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image 
ties of 
lensity 
D(z,y) of a photographic image and the exposure 
H [luz - sec] is 
D(z,y) — y log H -- Do (14) 
for normal exposure, where z, y are image coordina- 
tes of a pixel, y (= 1) is the gradation and Dy is a 
constant. Usually, y depends on the developer, the de- 
velopment time and temperature, and the photogra- 
phic material. The exposure H depends, first of all, on 
the reflection properties of surfaces. Many natural ter- 
rain features roughly approximate diffuse reflectors. A 
Lambertian surface is a perfect diffuse reflector with 
the property that the radiance L [cd -m7-?] is constant 
for any incident angle. 
The relation (14) is very important as it explains the 
physical meaning of image intensities. From (14) one 
can derive the following radiometric constraint (cf. 
Zheng, 1990): 
D(z, y)/y4- 2-log(c? -z?--y?) -n-- V (z, y) = 0, (15) 
where 7 can be considered as a constant for all pixels 
in the same image; but v is a local parameter which 
changes from pixel to pixel. The physical meaning of 
V in (15) is the logarithm of the luminance intercep- 
ted by the lens for a pixel. 
The image coordinates x and y in (15) are functions 
of the object coordinates of the surface element, ac- 
cording to the well known projection equation: 
x 1 X Xo 
v: Jodi ¥ Jad). 0) 
—e m Zz Zo 
where m is a scale factor, R is the rotation matrix con- 
taining three rotation angles ($,w, «), (X, Y, Z) are 
the corresponding ground coordinates of the image 
point (z, y), and Q — ($,w, &, Xo, Yo, Zo) are camera 
orientation parameters. Besides, D(z,y) has to be 
re-sampled from the neighboring digitized pixels by 
using, for instance, a bilinear interpolation 
D-GTL, (17) 
where L and G denote a set of intensities of neigh- 
boring pixels and a corresponding coefficient vec- 
tor, respectively. Thus, for the ground surface point 
(X,Y, Z), the left side of (15) is a function of many 
parameters: 
fOG Y, Z,Q, L,y,n, v) = 0. (18) 
Now, let us look at the problem of surface reconstruc- 
tion from multiple images. For the purpose of simpli- 
city, we discuss here only the solution of recovering the 
surface profile, which is represented using K discrete 
profile points, from J images, M;,j € J = [1,..., J], 
493 
  
  
  
Figure 2: Surface reconstruction from image data 
which are taken from different views and depict the 
same surface (cf. Fig. 2). Besides, we also suppose 
that the orientation parameters of these images are 
all known. For the i*" profile point, i € Z — [1, ..., K], 
we can write J constraints like (18). For K profile 
points in Z we can write totally J x K constraints 
like (18). Supposing a Lambertian surface and y ~ 1, 
these constraints can be further simplified (cf. Zheng, 
1990): 
cdi-4z24342 
D(z, ; D * ;)-F21 rr TA = 0, 
(21 V1)+P; (=; yj) og Gub +9; 
(19) 
where p; = —71/7j, ¢j = m—n;, j € [2,..., J], and we 
have a set of new parameters p; and q;, j € [2,..., J], 
which describe approximately the radiometric rela- 
tionship of the image M; with the other images 
Mj, j € [2,...,J]. Our a priori knowledge about p; 
and gj is that p; should be around 1 and q; should 
be around 0. Linearization of these now constraints 
gives 
UV+AAZ+BAQ+W(L,Z0,Qo)=0 (20) 
where L and V denote two vectors containing obser- 
vations, i.e. intensities of image pixels, and their resi- 
duals; Zo and AZ denote two vectors containing ap- 
proximate elevations of profile points in Z and their 
corrections; Qo and AQ denote two vectors containing 
approximate values of p; and qj, j € [2,...,J] and 
their corrections; and U, A, B, and W are correspon- 
ding coefficient matrices and the vector of constants, 
respectively. It is clear that (20) is strongly under- 
determined as V, AZ and AQ are all unknown. The 
total number of unknowns is much larger than that of 
the constraints, and one could generally hypothesize 
an infinite number of different solutions that would 
meet (20). So, we have to use criteria to restrict the 
space of acceptable solutions and to find a unique so- 
lution which will be a best one to interpret the image 
data.