Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces

Compressed sensing

Introduction to a Transient World. Fourier Kingdom. Discrete Revolution. Time Meets Frequency. Frames. Wavelet Zoom. Wavelet Bases. Wavelet Packet and Local Cosine Bases. An Approximation Tour. Estimations are Approximations. Transform Coding. Appendix A: Mathematical Complements. Appendix B: Software Toolboxes.

http://ahvazdownload.persiangig.com/document/article/39.pdf

A wavelet tour of signal processing

We propose a novel image denoising strategy based on an enhanced sparse representation in transform domain. The enhancement of the sparsity is achieved by grouping similar 2D image fragments (e.g., blocks) into 3D data arrays which we call "groups." Collaborative Altering is a special procedure developed to deal with these 3D groups. We realize it using the three successive steps: 3D transformation of a group, shrinkage of the transform spectrum, and inverse 3D transformation. The result is a 3D estimate that consists of the jointly filtered grouped image blocks. By attenuating the noise, the collaborative filtering reveals even the finest details shared by grouped blocks and, at the same time, it preserves the essential unique features of each individual block. The filtered blocks are then returned to their original positions. Because these blocks are overlapping, for each pixel, we obtain many different estimates which need to be combined. Aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy. A significant improvement is obtained by a specially developed collaborative Wiener filtering. An algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail; an extension to color-image denoising is also developed. The experimental results demonstrate that this computationally scalable algorithm achieves state-of-the-art denoising performance in terms of both peak signal-to-noise ratio and subjective visual quality.

Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering

We address the image denoising problem, where zero-mean white and homogeneous Gaussian additive noise is to be removed from a given image. The approach taken is based on sparse and redundant representations over trained dictionaries. Using the K-SVD algorithm, we obtain a dictionary that describes the image content effectively. Two training options are considered: using the corrupted image itself, or training on a corpus of high-quality image database. Since the K-SVD is limited in handling small image patches, we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image. We show how such Bayesian treatment leads to a simple and effective denoising algorithm. This leads to a state-of-the-art denoising performance, equivalent and sometimes surpassing recently published leading alternative denoising methods

Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries

Humans perceive the three-dimensional structure of the world with apparent ease. However, despite all of the recent advances in computer vision research, the dream of having a computer interpret an image at the same level as a two-year old remains elusive. Why is computer vision such a challenging problem and what is the current state of the art? Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images. It also describes challenging real-world applications where vision is being successfully used, both for specialized applications such as medical imaging, and for fun, consumer-level tasks such as image editing and stitching, which students can apply to their own personal photos and videos. More than just a source of recipes, this exceptionally authoritative and comprehensive textbook/reference also takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene. These problems are also analyzed using statistical models and solved using rigorous engineering techniques Topics and features: structured to support active curricula and project-oriented courses, with tips in the Introduction for using the book in a variety of customized courses; presents exercises at the end of each chapter with a heavy emphasis on testing algorithms and containing numerous suggestions for small mid-term projects; provides additional material and more detailed mathematical topics in the Appendices, which cover linear algebra, numerical techniques, and Bayesian estimation theory; suggests additional reading at the end of each chapter, including the latest research in each sub-field, in addition to a full Bibliography at the end of the book; supplies supplementary course material for students at the associated website, http://szeliski.org/Book/. Suitable for an upper-level undergraduate or graduate-level course in computer science or engineering, this textbook focuses on basic techniques that work under real-world conditions and encourages students to push their creative boundaries. Its design and exposition also make it eminently suitable as a unique reference to the fundamental techniques and current research literature in computer vision.

/pdf/computer-vision-algorithms-and-applications-25dn6wu83j.pdf

Computer Vision: Algorithms and Applications

We describe a method for removing noise from digital images, based on a statistical model of the coefficients of an overcomplete multiscale oriented basis. Neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables: a Gaussian vector and a hidden positive scalar multiplier. The latter modulates the local variance of the coefficients in the neighborhood, and is thus able to account for the empirically observed correlation between the coefficient amplitudes. Under this model, the Bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable. We demonstrate through simulations with images contaminated by additive white Gaussian noise that the performance of this method substantially surpasses that of previously published methods, both visually and in terms of mean squared error.

/pdf/image-denoising-using-scale-mixtures-of-gaussians-in-the-2qh5uqp80y.pdf

Image denoising using scale mixtures of Gaussians in the wavelet domain

Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrix-valued filterbanks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar two-channel wavelet systems. After reviewing this theory, we examine the use of multiwavelets in a filterbank setting for discrete-time signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filterbank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a one-dimensional (1-D) signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximation-based preprocessing. We describe an additional algorithm for multiwavelet processing of two-dimensional (2-D) signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is well-suited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via wavelet-shrinkage, and data compression. After developing the approach via model problems in one dimension, we apply multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.

/pdf/the-application-of-multiwavelet-filterbanks-to-image-1g5nj4t750.pdf

The application of multiwavelet filterbanks to image processing

Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a two-band orthogonal filter bank). For "multifilters" those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust (see J. Approx. Theory, vol.78, p.373-401, 1994) constructed two scaling functions that have extra properties not previously achieved. The functions /spl Phi//sub 1/ and /spl Phi//sub 2/ are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function /spl Phi/, apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2/spl times/2 matrix coefficients while retaining orthogonality of the multiwavelets. This note derives the properties of /spl Phi//sub 1/ and /spl Phi//sub 2/ from the matrix dilation equation that they satisfy. Then our main step is to construct associated wavelets: two wavelets for two scaling functions. The properties were derived by Geronimo, Hardin, and Massopust from the iterated interpolation that led to /spl Phi/1 and /spl Phi//sub 2/. One pair of wavelets was found earlier by direct solution of the orthogonality conditions (using Mathematica). Our construction is in parallel with recent progress by Hardin and Geronimo, to develop the underlying algebra from the matrix coefficients in the dilation equation-in another language, to build the 4/spl times/4 paraunitary polyphase matrix in the filter bank. The short support opens new possibilities for applications of filters and wavelets near boundaries. >

Short wavelets and matrix dilation equations

A function (t) is reenable if it satisses a dilation equation (t) = P k C k (2t ? k). A reenable function (scaling function) generates a multiresolution analysis (MRA): Set of T 1 j=?1 V j = f0g, and translates (t ? k) constitute a basis of V 0. Then a basis fw jk : w jk = w(2 j t ? k) j; k 2 Zg of L 2 (R) is generated by a wavelet w(t), whose translates w(t ? k) form a basis of W 0 , V 1 = V 0 W 0. A standard (scalar) MRA assumes that there is only one scaling function. We make a step forward and allow several scaling functions 0 (t); : : :; r?1 (t) to generate a basis of V 0. The vector (t) = 0 (t) : : : r?1 (t)] T satisses a dilation equation with matrix coeecients C k. Associated with (t) is a multiwavelet w(t) = w 0 (t) : : :w r?1 (t)] T. Unlike the scalar case, construction of a multiwavelet is a nontrivial design problem. We discuss the ways to solve it. This work extends wavelet theory to the vector case. Multiwavelets are a natural generalization of scalar wavelets. They have new features arising from the matrix structure of the dilation equation. For example, a multi-scaling function can combine orthogonality, symmetry and high approximation order while a scalar function cannot. The properties of an MRA can be formulated either in time or in frequency (Fourier) domain. We start with establishing conditions of orthogonality and approximation in the time domain. Also we nd a way to construct multi-scaling functions and present the nite element example. This method requires solution of a large system of nonlinear equations. We therefore switch to the frequency domain. Working in the Fourier domain one faces the necessity to deal with the reenement mask (two-scale symbol) P (!) = P k e ?ik! C k. In the scalar case P(!) is a trigonometric polynomial. Its factorization P(!) = (1+e ?i!) m P 0 (!) deenes regularity and approximation properties of the scaling function. In the vector case P (!) becomes a matrix trigonometric polynomial. Each factor (1+e ?i!) changes to a two-scale similarity transform (TST) which connects matrix polynomials P (!) and Q(!) by the relation Q(!) = M(2!)P (!)M ?1 (!). We study the properties of TST …

Multiwavelets--theory and applications

We describe a statistical model for images decomposed in an overcomplete wavelet pyramid. Each coefficient of the pyramid is modeled as the product of two independent random variables: an element of a Gaussian random field, and a hidden multiplier with a marginal log-normal prior. The latter modulates the local variance of the coefficients. We assume subband coefficients are contaminated with additive Gaussian noise of known covariance, and compute a MAP estimate of each multiplier variable based on observation of a local neighborhood of coefficients. Conditioned on this multiplier, we then estimate the subband coefficients with a local Wiener estimator. Unlike previous approaches, we (a) empirically motivate our choice for the prior on the multiplier; (b) use the full covariance of signal and noise in the estimation; (c) include adjacent scales in the conditioning neighborhood. To our knowledge, the results are the best in the literature, both visually and in terms of squared error.

Vasily Strela

Papers

Image denoising using scale mixtures of Gaussians in the wavelet domain

The application of multiwavelet filterbanks to image processing

Short wavelets and matrix dilation equations

Multiwavelets--theory and applications

Adaptive Wiener denoising using a Gaussian scale mixture model in the wavelet domain