Glenn R. Easley

Compressed Synthetic Aperture Radar

In this paper, we introduce a new synthetic aperture radar (SAR) imaging modality which can provide a high-resolution map of the spatial distribution of targets and terrain using a significantly reduced number of needed transmitted and/or received electromagnetic waveforms. This new imaging scheme, requires no new hardware components and allows the aperture to be compressed. It also presents many new applications and advantages which include strong resistance to countermesasures and interception, imaging much wider swaths and reduced on-board storage requirements.

A Shearlet Approach to Edge Analysis and Detection

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.

Shearlet-Based Total Variation Diffusion for Denoising

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.

Shearlet-Based Deconvolution

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.

Optimally Sparse Image Representations using Shearlets

It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. This paper presents a new discrete multiscale directional representation called the discrete shearlet transform. This approach, which is based on the shearlet transform previously developed by the authors and their colaborators, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. Numerical experiments demonstrate that the discrete shearlet transform is very competitive in denoising applications both in terms of performance and computational efficiency.

Compressed sensing for Synthetic Aperture Radar imaging

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 countermesasures and interception, and reduced on-board storage constraints.

Generalized Discrete Radon Transforms and Their Use in the Ridgelet Transform

We introduce and study a new class of Radon transforms in a discrete setting for the purpose of applying them to the ridgelet and curvelet transforms. We give a detailed analysis of the p-adic case and provide a closed-form formula for an inverse of the p-adic Radon transform. We give conditions for a scaled version of the generalized discrete Radon transform to yield a tight frame, and discuss a direct Radon matrix method for the implementation of a local ridgelet transform. We then study the effectiveness of some types of the generalized Radon transforms in reducing a type of noise known as speckle that is present in synthetic aperture radar (SAR) imagery.

Image Processing Using Shearlets

Since shearlets provide nearly optimally sparse representations for a large class of functions that are useful to model natural images, many image processing methods benefit from their use. In particular, the error rates of data estimation from noise are highly dependent on the sparsity properties of the representation, so that many successful applications of shearlets center around restoration tasks such as denoising and inverse problems. Other imaging problems, where also the application of the shearlet representation turns out to be very beneficial, include image enhancement, image separation, edge detection, and estimation of the geometric features of an object.

Edge detection and processing using shearlets

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.

Separated Component-Based Restoration of Speckled SAR Images

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.