the second term in (9) appropriately. Of course, de- 
signing E(X) is a skill. One needs knowledge about 
the physical meaning of the solution and the internal 
coherence of unknown parameters. 
Generally, the second term in (9) can be designed to 
have the form 
2 E(X)= ET E,! E, E=®X-+#, (10) 
where ® is an operator, ¥ is a vector, and X, is a 
matrix. They have to be determined using our a priort 
knowledge. If we, for instance, a priori know that the 
elements z; € X, i — 1,..., m, should have values 
around aj, i — 1,...,m, then we can construct 
o2 0 0 
T 0 - 02 0 
m3 l5 | 
0.0 c2, 
1 0 0 ay 
0 1. 0 a2 
P= E us : A= ; ; (11) 
0 0 4 am 
where o;, à = 1,..., 7m, denote the degree of the cer- 
tainty of our a priori knowledge. 
Let us solve the ill-posed inverse problem (1) again, 
but using the new criterion (9) which is equivalent to 
VTx-iy - ETY,!E — min. (12) 
This lead to the new normal equation 
ATS AUOT NS 19) YA YY +9710. 
(13) 
It is sure that the new normal matrix N = ATX-1A4 
QT Y:-1ó is no more singular even for underdetermi- 
ned ill-posed inverse problems, if ®, De and ® are all 
appropriately constructed. 
7 Surface Reconstruction 
There are, as indicated above, many problems in com- 
puter vision which can be generally formulated as in- 
verse problems. We have proposed approaches which 
provide a sound theoretical basis but offer few practi- 
cal computational methods for dealing with concrete 
tasks in computer vision. So, in this section, we go 
further into the application of the inverse problem 
theory to an elementary problem, i.e. the computing 
of the representation of visible surfaces from multiple 
images. 
  
I 12 — — 
  
— eee fe peers 
T. d M 
Figure 1: The meaning of the label lj; 
  
7.1 Representation of Visible Surfaces 
The role of a representation is to make certain infor- 
mation explicit at an appropriate point in the problem 
analysis as the abstract information must be expres- 
sed by concrete descriptions. Thus, the choice or de- 
sign of a representation affects the success of analysis. 
The representation of object surfaces deals with stra- 
tegies and techniques for describing their geometrical 
and physical properties in a way appropriate for nu- 
merical processing. 
Let S be a set of parameters which describe the geo- 
metrical and physical properties of an object sur- 
face. An element S € S can be a concrete measure, 
e.g. elevation (depth), deformation, reflectivity, etc.. 
Each element S € S can be mapped onto XY -plane 
in a 3D coordinate system and represented mathe- 
matically as S = S(X,Y),S € S. For computa- 
tional reasons, we rather represent S by a grid of 
square 1 x 1 elements, where each element is cen- 
tered at the coordinates (X;,Y;) of the it" element. 
Then, the object surface is described by m x n ele- 
ments: Si = S(N,Y),i € - [1,..., /], where 7 
can be thought of as a vector belonging to the set 
(1, ..., m) x (1, ..., n) which has totally m x n elements. 
Sometimes we may be also interested in the spatial 
coherence (continuity) of S. So we introduce a label 
set L whose element l;; represents the strength of the 
spatial coherence between two neighbor 5; and 5; (cf. 
Fig. 1). The label l;; can be binary: lj; — 1 for con- 
tinuity between 5; and S;, lj; — 0 for discontinuity 
between 5; and S;. l;; can also take the value between 
0 and 1, i.e. lj; € [0, 1], for continuously describing the 
coherence strength. 
7.2 Forward Modeling 
The purpose of forward modeling is to find constraints 
linking elements in S with observations, i.e. image 
densities (intensities), based on physical properties of 
imaging. The relationship between the image density 
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