L i n e a r S H G C s
Right Linear SHGCs have some additional properties that can aid in c o n t o u r analysis. First, recall
that equation (4-2) gives the condition satisfied by points on the contour generator of a Right Linear
SHGC.
This is a condition only on f, so the c o n t o u r generators are curves of constant f.
implied also by the Corresponding Normal Theorem, dN/ds
This is
= (0,0,0) for an LSHGC.
So, each contour generator is a straight line of constant t on the surface. These lines must all pass
through the apex of the shape, so any two such contours in the image must intersect at the image of
the apex. This is one point on the axis; by the the Pivot T h e o r e m , the axis can then be defined in any
direction passing through this point. On the other hand, it may be very difficult to determine a unless
the cross-section function C(t) is k n o w n in advance, since so little information is contained in the
image contours.
Interestingly, the c o n t o u r generators in an oblique view need not c o r r e s p o n d to relative maxima
44
side view
oblique view
Figu re 3 5 :
end view
Contour Generators Need Not Be At Vertical Extrema
and minima of v (t), i.e. the c o n t o u r generators need not be on the " t o p " and " b o t t o m " of t h e shape,
Q
but may o c c u r where dv /dt
Q
* 0. For example, figure 35 s h o w s a c o n e with a vertical stripe. In an
oblique or end view, parts of the surface will be visible on both sides of t h e stripe.
On a Right Linear SHGC, there will be no partially o c c l u d e d c o n t o u r s (except possibly at the near
end of the shape), since all c o n t o u r generators are linear.
4 . 5 . 2 C o n t o u r s of Right Polygonal SHGCs
A Right Polygonal SHGC has t w o types of c o n t o u r generators: faces tangent to the line of sight,
and creases. A c o n t o u r generator on a face obeys 0 = N ' F, s o
dr
m tan a
w
Figure 3 6 :
Crease Contours for a Right Polygonal SHGC
45
In addition, each vertex of the polygonal cross-section creates a crease on the surface w h i c h might
be visible in the image, so each crease is in effect a c o n t o u r generator (though not normally tangent
to the line of sight) (figure 36). If C(t )
= {u , v ) is a vertex, then
Q
P{sJ )
Q
= P
C Q
C G
Q
( s ) = ( t f r ( s ) , v r ( s ) , s)
0
/(s,f ) = / ( s ) = ( x
0
Q
C G
Q
,y
C G
) (s)
x y
= (u r(s)cosa
0
+ ss/na,
v r{s))
Q
xy
Since crease c o n t o u r s provide so m u c h information in the image of a Right Polygonal SHGC, we will
limit our attention to creases in the following discussion.
First, we note that tangents to crease c o n t o u r s for corresponding points intersect at a point on the
image of the axis, just as described above for c o n t o u r generators on an RSHGC. It is especially easy
to find pairs of corresponding points on a Polygonal RSHGC, as in figure 37. Suppose C{tJ
=
(u
v
and C ( f ) = ( u , v ) are vertices c o r r e s p o n d i n g to visible crease contours. Then for any s, the slope
2
2
2
of the image line joining /(s,^) and l(sJ )
2
is:
Ay
v r(s) - v r{s)
Ax
u r{s) cos a + s sin a - i / r(s) cos a - s sin a
2
2
v
A
1
(u
0
2
-
- u ) cos a
A
^—^
F i g u r e 3 7 : Tangents of Crease Contours Intersect on the Axis
w h i c h is independent of s. Thus, the slope of the line joining corresponding points on the two crease
contours is constant.
By examining either end of the PRSHGC, we can pick two vertices, find the
slope of the line joining these, and thus find pairs of c o r r e s p o n d i n g points all along the associated
crease contours. In this way, the image of the axis can be f o u n d . This will only fail in the event that
the crease c o n t o u r s have constant slope, i.e. the shape is also a Linear SHGC.
In s u c h a case,
however, as described above, the apex is easily f o u n d , and other points on the axis can be arbitrary.
46
Unfortunately, there seems to be little direct evidence in the crease c o n t o u r s to indicate the viewing
angle a .
Similarly, while the authors believe the images of the axis endpoints ought to be easily
determined, no direct method has yet been f o u n d . Perhaps this problem is deceptive in its apparent
simplicity. In any event, at the present time, outside k n o w l e d g e seems necessary in order to align t h e
image for contour analysis.
Supposing that the image has in fact been aligned in a c c o r d a n c e with our imaging model, t h e
analysis of the crease c o n t o u r s is straightforward, and is based on these formulae:
dx
—
=
ds
r r
n
0
~
ds
u
0
9
"n<
sin a
d r / d s
>
sin <j + U q COS a ( d r / d s )
C Q
So, at any point /
+
ds
ds
«frcfl
dx
dr
u cos a
C G
(s) = (*
_
C G
,y
C G
) (s),
VnXr.rM)-Unyc.rSs)c° °
s
v stn a
Q
r(s) =
Ç
Q
o
v
dr
d s
sin a ( d y
~ o~ ^o
v
r G
c o s o r ( d
/dx
C G
)
W CG
d x
)
A "special v i e w p o i n t " will not normally arise with crease c o n t o u r s , since a crease c o n t o u r is
tangent to the line of sight only if d x
C Q
/ d s = 0 and d y
C Q
/ d s = 0, i.e. in one of t w o situations:
• d r / d s = 0 and a = 0: an end view
• v
Q
= 0 and d r / d s = -tan
a / u\
Q
a vertex on the u-axis, w h i c h creates a horizontal
crease w h o s e image c a n n o t o c c l u d e (nor be o c c l u d e d by) that of any other crease.
47
4.6 Contour Generators in Two Views
generator 2
Figu re 3 8 :
Figu re 3 9 :
generator 1
C o n t o u r Generators From Two Points of View
Shadow Line is a Contour Generator for the Light S o u r c e
Suppose we have an object w h i c h is being viewed by t w o cameras, from two different directions.
What will the c o n t o u r generators from one point of view look like in t h e image as seen from the other
point of view? This is an important question for image understanding, since it bears on three different
imaging situations:
• Stereo
--
Commonly,
two
different
cameras
are
viewing
the
same
scene,
and
corresponding points in the two views must be f o u n d . However, the c o n t o u r generators
from one view generally d o not match the c o n t o u r generators from the other view (figure
38). So, w e need to determine how t h e c o n t o u r generators from one image appear in the
other image.
• Range Finders
-- A c o m m o n type of range-finder has a separate illuminator and camera,
48
and uses triangulation to determine the distance to various points in the scene. However,
points o c c l u d e d from illumination are generally not the same as the points o c c l u d e d from
the camera.
Since the boundaries of occluded areas are the projections of c o n t o u r
generators onto b a c k g r o u n d objects, w e want to know how the c o n t o u r generators as
pertains to the illuminator are viewed by the camera.
• Shadow Geometry
-- In images with strong light sources and low ambient ("diffuse") light
levels, shadows will frequently appear.
(usually
highly
visible)
"shadow
line"
If an object is illuminated, there will appear a
(also
called
"terminator")
separating
the
illuminated part of the object from the self-shaded part of the object (the part facing away
from the light source) (figure 39). This s h a d o w line is simply a c o n t o u r generator from the
point of view of t h e light source, a n d w e want to k n o w how to analyze its image as seen by
the camera.
eye
F i g u re 4 0 : T w o Angles Define the Illumination Direction
Since this last situation is highly intuitive, w e will c o u c h the following discussion in terms of
shadows. W e will use L = ( / , / , / ) to d e n o t e the direction pointing at t h e light source. T h e angle of
W
V
s
illumination, A, will represent the angle from L to S , and /? will be the dihedral angle at S from t h e S - F
plane to the S - L plane (figure 40). Then
L = ( / , / , / J = ( - cos fi sin \ , sin /? sin \ , COS \)
w
w
49
On a Right SHGC, the shadow lines must obey N ± L, i.e.
0 =
N'L
dv
= cos B sin X — +
dt
du
sin B sin X — +
dt
r
dr
r
h(t) cos X
ds
F i g u r e 4 1 : S h a d o w Line on a Right Circular SHGC
In the case of a Right Circular SHGC, as s h o w n in figure 4 1 , the shadow line condition simplifies to:
0 = sin X cos {2irt
i =
p
4-
2TT
1
cos'
- /?) + cos X
,
1
\ - cot X
2TT
dr
ds
dr
.
ds
)
This is similar to the relation between s and t for c o n t o u r generators seen from F, except for the
addition of the term $/2m
w h i c h represents the shift in f-coordinates d u e to the dihedral illumination
angle relative to the axis. On the s h a d o w line,
uJt)
= cos 2mt = - c o s /?
cot X —
ficotX
ds
v {t)
= sin 2irt = V
Q
= \lcos
2
1
- " (0
- sin ft
/8 V 1
- cot*
cot X i
2
(dr/ds)
4
2
c
fi - cos 2)3 c o f X ( d r / d s )
2
2
- s/n 2 £ c o f X ( d r / d s ) [1 - cot
2
2
2
X (dr/ds) ]
2
1 / 2
From this point, while derivation of P(s), /(s), and d x / d s is achievable, the derivation of d y / d s , and
hence d y / d x , s, r(s), and d r / d s seems to be very difficult.
The exhaustive analysis of s u c h image
contours is thus beyond our reach at present, even for the relatively simple case of Right Circular
SHGCs.
50
One c o n c l u s i o n , however, can be reached: since the interval of uncertainty of r(s) decreases with
an increase in the viewing angle a or t h e illumination angle X, w e get additional information from t h e
shadow line for Right Circular SHGC only w h e n X > a, i.e. the illumination direction is more nearly
perpendicular to the axis than is the line of sight.
Under perspective projection from the s e c o n d (illuminator) point of view, for example if t h e light
s o u r c e is a point s o u r c e rather than being infinitely far away, t h e illumination direction b e c o m e s L(s,/),
a function of s and f, and shadow-line analysis is far more difficult.
4.7 Analysis of Surface Normals
In an image understanding situation, information about surface normals may be available from
s u c h sources as photometric analysis, texture analysis, shadow geometry, or range-finder data.
Surface normals can be used to aid in the analysis of an image of an SHGC.
4 . 7 . 1 I n d i v i d u a l S u r f a c e N o r m a l s of a n R C S H G C
F i g u re 4 2 :
Knowledge of a Single Surface Normal
W e will begin with the simplest case: a single surface normal of a Right Circular SHGC (figure 42).
With the image aligned as above, let /(s,f) be the image point at w h i c h the surface normal is given.
Equation (4-4) above gives /(s,f) as a function of s, t, r(s), and a.
The surface normal itself has t w o degrees of freedom, specifying its direction, since its length is
irrelevant. The surface normal of an RCSHGC is:
51
N(s,t)
= (cos 2mt) W + (sin 2mt) V - ( d r / d s ) S
/
= ( c o s a c o s 2itt
- s/n a
dr
, sin 2mt
ds
ì
- s/n a c o s 27rf - c o s a
We can describe the direction of the normal by its gradient
n , n ) (s,f)
dr
ds
x
)
x y z
[15]: if the normal vector is Af(s,f) = (n ,
x
. then its gradient is
G(s,0 = (p,Q) (s,f)
p(s,f) =
n
n
=
z
s/n a ( d r / d s ) - cos a c o s 27rf
—
sin a cos 27rf + cos a ( d r / d s )
:
:
n ,
- sin 2irt
= -x- =
n
s/n a c o s 27rf + cos a ( d r / d s )
q(s t)
ì
(4-7)
(4-8)
z
When /(s,f) and G(s,0 are k n o w n , therefore, w e have four equations (x, y, p, q) in five u n k n o w n s (s,
f, r(s), d r / d s , a ) . Thus, if o n e of the u n k n o w n s is given, the other four can be d e t e r m i n e d .
Suppose, for example, that the viewing angle a has been determined by s o m e other means. Then
w e c a n solve for the four remaining u n k n o w n s : first p and q are used to determine t and d r / d s ; then y
is used to determine r(s); finally, x is used to solve for s. T h e resulting equations are:
s =
xq - y cos a (p cos a - sin a)
q sin a
t =
s/n
2m
r(s) =
1
—=Vk
yVk
Q
dr
ds
- 1 - sin a (p cos a - s/n a)
cos a / c
1 / 2
where
k = (p c o s a - s/n a )
2
+
q
2
Thus, from a single surface normal on a Right Circular SHGC, when the viewing angle is k n o w n , w e
can determine the position of the c o r r e s p o n d i n g point on the surface (s and f), the radius at that point
(r(s)), and the derivative of the radius function at that point ( d r / d s ) (figure 43).
52
description r(s)
image x.y.p.q
dr/ds
Figure 4 3 :
Analyzing a Single Surface Normal
4 . 7 . 2 Aligning t h e Image for A n a l y s i s of S u r f a c e Normals
The above analysis holds only w h e n t h e image is aligned in a c c o r d a n c e with these rules:
1. The image of the axis is on t h e h o r i z o n t a l x-axis in t h e image.
2. The viewing angle a is k n o w n .
3. The image of the far axis endpoint is at (0,0)
, and the near endpoint image is at (sin
xy
In general, condition (1) will not be difficult to establish, but conditions (2) and (3) may require
considerable analysis.
completely
aligned
We will call an image partially
aligned
if it c o n f o r m s to (1), and aligned
or
if it c o n f o r m s to (1)-(3). In addition, for partial alignment, w e will require that the
images of the axis endpoints are at ( 0 , 0 )
aligned image are denoted x
x p y p
and (1 > 0 )
x p y p
(figure 44). If the coordinates of a partially
and y , then they are related to aligned coordinates x and y by t h e
p
formula:
l
VVxpyp
=
*
s / n
a
'
s i n
a
The division of both coordinates is required to preserve the relative proportions of the shape and its
description.
To completely align an image w h i c h is already partially aligned, it is necessary to determine the
endpoints of the axis and the viewing angle. The axis endpoints may be completely invisible, thus
presenting an unavoidable ambiguity; also, in the event that the axis endpoints c a n n o t in fact be
determined, the only penalty is a translation and scaling of r(s) relative to s, w h i c h is probably only a
minor problem for shape recognition.
However, the determination of the viewing angle is an
interesting problem in its own right, and the penalty for guessing w r o n g may be a considerable
distortion of r(s). Therefore, w e will c o n c e n t r a t e on the problem of determining the viewing angle.
53
In real images, where surface normals of curved surfaces are being analyzed, one or more of the
following types of information are frequently being used (figure 45):
54
• texture gradients
Using texture gradients, including photometric analysis, a one-
dimensional constraint can be determined between the c o m p o n e n t s p and q of the
surface gradient [ 7 , 8 , 9 ] .
• shadow geometry
Using shadow geometry, the surface normal can be determined at
every point along the shadow line [17].
• range data
Using a range-finder, the raw distance data can be used to determine
approximate surface normals at many points (almost every point) in the image.
While analysis of t h e constraints p r o d u c e d by texture gradients is beyond the s c o p e of this paper,
the following sections explore the latter two kinds of information and how they can be used to
determine the viewing angle a.
The discussion herein will be limited to Right Circular SHGCs, for the sake of simplicity.
4 . 7 . 3 C r o s s - S e c t i o n s For a Given S u r f a c e N o r m a l
Suppose w e have a partially aligned image of a Right Circular SHGC, and s u p p o s e further that w e
are given a single surface normal. W e k n o w its position in the image, given by:
x(s,f) = r(s) cot a cos 2mt + s
r(s) sin
2irt
y(s,t) =
sin a
and the gradient of the surface normal, p and q as previously d e f i n e d . (Note that p and q are the same
for aligned and partially aligned images.)
Figu re 4 6 :
Cross-Section of a Right Circular SHGC T h r o u g h a Point
As shown in figure 46, the cross-section of t h e shape t h r o u g h this point, an arc of constant s on t h e
surface, is a circle whose projection in the image is an ellipse. We can determine its equation using x
and y. First, w e determine:
55
sin 2MT
=
y sin (J
and
r(s)
X cos 2MT
s
-
r(s) cot A
So
, (x - s) sin <x
1 = (
r{s) cos A
, y sin
+ (
x 2
)
A
r{s)
X
)
2
w h i c h is the equation of an ellipse with center at (s,0), vertical semimajor axis of r(s) / sin <x, and
horizontal semiminor axis of r(s) cos A / sin A.
We c a n n o t yet determine the specific paramaters of the ellipse yet, since w e d o not know s, r(s), or
a. However, w e do have a surface normal given by N = (p, q, 1) with p and q defined by equations
(4-7) and (4-8) above. Using x, y, p, and q, w e can solve for s, f, r(s), and d r / d s all as functions of t h e
u n k n o w n viewing angle, a, for a partially aligned image of an RCSHGC:
K
= (p cos a - s/n a )
s =
xq - y cos a
1
. -1
2TT
r(s) =
dr
ds
2
+
Q
2
(p cos a - sin
A)
«
VK
y sin A
VK
- 1 - sin a (p cos
cos
A
- s/n a )
AK}
/2
Thus, given a single surface normal in a partially aligned image, the image of the c o r r e s p o n d i n g
cross-section is an ellipse; this ellipse is determined as a function of the viewing angle, a.
4.7.4 Shadow Geometry for RCSHGCs
In figure 47, w e have a line drawing of a Right Circular SHGC, w h o s e shadow is visible and is cast
upon a flat surface. If the direction of illumination is k n o w n , w e can determine the surface normals
along the shadow line [17].
This information can be c o m b i n e d with the o c c l u d i n g c o n t o u r of the
RCSHGC, to determine the viewing angle.
We begin by partially aligning the image. Now, select a single point / = (x,y)
line. At this point, the surface gradient (p,q) can be obtained.
n
along the shadow
56
Figure 4 7 :
S h a d o w Geometry Provides Surface Normals Along S h a d o w Line
For each value of the viewing angle a, w e can c o m p u t e the implied values of s, t, r(s), and d r / d s at
/. Now w e can easily derive the following formulas for
'
C G x p y p
i
the image of the c o n t o u r generator in a
partially aligned image, from the previously seen equations for /
C Q
, the image of the c o n t o u r
generator in an aligned image:
'cGxpyp ^CG'^CG^xpyp
=
= - r(s) cos
2
d r
a
+ s
ds
ris)
CG
y
dx
cot
=
2
a (dr/ds)
2
sm a
= ( 1 / \Jsin
2
a - cos a (dr/ds) )
2
2
CG
—
ds
Now, if \tan a | > | d r / d s | , there must be some point on the c o n t o u r generator c o r r e s p o n d i n g to the
value of s at the given point / (figure 48). At this point, t h e values of x
C Q
, y
C Q
, and d y
determined by substituting for s, r(s), and d r / d s in the above equations, to yield:
y cos o (sin a + c o s a [sin a - p cos a])
2
CG ~
X
x
+
y ([q
z
+ 2] sin
2
a - 2p sin a cos a - 1) 1/2
q sin a
C Q
/dx
C G
can be
57
DYCC/DXCG
F i g u re 4 8 :
dy
dx
Corresponding Point on Tangency Contour Generator
- p sin a - cos or
C G
([q
2
C Q
+ 2] sin
2
a - 2p sin a cos a -
1)
1 / 2
with k defined as above.
possible points Peg
with slopes dy/dx indicated
F i g u re 4 9 :
At Most One Corresponding Point Exists
In an image of a Right Circular SHGC, it is guaranteed that at most one point on the o c c l u d i n g
contour (and, of course, its reflection about the x-axis) will c o r r e s p o n d to the same value of s as the
given point /. Thus, it is possible to search along the points ( x
C G
,y
C G
) for all values of a, as s h o w n in
figure 4 9 , looking for a point w h i c h lies on the outline of the shape in the image. There may be more
58
than one such point on the outline; however, at most o n e s u c h point will have the correct c o n t o u r
slope d y
C G
/dx
C G
in the image. When this point is f o u n d , the value of a w h i c h determined it i s i h e
viewing angle for image. Full alignment can n o w be c o m p l e t e d , and recovery of r(s) can p r o c e e d .
A summary of the process is this:
An image of an RCSHGC is partially aligned.
Now, given a
surface normal at a specific point, the c o r r e s p o n d i n g point (and slope) of the o c c l u d i n g c o n t o u r for
the same value of s can be determined as a function of t h e viewing angle cr. By searching the image
for these points for all values of a , looking for a crossing of the o c c l u d i n g c o n t o u r exhibiting t h e
predicted slope, the viewing angle a can be d e t e r m i n e d . The image can then be completely aligned
for more detailed analysis of the shape.
4.7.5 Range Data A n a l y s i s for RCSHGCs
F i g u re 5 0 :
Range Data Yields an Image of Surface Normals
When a range finder is used to p r o d u c e an image, the result is a set of distances to points in t h e
field of view. By fitting local tangent planes, surface orientations (e.g. gradients) can be determined
at many points in the image - almost at every pixel (but with some gaps, if a c o m m o n triangulationbased range finder is used) (figure 50). The r e d u n d a n c y provided by s u c h a large number of surface
gradients can be used to estimate the viewing angle in a partially aligned image. Here, as above, the
discussion will be limited to Right Circular SHGCs for simplicity.
4 . 7 . 5 . 1 M e t h o d of M i n i m i z i n g D e r i v a t i v e s
Suppose w e have used a range finder to determine the surface gradient at many points in a partially
aligned image of an RCSHGC.
For each value of t h e viewing angle a, w e can c o m p u t e t h e
corresponding values of s, f, r(s), and d r / d s at each surface normal, using the above formulae.
59
image x,y,p,q
description r(s)
dr/ds
F i g u re 5 1 : Two Surface Normals on the Same Cross-Section
If w e had any two surface normals c o r r e s p o n d i n g to exactly the same value of s, as in figure 5 1 , w e
could determine the viewing angle as that value of cr for w h i c h the two normals p r o d u c e d identical
values of s, r(s), and d r / d s .
(The authors speculate that this value will be unique.)
In general,
however, w e c a n n o t assume that s u c h a pair of surface normals will be present in the range data.
Instead, w e can use each pair of surface normals to indicate the value of a w h i c h produces the best
consistency between the c o r r e s p o n d i n g estimates of s, r(s), and d r / d s .
T h e n , using some voting
scheme, the most preferred value of a c a n be selected for performing complete alignment of the
image and further analysis of the shape.
image x,y,p,q
Figure 5 2 :
description r(s)
Two Surface Normals on Different Cross-Sections
Consider a pair of points ^ and / at w h i c h the surface gradients (i.e. normals) ( p
2
v
and ( p , q )
2
are k n o w n , as s h o w n in figure 52. For any value of the viewing angle a, w e can c o m p u t e s
v
d r / d s | , s , r ( s ) , and d r / d s |
s 1
2
2
2
r{sj,
. We w o u l d like to define a " c o m p l e x i t y " measurement for r(s); t h e n ,
we can determine the value of a w h i c h minimizes this complexity, and say that this value of cr is the
value indicated by our pair of surface normals.
Now, for any value of a, there exist an infinite number of functions r{s) w h i c h c o r r e s p o n d to the
60
constraints at s and s . A useful kind of complexity might be the highest-order non-zero derivative of
1
2
r(s): the function r(s) with the fewest non-zero derivatives might be called the " s i m p l e s t " .
With four constaints on r(s) (r(s) at t w o points, a n d d r / d s at t w o points), the f u n c t i o n with the fewest
non-zero derivatives will be a cubic polynomial, with three non-zero derivatives, of the f o r m :
r(s) = a s
3
+ bs
2
+ cs + d
with the coeffecients a, fc>, c, and d determined as in section 4.4.2. Thus, for each value of a, the
coefficients a, b, c, and d can be determined for the cubic polynomial function r(s) relating t w o
surface normals.
W e wish to select t h e value of a minimizing complexity, i.e.
minimizing the number of non-zero
derivatives. Thus, w e can c h o o s e a m o n g those c u b i c f u n c t i o n s s o m e "least c o m p l e x " f u n c t i o n , a n d
let the c o r r e s p o n d i n g value of a be the value indicated by our pair of surface normals. O n e measure
of complexity for these c u b i c s might be |a|; since the third derivative d r / d s
3
3
= 6a, minimizing |a| is
equivalent to minimizing the magnitude of the third derivative of r(s). Ideally, w e w o u l d hope to find a
value of a for w h i c h |a| = 0 , i.e. a quadratic function r(s) sufficient to a c c o u n t for our pair of s u r f a c e
normals.
Unfortunately, the expression for a is a c o m p l i c a t e d f u n c t i o n :
_ (s
2
- sj
(dr/ds|
s 1
+ d r / d s [ ) + 2(r{s )
s 2
A
-
r(s ))
2
Substituting for s, r(s), and d r / d s in terms of a does not yield any simplification of the above result.
Because it seems difficult to find a useful analytic f o r m u l a for a , some numeric t e c h n i q u e to minimize
|a| may be necessary.
On the other h a n d , not every pair of surface normals need be analyzed.
Only those surface
normals at points with similar values of s need be considered to provide meaningful constraint; this is
equivalent to saying that the assumption that r(s) is quadratic should only be applied locally, over
small intervals of s. As an approximation to this criterion, only surface normals at points with similar
values of x need be c o m p a r e d (figure 53).
The proposed method for the analysis of range finder images of Right Circular SHGCs is thus:
First, fit local surfaces to the range data, to yield an image of surface orientations. Partially align this
image. Now examine each pair of s u r f a c e normals at points within vertical bands of s o m e size (i.e.
with similar x-coordinates). For each s u c h pair, determine the value of a w h i c h minimizes |a| (i.e. the
61
A
>
<
V
-
- t
T /
\
N
t
V
F i g u re 5 3 :
Only C o m p a r e Surface Normals with Similar Values of x
magnitude of the third derivative of the c u b i c approximation to r(s)); that pair of surface normals will
" v o t e " for this value of a, possibly weighted by s o m e c o n f i d e n c e measure s u c h as the c o m p u t e d
value of | s
1
-
s |.
2
Find the value of a with the strongest support; assume this is the true viewing
angle. Now, the image can be completely aligned for more detailed reconstruction of the RCSHGC
description.
While this method involves considerable assumptions about the true nature of the object being
observed, it is still significant that the analysis is attempting to minimize a viewpoint-independent
measure of the complexity of the object.
4 . 7 . 5 . 2 M e t h o d of I n t e r p o l a t i n g N o r m a l s
There is an alternative a p p r o a c h to analyzng range data, w h i c h avoids the approximation to r(s) in
the previous method, relying instead on a smoothing assumption. In this t e c n h i q u e , processing relies
on a surface normal interpolation f u n c t i o n , capable of determining the (approximate) gradient of the
surface normal anywhere in the object's image, by interpolating between k n o w n values of the
gradient.
With s u c h a function available, w h o l e ellipses can be fit to the data. Consider a surface normal
w h o s e gradient (p,q) at a point (x,y) in a partially aligned image is k n o w n . Now, for each value of the
(unknown) viewing angle a, not only is the elliptic
image of the cross-section t h r o u g h
(x,y)
determined, but also the gradient at each point on this ellipse is determined (figure 54). For, at any
value of f, the c o r r e s p o n d i n g image point (x ,y ) and gradient ( p , 9 ) are given by:
t
t
t
t
62
F i g u r e 5 4 : Fitting Ellipses to Interpolated Surface Normals
y cos
(/c
A
1/2
cos
2IRT
+ sin
A -
p cos
A)
x =x +
t
yk
U2
p = -tan
t
sin 2MT
cos
1
cos
:
A A (1
-/c
+
TJZ
sin
A [P cos
w h e r e k = (p c o s a -
s/n a)
A -
sin
A +
k*
cos
u
2IRT])
sin 2MT
1 / 2
1 + sin A (p c o s a - sin A + / c
x
2G
2
+ q.
2
1 / 2
c o s 27rf)
(These are obtained from the formulae in section 4.7.3,
substituting t h e equations for s, r(s), and d r / d s into those for x, y, p, and qr.)
At any s u c h point (x ,y ), t h e interpolated gradient (p q.)
v
can be determined from the interpolation
function, using t h e image data in that n e i g h b o r h o o d .
An error measure c a n then be derived t o
t
t
indicate how far the (interpolated) d a t a is from the d a t a predicted from t h e given point a n d t h e
assumed viewing angle a . O n e s u c h error measure might be t h e angle between the predicted normal
N = (p ,Q ,1) and t h e interpolated normal N = ( p ^ , ! ) , given by:
-
{
t
}
t
*
E(a,F) = cos
N ' N.
— = cos'
1
l|Af ||
t
ll^ll
1
1
5
(P +
2
t
pn. + q q + 1
5— \ / o A
5
Q
+ D 1 / 2 ( P + <7
2
X
2
T7T
2
+D
1 / 2
63
For a given value of o~, an error measure E(a) might be c o m p u t e d by taking the average of E(a,f)
over some range of values of t between approx. 1/4 and 3 / 4 (i.e. on t h e near side of the object). Note
that, for s o m e values of f, the object's surface might be o c c l u d e d . T h e point (x,y) used to determine
all the values of £(cr) might then " v o t e " for that value of a w h i c h minimizes £(<r).
This method for analyzing range finder images is this: First, fit local surfaces to the range data, to
yield a (sparse) image of surface orientations.
Partially align this image.
Now, for each surface
normal in some set of surface normals, determine the value of <r w h i c h minimizes £ ( a ) , the average
angle between the surface normals prediced by the selected normal and the interpolated normals
determined from the surrounding data.
value of a.
Each of the selected normals will " v o t e " for the indicated
Select the value of a receiving the strongest support; assume this is the true viewing
angle. Now, the image can be completely aligned for more detailed analysis.
This method makes weaker assumptions about r(s) than the previous technique, and probably
requires less c o m p u t a t i o n . However, it does d e p e n d u p o n the existence of an interpolation function
for surface normals, and might also be more sensitive to noise in the data.
64
5. Conclusions
In this paper, w e have presented several kinds of results.
First, there have been specific
observations and formulae relating to:
• Parameterization of shapes, in particular Straight H o m o g e n e o u s Generalized Cylinders.
• Theorems about equivalence of descriptions, and behavior of surface normals, for
SHGCs.
• Formulae for several attributes of SHGCs, s u c h as coordinates of points, surface
normals, and images of these.
• Observations about c o n t o u r generators, s u c h as planarity and points of singularity.
These observations were derived for SHGCs, but in general will apply to more c o m p l e x
shapes as well.
• Analysis techniques for tangency c o n t o u r s and surface normals of Right Circular SHGCs
and Right Polygonal SHGCs, including line d r a w i n g s , images with s h a d o w s , and rangefinder data.
This w o r k has been theoretical, u n a c c o m p a n i e d so far by implementation; it might be t h o u g h t of as
preliminary, exploratory work aimed at outlining goals for eventual implementation efforts.
In addition to these specific observations, a methodology has been p r o d u c e d and demonstrated for
analyzing the imaging properties of shapes described parametrically:
1. Parameterize the shape using s o m e set of functions.
2. Parameterize the surface of the shape by s and t.
3. Assign a local (surface-based) c o o r d i n a t e system and find points P(sJ) a n d surface
normals A/(s,f).
4. Assign a global (object-centered) coordinate system and convert P and N to global
coordinates.
5. Invoke a tangency argument with viewing direction F ( i n global coordinates) to determine
the condition on c o n t o u r generators.
6. Imbed the object in the world coordinate system (x-y-z) with F = Z .
7. Determine equations for the images of points /(s,/), note the specialization to t a n g e n c y
contours, and solve for shape parameters in terms of image c o n t o u r s .
8. For surface normal analysis, convert Afto world coordinates, solve for gradient (or other
representation), solve for shape parameters at arbitrary point in terms of / and image of N.
This methodology is being applied to shapes other than SGCs, in work in progress by the authors.
Finally, it has been discovered that there are several problems w h o s e solutions are very difficult to
grasp intuitively; analysis has revealed some important facts not suspected by the authors based on
their own intuition. These problems include:
• The conditions for uniqueness or equivalence of shape descriptions.
65
• The planarity (or lack thereof) for c o n t o u r generators.
• The c o n d i t i o n s under w h i c h c o n t o u r generators exhibit singularities.
• The characterization of the image c o n t o u r s in pictures of even relatively simple shapes.
5.1 Future Work
Although intended applications must g u i d e the direction of future development of s u c h theoretical
w o r k as this, there are several open areas already revealed by this research. The analysis of c o n t o u r s
has been carried out in some detail for Right Circular SHGCs; it* w o u l d be useful to extend these
results to other types of Generalized Cylinders as well. In addition, no specific algorithms have been
suggested for analyzing arbitrary image c o n t o u r s , even if the object being viewed is of a k n o w n type.
T h e results will all be more useful w h e n extended to images under perspective projection.
The
analysis of c o n t o u r generators from t w o viewpoints s h o u l d be able to provide at least some cues as to
the solid shape. All of these suggestions pertain to generalizing the results in this paper; in situations
w h e r e the authors have been d a u n t e d by mathematical tractability, the use of some symbolmanipulation system s u c h as MACSYMA might allow for further achievement.
Of equal interest, some considerable s p a c e has been devoted in this paper to the description of
images and image analysis for Right Circular SHGCs (i.e. solids of revolution), yet several problems
are still unsolved.
These problems include characterizing and analyzing shading and highlights,
analyzing the position of the shadow line in detail, and finding the images of the axis endpoints in an
image. This kind of very detailed analysis of simple shapes may yield quite interesting insights for
image understanding in general.
Finally, implementation is important as an e m b o d i m e n t of these theoretical results, and for
providing feedback for directing further analysis. An important problem to deal with will be that of
d e c i d i n g w h e t h e r an object is in fact a member of a specific shape class, a problem w h i c h is not
addressed in this paper.
5.2 Acknowledgements
J o h n Kender at C o l u m b i a University provided several useful ideas and helped make the basic
definitions more rigorous. Chris B r o w n at Rochester suggested the torus problem, w h i c h led to the
formalization of the methodology outlined in the Conclusions.
66
6. Bibliography
[1]
Ballard, D. H. and B r o w n , C. M.
Computer
Vision.
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[2]
Binford,T.O.
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Visual perception by computer.
In Proc. IEEE Conf. on Systems and Control.
[3]
Miami, December, 1 9 7 1 .
Brooks, R. A.
Symbolic reasoning a m o n g 3-D models and 2-D images.
Artificial
Intelligence
17:285-348,1981.
Special volume on c o m p u t e r vision.
[4]
Davis, H. F. and Snider, A. D.
Introduction
to Vector Analysis, 3rd ed.
Allyn and B a c o n , Boston, 1975.
[5]
Hilbert, D. and Cohn-Vossen, S.
Geometry
and the
Imagination.
Chelsea Publishing Co., New York, 1952.
[6]
Hollerbach, J . M.
Hierarchical
Shape Description
of Objects by Selection
and Modification
of
Prototypes.
AI-TR 346, MIT, November, 1975.
[7]
Horn, B. K. P.
Understanding Image Intensities.
Artificial
[8]
Intelligence
8:201 - 2 3 1 , 1 9 7 7 .
Kanade, T. a n d Kender, J .
Mapping Image Properties into Shape Constraints: Skewed Symmetry, Affine-Transformable
Patterns, and the Shape-from-Texture Paradigm.
In IEEE Workshop
on Picture Data Description
and Management,
pages 130-135. Asilomar,
CA, August, 1980.
[9]
Kender, J .
Shape From Texture: A Computational Paradigm.
In B a u m a n n , L. S. (editor), Proc. ARPA IUS Workshop,
pages 134-138. April, 1979.
67
[10]
Kuhl, F. P. and Giardina, C. R.
Elliptic Fourier features of a closed c o n t o u r .
Computer
[11]
Graphics
and Image Processing
18:236-258,1982.
Marr, D.
Analysis of o c c l u d i n g contour.
Proc. Royal Society
[12]
of London
B-197:441-475,1977.
Marr, D. and Nishihara, H. K.
Representation and recognition of t h e spatial organization of three-dimensional shapes.
Proc. Royal Society of London
[13]
B-200:269-294,1978.
Nevatia, R.
Machine
Perception.
Prentice-Hall, E n g l e w o o d Cliffs, NJ, 1982.
[14]
Rubinstein, Z.
A Course in Ordinary
and Partial Differential
Equations.
Academic Press, New York, 1969.
[15]
Shafer, S. A., Kanade, T., and Kender, J . R.
Gradient S p a c e Under O r t h o g r a p h y and Perspective.
In IEEE Workshop
on Computer
Vision: Representation
and Control,
pages 26-33. August,
1982.
[16]
Shafer, S. A.
A T a x o n o m y of Generalized Cylinders.
IEEE Transactions
on Pattern Analysis
and Machine
Intelligence
, 1983.
Submitted for C o r r e s p o n d e n c e s .
[17]
Shafer, S. A. and Kanade, T.
Using S h a d o w s in Finding Surface Orientations.
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[18]
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Generalized cones from serial sections.
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15(2): 154-166, February, 1 9 8 1 .
68
I. Symbols
Constants and coordinates are s h o w n in italics (s, W), functions are in boldface italics (r, AO.
Scalar values are in lower-case (f, u ),
c
vector or n-tuple values in upper case (G, P), angles in Greek
letters (a, a).
Generalized Cylinders
G
Generalized Cylinder
Classes of Shapes
GC
Generalized Cylinders
LSHGC
Linear SHGCs
SGC
Straight Generalized Cylinders
RSHGC
Right SHGCs
SHGC
Straight H o m o g e n e o u s GCs
CSHGC
Circular SHGCs
PSHGC
Polygonal SHGCs
Components of a Generalized Cylinder
A(s)
axis f u n c t i o n
r(s)
radius function
a
angle of inclination
E(s,t)
envelope f u n c t i o n
C{t)
contour function
u
c c
c o m p o n e n t s of c o n t o u r
v
Coordinates of a Generalized Cylinder
s
distance along axis
S
axis vector
t
distance along c o n t o u r
U
horizontal on cross-section
u,v
cross-section coordinates
V
vertical on cross-section
w
normalized u
W
normalized horizontal
Measures of a Generalized Cylinder
P(s,f)
p ,p ,P
w
v
m
s
point on surface
W(s,f)
coordinates of point
n ,n /7
slope of r(s) for LSHGC
h(t)
x
yJ
surface normal
z
c o m p o n e n t s of normal
c o n t o u r Wronskian
Projections of Generalized Cylinders
viewing direction
X
horizontal in image
viewing direction coordinates
Y
vertical in image
a
viewing angle ( F t o S)
Z
direction of eye
L
illumination direction
x,y,z
w o r l d and image coordinates
'w'Vs
ilium, direction coordinates
/(s,0
image of point on SHGC
F
W
V
s
69
illumination angle
x(s,t),y(s,t)
image coordinates of SHGC point
dihedral illumination angle
G = (p,g)
surface gradient
70
II. Properties of SHGCs
11.1 Straight Homogeneous GC: SHGC
Definition
G = ( A C, r, a)
axis A(s) linear
c o n t o u r C(f) = {u ,v )(t)
c
Q
continuous
radius r(s) differentiable
inclination a constant
Description
GC with straight axis, all cross-section planes parallel, all
cross-sections of same shape but varying size.
Point on the Surface
P(s,f) = (u {t) r{s) sin a , v (t) r(s), s + u (t) r(s) cos a)
Q
Q
Q
Contour Wronskian
h{t) = u it)—£dt
c
c
- v
c
(t)—£df
hit) = 0 iff the SHGC can be imbedded in a plane.
Surface Normal
N(s,f) = ( hit) cos a
dr
dv
+ — ^
ds
df
n
du
- sin a — 7 * - ,
df
r
%
- h{t) sin a
dr
—
ds
71
Il.2 Right SHGC: RSHGC
Definition
G = ( A C, r, t t / 2 )
A, C, r as for SHGC.
Description
SHGC with cross-section planes perpendicular to axis.
Special Properties
There is no preferred direction for u and v axes on cross-section planes, so imaging
geometry can be simplified. Also, U = W.
Point on t h e Surface
P(s,0 = (u (0 r(s), v (t) r(s), s)
c
Q
Contour Wronskian
Same as SHGC - no simplification.
Surface Normal
A/(s,f) = (
Image of a Point
/(s,0 = (x,y) ( s , 0
xy
= ( i i ( f ) r(s) cos a + s sin a, v (t) r f s ) ) ^
c
c
World Coordinates
P = (w,
s)
w v s
= (w cos a + s s/'n a , v, - w s/n a + s cos a).
Oblique View Contour Generator Condition
0 = sin a
dt
+ h(t) cos a
dr
ds
Oblique View Contour Generator Domain
Intractable.
Oblique View Contour Generator Point
Intractable.
Oblique View Contour Generator Planarity
Intractable.
73
1.1.3 Linear SHGC: LSHGC
Definition
G = ( A C, r, a , 0)
r(s) = m ( s - s )
0
A C, a as for SHGC.
Description
SHGC with linear radius f u n c t i o n . Apex is at s = s .
Q
Special Properties:
• Slant Theorem:
Can describe an LSHGC with cross-section planes at any orientation (as
long as they cut completely t h r o u g h shape).
• Pivot Theorem:
Can describe an LSHGC with any axis passing t h r o u g h apex (excluding
projection of shape t h r o u g h apex).
• Corresponding
Normal Theorem:
dN/ds
= (0,0,0)
Point on t h e Surface
P(s,f) = (m (s - s ) u (t) sin a , m (s - s ) v (t),
Q
Q
Q
s + m (s-s )
Q
Q
u (t)
c
cos a)
Contour Wronskian
Same as SHGC
no simplification.
Surface Normal
A/(s,f) = \ m h{t) cos a + —
dt
sin a —
-
m h(t) sin a )
dt
Right LSHGC:
Image of a Point
/(s,0 = (x,y) ( s , 0
x y
= (m (s - s ) u ( f ) cos a + s s/n a, m (s - s ) ^ ( f ) ) ^
Q
c
Oblique View Contour Generator Condition
dv
0 = sin a —^— + m h(t) cos a
dt
r
Q
74
Oblique View Contour Generator Domain
i satisfying above c o n d i t i o n .
Q
Oblique View Contour Generator Point
Intractable.
Oblique View Contour Generator Planarity
Always.
75
11.4 Circular SHGC: CSHGC
Definition
G =
r, a)
(A, C,
C(t) = {u ,v )
Q
(t) = (cos 2mt, sin 2mt)
c
A r, a as for SHGC.
Description
SHGC with circular cross-section.
Point on the Surface
P(s,f) = (r(s) sin a cos 2mt, r(s) sin 2mt, s + r(s) cos a cos 2mt)
Contour Wronskian
M O = 2tt
Surface Normal
,
dr
A/(s,f) = ( 27T (cos 27rt + cos a
), 2tt sin a sin 2mt,
ds
- 2m sin a
dr
x
ds
)
Right CHGC:
Image of a Point
/(s,f) = (x,y) (s,f)
xy
= (r(s) cos a cos 2mt + s sin a, r(s) sin
2mt)^
xy
Oblique View Contour Generator Condition
i =
1
27T
cos
-1 /
( - cot a
d
r
ds
\
j
Oblique View Contour Generator Domain
dr
I—
ds
I < I tan a |
Oblique View Contour Generator Point
P(s,0 = P
C G
( s ) = {cota
r ( s ) — , r(s) V1
ds
Oblique View Contour Generator Planarity
Difficult to evaluate.
- cot
2
a (dr/dsf
, s )
76
Oblique View C o n t o u r
W >
S
( CG'W( >xy
=
X
S
cos a
,
= ( - r(s)
s/n a
dr
or
2
= ( 1 / \Jsin
I
T
a (dr/ds)
x
)
ds
a - cos a (dr/ds) )
2
dx
/
+ s sin a , r(s) V 1 - cot
2
2
C Q
—
ds
Oblique View C o n t o u r Analysis
o —*
s/n a
r{s) = y
—
ds
C G
(s) V l
= (sin
a/
+ cos a ( d y ^ / d x ^ )
2
V 1 + cos a ( d y
2
C
r r
G
/dx
C
2
r r
G
)
2
)
dx
Image of a S u r f a c e Normal
N(s,t) = ( c o s a c o s 2ttJ - s/n a
, sin 2tit,
ds
Surface Gradient
G(s,f) = (p,q) (s,f)
p(s,t)
=
g(s,f) =
s/'n a ( d r / d s ) - c o s <r cos 2-at
sin a cos 2irt
+ cos or ( d r / d s )
- sin 2mt
sin a cos 2mt + c o s a ( d r / d s )
- sin a c o s 2irt
- cos a
)
. ds
77
Surface Normal Analysis
k = (p cos a - sin a)
2
+ q
2
xg - y cos a (p cos a - s/n a)
s =
*
t =
q sin a
2m
r(s) =
dr
ds
• -1
1
sin '
q
Vk
—
yVk
- 1 - sin a (p cos a - sin a)
cos a
kV
2
78
11.5 Polygonal SHGC: PSHGC
Definition
G = ( A C, r, a)
C{t) = {u ,v )
c
(0 is piecewise linear
Q
Between vertices, C(t) - (u v )
Qi
(f) = (mj
c
+ 6 , m f +
y
v
b)
y
A r, a as for SHGC.
Description
SHGC with polygonal cross-section.
Special Properties
Corresponding
Normal
Theorem:
dN/dt
= (0,0,0) on face.
Point on the Surface
P(s,f) = ({mj
+ bj
r(s) sin a , {mj
+ bj
r(s), s + (mj
+ bj
r(s) cos a)
C o n t o u r Wronskian for a Face
h(t) = /77 d v
mb
u
u
y
Surface Normal for a Face
N(sj)
,
= ( m, + (m b
v
dr
- m b ) cos a — ,
u
u
v
- m sin a , ( m t o - m ù ) sin a
u
u
v
v
u
Right PSHGC:
Image of a Point
/(s,f) = (x,/) ( s , 0
= ((#n f + b ) r(s) cos a + s s/n a, ( m f + b )
xy
u
u
Oblique View C o n t o u r Generator C o n d i t i o n for a Face
dr
ds
m tan o
y
m ö
u
v
-
m ö
v
u
Oblique View Contour Generator Domain for a Face
s satisfying above c o n d i t i o n
Q
Oblique View Contour Generator Point for a Face
Intractable
y
y
r(s))^
dr
—
79
Oblique View Contour Generator Planarity for a Face
Difficult to determine
Image of a Point on a Crease
l{sj )
0
= /
C G
(s) = (*
C G
,y
C G
) (s)
x y
= {u r(s) cos a + s sin a, v
Oblique View Contour Analysis for a Crease
v sin a
Q
r(s) =
C
df
ds
G
s/n g ( d y
v
0
C G
/dx
C G
)
- u cos a ( d / c Q / d x ^ )
Q
Q
Q
r{s))^
A Taxonomy of
Generalized Cylinders
29 November 1982
Steven A. Shafer
Computer
Science
Department
Carnegie-Mellon
University
Abstract
A number
of researchers
representation,
but different
standardization
subclasses
used
people
of the meaning
a proposal
of interesting
corresponding
have defined
subclasses,
Finally, this terminology
describe
common
mathematical
these terms in different
terms,
for a definition
and
is presented,
classes.
(or generalized cones,) for
and
research
results
ways.
shape
There has been
are sometimes
limited
no
to
terminology.
is made
terminology
generalized cylinders
of these
for which there is no
In this paper,
number
have
for
of Generalized
the description
including
is used to compare
Cylinder,
of the
abbreviations
"ends"
for the definition
of such
shapes.
A
of these
shape
the shape classes used in past research,
and to
shapes such as "solid of
for the names
of a
revolution".
This research was sponsored by the Defense Advanced Research Projects Agency (DOD), ARPA
Order No. 3597, monitored by the Air Force Avionics Laboratory Under Contract F33615-81 -K-1539.
The views and conclusions contained in this d o c u m e n t are those of the authors and should not be
interpreted as representing the official policies, either expressed or implied, of the Defense Advanced
Research Projects Agency or the US Government.
T a b l e of Contents
1 . Introduction
2. The Taxonomy
2.1 Generalized Cylinders
2.2 Subsets of GC
2.2.1 Restricting t h e Axis
2.2.2 Restricting t h e Cross-Section Planes
2.2.3 Restricting t h e Cross-Section
2.2.4 Restricting t h e Transformation Rule
2.3 Supersets of GC
2.4 Describing t h e Ends of a Generalized Cylinder
3 . A p p l i c a t i o n of t h e T a x o n o m y
3.1 C o m m o n Mathematical Shape Classes
3.2 Shape Classes Used in Past Research
4. Summary
4.1 A c k n o w l e d g e m e n t s
5. Bibiography
List of Figures
F i g u re 1 : T a x o n o m y of Generalized Cylinders
1
1. Introduction
Since the introduction of generalized
cylinders
by Binford in 1971 [3], a number of researchers
have used this s c h e m e for representation of solid shapes [ 1 , 6 , 7, 8, 9 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 ] .
However,
different people have used the term to refer to somewhat different classes of shapes; indeed, an
alternate term, generalized
cone, has been used by some. There has been no clear-cut definition of
these terms w h i c h might be universally a c c e p t e d , nor has there even been a suitable terminology for
expressing the different shape classes referred to by different researchers using these terms.
In this paper, a set of terminology is introduced for describing generalized cylinders and several
subsets and supersets thereof, along with a notation for naming these shape classes.
The
presentation takes the form of a t a x o n o m y of shapes.
It is hoped and intended that this set of terms will be sufficiently general to allow the definition and
comparison of the various shape classes used in past and present research efforts. To illustrate the
sufficiency of this terminology, the shape classes used by several researchers are described using the
terms presented herein.
Finally, the terms for several c o m m o n mathematical classes, s u c h as
cylinders and solids of revolution, are presented, as defined in this taxonomy.
It is beyond the s c o p e of this w o r k to address s u c h issues as t h e parameterization of specific
shapes, or the existence of multiple descriptions of the same shape. However, this terminology may
contribute to the future investigation of these problems by facilitating a precise formulation of the
questions involved.
2
2. The Taxonomy
T h e taxonomy begins with the definition of Generalized Cylinders.
This definition involves four
parts: by imposing suitable restrictions on each part, a number of subclasses are defined. By allowing
generalizations, supersets are defined.
T h e terminology uses Generalized Cylinders as a basis; this class of shapes may be abbreviated
"GC".
Each subclass introduces a new adjective (e.g. " C i r c u l a r " ) ; the name of that shape class is
thus "Circular Generalized C y l i n d e r s " . Frequently, it will be desirable to cascade s u c h adjectives, as
in " R i g h t Circular Generalized C y l i n d e r s " . Each adjective is normally abbreviated with a single letter;
the name of the class may be abbreviated in part or in full, as desired, for example, " R C G C " , " R i g h t
C G C " , "Circular R G C " , and " R i g h t Circular Generalized Cylinders" are all terms for the same shape
class. (The s e c o n d and third terms might be useful for emphasizing a particular property of the class
in some situations.)
T h e names of shape classes will be capitalized w h e n used to refer to the
definitions presented herein.
As noted in the taxonomy, certain restrictions imply others.
In s u c h cases, the redundant terms
may be omitted. For example, all Circular GCs are also H o m o g e n e o u s GCs. Thus, "Circular G C " is
another name for "Circular H o m o g e n e o u s G C " .
In addition to the basic t a x o n o m y of shapes, definitions are presented for describing t h e ends of a
Generalized Cylinder. These allow a somewhat finer grain of description than the basic terminology.
2.1 Generalized Cylinders
A Generalized
Cylinder
(GC) is defined by four parts:
• There is a s p a c e curve, calfed the axis of the shape.
• At each point on the axis, at s o m e fixed angle to the tangent to the axis, there is a
cross-section
• O n . each
cross-section
plane defined.
such
cross-section
plane, there
a planar
curve w h i c h
constitutes
the
of the object on that plane.
• There is a transformation
rule w h i c h specifies the transformation of the cross-section as
the cross-section plane is swept along the axis. This rule always imposes (at least) the
constraint that the cross-section c h a n g e s smoothly.
3
The surface of the object is the union of the cross-sections.
than a descriptive
This is a generative
definition, rather
definition, of Generalized Cylinders.
T h e term Generalized
Cone may be considered s y n o n y m o u s with Generalized Cylinder, since there
has been no clear distinction between the t w o in the past.
2.2 Subsets of GC
The various subclasses of Generalized Cylinders are formed by imposing restrictions u p o n the four
parts of the definition above.
WGC
SGC
TGC
RGC
\
Skewed GC
GC
CGC
/ \
Layered GC
PGC
OGC
HGC
BLGC
V
I
LGC
UGC
F i g u r e 1 : Taxonomy of Generalized Cylinders
The taxonomy is illustrated in Figure 1, in w h i c h the nodes are shape classes. The lower classes
are subsets of those above t h e m . Adjacent branches are not generally mutually exclusive; thus, the
figure is a sort of abbreviated set diagram rather than a strict " t r e e " .
4
2.2.1 Restricting the Axis
Straight
GC (SGC) (opposite:
Curved)
A Straight Generalized Cylinder has a line segment in s p a c e for its axis.
Toroidal GC (TGC) A Toroidal Generalized Cylinder has a closed curve in s p a c e as its axis.
2.2.2 Restricting the Cross-Section Planes
Right GC (RGC) (opposite:
Oblique)
A Right Generalized Cylinder has cross-section planes perpendicular to the
tangent to the axis. (In a Straight GC, perpendicular to t h e axis itself.)
2.2.3 Restricting the Cross-Section
Circular
GC (CGC) A Circular Generalized Cylinder has cross-sections w h i c h are all circles: Circular
GCs are thus also R o u n d e d and Closed (see below). T h e axis is generally defined
to pass t h r o u g h the centers of the circles, in w h i c h case the Circular GC is also
H o m o g e n e o u s (see below).
Polygonal
GC (PGC) (opposite:
Rounded)
A Polygonal Generalized Cylinder has cross-sections w h i c h are piecewise linear.
Open GC (OGC) (opposite: Closed)
An Open Generalized Cylinder has cross-sections w h i c h are not Jordan curves,
i.e. the cross-section is not a simple closed curve. Usually, an Open GC will have
a cross-section w h i c h is s o m e arc on the cross-section plane; the shape is t h u s a
piece of a warped sheet in space.
2.2.4 Restricting t h e T r a n s f o r m a t i o n Rule
Homogeneous
GC (HGC) (opposite:
Heterogeneous)
A H o m o g e n e o u s Generalized Cylinder has a transformation rule w h i c h allows only
uniform scaling of the cross-section as it is swept along the axis; thus, all crosssections have the same shape, but may vary in size.
Linear GC (LGC)
A Linear Generalized Cylinder is a H o m o g e n e o u s Generalized Cylinder in w h i c h
the size of the cross-section is proportional to distance along the axis, measured
from some point called the apex of the shape.
Uniform GC (UGC) A Uniform Generalized Cylinder is a Linear Generalized Cylinder in w h i c h all
cross-sections are identical in size as well as shape.
5
Bilinear
GC
(BLGC)
A- Bilinear Generalized Cylinder is a Generalized Cylinder in w h i c h the crosssection size varies with distance from the apex, but the two orthogonal directions
on the cross-section planes have different factors of proportionality, i.e. the crosssection shape is scaled differently in t h e two directions.
2.3 Supersets of GC
The following are generalizations of Generalized Cylinders:
Warped GC (WGC) A Warped Generalized Cylinder is defined in a similar manner to a Generalized
Cylinder, but the cross-sections need not be planar. T h e cross-sections are thus
space curves defined relative to t h e cross-section planes, but need not be
contained within t h e m .
This is t h e class of shapes originally defined
as
"generalized c y l i n d e r s " by Binford [3]. It may be interesting to discover whether
each shape that is a Warped GC may also be described as a GC (aside from the
obvious difference in the planarity of the ends of the shape); if so, WGC = GC.
The author conjectures that this is true.
Skewed GC
A Skewed Generalized Cylinder is defined similarly to a Generalized Cylinder, but
the cross-section planes need not all be at the same angle with respect to t h e axis.
This allows "rotational s w e e p s " of the cross-section planes, as well as translation
of the cross-section planes along the axis.
Layered GC
A Layered Generalized Cylinder is a Skewed GC in w h i c h the cross-section
planes, while not at a fixed angle to the axis, are parallel to each other.
This
situation arises, for example, w h e n examining serial sections of Curved GCs.
Layered GCs d o not actually constitute a superset of GCs: a Layered GC with a
straight (linear) axis is a Straight GC; whereas a Layered GC with a non-linear axis
is a Skewed GC but not a GC.
2.4 Describing the Ends of a Generalized Cylinder
Each end of a Generalized Cylinder may be characterized, independently of the characterization of
the shape as a whole.
Blunt
An end of a GC is Blunt if the surface normals c o n v e r g e in orientation to be
perpendicular to the final cross-section plane at that end (i.e. the surface
smoothly a p p r o a c h e s the final cross-section).
Flat
An end of a GC is Flat if it is not Blunt, and the final cross-section is a closed
6
curve. The surface normals are d i s c o n t i n u o u s along the final cross-section, i.e.
between the side(s) of the shape and the Flat e n d .
Chiseled
An end of a GC is Chiseled if it is not Blunt, and the final cross-section is an arc.
T h e surface normals are discontinuous along the final cross-section arc.
Pointed
An end of a GC is Pointed if it is not Blunt and if the final cross-section consists of
a single point. The surface normal at this point is u n d e f i n e d .
It is especially interesting to note that a Linear GC always has one end Flat, and the other end may
be either Flat or Pointed (but not Blunt or Chiseled).
A Uniform GC always has two Flat ends.
A
Bilinear GC may have one Flat and the other Flat, Chiseled, or Pointed; or it may have both ends
Chiseled. The ends of other GCs may be of any type.
7
3. Application of the Taxonomy
3.1 Common Mathematical Shape Classes
This terminology can be used to define several c o m m o n mathematical shape classes. In general,
the terms used to restrict those mathematical classes are the same terms presented here for
restricting the class of Generalized Cylinders (i.e. Right, Circular).
cylinder
A cylinder
is a Straight Uniform Generalized Cylinder, i.e. SUGC.
Subclasses:
right, circular.
cone
A cone is a Non-Uniform Straight Linear GC (Non-Uniform SLGC), with one end
(the apex) Pointed. Subclasses: right, circular.
frustrum
A frustrum
is a Non-Uniform Straight Linear GC (Non-Uniform SLGC) with both
ends Flat.
solid of revolution
A solid of revolution
parallelopiped
A parallelopiped
is a Straight Right Circular Generalized Cylinder (SRCGC).
is a Straight Uniform Polygonal Generalized Cylinder (SUPGC).
Subclasses: right.
pyramid
A pyramid
is a Non-Uniform Linear Polygonal Straight Right Generalized Cylinder
(Non-Uniform LPSRGC), with o n e end Pointed.
torus
A torus is a Uniform Right Toroidal Generalized Cylinder (URTGC). T h e shape of
the axis and the cross-section must be described separately; in this terminology, a
"Circular
URTGC"
implies that the cross-section
is circular, but does
not
constrain the axis to be a circle.
2 - 1 / 2 D shape
The two-and-a-half-D
shape class used in C A D / C A M is somewhat loosely defined,
but usually consists of two polygonal faces linked by quadrilateral or triangular
facets [5].
This corresponds to Polygonal Straight GCs with Flat ends, usually
Right, but possibly Oblique and with nonparallel ends. Sometimes, shapes with
curved axes are also called 2 - 1 / 2 D shapes.
This type of shape should not be
confused with the 2 - 7 / 2 D sketch, a representational s c h e m e used by Marr [9].
8
3.2 Shape Classes Used in Past Research
T h e terms generalized
various shape classes.
cylinder
and generalized
cone have been used in past research, to refer to
Binford [3] de fined "generalized c y l i n d e r s " to be the class here called
W a r p e d GCs. Agin [1] defined the term similarly, but his results were limited to Right Circular Linear
GCs, i.e. those with circular cross-section perpendicular to the axis, a n d linear scaling of t h e crosssection, but with any s p a c e curve as axis. Hollerbach [7] defined "generalized c y l i n d e r " as RGC, and
limited his attention to HRGCs w h o s e axes were planar (but not necessarily straight).
Ballard and
Brown [2] defined "generalized cylinder" to be Closed GC, i.e. GCs with cross-sections w h i c h are
closed arcs; Shani [13] defined "generalized c y l i n d e r s " as Closed RGCs, i.e. GCs with cross-sections
w h i c h are closed arcs and lie at right angles with the axis.
Nevatia and Binford [11], and Miyamoto and Binford [10], used the term "generalized c o n e " to
indicate Right GCs, i.e. GCs with cross-sections perpendicular to t h e axis. Marr and Nishihara [9]
used the same term to indicate H o m o g e n e o u s GCs, those w h o s e cross-sections have constant shape
but varying size, but Marr in [8] a d d e d the restriction that t h e axis be straight (i.e. SHGCs). W o o d h a m
[15] also referred to SHGCs as "generalized c o n e s " .
In describing the A C R O N Y M p r o g r a m , Brooks [6] uses the term "generalized c o n e " , defined (as
here) to be GC. The p r o g r a m ' s repertoire of shapes, however, is limited to those GCs w h i c h are:
1. Either Circular or Polygonal.
2. Bilinear (including Linear and Uniform as special cases).
3. Either Straight; or Right Toroidal with circular axis.
However, Brooks' co-researcher
Binford [4] uses the term
"generalized
cylinder"
to refer
to
ACRONYM'S shape descriptions.
S o r o k a [ 1 4 ] defined "Elliptical C o n e s " to be S B L G C with elliptical cross-sections; i.e. GCs with
straight axes and elliptical cross-sections scaled independently (but linearly) in t w o
orthogonal
directions. Soroka allowed both Right and Oblique GCs.
In research related to this paper, Shafer and K a n a d e [ 1 2 ] use the terminology described herein.
Their attention is limited to Straight H o m o g e n e o u s GCs, for w h i c h they present fundamental theorems
derived from the definition, and an analysis of image c o n t o u r s and surface normals for SHGCs.
9
4. Summary
This paper is intended to be a step towards the development of a precise set of terms to be used for
describing shape classes. These might be used to c o m p a r e or describe the classes of shapes dealt
with by various researchers; they might even be used to help formulate some of the more difficult
problems involved in shape description.
To this e n d , a taxonomy and terminology have been presented for the shapes k n o w n
generalized
cylinders.
as
The taxonomy begins with the definition of Generalized Cylinders, a n d
includes subsets and supersets of this class of shapes.
T h e terminology introduces terms and
abbreviations for each shape class.
C o m m o n mathematical shape classes are equivalent to some of these classes, and the shapes
dealt with by other researchers can be described in this terminology. Doing so may allow the reader
to better understand what assumptions have been made in various research efforts.
4.1 Acknowledgements
Takeo Kanade provided g u i d a n c e for this work. T o m Binford c o n t r i b u t e d several useful ideas.
10
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