It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained l1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms l1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted l1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the l1 norm of the coefficient sequence as is common, but by reweighting the l1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing.

/pdf/enhancing-sparsity-by-reweighted-l-1-minimization-znwoz4koxr.pdf

Enhancing Sparsity by Reweighted ℓ 1 Minimization

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained L1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms L1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted L1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations--not by reweighting the L1 norm of the coefficient sequence as is common, but by reweighting the L1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as compressed sensing.

Enhancing Sparsity by Reweighted L1 Minimization

Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

/pdf/anisotropic-diffusion-in-image-processing-z13u09qvsm.pdf

Anisotropic diffusion in image processing

We propose, analyze, and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also the isotropic forms of TV discretizations. The per-iteration computational complexity of the algorithm is three fast Fourier transforms. We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or $q$-linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the lagged diffusivity algorithm for TV-based deblurring. Some extensions of our algorithm are also discussed.

/pdf/a-new-alternating-minimization-algorithm-for-total-variation-p0vrnncv59.pdf

A New Alternating Minimization Algorithm for Total Variation Image Reconstruction

We study a functional with variable exponent, $1\leq p(x)\leq 2$, which provides a model for image denoising, enhancement, and restoration. The diffusion resulting from the proposed model is a combination of total variation (TV)-based regularization and Gaussian smoothing. The existence, uniqueness, and long-time behavior of the proposed model are established. Experimental results illustrate the effectiveness of the model in image restoration.

Variable Exponent, Linear Growth Functionals in Image Restoration

We propose a new definition of the total variation (TV) norm for vector-valued functions that can be applied to restore color and other vector-valued images. The new TV norm has the desirable properties of (1) not penalizing discontinuities (edges) in the image, (2) being rotationally invariant in the image space, and (3) reducing to the usual TV norm in the scalar case. Some numerical experiments on denoising simple color images in red-green-blue (RGB) color space are presented.

Color TV: total variation methods for restoration of vector-valued images

The phenomenon of super-resolution in time-reversal acoustics is analyzed theoretically and with numerical simulations. A signal that is recorded and then retransmitted by an array of transducers, propagates back though the medium, and refocuses approximately on the source that emitted it. In a homogeneous medium, the refocusing resolution of the time-reversed signal is limited by diffraction. When the medium has random inhomogeneities the resolution of the refocused signal can in some circumstances beat the diffraction limit. This is super-resolution. A theoretical treatment of this phenomenon is given, and numerical simulations which confirm the theory are presented.

/pdf/super-resolution-in-time-reversal-acoustics-4d3zeqr1bb.pdf

Super-resolution in time-reversal acoustics

We describe some numerical techniques for the total variation image restoration method, namely a primal-dual linearization for the Euler-Lagrange equations and some preconditioning issues. We also highlight extension of this technique to color images, blind deconvolution and the staircasing effect.

Total variation image restoration: numerical methods and extensions

Variational PDE models and methods for image processing

In this paper, we study a multiple-input-single-output (MISO) underwater communication system that applies time reversal (TR) to transmit signals so that they focus spatially and compress temporally on the intended receiver. Our simulations model an underwater acoustic channel as a waveguide, and we investigate the cases of a waveguide both with and without random inhomogeneities. We investigate physical TR metrics and communications related performance indicators. The results of our simulations show that spatial focusing depends strongly on the delay spread ( DS ) , as has been seen in experiments. This physical property of TR could be exploited in communication systems where signal coherence is desired only at the receiver location. However, in the simulations, we find that while spatial compression increases with DS in a robust way (i.e., even when inhomogeneities exist), time compression does not increase with DS. Moreover, physical measures of the temporal compression (temporal peak-to-sidelobe ratio) do not improve with waveguide inhomogeneities. Nevertheless, TR reduces intersymbol interference (ISI) at the receiver as DS increases for both types of waveguides, which is an important effect for efficient, high-speed communication. In addition to TR, preequalization at the transmitter can ideally eliminate ISI without significantly affecting spatial compression. However, this preequalization causes a reduction of received power, which may be acceptable when the signal-to-noise ratio (SNR) at the receiver is high.

Spatial Focusing and Intersymbol Interference in Multiple-Input–Single-Output Time Reversal Communication Systems

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