Model of Random Field with Piece-Constant Values and Sampling-Restoration Algorithm of Its Realizations

Abstract

We propose a description of the model of a random piecewise constant field formed by the sum of realizations of two Markov processes with an arbitrary number of states and defined along mutually perpendicular axes. The number of field quantization levels can be arbitrary. Realizations of a random field model of the desired shape are created by appropriate selection of parameters for formative realization of Markov processes. For the proposed field model, we investigated the sampling and restoration algorithm of any selected realizations. As a result, we determined the optimal sampling and recovery algorithms. The resulting sampling is fundamentally non-periodic. Recovery errors are calculated. Two examples are considered.

Melanoma and Nevus Skin Lesion Classification Using Handcraft and Deep Learning Feature Fusion via Mutual Information Measures

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Two-Dimensional Sampling-Recovery Algorithm of a Realization of Gaussian Processes on the Input and Output of Linear Systems

“…The application of the CMM (see, for example, [ 10 , 11 , 12 ]) to the study of SRA of realizations of random processes has a number of advantages (see [ 13 , 14 , 15 , 16 , 17 ] and references therein) in comparison with the well-known Balakrishnan theorem (TB) [ 18 ] and many of its generalizations. Indeed, SRA based on CMM are distinguished by such positive qualities as: (1) Restoration of a sampled realization of a random process according to the CMM automatically provides a minimum of the root-mean-square error of restoration; (2) the restoring function, like the restoring error function in the general case, takes into account the main statistical characteristics of a random process: Probability density, covariance, and cumulant functions; spectrum (a process with a limited spectrum is a special case); (3) the considered algorithms are optimal for any number and location of samples (the variant of periodic samples is a special case); (4) general analytical expressions for the considered SRA cover stationary and non-stationary variants of stochastic processes; (5) sampled stochastic processes can be Gaussian and non-Gaussian, continuous and discontinuous, etc.…”

“…Indeed, SRA based on CMM are distinguished by such positive qualities as: (1) Restoration of a sampled realization of a random process according to the CMM automatically provides a minimum of the root-mean-square error of restoration; (2) the restoring function, like the restoring error function in the general case, takes into account the main statistical characteristics of a random process: Probability density, covariance, and cumulant functions; spectrum (a process with a limited spectrum is a special case); (3) the considered algorithms are optimal for any number and location of samples (the variant of periodic samples is a special case); (4) general analytical expressions for the considered SRA cover stationary and non-stationary variants of stochastic processes; (5) sampled stochastic processes can be Gaussian and non-Gaussian, continuous and discontinuous, etc. Moreover, the CMM has been productively applied to study the SRA of random fields, both Gaussian [ 19 , 20 ] and fields with jumps [ 17 , 21 ]. Of course, the version of a multidimensional Gaussian process turns out to be more convenient for analysis since there are simple analytical relations for it.…”