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9.1. OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES

Lecture



We first consider the discrete superposition operator of a finite array of samples (for simplicity, it is assumed that all arrays of samples are square)   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (Where   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ) with finite array   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (Where   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ), playing the role of impulse response. In the general case, the impulse response may vary depending on the coordinates   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES readout in the output array   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES . The operation of superposition in a limited area is determined by the ratio

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (9.1.1)

Where   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and arrays   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES have zero values ​​outside the ranges of the corresponding indices. Analyzing the limiting values ​​of the indices of the impulse response counts, we can be sure that   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and therefore the output array   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES It is larger than the original (Fig. 9.1.1).

If arrays   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES presented respectively as a vector   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES size   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and vectors   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES size   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES , then the transform (9.1.1) can be written as [1]

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES , (9.1.2)

Where   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES - size matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES containing impulse response counts. The matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES superposition operator conveniently divided into blocks   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES size   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES .

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES

Fig. 9.1.1. The superposition of the final arrays of the impulse response samples and the original image: A - passive from   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES image readouts; B - rotated 180 ° array of   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES impulse response counts.

Analyzing the limits of summation in expression (9.1.1), it can be shown that

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (9.1.3)

Arbitrary nonzero matrix element   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES has the appearance

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (9.1.4)

Where   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES , but   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES . It follows that the matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES It has a regular structure and is rather rarely filled, with nonzero blocks grouped as a band in the middle part of the matrix.   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES , contain zones of zero elements.

If the form of the impulse response is invariant with respect to the shift (that is, the same for all points of the output array), then the structure of the matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES independent of coordinates   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES output countdown. Then

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (9.1.5)

Thus, all columns of the matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES formed by shifting the first column. In this case, the superposition operator is called the convolution operator of finite arrays. In fig. 9.1.2; a shows the printouts of matrices obtained in the convolution of final arrays obtained on the digital computer, for the case when the input array has dimensions   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ) output array   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ), and an array of impulse response counts   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ). Pairs of integers   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES in the matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES designate   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES matrix element   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES .

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES

Fig. 9.1.2. Examples of matrices of convolution operators of finite arrays: a - the general case,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ; b - impulse response of the Gaussian form,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES .

Matrix structure   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES is better seen in the example of the larger matrix shown in Fig. 9.1.2, b. In the matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES ,   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES , and the impulse response is symmetric and has a Gaussian form. Note that in this example, the dimensions of the matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES equal to 256   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES 64.

Using the technique applied in deriving the relation (8.1.7), the superposition operator can be represented in the matrix form

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES (9.1.6)

If the impulse response is shift-invariant and separable, i.e.

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES , (9.1.7)

Where   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES - the column vector, describing, respectively, the nature of the change of the impulse response in columns and rows, then

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES . (9.1.8)

Matrices   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES have dimensions   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES and species structure

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES . (9.1.9)

The operation of two-dimensional convolution in this case is reduced to the sequential calculation of one-dimensional convolutions in rows and columns. In this way,

  9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES . (9.1.10)

To obtain the final convolution or superposition, in general, you must perform   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES arithmetic operations, and this number does not include multiplication by zero elements of the matrix   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES . If the operator is separable, that is, it satisfies equality (9.1.10), then it suffices to execute   9.1.  OPERATOR OF SUPERPOSITION OF FINITE MASSIFIES operations.


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Digital image processing

Terms: Digital image processing