Window Functions and Their Applications in Signal Processing

Window functions—otherwise known as weighting functions, tapering functions, or apodization functions—are mathematical functions that are zero-valued outside the chosen interval. They are well established as a vital part of digital signal processing. Window Functions and their Applications in Signal Processing presents an exhaustive and detailed account of window functions and their applications in signal processing, focusing on the areas of digital spectral analysis, design of FIR filters, pulse compression radar, and speech signal processing. Comprehensively reviewing previous research and recent developments, this book: Provides suggestions on how to choose a window function for particular applications Discusses Fourier analysis techniques and pitfalls in the computation of the DFT Introduces window functions in the continuous-time and discrete-time domains Considers two implementation strategies of window functions in the time- and frequency domain Explores well-known applications of window functions in the fields of radar, sonar, biomedical signal analysis, audio processing, and synthetic aperture rada

Window Functions and Their Applications in Signal Processing

Window functions—otherwise known as weighting functions, tapering functions, or apodization functions—are mathematical functions that are zero-valued outside the chosen interval. They are well established as a vital part of digital signal processing. Window Functions and their Applications in Signal Processing presents an exhaustive and detailed account of window functions and their applications in signal processing, focusing on the areas of digital spectral analysis, design of FIR filters, pulse compression radar, and speech signal processing. Comprehensively reviewing previous research and recent developments, this book: Provides suggestions on how to choose a window function for particular applications Discusses Fourier analysis techniques and pitfalls in the computation of the DFT Introduces window functions in the continuous-time and discrete-time domains Considers two implementation strategies of window functions in the time- and frequency domain Explores well-known applications of window functions in the fields of radar, sonar, biomedical signal analysis, audio processing, and synthetic aperture rada

Sparse Fast Trigonometric Transforms

Trigonometric transforms like the Fourier transform or the discrete cosine transform (DCT) are of immense importance in signal and image processing, physics, engineering, and data processing. The research of past decades has provided us with runtime optimal algorithms for these transforms. Significant runtime improvements are only possible if there is additional a priori information about the sparsity of the signal. In the first part of this thesis we develop sublinear algorithms for the fast Fourier transform for frequency sparse periodic functions. We investigate three classes of sparsity: short frequency support, polynomially structured sparsity and block sparsity. For all three classes we present new deterministic, sublinear algorithms that recover the Fourier coefficients of periodic functions from samples. We prove theoretical runtime and sampling bounds for all algorithms and also investigate their performance in numerical experiments. In the second part of this thesis we focus on the reconstruction of vectors with short support from their DCT of type II. We present two different new, deterministic and sublinear algorithms for this problem. The first method is based on inverse discrete Fourier transforms and uses complex arithmetic, whereas the second one utilizes properties of the DCT and only employs real arithmetic. We show theoretical runtime and sampling bounds for both algorithms and compare them numerically in experiments. Furthermore, we generalize the techniques for recovering vectors with short support from their DCT of type II using only real arithmetic to the two-dimensional setting of recovering matrices with block support, also providing theoretical runtime and sampling complexities for the obtained new two-dimensional algorithm

Reconstruction of undersampled signals and alignment in the frequency domain

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Data compression techniques applied to high resolution high frame rate video technology

An investigation is presented of video data compression applied to microgravity space experiments using High Resolution High Frame Rate Video Technology (HHVT). An extensive survey of methods of video data compression, described in the open literature, was conducted. The survey examines compression methods employing digital computing. The results of the survey are presented. They include a description of each method and assessment of image degradation and video data parameters. An assessment is made of present and near term future technology for implementation of video data compression in high speed imaging system. Results of the assessment are discussed and summarized. The results of a study of a baseline HHVT video system, and approaches for implementation of video data compression, are presented. Case studies of three microgravity experiments are presented and specific compression techniques and implementations are recommended

Optimal prefilters for display enhancement

Creating images from a set of discrete samples is arguably the most common operation in computer graphics and image processing, lying, for example, at the heart of rendering and image downscaling techniques. Traditional tools for this task are based on classic sampling theory and are modeled under mathematical conditions which are, in most cases, unrealistic; for example, sinc reconstruction – required by Shannon theorem in order to recover a signal exactly – is impossible to achieve in practice because LCD displays perform a box-like interpolation of the samples. Moreover, when an image is made for a human to look at, it will necessarily undergo some modifications due to the human optical system and all the neural processes involved in vision. Finally, image processing practitioners noticed that sinc prefiltering – also required by Shannon theorem – often leads to visually unpleasant images. From these facts, we can deduce that we cannot guarantee, via classic sampling theory, that the signal we see in a display is the best representation of the original image we had in first place. In this work, we propose a novel family of image prefilters based on modern sampling theory, and on a simple model of how the human visual system perceives an image on a display. The use of modern sampling theory guarantees us that the perceived image, based on this model, is indeed the best representation possible, and at virtually no computational overhead. We analyze the spectral properties of these prefilters, showing that they offer the possibility of trading-off aliasing and ringing, while guaranteeing that images look sharper then those generated with both classic and state-of-the-art filters. Finally, we compare it against other solutions in a selection of applications which include Monte Carlo rendering and image downscaling, also giving directions on how to apply it in different contexts.Exibir imagens a partir de um conjunto discreto de amostras é certamente uma das operações mais comuns em computação gráfica e processamento de imagens. Ferramentas tradicionais para essa tarefa são baseadas no teorema de Shannon e são modeladas em condições matemáticas que são, na maior parte dos casos, irrealistas; por exemplo, reconstrução com sinc – necessária pelo teorema de Shannon para recuperar um sinal exatamente – é impossível na prática, já que displays LCD realizam uma reconstrução mais próxima de uma interpolação com kernel box. Além disso, profissionais em processamento de imagem perceberam que prefiltragem com sinc – também requerida pelo teorema de Shannon – em geral leva a imagens visualmente desagradáveis devido ao fenômeno de ringing: oscilações próximas a regiões de descontinuidade nas imagens. Desses fatos, deduzimos que não é possível garantir, via ferramentas tradicionais de amostragem e reconstrução, que a imagem que observamos em um display digital é a melhor representação para a imagem original. Neste trabalho, propomos uma família de prefiltros baseada em teoria de amostragem generalizada e em um modelo de como o sistema ótico do olho humano modifica uma imagem. Proposta por Unser and Aldroubi (1994), a teoria de amostragem generalizada é mais geral que o teorema proposto por Shannon, e mostra como é possível pré-filtrar e reconstruir sinais usando kernels diferentes do sinc. Modelamos o sistema ótico do olho como uma câmera com abertura finita e uma lente delgada, o que apesar de ser simples é suficiente para os nossos propósitos. Além de garantir aproximação ótima quando reconstruindo as amostras por um display e filtrando a imagem com o modelo do sistema ótico humano, a teoria de amostragem generalizada garante que essas operações são extremamente eficientes, todas lineares no número de pixels de entrada. Também, analisamos as propriedades espectrais desses filtros e de técnicas semelhantes na literatura, mostrando que é possível obter um bom tradeoff entre aliasing e ringing (principais artefatos quando lidamos com amostragem e reconstrução de imagens), enquanto garantimos que as imagens finais são mais nítidas que aquelas geradas por técnicas existentes na literatura. Finalmente, mostramos algumas aplicações da nossa técnica em melhoria de imagens, adaptação à distâncias de visualização diferentes, redução de imagens e renderização de imagens sintéticas por método de Monte Carlo

Orthogonal transforms and their application to image coding

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