The reasoning method just illustrated is called per- 
fect induction. Here, the available information which 
is necessary to support the desired logical inference is 
perfect. 'This means that all the statements only have 
two values for their validities, either true or false, and 
we are able to check exhaustively all the possible com- 
binations. Unfortunately, in many practical problem- 
solving situations, especially in image analysis and 
understanding, the available knowledge is incomplete 
or inexact. The coordinates of the points in 7, for ex- 
ample, may contain measure-errors and only from the 
fact that some points of P do not fulfill y =a =z + b 
we can not come to the conclusion that P does not 
depict a straight line. So, the validity of Y (or ^Y) 
is not binary and not easy to prove if we do not have 
knowledge about measure-errors. In cases like this, we 
need reasoning methods in making just decisions. 
4 Inverse Problems 
Mathematically, the inverse problem can be described 
as follows. Given a mapping f from set A’ into set , 
ie. f : Y! — y. The solution of the inverse problem 
consists in the interpretation of data Y € yy in order 
to recover the original image X € À. This is exactly 
the same goal as that of an inductive inference men- 
tioned above. 
Let us now consider a linear mapping A : A — y. 
The inverse problem is to identify X from the data 
Y: 
AX Y. (1) 
The solvability of this inverse problem could be dis- 
cussed as follows: 
e If A is bijective and A-! is stable one can easily 
get an unique solution X = A-!Y. 
e If A is injective but not surjective, the inverse 
problem is overdetermined and has no solution. 
One can, however, get an unique pseudo solu- 
tion through minimizing the norm of the residual 
IV = lY — AXI. 
e If A is not injective, the inverse problem is under- 
determined and there is an unique pseudo solu- 
tion X — A*Y, where A* is the so called Moore- 
Penrose Inversion, which, unfortunately, is only 
stable if the domain of A is closed in Y. 
So, it is quite clear that the ambivalent non-injective 
inverse problem is practical not solvable through a 
numeric process, as any few errors in Y can destroy 
the solution totally. 
Schematically, there are two reasons for the ill- 
posedness of inverse problems: intrinsic lack of data, 
490 
and observation uncertainties. With additional infor- 
mation, for instance some a priori assumptions on 
model parameters X or an additional data set, many 
such problems can be reformulated into well-posed 
solvable problems. Now, the main question is how to 
integrate a priori knowledge to solve ill-posed inverse 
problems. We need criteria to impose constrains on 
the solution space and a framework to integrate a 
priori knowledge in order to select an unique solution 
(the so called best solution) for given data. Intuiti- 
vely, the best solution exists only in connection with 
criteria which are, of course, strongly task dependent. 
5 The MAP Criteria 
The first criterion which we would introduce in this 
section is the so called Maximum A Posteriori (MAP) 
criterion which is based on probability theory (Geman 
and Geman, 1984 ). It selects as the best solution the 
model parameters X that maximizes the conditional 
probability of X given the data Y: P(X | Y), subject 
to the inverse problem (1). The MAP criterion leads 
to three important estimation methods, namely the 
bayesian estimate method (BE), the maximum like- 
lihood method (ML) and least squares method (LS) 
(cf. Tab. 4), which are widely used in data processing. 
Using Bayes’ theorem gives 
P(X |Y) = P(Y | X)P(X)/P(Y), (2) 
where P(Y | X) is the conditional probability of get- 
ting data Y given the model parameters X, P(X) are 
the prior probability of X. The relation (2) shows how 
the prior probability P(X) changes to the posterior 
probability P(X | Y) asa result of acquiring new infor- 
mation Y. Intuitively, the MAP criterion will choose 
X that maximizes 
P(Y | X)P(X), (3) 
if P(Y) is constant. This is the principle of the 
Bayesian estimation. Further, under the specification 
that the prior probabilities P(X) are all the same, 
ie. P(X) is constant, the MAP criterion leads to 
the simpler maximum likelihood principle of selec- 
ting that X which maximizes P(Y | X). If the ran- 
dom variables to which the data Y refer are normally 
distributed, the maximum likelihood estimation will 
give the same results as the least squares estimation 
which has widely been used in different branches of 
science and engineering for over a century and a half. 
If V is the vector of observational residuals, for which 
E(V) = 0, and which is assumed to be normally dis- 
tributed, and X is the covariance matrix of the distri- 
bution, then we have 
P(Y | X) = P(V) = C - exp |-5v7=="v] ud) 
  
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