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.

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.

We consider the problem of registering (aligning) two images to sub-pixel accuracy by optimization of objective functions constructed from the images' intensity values. We show that some widely used interpolation methods can introduce multiple local optima in the energy of the interpolated image which, if not counter-balanced by other terms, can cause local optima in registration objective functions including the sum of squared differences, cross correlation, and mutual information. We discuss different solutions to address the problem based on high degree B-spline interpolation, low pass filtering the images, and stochastic integration. Numerical examples using synthetic and real signals and images are shown.

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.

We look at the problem of extracting geometric functions such as spatial transformations and curves delineating edges to subpixel accuracy from noisy, sampled data. We analyze the stochastic properties of continuous B-spline interpolation to show that, in general noisy circumstances, sub-pixel accuracy is not obtainable when using low degree B-splines. Results using magnetic resonance image (MRI) data are shown

We introduce a methodology for adaptive sequential sensing and processing in a classification setting. Our objective for sensor optimization is the back-end performance metric--in this case, misclassification rate. Our methodology, which we dub Integrated Sensing and Processing Decision Trees (ISPDT), optimizes adaptive sequential sensing for scenarios in which sensor and/or throughput constraints dictate that only a small subset of all measurable attributes can be measured at any one time. Our decision trees optimize misclassification rate by invoking a local dimensionality reduction-based partitioning metric in the early stages, focusing on classification only in the leaves of the tree. We present the ISPDT methodology and illustrative theoretical, simulation, and experimental results.

The theory of spectral analysis of a particular class of noncommutative groups-wreath products of cyclic groups-has been shown to have a group-based convolution that leads to a new class of noncommutative filters. These filters, with their group and scale-selective properties and their relationship to DFT filter banks, offer some intriguing possibilities in signal processing applications. We give a summary of some of the basic properties of convolution with wreath product cyclic groups and illustrate those properties through an example. Applications to some basic signal processing tasks are proposed.

We present a direct generalization of local trigonometric bases to decompositions of L/sup 2/(R/sup 2/) with non-rectangular support. We describe decompositions of L/sup 2/(R/sup 2/) into n subspaces supported on approximate equiangular sectors and work out the case for n=3 in detail. For prototypical decompositions, windowed circular harmonics serve as basis functions. The technique applies equally well to decompositions of higher dimensional space.

Let $\poly = \{P_0,\dots,P_{n-1}\}$ denote a set of polynomials with complex coefficients. Let $\pts = \{z_0,\dots,z_{n-1}\}\subset \cplx$ denote any set of {\it sample points}. For any $f = (f_0,\dots,f_{n-1}) \in \cplx^n$, the {\it discrete polynomial transform} of f (with respect to $\poly$ and $\pts$) is defined as the collection of sums, $\{\fhat(P_0),\dots,\fhat(P_{n-1})\}$, where $\fhat(P_j) = \langle f,P_j \rangle = \sum_{i=0}^{n-1} f_iP_j(z_i)w(i)$ for some associated weight function w. These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper, we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of N orthogonal polynomials of degree at most $N-1$, we give an $O(N\log^2 N)$ algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the $N^2$ operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a three-term recurrence and thus it may be applied to any such set of discretely sampled functions. In particular, sampled orthogonal polynomials generate the vector space of functions on a distance transitive graph. As a direct application of our work, we are able to give a fast algorithm for computing subspace decompositions of this vector space which respect the action of the symmetry group of such a graph. This has direct applications to treating computational bottlenecks in the spectral analysis of data on distance transitive graphs, and we discuss this in some detail.

We present a wreath product approach for matched filtering to detect rotated copies of a template in an image. We view the image as a homogeneous space for a wreath product, a noncommutative symmetry group. The corresponding Fourier analysis has a natural multiresolution structure and accompanying efficient algorithm which we explain and illustrate with an example. The associated matched filter is a new example of the use of a noncommutative convolution for image processing. Numerical experiments are described in which this noncommutative approach outperforms standard Fourier-based methods.

A fast algorithm for the computation of spherical harmonic expansions of bandlimited functions on the 2-sphere is described. The algorithm also provides for an efficient inverse transform (synthesis) as well and consequently fast convolution on the 2-sphere is also possible. We discuss applications to image processing and medical imaging as well as aspects of our working implementation.

Discusses the advantages and disadvantages of using a Karhunen-Loeve (K-L) expansion of a training set of images to reduce the number of encodes required for a magnetic resonance (MR) image of a new object. One form of this technique has been proposed [Cao and Levin, 1993] and another implemented [Zientara et al., 1994]. The authors evaluate the error likely to be achieved as a function of the number of encodes and two technical problems: reduced SNR in the images and smoothing of the K-L functions in practice. As an alternative, they propose the use of joint best bases [Wickerhauser, 1994] derived from the local trigonometric library as an approximation to the K-L basis. These bases approach the rate-distortion characteristic achieved by the K-L basis, but they are easier to use in MRI and can be applied with existing methods for fast acquisition.

We consider the problem of computing the shortest schedule of the intervals [j2−i,(j + 1)2−i), for 0 ⩽ j ⩽ 2i − 1 and 1 ⩽ i ⩽ k such that separation of intersecting intervals is at least R. This problem arises in an application of wavelets to medical imaging. It is a generalization of the graph separation problem for the intersection graph of the intervals, which is to assign the numbers 1 to 2k + 1 − 2 to the vertices, other than the root, of a complete binary tree of height k in such a way as to maximize the minimum difference between all ancestor descendent pairs. We give an efficient algorithm to construct optimal schedules.

We present a new technique for image contrast enhancement via multiscale gradient transformation. In contrast to histogram-based techniques which aim to change the global statistical distribution of pixel intensities, we improve image contrast by modifying the modulus of the gradient image at multiple scales. In computation, a multiscale gradient representation of the image is generated by a wavelet transform (WT). Linear or nonlinear transformation is applied to multiscale gradients. A contrast-enhanced image is obtained from the transformed multiscale gradients by the inverse wavelet transform. Experimental results demonstrate two advantages of the new method: its flexibility to selectively enhance features of different sizes and ability to control noise magnification.>

Examines the advantages and disadvantages of using a Karhunen-Loeve (K-L) expansion of a training set of images to reduce the number of encodes required for a magnetic resonance (MR) image of a new object. One form of this technique has been proposed (Yue Cao and Levin, 1993) and another implemented (Zientara et al., 1994). The present authors evaluate: a) the error likely to be achieved as a function of the number of encodes and b) two technical problems: reduced SNR in the images and smoothing of the K-L functions in practice. They propose the use of localized trigonometric bases developed by Coifman and Wickerhauser (1992) as an alternative to the K-L basis. The localized trigonometric bases approach the error achieved by the K-L basis, but they are easier to use and can be used with existing methods for fast acquisition.>

Wavelet transforms are multiresolution decompositions that can be used to analyze signals and images. They describe a signal by the power at each scale and position. Edges can be located very effectively in the wavelet transform domain. A spatially selective noise filtration technique based on the direct spatial correlation of the wavelet transform at several adjacent scales is introduced. A high correlation is used to infer that there is a significant feature at the position that should be passed through the filter. The authors have tested the technique on simulated signals, phantom images, and real MR images. It is found that the technique can reduce noise contents in signals and images by more than 80% while maintaining at least 80% of the value of the gradient at most edges. The authors did not observe any Gibbs' ringing or significant resolution loss on the filtered images. Artifacts that arose from the filtration are very small and local. The noise filtration technique is quite robust. There are many possible extensions of the technique. The authors see its applications in spatially dependent noise filtration, edge detection and enhancement, image restoration, and motion artifact removal. They have compared the performance of the technique to that of the Weiner filter and found it to be superior.

The authors describe two different applications of wavelet transforms in magnetic resonance imaging (MRI). In each case, the use of wavelets offers tangible benefits over the traditional Fourier-transform-based imaging. These stem from the simultaneous time and frequency localization properties of the wavelets. The first application utilizes the localization properties of wavelets to acquire single T/sub 2/ weighted images in times which are short relative to current methods. An interesting combinatorial scheduling problem arises from the multiscale structure inherent in the formulation of this procedure. Its solution is sketched and applied to the imaging process. The second example is a technique for potentially 'instant' imaging. Image acquisition is modeled as an inverse wavelet transform of the spin density in the region in question. Signal acquisition takes place in as few as one or two echoes after the excitation by RF pulses designed to excite a moving band of spins in a slice. These sweeping RF pulses excite strips of spins which rephase at different times during acquisition. They replace phase-encoding gradients; no fast pulsed gradients are required.>

The problem of computing the convolution of two functions on the sphere by means of a spherical transform is considered. Such convolutions are applicable to surface recognition and the location of both rotated and translated patterns in an image. The authors give convolution theorems that relate the spherical transform to convolution, sampling theorems that allow the exact computation of the transform for band-limited functions, and algorithms with asymptotically improved running time for the exact computation of the harmonic expansion. The net result is an O(n/sup 1.5/(log n)/sup 2/) algorithm for the exact computation of the convolution of two bandlimited functions sampled at n points in accordance with the sampling theorem. The techniques developed are applicable to computing other transforms, such as the Laguerre, Hermite, and Hankel transforms.>