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 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...

. This paper investigates structural properties of diffusive scalespaces and develops a Riemannian description based on electromagnetic (EM) field theory. The generalized diffusion equation defining photometric transitions is interpreted as a Lorentz gauge condition expressing the trace Lorentzinvariance of an EM quadripotential with covariant scalar and contravariant vector components, respectively related to photometric and geometric image properties. This gauge condition determines EM quadrifield and quadricharge which satisfy Maxwell equations. Deriving their general expressions as functions of scale-space geometric or energetic features yields Lorentz-invariants which synthetize intrinsic multiscale image properties. Keywords : Multiscale analysis, geodesic flows, deformable manifolds, variational methods, gauge theory. 1 Introduction Anisotropic diffusion is an efficient nonlinear filtering technique for deriving deterministic multiscale image descriptions. Extensive studies ba...

This paper addresses the problem of feature enhancement in noisy images when the feature is known to be constrained to a manifold. As an example, we approach the direction denoising problem in a general dimension via the geometric Beltrami framework for image processing. The spatial-direction space is a fiber bundle in which the spatial part is the base manifold and the direction space is the fiber. The feature (direction) field is represented accordingly as a section of the spatial-feature fiber bundle. The resulting Beltrami flow is a selective smoothing process that respects the bundle's structure, i.e., the feature constraint. Direction di#usion is treated as a canonical example of a non-Euclidean feature space. The structures of the fiber spaces of interest in this paper are the unit circle S , the unit sphere S , and the unit hypersphere S . Applications to color analysis are discussed, and numerical experiments demonstrate again the benefits of the Beltrami framework in comparison to other feature enhancement schemes for nontrivial geometries in image processing.

ontrol , control J(f) (here the prototypical J(f) = R [f 00 (x)] 2 dx, or control the residual sum of squares. J(f) is controlled by finding f in a certain Sobolev Hilbert space to minimize the residual sum of squares (RSS) under the constraint that J(f) C, then C is the regularization parameter. RSS is varied by finding f to minimize J(f) subject to RSS S, then S is the regularization parameter. Under some mild conditions J(f) = C; RSS = S, and ; C and S are equivalent in the sense that control

Numerous applications in statistics, signal processing, and machine learning regularize using Total Variation (TV) penalties. We study anisotropic (ℓ1-based) TV and also a related ℓ2-norm variant. We consider for both variants associated (1D) proximity operators, which lead to challenging optimization problems. We solvetheseproblemsbydevelopingNewton-type methods that outperform the state-of-the-art algorithms. More importantly, our 1D-TV algorithms serve as building blocks for solving the harder task of computing 2- (and higher)dimensional TV proximity. We illustrate the computationalbenefitsof ourmethodsby applying them to several applications: (i) image denoising;(ii)imagedeconvolution(bypluggingin ourTV solversinto publiclyavailablesoftware); and (iii) four variantsof fused-lasso. The results showlargespeedups—andtosupportourclaims, we providesoftwareaccompanyingthispaper. 1.