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9.4. RELATIONSHIP BETWEEN LINEAR OPERATORS

Lecture



From the matrix of the cyclic superposition operator 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS It is possible to obtain the matrices of the superposition operator 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS and discretized superposition operator 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . To do this, enter the selection matrix

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.1а)

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.1b)

Where 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS - unit size matrix 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . We give the relations connecting these matrices with matrices obtained from them by generalized inversion and transposition:

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.2а)

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.2b)

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.2в)

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.2g)

Analyzing the structure of various linear operators, we can show that

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.3а)

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.3b)

So the matrix 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS formed by highlighting the first 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS rows and 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS block columns 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS matrices 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . At the same time, in all other blocks, the first 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS rows and 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS columns. Similarly from the matrix 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS can form a matrix 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . Matrix elements 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS which matrices are formed 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS and 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS in fig. 9.3.1, and enclosed in frames.

From the definition (9.3.1) of the extended array of samples of the original image, it follows that the vector of samples of the final source image 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS formed from the vector of the extended image 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS using the allocation operation:

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.4а)

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.4b)

It can also be shown that the output vector of the superposition operator of finite arrays can be obtained from the output vector of the cyclic superposition operator using the extraction operation:

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.5a)

There is an inverse relationship between vectors.

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.5b)

For a discretized superposition operator

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS , (9.4.6)

however the reverse transition from 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS to 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS cannot be performed due to the underdetermined discretized superposition operator. Transforming 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS and 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS in the matrix form, from the relation (9.4.5а) it is possible to obtain the equality

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.7)

Since the selection operator has the property of separability, the formula (9.4.7) is simplified to the form

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.8)

Similarly, from equality (9.4.6) relating to the discretized superposition operator, one can obtain the relation

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS . (9.4.9)

In fig. 9.4.1 shows the location of the elements of the matrix 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS the matrices for the superposition operator of finite arrays are formed 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS and for the discretized superposition operator 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS .

9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS

Fig. 9.4.1. Location of the matrices 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS and 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS in the matrix 9.4.  RELATIONSHIP BETWEEN LINEAR OPERATORS .

a is a superposition operator of finite arrays; b - discretized superposition operator.

So, for both operators, output vectors can be obtained from the result of cyclic superposition using the operation of extracting a part of the elements. As shown in Ch. 11, this fact allows us to simplify the calculations.


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Digital image processing

Terms: Digital image processing