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can operationally define a set of some observable pa- 
rameters Y whose actual values hopefully are relata- 
ble to a set of the model parameters Jt. To solve the 
forward problem is to predict the values of the ob- 
servable parameters Y € y, given arbitrary values of 
the model parameters X € X. To solve the inverse 
problem is to infer the values of the model parame- 
ters X from given observed values of the observable 
parameters Y (cf. Tarantola, 1987). 
Obviously, many problems in computer vision can 
be formulated as such inverse problems (a particu- 
lar kind of inference process called induction). The 
scientific procedure to solve these inverse problems 
distinguishes the following three steps: 
1. Representation (parameterization) of System 
S: Designing a language to represent the cha- 
racteristic features of S. That is, establishing a 
minimal set of model parameters X whose values 
completely characterize the system (from a given 
point of view). 
2. Forward modeling: Identification of the phy- 
sical laws (constraints) which, for given values 
of model parameters X, allow predictions as to 
the results of measurements on some observable 
parameters Y. 
3. Inversion: Use of the actual results of some mea- 
surements of the observable parameters to infer 
(estimate) the actual values of the model para- 
meters. 
3 Inductive Inference 
The term inference refers generally to effective pro- 
cedures for deriving new facts from known ones. To 
draw an inference is to come to believe a new fact on 
the basis of other information. There are many kinds 
of inference. The best understood is deduction, which 
proceeds from a set of assumptions called axioms 
to new statements that are logically implied by the 
axioms. The deductive inference is logically correct as 
deduction from true premises is guaranteed to result 
in a true conclusion. The standard way to characte- 
rize deduction is by using a system called predicate 
calculus which consists of a language for expressing 
propositions and rules for how to infer new facts (pro- 
positions) from those we already have. To deduce new 
facts from the axioms, we use one or more so called 
rules of inference. 
À second kind of inference, on the other hand, is cal- 
led induction, which is a calculus for inferring gene- 
ralizations from particular observations. This induc- 
tive inference process could be thought of as having 
the form * from: if (X — Y) and Y , infer: X " and 
489 
  
  
  
  
  
  
Variables Premises Conclusions 
A, Y Ao m Y Lex d x - X 
T T T F T F 
T F F T T F 
F T T F F T 
F F T T F T 
  
  
  
  
  
  
  
  
  
  
Table 1: Truth table used to draw inference 
it performs abstraction, producing generalities from 
specifics. The inductive inference can be illustrated 
using a simple example of geometrical reasoning from 
which we wish to answer a question: 
Given a set of geometrical points P = 
(24, yi), $ — 1,..., n. Infer if this set of points 
depicts a straight line. 
To answer this question, we can use some statements 
that express information during inference: 
If P represents a straight line, then y — 
a t + b is valid for all points of P, where 
a and b are two constants. Some points of P 
do not fulfill y = a x + b. Does P depicts a 
straight line? 
In order to express these statements, we have to agree 
on a suitable set of atomic propositions like: 
e X: P depicts a straight line. 
e Y:y-—a z + bis valid for all points of 7. 
The original statements expressing information du- 
ring inference are called premises and can be descri- 
bed as follows: 
Ay oy 
So, the question would be answered if we could prove 
the proposition X from the premises, or alternatively 
if we could prove ^X. Since this is a small problem, 
we can easily employ an exhaustive examination of 
all possible assignments of truth values to the pro- 
positions X and Y to check for the validity of either 
possible conclusion. Using the so called truth table 
(cf. Tab. 1) we can list all the possible combinations. 
Let us first check the validity of X as a conclusion 
by examining every row in which all two premises are 
truth. In this example there is only one row where all 
premises are truth (the bottom row). It is intuitive 
that the potential conclusion X is false here whereas 
^X is true and this corresponds to the correct ans- 
wer: P does not depict a straight line.