You get a bonus - 1 coin for daily activity. Now you have 1 coin

1.8. GENERALIZED LINEAR SYSTEMS

Lecture



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,   1.8.  GENERALIZED LINEAR SYSTEMS and   1.8.  GENERALIZED LINEAR SYSTEMS which, interacting in some way   1.8.  GENERALIZED LINEAR SYSTEMS give function   1.8.  GENERALIZED LINEAR SYSTEMS :

  1.8.  GENERALIZED LINEAR SYSTEMS . (1.8.1)

Let be   1.8.  GENERALIZED LINEAR SYSTEMS - operator of the system that transforms   1.8.  GENERALIZED LINEAR SYSTEMS , which has the following properties:

  1.8.  GENERALIZED LINEAR SYSTEMS (1.8.2a)

and

  1.8.  GENERALIZED LINEAR SYSTEMS (1.8.2b)

Where   1.8.  GENERALIZED LINEAR SYSTEMS is a constant, and the colon denotes a generalized multiplication by a constant. In [4] it is shown that if the operation   1.8.  GENERALIZED LINEAR SYSTEMS is reduced to adding vectors, and the operation: - to multiplying a vector by a scalar, then the operator   1.8.  GENERALIZED LINEAR SYSTEMS can be represented as a chain of operators, called a homomorphic filter (Fig. 1.8.1). First operator   1.8.  GENERALIZED LINEAR SYSTEMS turns operations   1.8.  GENERALIZED LINEAR SYSTEMS and: in addition of vectors and multiplication of a vector by a scalar:

  1.8.  GENERALIZED LINEAR SYSTEMS (1.8.3a)

and

  1.8.  GENERALIZED LINEAR SYSTEMS (1.8.3b)

  1.8.  GENERALIZED LINEAR SYSTEMS

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   1.8.  GENERALIZED LINEAR SYSTEMS which is the reverse of the first operator, i.e.

  1.8.  GENERALIZED LINEAR SYSTEMS (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   1.8.  GENERALIZED LINEAR SYSTEMS is obtained by multiplying the functions   1.8.  GENERALIZED LINEAR SYSTEMS and   1.8.  GENERALIZED LINEAR SYSTEMS i.e.

  1.8.  GENERALIZED LINEAR SYSTEMS (1.8.5)

Prologized by both sides of equality (1.8.5), we obtain the sum of the logarithms of the functions   1.8.  GENERALIZED LINEAR SYSTEMS and   1.8.  GENERALIZED LINEAR SYSTEMS :

  1.8.  GENERALIZED LINEAR SYSTEMS (1.8.6)

Function   1.8.  GENERALIZED LINEAR SYSTEMS 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.  GENERALIZED LINEAR SYSTEMS (1.8.7)

Logging this equation gives

  1.8.  GENERALIZED LINEAR SYSTEMS (1.8.8)

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


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Digital image processing

Terms: Digital image processing