Two-dimensional Discrete Signals Research Articles

Approximate methods for solving amplitude-phase problem for discrete signals

Methods for solving amplitude and phase problems for one and two-dimensional discrete signals are proposed. Methods are based on using nonlinear singular integral equations. In the one-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear singular integral equations. In the two-dimensional case amplitude and phase problems are modeled by corresponding linear and nonlinear bisingular integral equations. Several approaches are presented for modeling two-dimensional problems: 1) reduction of amplitude and phase problems to systems of linear and nonlinear singular integral equations; 2) using methods of the theory of functions of many complex variables, problems are reduced to linear and nonlinear bisingular integral equations. To solve the constructed nonlinear singular integral equations, methods of collocation and mechanical quadrature are used. These methods lead to systems of nonlinear algebraic equations, which are solved by the continuous method for solution of nonlinear operator equations. The choice of this method is due to the fact that it is stable against perturbations of coefficients in the right-hand side of the system of equations. In addition, the method is realizable even in cases where the Frechet and Gateaux derivatives degenerate at a finite number of steps in the iterative process. Some model examples have shown effectiveness of proposed methods and numerical algorithms.

A Hybrid Model of 2d-DCT and 2d-Mycielski Algorithm for Hourly Global Solar Irradiation

In this work, hourly solar irradiation is defined as a two-dimensional discrete signal. This two-dimensional discrete signal is modeled by a novel hybrid approach, which includes both ...

TWO-DIMENSIONAL SIGNAL PROCESSING IN THE DISCRETE FOURIER BASIS

Исследуются проблемы обработки двумерных сигналов в пространственно-частотной области на базе двумерного дискретного преобразования Фурье. Рассмотрено общее определение и математическое описание двумерного дискретного сигнала в пространственной области. Дана алгебраическая форма двумерного дискретного преобразования Фурье, кратко рассмотрены основные свойства двумерного дискретного преобразования Фурье. Проведен (по отечественным и зарубежным информационным источникам) системный анализ дискретных двумерных сигналов (природы их происхождения, источников формирования), а также методов их обработки, основанных на двумерном дискретном преобразовании Фурье. Рассмотрены приложения дискретных двумерных сигналов в различных предметных областях. Рассмотрены достоинства и недостатки цифровых методов дискретной двумерной обработки на основе двумерного дискретного преобразования Фурье. Выдвинута рабочая гипотеза решения проблематики дискретной двумерной обработки сигналов в пространственно-частотной области путем разработки новых двумерных базисных систем, сочетающих преимущества двумерного дискретного базиса Фурье и снижающих (устраняющих) влияние его негативных эффектов. Показано, что двумерный вариант дискретного канонического разложения случайных сигналов, разработанный В. С. Пугачевым, предполагает (по умолчанию) видоизменение стандартной циклической двумерной корреляционной функции исходного сигнала. Предложены пути решения указанной проблемы.

Sliding signal processing in telecommunication networks based on two-dimensional discrete Fourier transform

A method of vertical sliding processing of two-dimensional discrete signals in the spatial frequency domain is proposed — a method of fast vertically sliding two-dimensional discrete Fourier transform. The mathematical representation of the two-dimensional discrete Fourier transform in algebraic and matrix form is considered. An effective method of vertically sliding two-dimensional discrete Fourier transform is proposed. The algorithm developed in the framework of the proposed method allows calculating the coefficients (bins) of this transformation in real time.

On the Role of Total Variation in Compressed Sensing

This paper considers the problem of recovering a one- or two-dimensional discrete signal which is approximately sparse in its gradient from an incomplete subset of its Fourier coefficients which have been corrupted with noise. We prove that in order to obtain a reconstruction which is robust to noise and stable to inexact gradient sparsity of order $s$ with high probability, it suffices to draw ${\cal O}$ $\!{(s {\rm log} N)}$ of the available Fourier coefficients uniformly at random. However, we also show that if one draws ${\cal O}$ ${\!(s {\rm log} N)}$ samples in accordance with a particular distribution which concentrates on the low Fourier frequencies, then the stability bounds which can be guaranteed are optimal up to log factors. Finally, we prove that in the one-dimensional case where the underlying signal is gradient sparse and its sparsity pattern satisfies a minimum separation condition, to guarantee exact recovery with high probability, for some $M<N$, it suffices to draw ${\cal O}$ ${\!(s{\rm log} M {\rm log} s)}$ samples uniformly at random from the Fourier coefficients whose frequencies are no greater than $M$.

Design and implementation in VHDL code of the two-dimensional fast Fourier transform for frequency filtering, convolution and correlation operations

The two-dimensional Fast Fourier Transform (FFT 2D) is an essential tool in the two-dimensional discrete signals analysis and processing, which allows developing a large number of applications. This article shows the description and synthesis in VHDL code of the FFT 2D with fixed point binary representation using the programming tool Simulink HDL Coder of Matlab; showing a quick and easy way to handle overflow, underflow and the creation registers, adders and multipliers of complex data in VHDL and as well as the generation of test bench for verification of the codes generated in the ModelSim tool. The main objective of development of the hardware architecture of the FFT 2D focuses on the subsequent completion of the following operations applied to images: frequency filtering, convolution and correlation. The description and synthesis of the hardware architecture uses the XC3S1200E family Spartan 3E FPGA from Xilinx Manufacturer.

A fast implementation of the discrete 2-D Gabor transform

An FFT-based gradient descent method for computing the non-orthogonal Gabor transform of a two-dimensional discrete signal I[ x, y] is described. When operating on images consisting of P × Q pixels divided into sub-images with M × N pixels, the estimated gain in computational speed over a neural network method proposed by Daugman is by a factor of kMN, where k = 10/[3 log 2(4 MN) +4].

Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression

A three-layered neural network is described for transforming two-dimensional discrete signals into generalized nonorthogonal 2-D Gabor representations for image analysis, segmentation, and compression. These transforms are conjoint spatial/spectral representations, which provide a complete image description in terms of locally windowed 2-D spectral coordinates embedded within global 2-D spatial coordinates. In the present neural network approach, based on interlaminar interactions involving two layers with fixed weights and one layer with adjustable weights, the network finds coefficients for complete conjoint 2-D Gabor transforms without restrictive conditions. In wavelet expansions based on a biologically inspired log-polar ensemble of dilations, rotations, and translations of a single underlying 2-D Gabor wavelet template, image compression is illustrated with ratios up to 20:1. Also demonstrated is image segmentation based on the clustering of coefficients in the complete 2-D Gabor transform. >

A new direct algorithm for image reconstruction from Fourier transform magnitude

In this paper we present a new direct solution to the problem of reconstructing a two-dimensional discrete signal of finite support from knowledge of only its Fourier transform magnitude and support. Using the autocorrelation function of the unknown signal, zeros are calculated of a polynomial whose coefficients correspond to the unknown image or the image reversed. From these zeros, a set of linear equations is developed whose solution yields either the original image or the image rotated by 180°.