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A generalized Walsh system for arbitrary matrices

Harding Steven N., Picioroaga Gabriel

Demonstratio Mathematica

2019

Vol. 52, nr 1

40--55

In this paper we study in detail a variation of the orthonormal bases (ONB) of L2 [0,1] introduced in [Dutkay D. E., Picioroaga G., Song M. S., Orthonormal bases generated by Cuntz algebras, J. Math. Anal. Appl., 2014, 409(2),1128-1139] by means of representations of the Cuntz algebra ON on L2 [0,1]. For N = 2 one obtains the classic Walsh system which serves as a discrete analog of the Fourier system. We prove that the generalized Walsh system does not always display periodicity, or invertibility, with respect to function multiplication. After characterizing these two properties we also show that the transform implementing the generalized Walsh system is continuous with respect to filter variation. We consider such transforms in the case when the orthogonality conditions in Cuntz relations are removed. We show that these transforms which still recover information (due to remaining parts of the Cuntz relations) are suitable to use for signal compression, similar to the discrete wavelet transform.

Biomedical images enhancement based on the properties of morphological spectra

Przytulska M., Kulikowski J. L., Wierzbicka D.

Biocybernetics and Biomedical Engineering

2015

Vol. 35, no. 3

206--215

The method enhancing distinctiveness of the micro-morphological structures, developed using the properties of morphological spectra of their monochromatic 2D images, is presented and its effects on the bone section image are statistically compared with enhancements by Sobel, Roberts and Laplace high-pass filters. Comparison of different filters based on statistical parameters of the classes of selected image details is presented. The preferable method for choosing filtering weight coefficients is described and illustrated by an example of processing an electron-microscope image of a biotechnological specimen. The applicability of this approach and possible development directions are discussed.