Inverse Problems

Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network (NN) architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward NNs, recurrent NNs, or convolutional neural networks. This has had a great impact in the area of mathematical modelling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We also show their relevance in various industrial applications.

Recovering a function or high-dimensional parameter vector from indirect measurements is a central task in various scientific areas. Several methods for solving such inverse problems are well developed and well understood. Recently, novel algorithms using deep learning and neural networks for inverse problems appeared. While still in their infancy, these techniques show astonishing performance for applications like low-dose CT or various sparse data problems. However, there are few theoretical results for deep learning in inverse problems. In this paper, we establish a complete convergence analysis for the proposed NETT (network Tikhonov) approach to inverse problems. NETT considers nearly data-consistent solutions having small value of a regularizer defined by a trained neural network. We derive well-posedness results and quantitative error estimates, and propose a possible strategy for training the regularizer. Our theoretical results and framework are different from any previous work using neural networks for solving inverse problems. A possible data driven regularizer is proposed. Numerical results are presented for a tomographic sparse data problem, which demonstrate good performance of NETT even for unknowns of different type from the training data. To derive the convergence and convergence rates results we introduce a new framework based on the absolute Bregman distance generalizing the standard Bregman distance from the convex to the non-convex case.

Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. Although DuNets have been successfully applied to many linear inverse problems, their performance tends to be impaired by nonlinear problems. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets—the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA, respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

This work develops a machine learned structural design model for continuous beam systems from the inverse problem perspective. After demarcating between forward, optimisation and inverse machine learned operators, the investigation proposes a novel methodology based on the recently developed influence zone concept which represents a fundamental shift in approach compared to traditional structural design methods. The aim of this approach is to conceptualise a non-iterative structural design model that predicts cross-section requirements for continuous beam systems of arbitrary system size. After generating a dataset of known solutions, an appropriate neural network architecture is identified, trained, and tested against unseen data. The results show a mean absolute percentage testing error of 1.6% for cross-section property predictions, along with a good ability of the neural network to generalise well to structural systems of variable size. The CBeamXP dataset generated in this work and an associated python-based neural network training script are available at an open-source data repository to allow for the reproducibility of results and to encourage further investigations.

In this paper we describe an investigation into the application of deep learning methods for low-dose and sparse angle computed tomography using small training datasets. To motivate our work we review some of the existing approaches and obtain quantitative results after training them with different amounts of data. We find that the learned primal-dual method has an outstanding performance in terms of reconstruction quality and data efficiency. However, in general, end-to-end learned methods have two deficiencies: (a) a lack of classical guarantees in inverse problems and (b) the lack of generalization after training with insufficient data. To overcome these problems, we introduce the deep image prior approach in combination with classical regularization and an initial reconstruction. The proposed methods achieve the best results in the low-data regime in three challenging scenarios.

Tikhonov regularization with square-norm penalty for linear forward operators has been studied extensively in the literature. However, the results on convergence theory are based on technical proofs and sometimes difficult to interpret. It is also often not clear how those results translate into the discrete, numerical setting. In this paper we present a new strategy to study the properties of a regularization method on the example of Tikhonov regularization. The technique is based on the well-known observation that Tikhonov regularization approximates the unknown exact solution in the range of the adjoint of the forward operator. This is closely related to the concept of approximate source conditions, which we generalize to describe not only the approximation of the unknown solution, but also noise-free and noisy data; all from the same source space. Combining these three approximation results we derive the well-known convergence results in a concise way and improve the understanding by tightening the relation between concepts such as convergence rates, parameter choice, and saturation. The new technique is not limited to Tikhonov regularization, it can be applied also to iterative regularization, which we demonstrate by relating Tikhonov regularization and Landweber iteration. All results are accompanied by numerical examples.

In passive imaging, one attempts to reconstruct some coefficients in a wave equation from correlations of observed randomly excited solutions to this wave equation. Many methods proposed for this class of inverse problem so far are only qualitative, e.g. trying to identify the support of a perturbation. Major challenges are the increase in dimensionality when computing correlations from primary data in a preprocessing step, and often very poor pointwise signal-to-noise ratios. In this paper, we propose an approach that addresses both of these challenges: it works only on the primary data while implicitly using the full information contained in the correlation data, and it provides quantitative estimates and convergence by iteration. Our work is motivated by helioseismic holography, a well-established imaging method to map heterogenities and flows in the solar interior. We show that the back-propagation used in classical helioseismic holography can be interpreted as the adjoint of the Fréchet derivative of the operator which maps the properties of the solar interior to the correlation data on the solar surface. The theoretical and numerical framework for passive imaging problems developed in this paper extends helioseismic holography to nonlinear problems and allows for quantitative reconstructions. We present a proof of concept in uniform media.

This paper introduces CUQIpy, a versatile open-source Python package for computational uncertainty quantification (UQ) in inverse problems, presented as Part I of a two-part series. CUQIpy employs a Bayesian framework, integrating prior knowledge with observed data to produce posterior probability distributions that characterize the uncertainty in computed solutions to inverse problems. The package offers a high-level modeling framework with concise syntax, allowing users to easily specify their inverse problems, prior information, and statistical assumptions. CUQIpy supports a range of efficient sampling strategies and is designed to handle large-scale problems. Notably, the automatic sampler selection feature analyzes the problem structure and chooses a suitable sampler without user intervention, streamlining the process. With a selection of probability distributions, test problems, computational methods, and visualization tools, CUQIpy serves as a powerful, flexible, and adaptable tool for UQ in a wide selection of inverse problems. Part II of the series focuses on the use of CUQIpy for UQ in inverse problems with partial differential equations.

Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Since inverse problems typically suffer from instability with respect to data perturbations, a variety of regularization techniques have been proposed. In particular, the use of filtered diagonal frame decompositions (DFDs) has proven to be effective and computationally efficient. However, existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze filtered DFDs with general non-linear filters. In particular, our results generalize singular value decomposition-based spectral filtering from linear to non-linear filters as a special case. As a first approach, we establish a connection between non-linear diagonal frame filtering and variational regularization, allowing us to use results from variational regularization to derive the convergence of non-linear spectral filtering. In the second approach, as our main theoretical results, we relax the assumptions involved in the variational case while still deriving convergence. Furthermore, we discuss connections between non-linear filtering and plug-and-play regularization and explore potential benefits of this relationship.

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

In multi-photon microscopy (MPM), a recent in-vivo fluorescence microscopy system, the task of image restoration can be decomposed into two interlinked inverse problems: firstly, the characterization of the point spread function (PSF) and subsequently, the deconvolution (i.e. deblurring) to remove the PSF effect, and reduce noise. The acquired MPM image quality is critically affected by PSF blurring and intense noise. The PSF in MPM is highly spread in 3D and is not well characterized, presenting high variability with respect to the observed objects. This makes the restoration of MPM images challenging. Common PSF estimation methods in fluorescence microscopy, including MPM, involve capturing images of sub-resolution beads, followed by quantifying the resulting ellipsoidal 3D spot. In this work, we revisit this approach, coping with its inherent limitations in terms of accuracy and practicality. We estimate the PSF from the observation of relatively large beads (approximately 1 $\mu \mathrm{m}$ in diameter). This goes through the formulation and resolution of an original non-convex minimization problem, for which we propose a proximal alternating method along with convergence guarantees. Following the PSF estimation step, we then introduce an innovative strategy to deal with the high level multiplicative noise degrading the acquisitions. We rely on a heteroscedastic noise model for which we estimate the parameters. We then solve a constrained optimization problem to restore the image, accounting for the estimated PSF and noise, while allowing a minimal hyper-parameter tuning. Theoretical guarantees are given for the restoration algorithm. These algorithmic contributions lead to an end-to-end pipeline for 3D image restoration in MPM, that we share as a publicly available Python software. We demonstrate its effectiveness through several experiments on both simulated and real data.

In this paper, we consider the low-rank matrix recovery problem. We propose the nonconvex regularized least absolute deviations model via $\ell_1-\alpha\ell_2 \ (0\lt\alpha\lt 1)$ minimization. We establish the theoretical analysis of the proposed model and obtain a stable error estimation. Our result is a nontrivial extension of some previous work. Different from most of the state-of-the-art methods, our method does not need any knowledge of standard deviation or any moment assumption of the noise. Numerical experiments show that our method is effective for many types of noise distributions.

The retrieval of non-Born scatterers is addressed within the contrast source inversion (CSI) framework by means of a novel multi-step inverse scattering method that jointly exploits prior information on the class of targets under investigation and progressively-acquired knowledge on the domain under investigation. The multi-resolution (MR) representation of the unknown contrast sources is iteratively retrieved by applying a Bayesian compressive sensing (BCS) sparsity-promoting approach based on a constrained relevance vector machine solver. Representative examples of inversions from synthetic and experimental data are reported to give some indications on the reliability and the robustness of the proposed MR-BCS-CSI method. Comparisons with recent and competitive state-of-the-art alternatives are reported, as well.

Magnetic resonance imaging (MRI) is a widely used medical imaging technique, but its long acquisition time can be a limiting factor in clinical settings. To address this issue, researchers have been exploring ways to reduce the acquisition time while maintaining the reconstruction quality. Previous works have focused on finding either sparse samplers with a fixed reconstructor or finding reconstructors with a fixed sampler. However, these approaches do not fully utilize the potential of joint learning of samplers and reconstructors. In this paper, we propose an alternating training framework for jointly learning a good pair of samplers and reconstructors via deep reinforcement learning. In particular, we consider the process of MRI sampling as a sampling trajectory controlled by a sampler, and introduce a novel sparse-reward partially observed Markov decision process (POMDP) to formulate the MRI sampling trajectory. Compared to the dense-reward POMDP used in existing works, the proposed sparse-reward POMDP is more computationally efficient and has a provable advantage. Moreover, the proposed framework, called learning to sample and reconstruct (L2SR), overcomes the training mismatch problem that arises in previous methods that use dense-reward POMDP. By alternately updating samplers and reconstructors, L2SR learns a pair of samplers and reconstructors that achieve state-of-the-art reconstruction performances on the fastMRI dataset. Codes are available at https://github.com/yangpuPKU/L2SR-Learning-to-Sample-and-Reconstruct.

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator.

Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network (NN) architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward NNs, recurrent NNs, or convolutional neural networks. This has had a great impact in the area of mathematical modelling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We also show their relevance in various industrial applications.

We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters. The focus is on problems where one seeks to optimize the placement of measurement points, at which data are collected, such that the uncertainty in the estimated parameters is minimized. We present the mathematical foundations of OED in this context and survey the computational methods for the class of OED problems under study. We also outline some directions for future research in this area.

Over the last decades, the total variation (TV) has evolved to be one of the most broadly-used regularisation functionals for inverse problems, in particular for imaging applications. When first introduced as a regulariser, higher-order generalisations of TV were soon proposed and studied with increasing interest, which led to a variety of different approaches being available today. We review several of these approaches, discussing aspects ranging from functional-analytic foundations to regularisation theory for linear inverse problems in Banach space, and provide a unified framework concerning well-posedness and convergence for vanishing noise level for respective Tikhonov regularisation. This includes general higher orders of TV, additive and infimal-convolution multi-order total variation, total generalised variation, and beyond. Further, numerical optimisation algorithms are developed and discussed that are suitable for solving the Tikhonov minimisation problem for all presented models. Focus is laid in particular on covering the whole pipeline starting at the discretisation of the problem and ending at concrete, implementable iterative procedures. A major part of this review is finally concerned with presenting examples and applications where higher-order TV approaches turned out to be beneficial. These applications range from classical inverse problems in imaging such as denoising, deconvolution, compressed sensing, optical-flow estimation and decompression, to image reconstruction in medical imaging and beyond, including magnetic resonance imaging, computed tomography, magnetic-resonance positron emission tomography, and electron tomography.

Electrical impedance tomography (EIT) is an imaging modality where a patient or object is probed using harmless electric currents. The currents are fed through electrodes placed on the surface of the target, and the data consists of voltages measured at the electrodes resulting from a linearly independent set of current injection patterns. EIT aims to recover the internal distribution of electrical conductivity inside the target. The inverse problem underlying the EIT image formation task is nonlinear and severely ill-posed, and hence sensitive to modeling errors and measurement noise. Therefore, the inversion process needs to be regularized. However, traditional variational regularization methods, based on optimization, often suffer from local minima because of nonlinearity. This is what makes regularized direct (non-iterative) methods attractive for EIT. The most developed direct EIT algorithm is the D-bar method, based on complex geometric optics solutions and a nonlinear Fourier transform. Variants and recent developments of D-bar methods are reviewed, and their practical numerical implementation is explained.

Magnetic particle imaging (MPI) is a relatively new imaging modality. The nonlinear magnetization behavior of nanoparticles in an applied magnetic field is exploited to reconstruct an image of the concentration of nanoparticles. Finding a sufficiently accurate model to reflect the behavior of large numbers of particles for MPI remains an open problem. As such, reconstruction is still computed using a measured forward operator obtained in a time-consuming calibration process. The model commonly used to illustrate the imaging methodology and obtain first model-based reconstructions relies on substantial model simplifications. By neglecting particle–particle interactions, the forward operator can be expressed by a Fredholm integral operator of the first kind when describing the inverse problem. Here, we review previously proposed models derived from single-particle behavior in the MPI context and consider future research on linear and nonlinear problems beyond concentration reconstruction applications. This survey complements a recent topical review on MPI (Knopp et al 2017 Phys. Med. Biol. 62 R124).

In medical and biological image processing, multi-dimensional images are often corrupted by blur and Poisson noise. In this paper, we first propose a new tensor logarithmic Schatten-$p$ (t-log-$S_p$) low-rank measure and a tensor iteratively reweighted Schatten-$p$ minimization (t-IRSpM) algorithm for minimizing such measure. Furthermore, we adopt this low-rank measure to regularize the non-local tensors formed by similar 3D image patches and develop a patch-based non-local low-rank model. The data fidelity term of the model characterizes the Poisson noise distribution and blur operator. The optimization model is further solved by an alternating minimization technique combined with variable splitting. Experimental results tested on 3D fluorescence microscope images show that the proposed patch-based tensor logarithmic Schatten-$p$ minimization (TLSpM) method outperforms state-of-the-art methods in terms of image evaluation metrics and visual quality.

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator.

This paper introduces a direct sampling method tailored for identifying the location of the source term within a time-fractional diffusion equation (TFDE). The key aspect of our approach involves the utilization of a versatile family of index functions, which can be chosen according to the specific characteristics of the source term. Recognizing the key role of the TFDE's fundamental solution within the index function, we further enhance our method by deriving its asymptotic expansions. This advancement not only enhances the accuracy, but also significantly improves the computational efficiency of our method. To validate the effectiveness and robustness of the proposed sampling method, we conduct a series of comprehensive numerical experiments.

A mild data-driven approach for microwave imaging is considered in this paper. In particular, the developed technique relies upon the use of a Newton-type inversion scheme in variable-exponent Lebesgue spaces, which has been modified by including a data-driven operator to enforce the available a-priori information about the class of targets to be investigated. In this way, the performance of the method is improved, and the problems related to the possible convergence to local minima are mitigated. The effectiveness of the approach has been evaluated through numerical simulations involving the detection of defects inside (partially) known objects, showing good results.

This work develops a machine learned structural design model for continuous beam systems from the inverse problem perspective. After demarcating between forward, optimisation and inverse machine learned operators, the investigation proposes a novel methodology based on the recently developed influence zone concept which represents a fundamental shift in approach compared to traditional structural design methods. The aim of this approach is to conceptualise a non-iterative structural design model that predicts cross-section requirements for continuous beam systems of arbitrary system size. After generating a dataset of known solutions, an appropriate neural network architecture is identified, trained, and tested against unseen data. The results show a mean absolute percentage testing error of 1.6% for cross-section property predictions, along with a good ability of the neural network to generalise well to structural systems of variable size. The CBeamXP dataset generated in this work and an associated python-based neural network training script are available at an open-source data repository to allow for the reproducibility of results and to encourage further investigations.

Combining the strengths of model-based iterative algorithms and data-driven deep learning solutions, deep unrolling networks (DuNets) have become a popular tool to solve inverse imaging problems. Although DuNets have been successfully applied to many linear inverse problems, their performance tends to be impaired by nonlinear problems. Inspired by momentum acceleration techniques that are often used in optimization algorithms, we propose a recurrent momentum acceleration (RMA) framework that uses a long short-term memory recurrent neural network (LSTM-RNN) to simulate the momentum acceleration process. The RMA module leverages the ability of the LSTM-RNN to learn and retain knowledge from the previous gradients. We apply RMA to two popular DuNets—the learned proximal gradient descent (LPGD) and the learned primal-dual (LPD) methods, resulting in LPGD-RMA and LPD-RMA, respectively. We provide experimental results on two nonlinear inverse problems: a nonlinear deconvolution problem, and an electrical impedance tomography problem with limited boundary measurements. In the first experiment we have observed that the improvement due to RMA largely increases with respect to the nonlinearity of the problem. The results of the second example further demonstrate that the RMA schemes can significantly improve the performance of DuNets in strongly ill-posed problems.

Microtexture regions (MTRs) are collections of grains with similar crystallographic orientation. Because their presence in titanium alloys can significantly impact aerospace component life, a nondestructive method to detect and characterize MTR is needed. In this work, we propose to use data from two nondestructive evaluation methods, eddy current testing (ECT) and scanning acoustic microscopy (SAM), in order to recover the boundary and dominant crystallographic orientation of each MTR in a specimen. ECT is an electromagnetic method that is sensitive to changes in crystallographic orientation associated with MTR; however, its low resolution prevents it from resolving MTR boundaries well. In contrast, SAM is a high frequency ultrasound method that is able to resolve MTR boundaries but is not sensitive to orientation. This paper proposes an algorithm to characterize MTR that makes use of a method known as covariance generalized matching component analysis. This method is used to build a surrogate linear forward model that relates MTR boundaries and orientation to ECT data. The model is inverted using the SAM data as a structural prior. We demonstrate this technique using simulated ECT and experimental SAM data from a large grain titanium specimen.

Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Since inverse problems typically suffer from instability with respect to data perturbations, a variety of regularization techniques have been proposed. In particular, the use of filtered diagonal frame decompositions (DFDs) has proven to be effective and computationally efficient. However, existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze filtered DFDs with general non-linear filters. In particular, our results generalize singular value decomposition-based spectral filtering from linear to non-linear filters as a special case. As a first approach, we establish a connection between non-linear diagonal frame filtering and variational regularization, allowing us to use results from variational regularization to derive the convergence of non-linear spectral filtering. In the second approach, as our main theoretical results, we relax the assumptions involved in the variational case while still deriving convergence. Furthermore, we discuss connections between non-linear filtering and plug-and-play regularization and explore potential benefits of this relationship.

The objective of this work is to quantify the reconstruction error in sparse inverse problems with measures and stochastic noise, motivated by optimal sensor placement. To be useful in this context, the error quantities must be explicit in the sensor configuration and robust with respect to the source, yet relatively easy to compute in practice, compared to a direct evaluation of the error by a large number of samples. In particular, we consider the identification of a measure consisting of an unknown linear combination of point sources from a finite number of measurements contaminated by Gaussian noise. The statistical framework for recovery relies on two main ingredients: first, a convex but non-smooth variational Tikhonov point estimator over the space of Radon measures and, second, a suitable mean-squared error based on its Hellinger–Kantorovich distance to the ground truth. To quantify the error, we employ a non-degenerate source condition as well as careful linearization arguments to derive a computable upper bound. This leads to asymptotically sharp error estimates in expectation that are explicit in the sensor configuration. Thus they can be used to estimate the expected reconstruction error for a given sensor configuration and guide the placement of sensors in sparse inverse problems.

This paper follows a detection-theoretic approach for using synthetic-aperture measurements, made at multiple moving passive receivers, in order to form an image showing the locations of stationary sources that are radiating unknown electromagnetic or acoustic waves. The paper starts with a physics-based model for the propagating fields, and, following the general approach of McWhorter et al (2023 arXiv:2302.06816, IEEE Open J. Signal Process.4 437–51), derives a detection statistic that is used for the image formation. This detection statistic is a quadratic function of the data. Each point in the scene is tested as a possible hypothesized location for a source, and the detection statistic is plotted as a function of location. Because this image formation process is nonlinear, the standard linear methods for determining resolution cannot be applied. This paper shows how to analyze the detection image by first writing the noiseless image as a coherent sum of shifted complex ambiguity functions of the source waveform. The paper then develops a technique for calculating image resolution; resolution is found to depend on the sensor-source geometry and also on the properties (bandwidth and temporal duration) of the source waveform. Optimal filtering of the image is given, but a simple example suggests that optimal filtering may have little effect. Analysis is also given for the case in which multiple sources are present.