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9.3. OPERATOR OF CYCLIC SUPERPOSITION

Lecture



When using the cyclic superposition operator, the arrays of the input and output images, as well as the impulse response, must be periodic in spatial variables with the same period. For the sake of unity, we assume, as before, that all these arrays have finite dimensions. Assume also that the array of source samples   9.3.  OPERATOR OF CYCLIC SUPERPOSITION by size   9.3.  OPERATOR OF CYCLIC SUPERPOSITION placed in the upper left corner of the array containing   9.3.  OPERATOR OF CYCLIC SUPERPOSITION zeros   9.3.  OPERATOR OF CYCLIC SUPERPOSITION i.e. an extended array is formed

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION at   9.3.  OPERATOR OF CYCLIC SUPERPOSITION , (9.3.1а)

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION at   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . (9.3.1b)

Similarly, an extended array of impulse response samples is formed:

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION at   9.3.  OPERATOR OF CYCLIC SUPERPOSITION , (9.3.2а)

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION at   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . (9.3.2b)

Next, we form periodically continued arrays   9.3.  OPERATOR OF CYCLIC SUPERPOSITION and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION , repeating (propagating) extended arrays with a period of   9.3.  OPERATOR OF CYCLIC SUPERPOSITION counts. The cyclic convolution of these arrays is by definition equal to

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION . (9.3.3)

The similarity of this expression with equality (9.1.1), describing the superposition of finite arrays, is striking. In fact, if   9.3.  OPERATOR OF CYCLIC SUPERPOSITION chosen so that   9.3.  OPERATOR OF CYCLIC SUPERPOSITION then   9.3.  OPERATOR OF CYCLIC SUPERPOSITION at   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . It should also be noted the similarity of cyclic superposition with the superposition of discretized arrays. These relationships become clearer when using the vector representation of cyclic superposition.

Assume that arrays   9.3.  OPERATOR OF CYCLIC SUPERPOSITION and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION presented accordingly   9.3.  OPERATOR OF CYCLIC SUPERPOSITION -component vectors   9.3.  OPERATOR OF CYCLIC SUPERPOSITION and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . Then for the operation of cyclic superposition, you can write the ratio

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION , (9.3.4)

Where   9.3.  OPERATOR OF CYCLIC SUPERPOSITION - matrix of array elements   9.3.  OPERATOR OF CYCLIC SUPERPOSITION size   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . The cyclic superposition operator can be conveniently expressed as a block matrix with blocks   9.3.  OPERATOR OF CYCLIC SUPERPOSITION size   9.3.  OPERATOR OF CYCLIC SUPERPOSITION :

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION , (9.3.5)

Where

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION , (9.3.6)

and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION a   9.3.  OPERATOR OF CYCLIC SUPERPOSITION and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . It should be noted that each row and each column of the block matrix   9.3.  OPERATOR OF CYCLIC SUPERPOSITION contain   9.3.  OPERATOR OF CYCLIC SUPERPOSITION nonzero blocks. If the array of impulse response samples is spatially invariant, then

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION (9.3.7)

and any row (or column) can be obtained by cyclically rearranging the blocks of the first row (or the first column). In fig. 9.3.1, and the example of the cyclic convolution operator is given, when the input and output arrays have dimensions   9.3.  OPERATOR OF CYCLIC SUPERPOSITION   9.3.  OPERATOR OF CYCLIC SUPERPOSITION , and the size of the array of samples of the impulse response is equal to   9.3.  OPERATOR OF CYCLIC SUPERPOSITION   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . In fig. 9.3.1, b shows the matrix structure of the same operator with   9.3.  OPERATOR OF CYCLIC SUPERPOSITION and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION when the impulse response is Gaussian.

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION

Fig. 9.3.1. Examples of matrices of the cyclic convolution operator.

a - the general case   9.3.  OPERATOR OF CYCLIC SUPERPOSITION ; b - impulse response of the Gaussian form,   9.3.  OPERATOR OF CYCLIC SUPERPOSITION .

If the impulse response is spatially invariant and separable, then

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION , (9.3.8)

Where   9.3.  OPERATOR OF CYCLIC SUPERPOSITION and   9.3.  OPERATOR OF CYCLIC SUPERPOSITION - view matrix

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION , (9.3.9)

which size is equal   9.3.  OPERATOR OF CYCLIC SUPERPOSITION . In this case, the two-dimensional cyclic convolution is calculated in accordance with the ratio

  9.3.  OPERATOR OF CYCLIC SUPERPOSITION . (9.3.10)


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Digital image processing

Terms: Digital image processing