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8.1. GENERALIZED LINEAR OPERATOR

Lecture



Consider an array   8.1.  GENERALIZED LINEAR OPERATOR of   8.1.  GENERALIZED LINEAR OPERATOR elements representing the source (input) image. When exposed to it by a generalized linear operator, an array is obtained from   8.1.  GENERALIZED LINEAR OPERATOR elements describing the transformed (output) image

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.1)

where is the core operator   8.1.  GENERALIZED LINEAR OPERATOR is a set of weighting factors that, in general, depend on the coordinates of elements of both the input and output images.

When analyzing linear image processing operations, it is convenient to use the vector representations described in Ch. 5 [1]. Therefore, we assume that the input array   8.1.  GENERALIZED LINEAR OPERATOR presented or as a matrix   8.1.  GENERALIZED LINEAR OPERATOR or as a vector   8.1.  GENERALIZED LINEAR OPERATOR obtained by scanning the matrix   8.1.  GENERALIZED LINEAR OPERATOR by columns. Similarly, assume that the output array   8.1.  GENERALIZED LINEAR OPERATOR can be represented by either a matrix   8.1.  GENERALIZED LINEAR OPERATOR or in the form of its sweep by columns, i.e. by vector   8.1.  GENERALIZED LINEAR OPERATOR . To simplify the notation, it will be assumed below that the matrices representing the input and output images are square with dimensions   8.1.  GENERALIZED LINEAR OPERATOR   8.1.  GENERALIZED LINEAR OPERATOR respectively. Now suppose that the symbol   8.1.  GENERALIZED LINEAR OPERATOR denotes a size matrix   8.1.  GENERALIZED LINEAR OPERATOR with which the vector of the original image   8.1.  GENERALIZED LINEAR OPERATOR size   8.1.  GENERALIZED LINEAR OPERATOR linearly converted to vector

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.2)

output image size. The matrix   8.1.  GENERALIZED LINEAR OPERATOR can be divided into blocks - matrix   8.1.  GENERALIZED LINEAR OPERATOR size   8.1.  GENERALIZED LINEAR OPERATOR (the number of which is also equal to   8.1.  GENERALIZED LINEAR OPERATOR ) - and present it as follows:

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.3)

In accordance with the relation (5.3.3), the output image vector   8.1.  GENERALIZED LINEAR OPERATOR can be expressed through the input image matrix   8.1.  GENERALIZED LINEAR OPERATOR :

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.4)

In addition, using equality (5.3.4), the matrix   8.1.  GENERALIZED LINEAR OPERATOR output image can be expressed through the vector p of the same image:

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.5)

From these formulas, we obtain an expression connecting the input and output matrices:

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.6)

Note that the operators   8.1.  GENERALIZED LINEAR OPERATOR and   8.1.  GENERALIZED LINEAR OPERATOR just isolated from the matrix   8.1.  GENERALIZED LINEAR OPERATOR block   8.1.  GENERALIZED LINEAR OPERATOR . Consequently,

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.7)

Let the linear transformation be separable, i.e. the matrix   8.1.  GENERALIZED LINEAR OPERATOR can be represented as a direct work

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.8)

Where   8.1.  GENERALIZED LINEAR OPERATOR and   8.1.  GENERALIZED LINEAR OPERATOR - operators to convert columns and rows of the image matrix   8.1.  GENERALIZED LINEAR OPERATOR . In this case

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.9)

Consequently,

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.10)

Thus, the output image matrix   8.1.  GENERALIZED LINEAR OPERATOR can be obtained by sequential processing of the matrix   8.1.  GENERALIZED LINEAR OPERATOR by rows and columns.

When processing images in many cases, it turns out that the linear transformation operator   8.1.  GENERALIZED LINEAR OPERATOR has a specific structure that allows to simplify computational operations. The following are important special cases illustrated in Fig. 8.1.1, when the dimensions of the input and output images are the same, i.e.   8.1.  GENERALIZED LINEAR OPERATOR .

a) When processing the matrix   8.1.  GENERALIZED LINEAR OPERATOR column only

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.11)

Where   8.1.  GENERALIZED LINEAR OPERATOR - conversion matrix for   8.1.  GENERALIZED LINEAR OPERATOR th column.

b) With the same processing of each column of the matrix   8.1.  GENERALIZED LINEAR OPERATOR

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.12)

c) When processing the matrix   8.1.  GENERALIZED LINEAR OPERATOR only in rows

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.13)

Where   8.1.  GENERALIZED LINEAR OPERATOR - conversion matrix for   8.1.  GENERALIZED LINEAR OPERATOR line

d) With the same processing of each row of the matrix   8.1.  GENERALIZED LINEAR OPERATOR

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.14a)

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.14b)

e) With the same processing of columns and the same processing of rows of the matrix   8.1.  GENERALIZED LINEAR OPERATOR

  8.1.  GENERALIZED LINEAR OPERATOR (8.1.15)

The number of arithmetic operations performed in each of these cases is shown in Table. 8.1.1.

  8.1.  GENERALIZED LINEAR OPERATOR

Fig. 8.1.1. The matrix structure of a linear operator: a - the general case; b - processing only by columns; in - processing only lines; d - processing only rows and columns.

Table 8.1.1. The number of arithmetic operations for linear transformation

Happening

Number of multiplications

and additions

Overall

Column Processing

Row processing

Processing rows and columns

Processing with a splittable matrix

  8.1.  GENERALIZED LINEAR OPERATOR

  8.1.  GENERALIZED LINEAR OPERATOR

  8.1.  GENERALIZED LINEAR OPERATOR

  8.1.  GENERALIZED LINEAR OPERATOR

  8.1.  GENERALIZED LINEAR OPERATOR

From relation (8.1.10) it can be seen that if a two-dimensional linear transformation has a separable matrix, then it can be performed by sequential one-dimensional processing of rows and columns of the sample array. As follows from the table. 8.1.1, for such transformations it is possible to significantly reduce the number of required computational operations: in the general case, when calculating by the formula (8.1.2),   8.1.  GENERALIZED LINEAR OPERATOR operations, but if you can use the formula (8.1.10), it is enough   8.1.  GENERALIZED LINEAR OPERATOR operations. Moreover, in this case the matrix   8.1.  GENERALIZED LINEAR OPERATOR can be stored in a storage device (memory) with sequential access, for example, on a disk or on a drum, and read line by line, i.e. there is no need to store the matrix   8.1.  GENERALIZED LINEAR OPERATOR in a more expensive random access memory. It is necessary, however, to transpose the results of the transformations in columns in order to perform row-wise transformations. In the works [2, 3] algorithms are described for transposing matrices recorded in a memory with sequential access.


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Digital image processing

Terms: Digital image processing