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11.5. Pseudo-reincarnation

Lecture



Transformation processing can be successfully applied to obtain generalized inverse matrices. As can be seen from the relation (11.1.6b), the matrix 11.5.  Pseudo-reincarnation linear operator in spectral space is associated with an arbitrary matrix 11.5.  Pseudo-reincarnation size 11.5.  Pseudo-reincarnation in the following way:

11.5.  Pseudo-reincarnation . (11.5.1)

In the same notation, the generalized inverse matrix 11.5.  Pseudo-reincarnation defined as

11.5.  Pseudo-reincarnation , (11.5.2)

Where 11.5.  Pseudo-reincarnation - size matrix 11.5.  Pseudo-reincarnation . Relations (11.5.1) and (11.5.2) are self-consistent, since it is known that for an arbitrary matrix 11.5.  Pseudo-reincarnation and unitary matrices 11.5.  Pseudo-reincarnation and 11.5.  Pseudo-reincarnation equality is fulfilled 11.5.  Pseudo-reincarnation [13, p. 100].

If the rank of the matrix 11.5.  Pseudo-reincarnation equals 11.5.  Pseudo-reincarnation , then, according to the formula (8.3.5),

11.5.  Pseudo-reincarnation (11.5.3)

and it is not difficult to show that

11.5.  Pseudo-reincarnation . (11.5.4)

In the opposite case, when the rank of the matrix 11.5.  Pseudo-reincarnation equals 11.5.  Pseudo-reincarnation , from the formula (8.3.6) it follows that

11.5.  Pseudo-reincarnation (11.5.5)

and the generalized inverse transformation matrix in the spectral space satisfies the relation

11.5.  Pseudo-reincarnation . (11.5.6)

We indicate, for example, that the generalized inverse matrices with respect to matrices of maximal rank corresponding to the superposition operators 11.5.  Pseudo-reincarnation and 11.5.  Pseudo-reincarnation determined by equalities

11.5.  Pseudo-reincarnation , (11.5.7а)

11.5.  Pseudo-reincarnation , (11.5.7b)

11.5.  Pseudo-reincarnation . (11.5.7b)

In fig. 11.5.1 shows printouts of generalized inverse matrices for convolution operators of one-dimensional signals using Fourier and Hadamard transforms. It is well seen that these matrices are more sparse than the original matrices. Moreover, the generalized inverse matrix of the cyclic convolution operator with the Fourier transform is diagonal, since, as follows from (11.2.13), the matrix 11.5.  Pseudo-reincarnation - diagonal.

11.5.  Pseudo-reincarnation

Fig. 11.5.1. Generalized inverse matrices of convolution operators of one-dimensional signals using Fourier and Hadamard transforms.

a is the final convolution; b - discretized integral convolution; c is a cyclic convolution.


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Digital image processing

Terms: Digital image processing