text: 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.
  @cite_30: 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.
  @cite_38: This paper describes a simple algorithm for reconstructing the complex index of refraction of a bounded object immersed in a known background from a knowledge of how the object scatters known incident radiation. The method described here is versatile accommodating both spatially and frequency varying incident fields and allowing a priori information about the scatterer to be introduced in a simple fashion. Numerical results show that this new algorithm outperforms the modified gradient approach which until now has been one of the most effective reconstruction algorithms available.
  @cite_31: 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)
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  @cite_4: Optical tomography has been widely investigated for biomedical imaging applications. In recent years optical tomography has been combined with digital holography and has been employed to produce high-quality images of phase objects such as cells. In this paper we describe a method for imaging 3D phase objects in a tomographic configuration implemented by training an artificial neural network to reproduce the complex amplitude of the experimentally measured scattered light. The network is designed such that the voxel values of the refractive index of the 3D object are the variables that are adapted during the training process. We demonstrate the method experimentally by forming images of the 3D refractive index distribution of Hela cells.
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  @cite_3: 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.
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  @cite_40: In an optical diffraction microscopy experiment, one measures the phase and amplitude of the field diffracted by the sample and uses an inversion algorithm to reconstruct its map of permittivity. We show that with an iterative procedure accounting for multiple scattering, it is possible to visualize details smaller than λ 4 with relatively few illumination and observation angles. The roles of incident evanescent waves and noise are also investigated.
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summary: 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.