The problem of reconstructing an object from the measurements of the light it scatters is common in numerous imaging applications. While the most popular formulations of the problem are based on linearizing the object-light relationship, there is an increased interest in considering nonlinear formulations that can account for multiple light scattering. In this paper, we propose an image reconstruction method, called CISOR, for nonlinear diffractive imaging, based on our new variant of fast iterative shrinkage thresholding algorithm (FISTA) and total variation (TV) regularization. We prove that CISOR reliably converges for our nonconvex optimization problem, and systematically compare our method with other state-of-the-art methods on simulated as well as experimentally measured data.
Many methods that attempt to integrate the nonlinearity of wave scattering have been proposed in the literature. The iterative linearization (IL) method @cite_15 @cite_30 iteratively computes the forward model using the current estimated permittivity, and estimates the permittivity using the field from the previously computed forward model. Hence, each sub-problem at each iteration is a linear problem. Contrast source inversion (CSI) @cite_38 @cite_43 @cite_3 defines an auxiliary variable called the contrast source, which is the product of the permittivity contrast and the field. CSI alternates between estimating the contrast source and the permittivity. Hybrid methods (HM) @cite_40 @cite_26 @cite_52 combine IL and CSI, aiming to benefit from each of their advantages. A comprehensive comparison of these three methods can be found in the review paper @cite_31 . Recently, the idea of neural network unfolding has inspired a class of methods that updates the estimates using error backpropagation @cite_4 @cite_50 @cite_29 @cite_28 @cite_42 . While such methods can in principle model the precise nonlinearity, in practice, the accuracy may be limited by the availability of memory to store the iterates needed to perform unfolding.
@cite_30 @cite_38 @cite_31 @cite_26 @cite_4 @cite_28 @cite_29 @cite_42 @cite_52 @cite_3 @cite_43 @cite_40 @cite_50 @cite_15
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In microwave or optical wave imaging set-ups, the measured scattered field is usually not linearly linked to the target parameter of interest. To reconstruct the latter, nonlinearized or linearized iterative inversion techniques, such as the contrast source inversion method (CSI) or the conjugate gradient method (CGM) have been proposed (Litman and Crocco 2009 Inverse Problems 25 020201–5). In this paper, we adapt to the three-dimensional vectorial case, a hybrid method (HM) that combines the advantages of the CSI and CGM, and compare the performances of the different approaches using microwave experimental data. The HM appears to be the best compromise in terms of reconstruction accuracy, computation time and robustness to noise. (Some figures may appear in colour only in the online journal) The aim of the present work is to validate a full vectorial electromagnetic inverse scattering algorithm against experimental data. Data were provided courtesy of Institut Fresnel, Marseille, France. These data were carried out in an anechoic chamber and correspond to different canonical targets as well as one mysterious object which is known only by experimentalists who measured the associated scattered field. The inverse algorithm was first developed in the optical domain and is adapted herein to the microwave domain. It is an iterative approach where the parameter of interest, namely the relative permittivity distribution, is updated gradually by minimizing a cost function describing the discrepancy between data and those that would be obtained via a forward solver for the best available estimate of the relative permittivity. The forward solver is based on the coupled dipole method which was introduced in the seventies to study the scattering of light by non-spherical dielectric grains. The forward and inverse schemes are briefly described and various examples are presented that demonstrate the efficiency of the inverse algorithm. In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection. The starting point of this paper is the observation that unrolled iterative methods have the form of a CNN (filtering followed by pointwise non-linearity) when the normal operator ( @math , where @math is the adjoint of the forward imaging operator, @math ) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a @math image on the GPU. Two compressive sensing inspired approaches for the solution of non-linear inverse scattering problems are introduced and discussed. Differently from the sparsity promoting approaches proposed in most of the papers published in the literature, the two methods here tackle the problem in its full non-linearity, by adopting a contrast source inversion scheme. In the first approach, the @math -norm of the unknown is added as a weighted penalty term to the contrast source cost functional. The second, and (to the best of our knowledge) completely original, approach enforces sparsity by constraining the solution of the non-linear problem into a convex set defined by the @math -norm of the unknown. A numerical assessment against a widely used benchmark example (the “Austria” profile) is given to assess the capabilities of the proposed approaches. Notably, the two approaches can be applied to any kind of basis functions and they can successfully tackle both reduced number of data (with respect to Nyquist sampling) and or overcomplete dictionaries.