1.8. GENERALIZED LINEAR SYSTEMS

In section 1.4, the concepts of linearity and superposition were introduced in order to extend the concept of linearity to a wider class of systems.

Consider two functions describing images, and which, interacting in some way give function :

. (1.8.1)

Let be - operator of the system that transforms , which has the following properties:

(1.8.2a)

and

(1.8.2b)

Where is a constant, and the colon denotes a generalized multiplication by a constant. In [4] it is shown that if the operation is reduced to adding vectors, and the operation: - to multiplying a vector by a scalar, then the operator can be represented as a chain of operators, called a homomorphic filter (Fig. 1.8.1). First operator turns operations and: in addition of vectors and multiplication of a vector by a scalar:

(1.8.3a)

and

(1.8.3b)

Fig. 1.8.1. Generalized linear systems: a - generalized system; b - representation of the generalized system as a homomorphic filter; в - multiplicative homomorphic filter.

The second stage of a homomorphic filter is a regular linear system. Third Stage - Operator which is the reverse of the first operator, i.e.

(1.8.4)

Fig. 1.8.1, in illustrates a particular case of a homomorphic filter for a multiplicative system [5], in which the function is obtained by multiplying the functions and i.e.

(1.8.5)

Prologized by both sides of equality (1.8.5), we obtain the sum of the logarithms of the functions and :

(1.8.6)

Function is transformed by some linear system, and then returns to the source image space by means of an exponential transformation. The operation of generalized multiplication of a vector by a scalar is defined as exponentiation

(1.8.7)

Logging this equation gives

(1.8.8)

The use of homomorphic filtering for image restoration is discussed in Ch. 15.