infor- 
ns on 
many 
posed 
ow to 
nverse 
ns on 
rate a 
lution 
1tuiti- 
| with 
ident. 
n this 
MAP) 
eman 
n the 
tional 
ibject 
leads 
y the 
1 like- 
| (LS) 
ssing. 
(2) 
f get- 
() are 
s how 
tertor 
infor- 
hoose 
(3) 
f the 
ation 
same, 
ds to 
selec- 
> ran- 
mally 
n will 
lation 
nes of 
» half. 
which 
y dis- 
listri- 
(4) 
  
criterion expression supposition 
  
MAP P(X | Y) — maz 
  
BE P(Y | X)P(X) 5 maz | * 
  
ML P(Y | X) 2 maz «m 
  
  
  
LS V4^Y-!V — min RE 
  
* P(Y) constant 
** P(X) =constant and * 
4X V ~ N(0,X) and ** 
  
  
  
Table 2: The MAP criterion and its progenies 
where C is constant. It is to see that the least squares 
criterion is to minimize VTX~1V which is equivalent 
to using a maximum likelihood estimation to maxi- 
mize P(Y | X). 
So far, we have discussed the MAP criterion and its 
progenies. Obviously, each criterion has its own sup- 
position (cf. Tab. 4). The LS criterion which is so wi- 
dely used in data processing as a general framework 
for problem solving is, for instance, only suitable for 
dealing with over-constrained inverse problems. For 
under-constrained inverse problems the MAP crite- 
rion is more appropriate as it provides a flexible fra- 
mework to integrate a priori knowledge to restrict 
the solution space and one can take the probability 
behavior of both the data and the desired solutions 
into account. There are many problems in compu- 
ter vision, especially in the low-level image proces- 
sing, including edge detection, spatial-temporal ap- 
proximation, image segmentation, image registration, 
and surface reconstruction (cf. Poggio et al.,1985), 
which are unfortunately of under-constrained nature 
and whose solutions demands on new inference tech- 
niques beyond the LS estimation. 
6 Restricting Solution Space 
The MAP criterion provides a general approach to 
handle the inverse problem in an uncertain environ- 
ment. It gives a mechanisms to restrict the solution 
space and to integrate a priori knowledge by specify- 
ing the appropriate prior probabilities P(X). Howe- 
ver, the MAP criterion doesn’t tell how to construct 
P(X). In this section we look at this issue. 
The parameter set X', as mentioned earlier, represents 
a physical system and can be considered as a parame- 
ter space. In principle, every point X € .t represents 
a possible solution. It can be easily imagined that not 
all points in the solution space are meaningful. Our 
job is to explore the solution space to find an appro- 
priate point (solution). So, the first problem is how to 
measure the appropriateness of a solution and how to 
491 
describe the solution space. A general way to do this 
is to define a probability distribution of the solution 
space P(X) (Tarantola, 1987). 
Let X be a set of parameters representing the 
state of a Markov random field. According to the 
Hammersley-Clifford theorem (cf. Geman and Ge- 
man, 1984; Chou and Brown, 1988), this random field 
can be described by a probability distribution of the 
Gibbs form: 
P(X)z 5 exp [500 , X € À, (5) 
where C is a normalizing constant, T is the so called 
temperature of the field that controls the flatness of 
the distribution of the configurations X and E(X) is 
the energy of X which consists of the sum of the local 
potential 
S(X)zm S V. (6) 
TeX 
The relation (5) suggests that the point in V with a 
higher energy occurs less likely. 
Now, let us look at the ill-posed inverse problem (1). 
According to the the least squares criterion (cf. Tab. 
4), one can get an unique pseudo solution through 
minimizing VTY;-!V, with respect to 
V=AX-Y. (7) 
This leads to solving the normal equation 
(4TN 1 A) X = ATH 1Y (8) 
Certainly, the normal matrix N — ATY-14 is regu- 
lar only if the problem (1) is overdetermined. This 
suggests that the least squares criterion can only be 
used to deal with overdetermined ill-posed inverse 
problems. For underdetermined ill-posed inverse pro- 
blems, which emerge so often in image understanding, 
the least squares criterion can not help us to find a 
satisfying solution, as it does not have a mechanism 
to restrict the solution space. 
Using the bayesian estimate method (BE) (cf. Tab. 
4), we have the following optimizing problem (cf. (3), 
(4), and (5)): 
ylyciy.4 2 E(X) — min, (9) 
with respect to (7). Intuitively, this criterion, in com- 
parison with the least squares criterion, is more po- 
werful to deal with underdetermined ill-posed inverse 
problems, as it gives not only a measure for the quality 
of the fitting , through the first term in (9), but also 
a measure for the probability of the solution, through 
the second term in (9). So we can integrate our a 
priori knowledge into the inverse process by designing