Abstract. The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries — stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), Matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis Pursuit (BP) is a principle for decomposing a signal into an “optimal ” superposition of dictionary elements, where optimal means having the smallest l 1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, in abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

The need for efficient content-based image retrieval has increased tremendously in many application areas such as biomedicine, military, commerce, education, and Web image classification and searching. We present here SIMPLIcity (Semanticssensitive Integrated Matching for Picture LIbraries), an image retrieval system, which uses semantics classification methods, a wavelet-based approach for feature extraction, and integrated region matching based upon image segmentation. As in other regionbased retrieval systems, an image is represented by a set of regions, roughly corresponding to objects, which are characterized by color, texture, shape, and location. The system classifies images into semantic categories, such as textured-nontextured, graphphotograph. Potentially, the categorization enhances retrieval by permitting semantically-adaptive searching methods and narrowing down the searching range in a database. A measure for the overall similarity between images is developed using a region-matching scheme that integrates properties of all the regions in the images. Compared with retrieval based on individual regions, the overall similarity approach 1) reduces the adverse effect of inaccurate segmentation, 2) helps to clarify the semantics of a particular region, and 3) enables a simple querying interface for region-based image retrieval systems. The application of SIMPLIcity to several databases, including a database of about 200,000 general-purpose images, has demonstrated that our system performs significantly better and faster than existing ones. The system is fairly robust to image alterations.

This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along C2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2−j, each element has an envelope which is aligned along a ‘ridge ’ of length 2−j/2 and width 2−j. We prove that curvelets provide an essentially optimal representation of typical objects f which are C2 except for discontinuities along C2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ‖f − f C n ‖ 2 L2 ≤ C · n−2 · (log n) 3, n → ∞. This rate of convergence holds uniformly over a class of functions which are C 2 except for discontinuities along C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n −1 as n → ∞, which is considerably worst than the optimal behavior.

ABSTRACT. Wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains

We discuss and examine Weierstrass' main contributions to approximation theory.

The Time-Frequency and Time-Scale communities have recently developed an enormous number of overcomplete signal dictionaries -- wavelets, wavelet packets, cosine packets, wilson bases, chirplets, warped bases, and hyperbolic cross bases being a few examples. Basis Pursuit is a technique for decomposing a signal into an "optimal" superposition of dictionary elements. The optimization criterion is the l 1 norm of coefficients. The method has several advantages over Matching Pursuit and Best Ortho Basis, including super-resolution and stability. 1 Introduction Over the last five years or so, there has been an explosion of awareness of alternatives to traditional signal representations. Instead of just representing objects as superpositions of sinusoids (the traditional Fourier representation) we now have available alternate dictionaries -- signal representation schemes -- of which the Wavelets dictionary is only the most well-known. Wavelet dictionaries, Gabor dictionaries, Multi-scale...

We study a simple “Horizon Model ” for the problem of recovering an image from noisy data; in this model the image has an edge with α-Hölder regularity. Adopting the viewpoint of computational harmonic analysis, we develop an overcomplete collection of atoms called wedgelets, dyadically organized indicator functions with a variety of locations, scales, and orientations. The wedgelet representation provides nearly-optimal representations of objects in the Horizon model, as measured by minimax description length. We show how to rapidly compute a wedgelet approximation to noisy data by finding a special edgelet-decorated recursive partition which minimizes a complexity-penalized sum of squares. This estimate, using sufficient sub-pixel resolution, achieves nearly the minimax mean-squared error in the Horizon Model. In fact, the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the Horizon, 1 ≤ α ≤ 2. Wedgelet analysis and de-noising may be used successfully outside the Horizon model. We study images modelled as indicators of star-shaped sets with smooth boundaries and show that complexity-penalized wedgelet partitioning achieves nearly the minimax risk in that setting also.

Content-based image retrieval using region segmentation has been an active research area. We present IRM (Integrated Region Matching), a novel similarity measure for regionbased image similarity comparison. The targeted image retrieval systems represent an image by a set of regions, roughly corresponding to objects, which are characterized by features reflecting color, texture, shape, and location properties. The IRM measure for evaluating overall similarity between images incorporates properties of all the regions in the images by a region-matching scheme. Compared with retrieval based on individual regions, the overall similarity approach reduces the influence of inaccurate segmentation, helps to clarify the semantics of a particular region, and enables a simple querying interface for region-based image retrieval systems. The IRM has been implemented as a part of our experimental SIMPLIcity image retrieval system. The application to a database of about 200,000 general-purpose images ...

is a library of routines for wavelet analysis, wavelet-packet analysis, cosine-packet analysis and matching pursuit. The library is available free of charge over the Internet. Versions are provided for Macintosh, UNIX and Windows machines.

This paper describes newly invented multiscale transforms known under the name of the ridgelet [6] and the curvelet transforms [9, 8]. These systems combine ideas of multiscale analysis and geometry. Inspired by some recent work on digital Radon transforms [1], we then present very effective and accurate numerical implementations with computational complexities of at most N log N. In the second part of the paper, we propose to combine these new expansions with the Total Variation minimization principle for the reconstruction of an object whose curvelet coefficients are known only approximately: quantized, thresholded, noisy coefficients, etc. We set up a convex optimization problem and seek a reconstruction that has minimum Total Variation under the constraint that its coefficients do not exhibit a large discrepancy from the the data available on the coefficients of the unknown object. We will present a series of numerical experiments which clearly demonstrate the remarkable potential of this new methodology for image compression, image reconstruction and image ‘de-noising.’