In this paper, we develop a novel framework to guide the design of initial estimates for phase retrieval given measurements realized from an arbitrary forward model. Particularly, we propose a general formalism for spectral initialization as an approximate Bregman loss minimization procedure in the range of the lifted forward model, such that a search over rank-1, positive semidefinite matrices is tractable by the synthesis of a quadratic form using elementwise processing of the intensity-only measurements. Via the Bregman loss approach we transcend the Euclidean sense alignment based similarity measure between phaseless measurements that is inherent in the state-of-the art techniques in the literature, in favor of information theory inspired divergence metrics over the positive reals. We derive spectral methods that perform approximate minimization of Kullback--Leibler and Itakura--Saito divergences over phaseless measurements by using elementwise sample processing functions which are designed under a minimal distortion principle. Our formulation relates and extends existing results on model dependent design of optimal sample processing functions in the literature to a model independent sense of metric-based optimality. Numerical simulations confirm the effectiveness of our approach in imaging problems using synthetic and real data sets.

Interferometric inversion involves recovery of a signal from cross-correlations of its linear transformations. A close relative of interferometric inversion is the generalized phase retrieval problem, for which significant advancements were made in recent years despite the ill-posed and nonconvex nature of the problem. One such prominent phase retrieval method is Wirtinger flow (WF) [E. J. Candes, X. Li, and M. Soltanolkotabi, IEEE Trans. Inform. Theory, 61 (2015), pp. 1985--2007], a computationally efficient nonconvex optimization framework that provides high probability guarantees for exact recovery under certain measurement models, specifically coded diffraction patterns, and Gaussian sampling vectors. In this paper, we develop a generalization of WF for interferometric inversion, which we refer to as generalized Wirtinger flow (GWF). Our approach treats the theory of low rank matrix recovery and the nonconvex optimization approach of WF in a unified framework. Such a treatment facilitates the identification of a new sufficient condition on the lifted forward model for exact recovery via GWF and results in a deterministic framework based on geometric arguments for convergence. Thereby, GWF extends the model specific probabilistic guarantees in [12] to arbitrary measurement maps characterized over the equivalent lifted domain in the context of interferometric inversion, covering both random and deterministic measurement models. We then establish our sufficient condition for the cross-correlations of linear measurements collected by complex Gaussian sampling vectors. In the particular case of interferometric inversion with the Gaussian model, we show that the exact recovery theory of standard WF implies our sufficient condition when we have cross-correlations, and the regularity condition of WF is redundant. Finally, we demonstrate the effectiveness of GWF numerically in a deterministic, interferometric multistatic radar imaging scenario.

We consider the problem of reconstructing an object given magnitudes of linear measurements. We follow the `lifting' approach, but unlike previous work which use convex relaxations of the unit rank constraint, we use a weakly-convex matrix penalty. We derive a convergent algorithm and show that it is computationally more feasible than those obtained under convex relaxations. We demonstrate numerically that when the signal to noise ratio is high, the proposed algorithm can achieve almost error-free reconstruction with fewer measurements than when convex relaxation is employed.

Layover is an artifact observed in synthetic aperture radar (SAR) images. This artifact manifests as geometric distortions or positioning errors due to unknown or inaccurate ground topography information. Layover artifact results in erroneous distance between reconstructed scatterers, particularly in elevated regions. We develop a quantitative analysis of positioning errors due to incorrect height information in backprojection based (BP) SAR image formation. Our analysis is based on microlocal techniques and provides an explicit algebraic relationship between the two-dimensional positioning error and height error. Our analysis is applicable to arbitrary imaging geometries including bistatic configuration, arbitrary antenna trajectories, wide apertures, and large scenes. While we focus primarily on BP based image formation, the results are also applicable to range-Doppler type image formation methods. Our results can be used to interpret and identify layover regions in SAR images and extract valuable information from layover artifacts.

We introduce a class of nonstationary stochastic processes suitable for modeling nonstationary radar targets and clutter. This class of processes can be characterized by pseudodifferential operators driven by the Wiener process. We refer to these processes as pseudostationary processes and define a concept of space-varying spectral density function using the symbol of the underlying operators. We then derive an analytic filtered-backprojection- and backprojection-filtering-type formulas based on the minimum mean square error criterion for synthetic aperture image reconstruction. The inversion formulas depend on the space-varying spectral density function of target and clutter. We present an algorithm for the simultaneous estimation of target a priori information and reconstruction of SAR images. The algorithm is computationally efficient and can be implemented with the computational complexity of fast-backprojection algorithms. Extensive numerical simulations demonstrate the performance of the method.

We consider a synthetic aperture radar (SAR) system that uses ultra-narrowband continuous waveforms (CWs) as an illumination source. Such a system has many practical advantages, such as the use of relatively simple low-cost and low-power transmitters, and in some cases, using the transmitters of opportunity, such as television and radio stations. Additionally, ultra-narrowband CW signals are suitable for motion estimation due to their ability to acquire high resolution Doppler information. In this paper, we present a novel synthetic aperture imaging method for moving targets using a bistatic SAR system transmitting ultra-narrowband CWs. Our method exploits the high Doppler resolution provided by ultra-narrowband CW signals both to image the scene reflectivity and to determine the velocity of multiple moving targets. Starting from the first principle, we develop a novel forward model based on the temporal Doppler induced by the movement of antennas and moving targets. We form the reflectivity image of the scene and estimate the motion parameters using a filtered-backprojection technique combined with a contrast optimization method. Analysis of the point spread function of our image formation method shows that reflectivity images are focused when the motion parameters are estimated correctly. We present analysis of the velocity resolution and the resolution of reconstructed reflectivity images. We analyze the error between the correct and reconstructed positions of targets due to errors in velocity estimation. Extensive numerical simulations demonstrate the performance of our method and validate the theoretical results.

The first-generation time-series synthetic aperture radar interferometry (TSInSAR) technique persistent-scatterer (PS) InSAR has been proven effective in ground deformation measurement over areas with high reflectivity by taking advantage of coregistered temporally coherent pointwise scatterers. In order to increase the spatial density of measurement points and quality of displacement time series over moderate reflectivity scenes, a second-generation TSInSAR called SqueeSAR was developed to extract displacement information from both PSs and distributed scatterers, by taking into account their temporal coherence and their spatial statistical behavior. In this paper, we propose a new second-generation TSInSAR, which is referred to as joint-scatterer (JS) InSAR, to measure the line-of-sight surface displacement using the neighboring pixel stacks. A novel goodness-of-fit testing approach is proposed to analyze the similarity between two JS vectors based on time-series likelihood ratios. By taking advantage of the proposed test, a new spatially adaptive filter is developed to estimate the covariance matrix. Based on the estimated covariance matrix, the projection of the joint signal subspace onto the corresponding joint noise subspace is applied to retrieve phase history. With coherence information of neighboring pixel stacks, JSInSAR is able to provide reliable geophysical parameters in the presence of large coregistration errors. The effectiveness of the proposed technique is verified with a time series of high-resolution SAR data from the TerraSAR-X satellite.

Fluorescence diffuse optical tomography (FDOT) is an emerging molecular imaging modality that uses near infrared light to excite the fluorophore injected into tissue; and to reconstruct the fluorophore concentration from boundary measurements. The FDOT image reconstruction is a highly ill-posed inverse problem due to a large number of unknowns and limited number of measurements. However, the fluorophore distribution is often very sparse in the imaging domain since fluorophores are typically designed to accumulate in relatively small regions. In this paper, we use compressive sensing (CS) framework to design light illumination and detection patterns to improve the reconstruction of sparse fluorophore concentration. Unlike the conventional FDOT imaging where spatially distributed light sources illuminate the imaging domain one at a time and the corresponding boundary measurements are used for image reconstruction, we assume that the light sources illuminate the imaging domain simultaneously several times and the corresponding boundary measurements are linearly filtered prior to image reconstruction. We design a set of optical intensities (illumination patterns) and a linear filter (detection pattern) applied to the boundary measurements to improve the reconstruction of sparse fluorophore concentration maps. We show that the FDOT sensing matrix can be expressed as a columnwise Kronecker product of two matrices determined by the excitation and emission light fields. We derive relationships between the incoherence of the FDOT forward matrix and these two matrices, and use these results to reduce the incoherence of the FDOT forward matrix. We present extensive numerical simulation and the results of a real phantom experiment to demonstrate the improvements in image reconstruction due to the CS-based light illumination and detection patterns in conjunction with relaxation and greedy-type reconstruction algorithms.

In the Part 1 of this two-part study, we present a method of imaging and velocity estimation of ground moving targets using passive synthetic aperture radar. Such a system uses a network of small, mobile receivers that collect scattered waves due to transmitters of opportunity, such as commercial television, radio, and cell phone towers. Therefore, passive imaging systems have significant cost, manufacturing, and stealth advantages over active systems. We describe a novel generalized Radon transform-type forward model and a corresponding filtered-backprojection-type image formation and velocity estimation method. We form a stack of position images over a range of hypothesized velocities, and show that the targets can be reconstructed at the correct position whenever the hypothesized velocity is equal to the true velocity of targets. We then use entropy to determine the most accurate velocity and image pair for each moving target. We present extensive numerical simulations to verify the reconstruction method. Our method does not require a priori knowledge of transmitter locations and transmitted waveforms. It can determine the location and velocity of multiple targets moving at different velocities. Furthermore, it can accommodate arbitrary imaging geometries. In Part 2, we present the resolution analysis and analysis of positioning errors in passive SAR images due to erroneous velocity estimation.

In Part 1 of this work, we present a passive synthetic aperture imaging and velocity estimation method for ground moving targets using a network of passive receivers. The method involves inversion of a Radon transform type forward model via a novel filtered backprojection approach combined with entropy optimization. The method is applicable to noncooperative transmitters of opportunity where the transmitter locations and transmitted waveforms are unknown. Furthermore, it can image multiple targets moving at different velocities in arbitrary imaging geometries. In this paper, we present a detailed analysis of the performance of our method. First the resolution analysis in position and velocity spaces is presented. The analysis identifies several factors that contribute positively or negativity towards position and velocity resolution. Next, we present a novel theory to analyze and predict smearing artifacts in position images due to error in velocity estimation of moving targets. Specifically, we show that small errors in the velocity estimation result in small positioning errors. We present extensive numerical simulations to demonstrate the theoretical results. While our primary interest lies in radar, the theory, methods and algorithms introduced in our work are also applicable to passive acoustic, seismic, and microwave imaging.

We present a novel method for ground moving target imaging using a synthetic aperture radar system transmitting ultranarrowband continuous waveforms (CW). Our method exploits the high Doppler resolution provided by ultranarrowband CW signals to image both the scene reflectivity and to determine the velocity of multiple moving targets. We develop a new forward model based on the temporal Doppler induced by the movement of antennas and moving targets. The forward model relates reflectivity and velocity information at each location to a correlated received signal. We form the reflectivity images of the moving targets and estimate their motion parameters using a filtered-backprojection (FBP) technique combined with the contrast or gradient optimization method. The method results in focused reflectivity images of moving targets and their velocity estimates, regardless of the target location, speed, and velocity direction. We show that the amplitude and visible edges of the targets can be correctly reconstructed when the correct target velocity estimate is used in the FBP imaging. We present the resolution analysis of the reflectivity images. Extensive numerical simulations demonstrate the performance of our method and validate the theoretical results.

We develop a new passive imaging method for moving targets in free space using measurements from a sparse array of receivers that rely on illumination sources of opportunity. Our imaging method consists of a novel passive measurement model for moving targets and an associated image formation method. The passive measurement model for moving targets relates measurements at a given receiver to measurements at other receivers in terms of Doppler and delay based on the physics of wave propagation as well as the statistics of noise. Next, we use this model to address the image formation as a generalized likelihood ratio test for an unknown target position and velocity. The image is formed by using the position- and velocity-resolved test-statistic that is obtained by maximizing the signal-to-noise ratio of the test-statistic. When the discriminant functional is constrained to be linear, the test-statistic can be viewed as the superposition of the filtered, scaled, and delayed correlations of the measurements obtained at different receivers. We analyze the spatial and velocity resolution of the four-dimensional point spread function of our imaging method in terms of the number of receivers and transmitters and the nature of the waveforms of opportunity. We present extensive numerical simulations to demonstrate the performance of our passive moving target imaging method for different numbers of receivers and different types of waveforms of opportunity available in the real world.

We consider synthetic aperture radar (SAR) imaging using ultra-narrowband continuous waveforms (CW). Due to the high Doppler resolution of CW signals, we refer to this imaging modality as Doppler Synthetic Aperture Radar (DSAR). We present a novel model and an image formation method for the bistatic DSAR for arbitrary imaging geometries. Our bistatic DSAR model is formed by correlating the translated version of the received signal with a scaled or frequencyshifted version of the transmitted CW signal over a finite time window. High frequency analysis of the resulting model shows that the correlated signal is the projections of the scene reflectivity onto the bistatic iso-Doppler curves. We next use microlocal techniques to develop a filtered-backprojection (FBP) type image reconstruction method. The FBP inversion results in backprojection of the correlated signal onto the bistatic iso- Doppler curves as opposed to the bistatic iso-range curves, performed in the traditional wideband SAR imaging. We show that our method takes advantage of the velocity, as well as the acceleration of the antennas in certain directions to form a high resolution SAR image. Our bistatic DSAR imaging method is applicable for arbitrary flight trajectories, nonflat topography, and can accommodate system related parameters. We present resolution analysis and extensive numerical experiments to demonstrate the performance of our imaging method.

In this paper, we present a novel synthetic aperture radar imaging modality that uses ultranarrowband sources of opportunity and passive airborne receivers to form an image of the ground. Due to its combined passive synthetic aperture and high Doppler resolution of the transmitted waveforms, we refer to this modality as the Doppler Synthetic Aperture Hitchhiker or Doppler-hitchhiker for short. Our imaging method first correlates the windowed signal obtained from one receiver with the scaled and translated version of the received signal in another window from the same or another receiver. We show that this correlation processing removes the transmitter-related variables from the phase of the resulting operator that maps the radiance of the scene to the correlated signals. We define a concept of passive Doppler scale factor using the radial velocities of the receivers. Next, we show that the scaled, translated, and correlated signal is the projection of the scene radiance onto the contours that are formed by the intersection of the surfaces of constant passive Doppler scale factor and ground topography. We use microlocal analysis to design a generalized filtered-backprojection operator to reconstruct the scene radiance from its projections. Our analysis shows that the resolution of the reconstructed images improves with the increased time duration and center frequency of the transmitted ultranarrowband signals. Our reconstruction method is analytic and therefore can be made computationally efficient. Furthermore, it easily accommodates arbitrary flight trajectories, nonflat topography, and system-related parameters. We present numerical simulations to demonstrate the performance of our imaging method.

Quantitatively accurate fluorescence diffuse optical tomographic (FDOT) image reconstruction is a computationally demanding problem that requires repeated numerical solutions of two coupled partial differential equations and an associated inverse problem. Recently, adaptive finite element methods have been explored to reduce the computation requirements of the FDOT image reconstruction. However, existing approaches ignore the ubiquitous presence of noise in boundary measurements. In this paper, we analyze the effect of finite element discretization on the FDOT forward and inverse problems in the presence of measurement noise and develop novel adaptive meshing algorithms for FDOT that take into account noise statistics. We formulate the FDOT inverse problem as an optimization problem in the maximum a posteriori framework to estimate the fluorophore concentration in a bounded domain. We use the mean-square-error (MSE) between the exact solution and the discretized solution as a figure of merit to evaluate the image reconstruction accuracy, and derive an upper bound on the MSE which depends upon the forward and inverse problem discretization parameters, noise statistics, a priori information of fluorophore concentration, source and detector geometry, as well as background optical properties. Next, we use this error bound to develop adaptive meshing algorithms for the FDOT forward and inverse problems to reduce the MSE due to discretization in the reconstructed images. Finally, we present a set of numerical simulations to illustrate the practical advantages of our adaptive meshing algorithms for FDOT image reconstruction.

We develop a new passive image formation method capable of exploiting information about multiple scattering in the environment using measurements from a sparse array of receivers that rely on illumination sources of opportunity. We use a physics-based approach to model wave propagation and develop a statistical model that relates measurements at a given receiver to measurements at other receivers. We formulate the imaging problem as a spatially resolved binary hypothesis testing problem using the model between the measurements at different receivers, statistics of the objects to be imaged and statistics of the additive noise and clutter. We address the spatially resolved hypothesis testing problem by constraining the associated discriminant functional to be linear and by maximizing the signal-to-noise-ratio of the test-statistic, and use the resulting spatially resolved test-statistics to form the image. We present numerical simulations to demonstrate the performance of the passive imaging algorithm.

In this paper, we consider a multistatic synthetic aperture radar (SAR) imaging scenario where a swarm of airborne antennas, some of which are transmitting, receiving or both, are traversing arbitrary flight trajectories and transmitting arbitrary waveforms without any form of multiplexing. The received signal at each receiving antenna may be interfered by the scattered signal due to multiple transmitters and additive thermal noise at the receiver. In this scenario, standard bistatic SAR image reconstruction algorithms result in artifacts in reconstructed images due to these interferences. In this paper, we use microlocal analysis in a statistical setting to develop a filtered-backprojection (FBP) type analytic image formation method that suppresses artifacts due to interference while preserving the location and orientation of edges of the scene in the reconstructed image. Our FBP-type algorithm exploits the second-order statistics of the target and noise to suppress the artifacts due to interference in a mean-square sense. We present numerical simulations to demonstrate the performance of our multistatic SAR image formation algorithm with the FBP-type bistatic SAR image reconstruction algorithm. While we mainly focus on radar applications, our image formation method is also applicable to other problems arising in fields such as acoustic, geophysical and medical imaging.

For imaging problems in which numerical solutions need to be computed for both the inverse and the underlying forward problems, discretization can be a major factor that determines the accuracy of imaging. In this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography. We model the forward problem by a pair of diffusion equations at the excitation and emission wavelengths and consider a finite element discretization method for the numerical solution of the forward problem. For the inverse problem, we use an optimization framework which allows incorporation of a priori information in the form of zeroth- and first-order Tikhonov regularization terms. Next, we convert the inverse problem into a variational problem and use Galerkin projection to discretize the inverse problem. Following the discretization, we analyze the error in reconstructed images due to the discretization of the forward and inverse problems and present two theorems which point out the factors that may lead to high error such as the mutual dependence of the forward and inverse problems, the number of sources and detectors, their configuration and their positions with respect to fluorophore concentration, and the formulation of the inverse problem. Finally, we demonstrate the results and implications of our error analysis by numerical experiments. In the second part of the paper, we apply our results to design novel adaptive discretization algorithms.

In the first part of this work, we analyze the effect of discretization on the accuracy of fluorescence diffuse optical tomography (FDOT). Our error analysis provides two new error estimates which present a direct relationship between the error in the reconstructed fluorophore concentration and the discretization of the forward and inverse problems. In this paper, based on these error estimates, we develop two new adaptive mesh generation algorithms for the numerical solutions of the forward and inverse problems in FDOT, with the objective of error reduction in the reconstructed optical images due to discretization while keeping the size of the discretized forward and inverse problems within the allowable limits. We present three-dimensional numerical simulations to demonstrate the improvements in accuracy, resolution and detectability of small heterogeneities in reconstructed images provided by the use of the new adaptive mesh generation algorithms. Finally, we compare our algorithms both analytically and numerically with the existing conventional adaptive mesh generation algorithms.

This paper addresses fluorescence diffuse optical tomography (FDOT) reconstruction problem with a priori information. We assume that approximate location of the fluorophore concentration is known from physiology of the disease and nature of the fluorophore injected. In addition, we assume the anatomical edge structures from an anatomical image modality partially coincident with the edges of the fluorophore concentration image. We formulate FDOT reconstruction in Bayesian framework and model a priori information of FDOT into a Gaussian Markov random field (GMRF) with several unknown parameters. We simultaneously estimate the optical image and the unknown a priori model parameters. Numerical simulations demonstrate that the a priori information could effectively improve the image reconstruction results.

In this paper, we present a new method to form pharmacokinetic-rate images of optical fluorophores directly from near infra-red (NIR) boundary measurements. We first derive a mapping from spatially resolved pharmacokinetic rates to NIR boundary measurements by combining compartmental modeling with a diffusion based NIR photon propagation model. We express this mapping as a state-space equation. Next, we introduce a spatio-temporal prior model for the pharmacokinetic-rate images and combine it with the state-space equation. We address the image formation problem using the extended Kalman filtering framework. We analyze the computational complexity of the resulting algorithms and evaluate their performance in numerical simulations. An important feature of our approach is that the reconstruction of fluorescence concentrations and compartmental modeling are combined into a single step 1) to take advantage of the inherent temporal correlations in dynamic NIR measurements, and 2) to incorporate spatio-temporal a priori information on pharmacokinetic-rate images. Simulation results show that the resulting algorithms are more robust and lead to higher signal-to-noise ratio as compared to existing approaches where the reconstruction of concentrations and compartmental modeling are treated separately. Additionally, we reconstructed pharmacokinetic-rate images using in vivo data obtained from three patients with breast tumors. The reconstruction results show that the pharmacokinetic rates of indocyanine green are higher inside the tumor region as compared to the surrounding tissue.

We introduce a novel synthetic-aperture imaging method for radar systems that rely on sources of opportunity. We consider receivers that fly along arbitrary, but known, flight trajectories and develop a spatio-temporal correlation-based filtered-backprojection-type image reconstruction method. The method involves first correlating the measurements from two different receiver locations. This leads to a forward model where the radiance of the target scene is projected onto the intersection of certain hyperboloids with the surface topography. We next use microlocal techniques to develop a filtered-backprojection-type inversion method to recover the scene radiance. The method is applicable to both stationary and mobile, and cooperative and noncooperative sources of opportunity. Additionally, it is applicable to nonideal imaging scenarios such as those involving arbitrary flight trajectories, and has the desirable property of preserving the visible edges of the scene radiance. We present an analysis of the computational complexity of the image reconstruction method and demonstrate its performance in numerical simulations for single and multiple transmitters of opportunity.

In this paper, we present an analytic, filtered backprojection (FBP) type inversion method for bistatic synthetic aperture radar (BISAR). We consider a BISAR system where a scene of interest is illuminated by electromagnetic waves that are transmitted, at known times, from positions along an arbitrary, but known, flight trajectory and the scattered waves are measured from positions along a different flight trajectory which is also arbitrary, but known. We assume a single-scattering model for the radar data, and we assume that the ground topography is known but not necessarily flat. We use microlocal analysis to develop the FBP-type reconstruction method. We analyze the computational complexity of the numerical implementation of the method and present numerical simulations to demonstrate its performance.

In radar, when the wavelength of the transmitted electromagnetic wave is considerably larger than the dimension of the antenna, the received signal is modeled as the integral of the mean reflectivity function of the illuminated scene over the intersection of spheres centered at the transmitter location and the surface topography. When the surface topography is flat the received signal becomes integral of the mean reflectivity function over circles which is also referred to as circular averages of the mean reflectivity function. Thus, reconstruction of the ground reflectivity from synthetic aperture radar data requires inversion of the circular averages. Apart from radar, circular averages inversion also arises in thermo-acoustic tomography and sonar. In this paper, we present a new inversion method for the circular averages that uses the relationship between the circular averages, hyperbolic X-ray and the Funk transforms. The method is exact and numerically efficient as compared to standard filtered backprojection algorithms. Numerical simulations demonstrate that the approach is practical.

We introduce a new bistatic synthetic-aperture imaging method for a radar system consisting of two receivers, which will be referred to as hitchhikers, and a source of opportunity. We assume the receivers fly along arbitrary, but known, flight trajectories. We develop a correlation-based filtered-backprojection reconstruction method that preserves the visible edges of the target scene in the reconstructed image. We present an analysis of the computational complexity of the introduced method and demonstrate its applicability in numerical simulations. Potential applications of the proposed method include image formation using low earth orbiting and space-borne satellites as sources of opportunity.

This paper presents a group-theoretic approach to address the wideband extended range-Doppler target imaging and design of clutter rejecting waveforms. An exact imaging method based on the inverse Fourier transform of the affine group is presented. A Wiener filter is designed in the affine group Fourier transform domain to minimize wideband clutter range-Doppler reflectivity. The Wiener filter is then used to form an operator to precondition transmitted waveforms to reject clutter. Alternatively, the imaging and clutter rejection methods are equivalently re-expressed to perform clutter suppression upon reception. These methods are coupled with noise suppression upon reception. Numerical simulations are performed to demonstrate the performance of the proposed approach. Our study shows that the framework introduced in this paper can address the joint design of receive and transmit processing, design of clutter rejecting waveforms, suppression of noise, and reduction of computational complexity in receive processing

We present a new derivation of the spherical harmonic decomposition of the projection slice theorem using harmonic analysis of the Euclidean motion group, M(N). The Radon transform is formulated as a convolution integral over M(N). Deconvolution using harmonic analysis of M(N) leads to spherical harmonic decomposition of the projection slice theorem. The proposed method of decomposition leads to new algorithms for the inversion of the Radon transform.

This paper presents a new method for the exponential Radon transform inversion based on harmonic analysis of the Euclidean motion group (M(2)). The exponential Radon transform is modified to be formulated as a convolution over M(2). The convolution representation leads to a block diagonalization of the modified exponential Radon transform in the Euclidean motion group Fourier domain, which provides a deconvolution type inversion for the exponential Radon transform. Numerical examples are presented to show the viability of the proposed method.

The paper presents a fundamentally new mathematical framework based on group representation theory for the modeling and processing of signals in the ultrawideband (UWB) regime. A group theoretical approach is motivated by the fact that in the UWB regime, the underlying mathematical structure of the inverse scattering is governed by the affine group. In particular, a received echo can be viewed as the affine Fourier transform of the range-Doppler image evaluated at the transmitted waveform. Based on group representation theory and a novel Wiener filtering method over the affine group, we derive a regularized closed form analytical estimate of the range-Doppler target reflectivity density in the presence of nonstationary noise and clutter in the UWB regime. This estimate also leads to a method of data fusion to form synthetic UWB high resolution images from multiple radars with narrowband transmission.

In this paper, we explore domain decomposition algorithms for the inverse DOT problem in order to reduce the computational complexity and accelerate the convergence of the optical image reconstruction. We propose a combination of a two-level multigrid algorithm with a modified multiplicative Schwarz algorithm, where a conjugate gradient is used as an accelerator to solve each sub-problem formulated on each of the partitioned sub-domains. For our experiments, simulated phantom configuration with two rectangular inclusions is used as a testbed to measure the computational efficiency of our algorithms. No a priori information about the configuration is assumed except for the source and detector locations. For the application of our modified Schwarz algorithm alone, we observe an increase in efficiency of 100% as compared to the conjugate gradient solution obtained for the full domain. With the addition of the coarse grid, this efficiency rises to 400%. The coarse grid also serves to improve the overall appearance of the reconstructed image at the boundaries of the inclusions.

In this paper, we address a class of inverse problems that are formulated as group convolutions. This is a rich area of research with applications to Radon transform inversion for tomography, wide-band and narrow-band signal processing, inverse rendering in computer graphics, and channel estimation in communications, as well as robotics and polymer science. We present a group-theoretic framework for signal modeling and analysis for such problems and propose a minimum mean-square error (MMSE) deconvolution method in a probabilistic setting. Key components of our approach are group representation theory and the concept of group stationarity. The proposed deconvolution method incorporates a priori information and noise statistics into the inversion process, which leads to a natural regularized solution. We present recovery of self-similar processes that are "blurred" and embedded in noise as a demonstration example. The method is applicable to a wide range of inverse problems involving both commutative and noncommutative groups including finite, compact, and majority of well-behaved locally compact groups.

In this paper, we present a stochastic deconvolution method for a class of inverse problems that arc naturally formulated as group convolutions. Examples of such problems include Radon transform inversion for tomography, radar and sonar imaging, as well as channel estimation in communications. Key components of our approach are group representation theory and the concept of group stationarity. We formulate a minimum mean square solution to the deconvolution problem in the presence of nonstationary measurement noise. Our approach incorporates a priori information about the noise and the unknown signal into the inversion problem, which leads to a natural regularized solution.

Due to diffuse nature of light photons, diffuse optical tomography (DOT) image reconstruction is a challenging 3D problem with a relatively large number of unknowns and limited measurements. As a result, the computational complexity of the existing DOT image reconstruction algorithms remains prohibitive. In this work, we investigate an adaptive multigrid approach to improve the computational efficiency and the quantitative accuracy of DOT image reconstruction. The key idea is based on locally refined grid structure for region of interest (ROI). The ROI may be defined as diagnostically significant regions, strong background heterogeneities and/or deep optical edges, A 2-level mesh is generated to provide high resolution for ROI and sufficiently high resolution for the rest of the image. A least squares (LS) solution is formulated for the inverse problem. Fast adaptive composite (FAC) 2-grid algorithm is employed to solve the inverse problem. Conjugate gradient (CG) is used at the relaxation stage of FAC 2-grid. Same problem is also solved using direct CG and standard 2-grid method for globally fine grid structure. Our numerical studies demonstrate that the proposed FAC based adaptive 2-grid approach provides up to 90% reduction in computational requirements as compared to the direct iterative and standard 2-grid methods while providing better image quality. The fundamental ideas introduced in this study are directly applicable to other linear and nonlinear inverse problems with Newton type global linearization.

In this paper we formulate the Radon transform as a convolution integral over the Euclidean motion group (SE(2)) and provide a minimum mean square error (MMSE) stochastic deconvolution method for the Radon transform inversion. Proposed approach provides a fundamentally new formulation that can model nonstationary signal and noise fields. Key components of our development are the Fourier transform over SE(2), stochastic processes indexed by groups and fast implementation of the SE(2) Fourier transform. Numerical studies presented here demonstrate that the method yields image quality that is comparable or better than the filtered backprojection algorithm. Apart from X-ray tomographic image reconstruction, the proposed deconvolution method is directly applicable to inverse radiotherapy, and broad range of science and engineering problems in computer vision, pattern recognition, robotics as well as protein science.

In this paper, we formulate an optimal Wiener filter for the restoration of self-similar processes from time-varying and nonstationary distortions. Fourier techniques and classical Wiener filtering are not applicable for the elimination of such distortions. We derive an optimal Wiener filtering algorithm by modeling time-varying degradation within the scale-invariant system theory and nonstationary, long-term correlated noise within the scale stationarity framework. Additionally, we develop fast and efficient estimation techniques for scale spectra in Mellin domain for the implementation of the proposed Wiener filter. The utility of the proposed techniques are demonstrated on simulation data.

In our previous work, we introduced a new class of nonstationary stochastic processes whose spectral representation is associated with the wavelet transforms and established a mathematical framework for the analysis of such processes. We refer to these processes as affine stationary processes. These processes are indexed by the affine group, or ax+b group, which can be thought of as a group of shifts and scalings. Affine stationary processes are nonstationary in the classical sense. However, their second order statistical properties are invariant under the affine group composition law. We show that any physically realizable affine stationary process is a wavelet transform of the white noise process. As a result, we derive a spectral decomposition of the affine stationary processes using wavelet transform. Additionally, we apply the results to the fractional Brownian motion (fBm). We show that fBm is an affine stationary process and the filter associated with the fBm is a continuous time analyzing wavelet. Finally, we apply our results to choose an optimal wavelet filter in the development of a spectral representation of fBm via wavelet transforms.

In our previous work, we proposed two classes of self-similar models for 1/f processes which we referred to as scale stationary and p-self similar models. We introduced a new mathematical framework and several new concepts, such as periodicity, autocorrelation, and spectral density function to analyze scale stationary and p-self similar processes. In particular, we introduced a family of finite parameter scale stationary models, similar in spirit to ARMA models by which any scale stationary processes can be approximated. In this work, we utilized the framework of scale stationary processes and introduced novel methods of 1/f signal modeling and parameter estimation techniques. These include a sampling theorem, a mathematically consistent estimator for the self-similarity parameter, an unbiased estimator for the scale autocorrelation function and a maximum likelihood estimator for scale stationary autoregressive models. Results from our study suggest that scale stationary processes provide a powerful framework for practical 1/f signal processing problems.

We introduce a class of stochastic processes whose correlation function obeys a structure of the form, E[X(t)X(/spl lambda/t)]=t/sup 2H//spl lambda//sup H/R(/spl lambda/), /spl lambda/,t>0. We refer to these processes as second order self-similar processes. This class of processes include fractional Brownian motion as a special case. We define a concept of autocorrelation and develop a spectral analysis framework via generalized Mellin transform for the proposed class. Additionally, we establish a relationship between the proposed self-similar processes and generalized linear scale invariant system theory. We give specific models and demonstrate their ability to model 1/f phenomena.

Multiresolution image decompositions (e.g., wavelets), in conjunction with a variety of quantization schemes, have been shown to be very effective for image compression. Recently, several promising tree-structured quantization schemes that exploit the correlation across scales have been proposed. In this paper, we present an image compression algorithm based on a multiresolution Markov random field model used to model the correlations of wavelet coefficients across the scales. We also present experimental results obtained using the algorithm.>