Methods for generating 3D exact views of convex polyhedra
for visual identification
Part II: Non-iterative methods,
implementation and test results
Monika Kowalczyk, Wojciech S. Mokrzycki
ICS PAS, 21, Ordona st., 01-237 Warsaw, Poland
{monika,wmokrzyc}@ipipan.waw.pl
Abstract. This paper is the second part of our study describing methods for obtaining a 3D multiview
exact and complete model of convex polyhedra used for visual identification. Non-iterative methods,
like the iterative ones [21], explore the concept of the view sphere with perspective projection and
the view sphere covering as a mechanism for representation completeness.
The non-iterative methods ( [19]) consist in calculating a view, determining the corresponding
single-view area (the so-called seedling single-view area) and then searching for the neighbouring
single-view areas (generating the views at the same time) in a spiral way until the whole view sphere is
covered by the latter. Having a complete set of the single-view areas (complete view sphere covering),
we get a complete set of views as well.
Test results and computational complexity estimation are also included.
Key words: object representation, distinguished features, 3D exact multiview model, visual object
identification, view sphere with perspective projection, standardised 3D views.
1. Introduction
In the first part of the study ( [21]) we have described a new concept of a view space
— a view sphere with perspective projection, a concept of verifying and correcting completeness of a multiview representation as well as iterative methods for generating a 3D
exact multiview representation of convex polyhedra.
In this paper we describe the so-called non-iterative methods for generating a 3D
exact multiview representation of convex polyhedra, which are based on the same concept of a view sphere with perspective projection and a view sphere covering as iterative
methods. Finally, we present the computational complexity and the results of implementation and tests of the latest method with a seedling single-view area, which is the
most compact and the best among all the presented methods.
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2. Non-iterative methods
2.1. Concept of non-iterative view model’s generation
The schemes of the two non-iterative methods presented below are similar: at the beginning an arbitrary viewpoint on the view sphere is chosen, and a view is generated. A
suitable single-view area is determined (the so-called seedling single-view area), and
then other tightly neighbouring (having at least one common edge) single-view areas are
found. Each new adjacent single-view area is combined with the seedling single-view
area in such a way that the combined area (called the known area) spreads over the
view sphere.
These methods are based on the statement that a multiview representation of an
object does not depend on consecutive viewpoint locations (on the view sphere) at the
time of view generation, but only on the distinguished object features and their mutual
arrangement. The partition of the view sphere into single-view areas depends on the visibility of the object distinguished features from particular areas and allows for obtaining
a set of 3D standardised views of the object.
As one can notice, the role of single-view areas in these methods increases. Actually,
non-iterative methods are focused on proper determination of single-view areas and on
a complete covering of the view sphere (on spreading of the known area to the whole
view sphere).
In these methods, the boundary register (BR) contains the outer edges of the
known area of the view sphere. It also acts as a mechanism for verifying representation completeness. If the BR is empty at the end of the algorithm, it means that the
known area has no edges, i.e. it already covers the whole view sphere. This implies
the completeness of the model. Therefore, emptiness of the BR is also a condition for
terminating the algorithm.
Thanks to that, these methods do not need any additional completing procedure.
This is one of the most important differences between iterative and non-iterative methods. Iterative methods need a completing loop at the end of algorithm which finds and
generates the missing views, and in this way ensures completeness of representation —
that is the reason of giving such methods the name ”iterative” methods. In opposition
to them, the group of methods presented here are called ”non-iterative”.
2.2. Method with single-view area as homogeneous patch
2.2.1. The concept
Let us notice that each single-view area is homogeneous — it is not divided by any plane
of any face of the polyhedron ( [17]). Hence, having a viewpoint and one boundary
face, f b1 , of the sought-after single-view area we can find the rest of boundary faces by
moving along the f b1 plane in a selected direction and searching for a plane intersecting
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M. Kowalczyk, W.S. Mokrzycki
with f b1 , which is the second boundary face f b2 , then moving along f b2 , and searching
for another plane intersecting with it, and so on until the contour around the viewpoint
closes. This process is described more precisely below.
2.2.2. Finding boundary of single-view area
The first viewpoint is positioned in any place of the view sphere. The first view is
generated, and the visible and invisible rings of faces are determined fig. 1, [11]. For the
first face f v1 from the visible ring a halving plane (HP)1 is constructed and we move
along HP in the direction opposite to the f v1 face. We stop when we meet any plane (of
a polyhedron face; in fact it can be only, a face from the rings, since the faces from the
rings are the only potential boundary faces) intersecting HP. This plane is the boundary
of the the single-view area — let it be the first boundary face f b1 . We refer to it as
the testing plane (TP), and the point of intersection (of HP and the f b1 plane), Pr , is
called the reference point.
TP
Pr
wk’
fb1
TP’
VP
seedling
single-view
area
P1
fb2
fb3
P1’’
HP
P1’
TP’’
Fig. 1. Finding a single-view area by moving along its edges.
Next, we move in the clockwise direction2 along the TP plane searching for the
nearest intersecting plane, which would be the second boundary face f b2 (one of the
rings faces, of course). In order to do that, we follow the construction shown in fig. 2:
1 This plane is defined by a view direction and a centre of the selected face, see part I of the
elaboration
2 The direction is clockwise according to the VP looking from above. It is not important which of
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a)
Methods for generating 3D exact views of convex polyhedra... Part II
b)
testing
plane TP
testing
plane TP
P0
RTP
P0
RTP
Pi
Pi
gi
nTP
STP
nTP
STP
OTP
OTP
Fig. 2. Searching for the next boundary face by determining the nearest plane intersecting with TP. The
figure illustrates that if the TP is the plane of an invisible face (b) its normal vector has a direction
opposite to that for a visible face (a), and also the direction of calculating the γi angles is different
in each case.
The TP plane intersecting with the view sphere forms a circle OT P (which is not a
great circle of the sphere). Its centre is in the point ST P (which is the point of intersection
→
of the T P and its normal vector nT P beginning in the view sphere centre O(0, 0, 0)),
and its radius is RT P (the distance between ST P and the reference point Pr ).
p
RT P = (xPr − xST P )2 + (yPr − yST P )2 + (zPr − zST P )2 .
→
Now, we can lead a versor wk which is parallel to the radius RT P , begins at the
point Pr and is directed outside the circle OT P . Turning it by π2 in the counterclockwise
→ ′
direction around a line perpendicular to the T P plane, we obtain a versor wk indicating
the direction of searching for the points Pi (i = 1..k) of intersection of the TP plane with
the rest of potential boundary planes (except TP). The point Pi which is the nearest to
the reference point Pr is the vertex of the single-view area, and the face related to it is
the sought-after second boundary face f b2 . To find it, we can calculate the distances li
between the point Pr and
p the consecutive points Pi :
li = (xPr − xPi )2 + (yPr − yPi )2 + (zPr − zPi )2 .
Instead of li , we calculate the angles γi between the vector RT Pr (for Pr (xr , yr , zr ))
and the vector RT Pi (for Pi (xi , yi , zi )):
xr xi + yr yi + zr zi
γi = 2π − arccos
2RT P
direction we choose, but it has to be the same direction for the whole process of view model generation.
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when the T P is the plane of a visible face, and
xr xi + yr yi + zr zi
γi = arccos
2RT P
when the T P is the plane of an invisible face. The smallest angle γi corresponds to the
nearest Pi point, P1 .
In order to find the subsequent boundary faces, the plane of the f b2 face becomes the
→
testing plane T P , the P1 point becomes the reference point Pr , the versor wk is placed
→′
there, the versor wk is found and we start moving along the current T P . Starting from
that moment we have to check if the face found as a consecutive boundary face is the
face f b1 which would mean that the contour of the single-view area is closed, and the
computation of this area is finished.
Other single-view areas are found in a similar way, except for determination of a view
and the first boundary face. A more detailed description of the procedure is given below.
2.2.3. Tight spreading of single-view areas
In general, spreading the single-view areas consists in finding the single-view areas neighbouring to the already known ones, and computing the corresponding view for each of
them (for the viewpoint placed in the centre of the given single-view area). In order to
ensure that the single-view areas adhere to each other tightly (i.e. without any possibility of missing any single-view area between them), the search for a new single-view area
starts from one of the boundary planes of the known area which is common for these
two areas (the known one and the new one). Such a common boundary edge is called
an exit edge of the known area. This search is conducted along in a spiral around the
known area and provides a full view sphere covering.
At the beginning, the known area is equal to the seedling single-view area. Fig. 3
illustrates the order of searching for the neighbouring areas in such an initial situation.
→
The versors vi point to the consecutive exit edges of the spreading known area. The arc
arrows show the direction of searching for the single-view area edges.
Completeness of the view sphere covering by single-view areas is ensured with the help
of the boundary register (BR). The BR contains the numbers of the outer boundary face
of the known area (the already determined and joined single-view areas). If the known
area covers the view sphere completely, it has no edges and BR is empty. This is a sign
that the representation is complete, and that the calculations should stop.
2.3. Method with view modification
2.3.1. The concept
The proposed method, which includes the tight spiral spreading of single-view areas,
bases on the observations following (fig. 3, [17–19]):
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v2
v3
v1
fb5
v10
fb4
fb1
v4
seedling
single-view
area
fb2
v9
v5
fb3
v8
v7
v6
Fig. 3. The concept of tight spreading of the known area beginning with the seedling single-view area.
• a single-view area borders on only one other single-view area along a single boundary
face;
• a visual event occurs when the view changes after the viewpoint movement; it happens
when the viewpoint crosses the contour of a single-view area; a change in the view
refers to either one (when VP crosses an edge of the single-view area) or two faces
(when VP crosses the vertex) from the rings which have been crossed;
• if a boundary face f b belongs to:
• the visible ring, then it disappears when the viewpoint VP crosses the plane of
this face (i.e., the VP passes from the internal side of the face to the external side)
— the view on the other side of the face is decreased by this face (it contains the
same faces as the current view except the face f b);
• the invisible ring, then it appears when VP crosses the plane of this face (i.e. VP
passes from the external side of the face to the internal side) — the view on the
other side of the face is enlarged by this face (it contains the same faces as the
current view plus the face f b).
We can use these dependencies to determine the neighbouring view and its rings,
which can be used for computing the single-view area3 .
3 A view is not necessary for computing the single-view area. Without a view we should check the
intersections of the current TP plane with all the planes of the polyhedron faces — having a view we
can limit the calculations to intersections with the faces from the rings only.
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2.3.2. The algorithm
Here we present an entire algorithm for generating a complete view set for a specified
polyhedron. It contains calculation of single-view areas (as in the previous method) and
modification of the consecutive views.
ALG. VC (generating a complete set of views using single-view areas)
1. Generate the first view for an arbitrary viewpoint VP, determine the rings of faces.
2. Set the halving plane HP for the first face from the visible ring (f v1 ).
3. Move along the great circle (formed by HP on the view sphere, in the opposite direction to the face f v1 ) until the plane of a face from the rings is intersected; the
intersected plane is the first boundary plane, f b1 , of the seedling single-view area.
4. Denote the intersected plane as the testing plane TP, and the point of intersection as
the reference point Pr .
→′
5. Determine the direction of searching for edges of single-view
area — set the wk versor.
→′
6. Starting from the point Pr , find the nearest (in the wk direction) point P1 of intersection of the plane TP with another face plane — this is a vertex of the single-view
area; the intersected plane refers to the second (consecutive) boundary face, f b2 (f bi ).
7. Repeat steps 4–6 once more; there are three boundary faces of the sought-after singleview area already.
8. Set the face f b3 (f bi ) as the new TP, and the point P1 as the new point Pr ; set the
→′
versor wk .
→′
9. Starting from the current point Pr , find the nearest (in the direction wk ) point P1 of
intersection of the actual TP plane with another face plane — this is a vertex of the
single-view area; before nominating the intersected plane as a boundary face check:
• if the intersected plane is the same as the first boundary plane (f b1 ); in this case
ignore it — the contour of the single-view area is closed;
• otherwise — nominate the intersected plane as a boundary plane, and go to step
8.
10. Register the boundary faces in the BR:
• if the single-view area just found is the seedling single-view area, register all of
its boundary faces as the boundary faces of the known area (insert them into the
boundary register BR);
• otherwise delete from BR (the boundary faces of the known area) and from the
sequence of the boundary faces of the new single-view area all the common faces;
insert all the remaining boundary faces of the new area at the end of BR.
11. If BR is empty, then jump to END.
12. Denote the last boundary face from the BR as an exit edge, f be ; determine and
register a new view, which is on the other side of this boundary face, using the
dependencies below:
• if the boundary face f be belongs to the visible ring of the last computed single-view
area, then the new view contains the same faces as the current view except the
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face f be ;
• if the boundary face f be belongs to the invisible ring of the last computed singleview area, then the new view contains the same faces as the current view and the
face f be ;
13. Find the rings of the new view.
14. Denote the last boundary face of the known area (an exit edge) as the first boundary
face f b1 of the new single-view area; find an already known point on the view sphere,
where the plane f b1 intersects the last but one boundary face of the known area —
set this point as the current reference point Pr , the intersected face as the second
boundary face f b2 of the new single-view area and as the current TP plane.
→′
15. Change the direction of the versor wk to the opposite one.
→′
16. Starting from the Pr point, find the nearest (in the direction wk ) point P1 of intersection of the TP plane with another face plane — this is a vertex of the single-view
area; the intersected plane refers to the third boundary face, f b3 .
Go back to step 8.
END
In the last two methods, the views are generated in an additional action. The most
important operations concern computation of the seedling single-view area and spreading
it tightly across the view sphere.
2.3.3. Computational complexity
The computational complexity Z of the view representation generation according to
the last method is determined ( [18, 19]) by the number of operations needed for the
computation of views and single-view areas. Thus it depends on the number of views
(m) and single-views areas (m). This is a function of the number of the polyhedron faces
(n), regularity of the object shape, mutual dependencies between the shape of the solid
and the size of the view cone angle (more precisely, the proportion between the surface
of the circumscribed sphere and its part ”covered” by the cone).
Denote by n the number of the polyhedron faces, by m the number of all different
standardised views, by lw1 the average number of operations in the first view computation, by lo1 the average number of operations in the seedling single-view area calculation,
by lw the average number of operations in computation of a single view and by lo the
average number of operations in computation of one single-view area. Then the computational complexity of generating the polyhedron view representation using the presented
algorithm can be estimated as follows:
• the number of views (as well as single-view areas) is
m = c · n2 ,
where c is a factor depending on the polyhedron’s regularity and the width of the
view cone angle; for the five tested solids, it varied over the range 0, 72 ÷ 0.88, 2α = π4
(for the regular cube c = 0.72);
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M. Kowalczyk, W.S. Mokrzycki
• the formula for the computational complexity Z of generating the view representation
is
Z = lw1 + lo1 = (c · n2 − 1)(lw + lo ).
Computation of a single view as well as one single-view area is a function of the
number of polyhedron faces (n). A view computation lw has the unity cost n. So, the
computational cost of the first view lw1 is a little bigger as the visibility of each face is
computed, but the cost is still linear.
Next, the cost of a single-view area determination is found for each boundary face
(which is no more than (n − 1)). The number of points of intersection with the rest of
the polyhedron faces is again (n − 1). So, the average cost of computing one single-view
area is about the order of n2 . In this way, we can say that the computational complexity
of a view representation for a convex polyhedron with n faces is a fourth degree function
of n:
Z = O(n4 ).
2.3.4. Implementation and test results
Sample solids
Five solids were used as the testing objects: a regular cube (fig. 4) and the following irregular solids: pentahedron (fig. 5), hexahedron (fig. 6), heptahedron (fig. 7) and
octahedron (fig. 8).
Z
Z
5 (-1.5, 1.5, 1.5)
6 (1.5, 1.5, 1.5)
4 (-1.5, -1.5, 1.5)
7 (1.5, -1.5, 1.5)
Y
Y
1 (-1.5, 1.5, -1.5)
X
0 (-1.5, -1.5, -1.5)
X
2 (1.5, 1.5, -1.5)
3 (1.5, -1.5, -1.5)
Fig. 4. A sample solid — a cube (6 faces, 12 edges, 8 vertices).
Results
The tests consisted in generation of the view model for each solid, for a hundred
of random initial viewpoints. For the regular cube, 26 views were generated, for the
irregular pentahedron — 20 views, for the irregular hexahedron — 32 views, for the
heptahedron — 40 views and for the octahedron — 52 views.
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5 (0, 1.5, 1.491)
Z
4 (0, -1.5, 1.491)
Z
Y
Y
1 (-2, 1.5, -0.745)
2 (2, 1.5, -0.745)
0 (-2, -1.5, -0.745)
X
X
3 (2, -1.5, -0.745)
Fig. 5. A sample solid — a pentahedron (5 faces, 9 edges, 6 vertices).
Z
Z
1 (0, 0, 7.8)
Y
Y
4 (0, 3, -0.7)
0 (-4.5, 0, -3.2)
3 (0, -3, -0.7)
X
X
2 (4.5, 0, -3.2)
Fig. 6. A sample solid — a hexahedron (6 faces, 9 edges, 5 vertices).
The results obtained (see figs. 9–13) confirm that the presented algorithm produces
a complete view representation, which is independent of the initial position of the viewpoint.
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Z
Z
5 (-2, 0.25, 2)
4 (-2, -1.75, 2)
6 (0, 2.25, 2)
7 (2, -1.75, 2)
Y
Y
1 (-2, 2.25, -2)
0 (-2, -1.75, -2)
2 (2, 2.25, -2)
X
X
3 (4, -1.75, -2)
Fig. 7. A sample solid — a heptahedron (7 faces, 13 edges, 8 vertices).
9 (0, 2.2, 3.2)
Z
Z
8 (0, -1.8, 3.2)
5 (-2, 2.2, 1.2)
6 (2, 2.2, 1.2)
4 (-2, -1.8, 1.2)
Y
Y
7 (2, -1.8, 1.2)
1 (-2, 1.2, -2.8)
0 (-2, -1.8, -2.8)
2 (2, 1.2, -2.8)
X
X
3 (2, -1.8, -2.8)
Fig. 8. A sample solid — an octahedron (8 faces, 16 edges, 10 vertices).
3. Summary
In the two parts of the study (Part I — [21]) new ideas, methods, algoritms, implementation and test results connected with generating standard and complete, exact multiview
representation of a convex polyhedron have been presented.
The idea of non-iterative methods presented in this paper constitutes a basis for
assuring completeness of the representation by tight covering of the view sphere with
single-view areas. It leads to faster, simpler and more clear algorithms.
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Fig. 9. The generated view representation for the cube from fig. 4, 26 views.
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Fig. 10. The generated view representation for the pentahedron from fig. 5, 20 views.
M. Kowalczyk, W.S. Mokrzycki
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Fig. 11. The generated view representation for the hexahedron from fig. 6, 32 views.
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Fig. 12. The generated view representation for the heptahedron from fig. 7, 40 views.
M. Kowalczyk, W.S. Mokrzycki
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Fig. 13. The generated view representation for the octahedron from fig. 8, 52 views.
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All the ideas came from our experience acquired during research on a model-based
visual identification system; the methods for generating a view representation of an
object are the main part of it.
We are convinced that the described algorithm ALG. VC allows for efficient solving
of the problem of generating multiview models not only for a convex polyhedron (for
which it was developed) but also (after some modifications) for other complex solids.
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