Many coherent imaging modalities such as synthetic aperture radar suffer from a multiplicative noise, commonly referred to as speckle, which often makes the interpretation of data difficult. An effective strategy for speckle reduction is to use a dictionary that can sparsely represent the features in the speckled image. However, such approaches fail to capture important salient features such as texture. In this paper, we present a speckle reduction algorithm that handles this issue by formulating the restoration problem so that the structure and texture components can be separately estimated with different dictionaries. To solve this formulation, an iterative algorithm based on surrogate functionals is proposed. Experiments indicate the proposed method performs favorably compared to state-of-the-art speckle reduction methods.

Wavelets with composite dilations extend the traditional wavelet approach by allowing for the construction of waveforms defined not only at various scales and locations but also according to various orthogonal transformations. The shearlets, which yield optimally sparse representations for a large class of 2D and 3D data is the most widely known example of wavelets with composite dilations. However, many other useful constructions are obtained within this framework. In this paper, we examine the hyperbolic shearlets, a variant of the shearlet construction obtained as a system of well localized waveforms defined at various scales, locations and orientations, where the directionality is controlled by orthogonal transformations producing a sort of shearing along hyperbolic curves. The effectiveness of this new representation is illustrated by applications to image denoising. Our results compare favorably against similar denoising algorithms based on wavelets, curvelets and other sophisticated multiscale representations.

Wavelets with composite dilations provide a general framework for the construction of waveforms defined not only at various scales and locations, as traditional wavelets, but also at various orientations and with different scaling factors in each coordinate. As a result, they are useful to analyze the geometric information that often dominate multidimensional data much more efficiently than traditional wavelets. The shearlet system, for example, is a particular well-known realization of this framework, which provides optimally sparse representations of images with edges. In this paper, we further investigate the constructions derived from this approach to develop critically sampled wavelets with composite dilations for the purpose of image coding. Not only do we show that many nonredundant directional constructions recently introduced in the literature can be derived within this setting, but we also introduce new critically sampled discrete transforms that achieve much better nonlinear approximation rates than traditional discrete wavelet transforms and outperform the other critically sampled multiscale transforms recently proposed.

Many coherent imaging modalities are often characterized by a multiplicative noise, known as speckle which often makes the interpretation of data difficult. In this paper, we present a speckle reduction algorithm based on separating the structure and texture components of SAR images. An iterative algorithm based on surrogate functionals is presented that solves the component optimization formulation. Experiments indicate this proposed method performs favorably compared to state-of-the-art speckle reduction methods.

In this paper, we propose a reconstruction method that recovers images assumed to have a sparse representation in a gradient domain by using partial measurement samples that are collected in the Fourier domain. A key improvement of this technique is that it makes use of a robust generalized Poisson solver that greatly aids in achieving a significantly improved performance over similar proposed methods. Experiments provided also demonstrate that this new technique is more flexible to work with either random or restricted sampling scenarios better than its competitors.

In this paper, we introduce a new Synthetic Aperture Radar (SAR) imaging modality that provides a high resolution map of the spatial distribution of targets and terrain based on a significant reduction in the number of transmitted and/or received electromagnetic waveforms. This new imaging scheme, which requires no new hardware components, allows the aperture to be compressed and presents many important applications and advantages among which include resolving ambiguities, strong resistance to countermeasures and interception, and reduced on-board storage constraints.

We carry out deconvolution by transforming the data into a new general discrete Radon domain that can handle any assumed boundary conditions for the associated matrix inversion problem. For each associated component (projection), one can then apply deconvolution routines to smaller (and possibly better) conditioned matrix inversion problems than the matrix inversion problem for the entire image. We demonstrate this new scheme by adaptively deconvolving these components using a combination of regularized inversion and wavelet filtering techniques. This procedure allows us to provide image estimates based on a generalized ridgelet frame. We then devise methods for carrying out this scheme locally to provide estimates based on generalized multiscaled ridgelets which are then filtered and combined to form an estimate from a curvelet-like domain. The techniques presented here suggest a whole new paradigm for developing deconvolution algorithms that incorporate leading deconvolution schemes. Various experimental results show that our methods can perform significantly better than standard deconvolution techniques.

We propose a shearlet formulation of the total variation (TV) method for denoising images. Shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization. Common approaches in combining wavelet-like representations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts after obtaining a nearly optimal estimate. We show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain. We also analyze the performance of a shearlet-based diffusion method. Numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform. Furthermore, the shearlet-TV scheme requires far fewer iterations than similar competitors.

In this paper, a new type of deconvolution algorithm is proposed that is based on estimating the image from a shearlet decomposition. Shearlets provide a multidirectional and multiscale decomposition that has been mathematically shown to represent distributed discontinuities such as edges better than traditional wavelets. Constructions such as curvelets and contourlets share similar properties, yet their implementations are significantly different from that of shearlets. Taking advantage of unique properties of a new M-channel implementation of the shearlet transform, we develop an algorithm that allows for the approximation inversion operator to be controlled on a multiscale and multidirectional basis. A key improvement over closely related approaches such as ForWaRD is the automatic determination of the threshold values for the noise shrinkage for each scale and direction without explicit knowledge of the noise variance using a generalized cross validation (GCV). Various tests show that this method can perform significantly better than many competitive deconvolution algorithms.

It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions

In this paper, we present an image enhancement technique based on a new multiscale generalized directional filter bank design. The design presented is a shift-invariant overcomplete representation, which is well suited to extracting geometric features such as edges. Special cases of this design method can be made to reduce to different and improved implementations of the shearlet and the contourlet transforms, which are known to represent certain classes of images optimally. Use of this new filter bank design has proven itself competitive in image restoration for noisy images and is well suited for distinguishing noise from weak edges. Experimental results show that our unique image enhancement technique out-performs wavelet and contourlet based enhancement methods.

Mathematically wavelets are not very effective in representing images containing distributed discontinuities such as edges. This paper deals with a new multiscale directional representation called the shearlet transform that has been shown to represent specific classes of images with edges optimally. Techniques based on this transform for edge detection and analysis are presented. Unlike previously developed directional filter based techniques for edge detection, shearlets provide a theoretical basis for characterizing how edges will behave in such representations. Experiments demonstrate that this novel approach is very competitive for the purpose of edge detection and analysis.