多尺度变换及其在图像纹理分类中的应用（英文版）

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Chapter 1 Introduction
　　1.1 Multiscale Methods
　　Signals (e.g. images/textures and videos) carry a lot of data in which relevant information is difficult to find. Processing signals can be faster and simpler if a sparse representation has been built for signals and few coefficients in the sparse representation can reveal the information we are looking for. Such representations can be constructed by decomposing signals over elementary waveforms chosen in a family called a dictionary. But the search for the Holy Grail of an ideal sparse transform adapted to all signals is a hopeless quest. The discovery of wavelet orthogonal bases and local time-frequency dictionaries have opened the door to a huge jungle of new transforms. Adapting sparse representations to signal properties，and deriving efficient processing operators， is therefore a necessary survival strategy [1].
　　An orthogonal basis is a dictionary of minimum size that can yield a sparse representation if designed to concentrate the signal energy over a set of few vectors. This set gives a geometric signal description. Efficient signal compression and noise reduction algorithms are then implemented with diagonal operators computed with fast algorithms. But this is not always optimal [1].
　　The Fourier transform is everywhere in physics and mathematics because it diagonalizes time-invariant convolution operators. It rules over linear timeinvariant signal processing， the building blocks of which are frequency filtering operators. Fourier analysis represents any finite energy function f(t) as a sum of sinusoidal waves. The amplitude of each sinusoidal wave is equal to its correlation with f， also called Fourier transform. For discrete signals， the Fourier transform is a decomposition in a discrete orthogonal Fourier basis [1].
　　A. Wavelets
　　Wavelet bases， like Fourier bases， reveal the signal regularity through the amplitude of coefficients， and their structure leads to a fast computational algorithm. However， wavelets are well localized and few coefficients are needed to represent local transient structures. As opposed to a Fourier basis， a wavelet basis defines a sparse representation of piecewise regular signals，which may include transients and singularities. In images， large wavelet coefficients are located in the neighborhood of edges and irregular textures. More details， developments and applications of wavelets can be found in [1].
　　B. Contourlets
　　The contourlet transform was recently developed by Do and Vetterli [5] in order to get rid of the limitations of wavelets. Actually， they utilized a double filter bank structure in which at first the Laplacian Pyramid (LP) [6] is used to capture the point discontinuities， and then a Directional Filter Bank (DFB) [7] is used to link point discontinuities into linear structure. So， the overall result of such a transform is based on an image expansion with basis elements like contour segments， and thus it is named the contourlet transform. More recent developments and applications on the contourlet transform can be found in [8]—[11].
　　Due to its cascade structure accomplished by combining the Laplacian pyramid with a directional filter bank at each scale， multiscale and directional decomposition stages in the contourlet transform are independent of each other. Therefore， one can decompose each scale into any arbitrary power of two’s number of directions， and different scales can be decomposed into different numbers of directions. For simplicity， we impose that in the pyramid DFB， the number of DFB decomposition levels is three at each scale of the pyramid to capture the directional information efficiently， that is， the number of directions at each scale is eight.
　　C. Shearlets
　　The shearlet transform was developed to overcome limitations inherent in wavelets. The shearlet representation is based on a simple but rigorous mathematical framework. This framework not only provides a more flexible theoretical tool for the geometrical representation of multidimensional data， but is also more natural for implementation. In one sense， the theory of shearlets can be seen as a theoretical justification for contourlets [12]. Moreover， the shearlet transform can be implemented using the succession of a Laplacian pyramid and directional filtering. For clarity， we briefly present some analysis about the development of shearlets as follows [13].
　　The notion of efficient representation of data plays an increasingly important role in areas across applied mathematics， data science and engineering. Over the past decade， there has been a rapidly increasing pressure to handle ever larger and higher-dimensional data sets， with the challenge of providing representations of these data that are sparse and computationally fast. Sparse representations have implications reaching beyond data compression. Understanding the compression problem for a given data type entails a precise