E V E R E T TL . M E R R I T T
Raytheon Co., Autometric Operation
Alexandria, V a . 22304
Camera Orientation with
Peripheral Circles
Images of circles such as storage tanks, sewer treatment
plants, etc., offer solutions, but most oblique exposures
do not generate right cones.
APPLICATION
C
HANCE
PHOTOGRAPHY
Of
events frequently provide the only record
that may be examined and analyzed after the
event or events have transpired. In such a
situation metrical analysis of a photograph
with less than optimum parameters is not a
matter of choice but necessity. Generally such
records contain no control data. T h e expedient procedure then is to determine the orientation of the record with condition equations
solved inasmuch as any peripheral image and
the axis of the cone define a constant angle k
regardless of the inclination of the focal plane
to the conic generated. T h e cosine of the constant angle k is expressed with the angle between
lines equation:
Specifically a 3 X3 matrix of the form is
solved for a , b and C :
ABSTRACT:Camera orientation i s treated with geometric rigor using image
coordinates of a n object circle generating a n oblique cone with the lens
nodal point as the vertex. I n contrast, the solution for camera orientation i s that
of a single conic where the lens nodal point and the object circle generate a right
cone. T h e equations for a right conic however are not valid for a n oblique cone.
Yet most oblique exposures of circles do not generate right cones because the lens
nodal point seldom lies o n a perpendicular to the center of the object circle.
t h a t assume certain geometric properties inherent in the object or objects photographed.
Vanishing points associated with systems of
parallel lines in object space and orthogonality associated with man-made structures have
been employed since the beginning of photogrammetry. T o some extent, images of circles
such as storage tanks, sewer treatment plants,
and numerous other man-made mechanisms
have been employed. Historically, the application of circular geometry to camera orientation has been the application of approximate right-cone equations to full circles.
? a + 3 1 2 b + - c =f
12
12
1
12
-aa + - b Y3
+ - c = lf
13
13
13
where subscripts 1, 2 and 3 denote peripheral
image points and
x,=!f
c
b
y. = -f.
THE RIGHTCONE
A right cone is generated wherever the lens
nodal point and the center of the object circle
define a perpendicular to the lane of the
circle. A photogrammetric right cone is illustrated in Figure 1. This case is easily
-fc = l
11
-x al + - b +
YI
11
11
These equations are not applicable to a n
oblique conic which is the usual case. An
oblique cone is generated wherever the line
connecting and lens nodal ~ o i n and
t
the center of the circle does not define a perpen-
245
PHOTOGRAMMETRIC ENGINEERING,
L
K,=KZ*KII
1972
ject circle are systematically unequal. The
angles subtended by a chord are unequal if
projected to any plane not coincident with the
plane of the object circle.
If the perpendicular t o the plane is the z
axis, the corresponding horizon plane passing
through the lens is parallel to the plane of the
circle. In this plane the inscribed angles are
equal. The image points on the horizon line
are defined by the image lines passing from
the inscribed points through the terminals of
the chord. Plane allbl is defined by the direction of z. Planes a l l and bll are defined by
images a , b and 1 with the lens nodal point.
The solution is the determination of what
values of x, and y, that will generate a plane
producing equal values of k for all pairs of
planes defined by the inscribed image points
and the imaged terminal of a selected chord.
The sine and cosine k may be expressed in the
following analytic form:
sin k =
(xa~yb;~;:hyal
xalxbl
cosk = -
L
)T.
+ yalybl +f
lallbl
Whence
FIG.1. Right cone intercepted by a focal plane.
-=
tan kf
1,
dicular to the plane of the circle. The various
approximate treatments of camera orientation with images of circles have not distinguished between right and oblique cones.
These treatments have varied from the ratios
of the minimum and maximum image diameters to anharmonic ratios if the center of the
circle is displayed and permits equal segments
to be assumed. I t has been shown t h a t inherent in the geometry of the photograph is
an exact solution to camera orientation if a
right cone is generated. I n the special case,
therefore, there is no justification for loose
geometry.
The geometric properties of a n oblique
conic are shown in Figure 2. The solution for a
right cone is developed from the property t h a t
any peripheral image subtends a constant
angle with the center of the object circle a t
the lens. This property does not hold for an
oblique cone. The oblique cone solution is
based on the theorem that a chord of a circle
subtends a constant angle k for all points on
its circumference. These angles projected to
any plane not parallel to the plane of the ob-
xalybl - xblyal
xalxbl
yalybl+ f
+
The normals to planes a l l and bll are also
normal to la1 and Lbl. These normals are expressed as follows:
f(ya - ~ 1 )
xayl - xlya
f ( x 1 - xa)
ynal =
xayl - xlya
xnal =
The normal I, is also normal to lines lal and
Lbl, whence coordinates of the horizon points
al and bl may similarly be expressed:
CAMERA ORIENTATION WITH PERIPHERAI, CIRCLES
247
FIG.2. Geometry of the general equation of camera orientation with
peripheral coordinates referred to an arbitrary chord ab.
E = - [ma1
+ xnbl]
F = - [ ~ n a l +~ n b l ]
G = - [ynal - ynbl]/f
H = [xnal - xnbl]/f
I = [xnaynb - xnb - yna]/f.
W i t h approximate values of x,' a n d y,' a n approximate average value of k' can be computed. Letting
Substituting i n t h e equation for t a n kf/l, a n
equation of t h e form is obtained:
A
+ xSt2B+ yi2C + z,'y,'D + x,'E + Y
a n approximate value of I' can be computed:
I'
tan k
tan k
+ (- xsys) D + ( ~ ~ . ) E + ( ? ~ . ) F
~ =
FM,
=
tan k'
-- M
1'
+ x,'G + y,'H,
whence
dl
where
I - I'
tan k'
= d ----Af
=
',I
+ xiG + yStB
which reduces t o
Ax,P
where
+ Ay,Q + A ~ " R= dl
248
PHOTOGRAMMETRIC ENGINEERING,
1972
equations formed with estimates of x,' and
y,' a r e normalized a n d solved iteratively until
ZdZ2 = minimum.
T h e number of iterations seldom exceed three.
TESTCASE
FIG.3. A test circle.
a = [ tan
, ( k'Z ~ ; C + ~ D + P -
y?)
+H
]
R = sec2k'M sin 1''
I',
T h e above equation has three unknowns a n d
therefore requires three o r more image coordinates of inscribed points t o form a 3 x 3
matrix. I n practice n points a r e used. T h e
T h e test of t h e equations was accomplished
with image coordinates measured o n a n
oblique exposure of a circle accurately scribed
o n a sheet of mylar. T h e exposure camera was
a Hasselbald 10 feet above t h e circle. T h i r t y four points were selected. T h e t e s t (Figure 3)
demonstrated t h e following:
(1) Points may be selected from either or both
sides of the chord as lone as the direction of the
angles are preserved. ~ h ; s , if in the use of Point
1 the coefficients are formed from a to b, then for
Point 15 on the opposite side of the chord the
coefficient must be formed from b to a.
(2) No significant accuracy is gained in using
more than 15 points.
(3) An accurate solution is obtainable with as
little as one-fourth of a circle.
(4) The solution is unreliable if the diameter
of the circular image is less than one-tenth the
focal length.
(5) Finally, the chord selected cannot fail t o
be parallel to the image of the horizon by more
than 30". For example, if a chord is selected 90"
to the horizon, one of the points of each pair will
not cut the horizon.
Symposium Proceedings Available
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to Members
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Annual March Meetings
1968-428
1970-769
pages, 1969-q15
pages, 1971-817
pages,
pages.
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Fall Technical Meetings
Portland, Oregon, 1969. 363 pages, 39 papers
Denver, Colorado, 1970. 542 pages, 33 papers
S a n Francisco, 1971, 770 pages, 71 papers*
Remote Sensing
21 selected papers, 1966, 290 pages
Third Biennial Workshop
Color Aerial Photography i n t h e plant Sciences a n d Related Fields, 20 papers, 288
pages
5 .oo
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Close-Range Photogrammetry, 1 97 1
33 papers, 433 pages
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American Society of Photogrammetry, 105 N. Virginia Ave., Falls Church, Virginia
22046.
* Includes papers from Symposium on Computational Photogrammetry.