Contrast Source Inversion Method: State of Art
TL;DR: Van den Berg and Abubakar as discussed by the authors discussed the possibility of the presence of local minima of the nonlinear cost functional and under which conditions they can exist, and introduced a new type of regularization, based on a weighted L 2 total variation norm.
Abstract: We discuss the problem of the reconstruction of the profile of an inhomogeneous object from scattered field data. Our starting point is the contrast source inversion method, where the unknown contrast sources and the unknown contrast are updated by an iterative minimization of a cost functional. We discuss the possibility of the presence of local minima of the nonlinear cost functional and under which conditions they can exist. Inspired by the successful implementation of the minimization of total variation and other edgepreserving algorithms in image restoration and inverse scattering, we have explored the use of these image-enhancement techniques as an extra regularization. The drawback of adding a regularization term to the cost functional is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation. Therefore, we first discuss the regularization as a multiplicative constraint and show that the weighting parameter is now completely prescribed by the error norm of the data equation and the object equation. Secondly, inspired by the edge-preserving algorithms, we introduce a new type of regularization, based on a weighted L2 total variation norm. The advantage is that the updating parameters in the contrast source inversion method can be determined explicitly, without the usual line minimization. In addition this new regularization shows excellent edge-preserving properties. Numerical experiments illustrate that the present multiplicative regularized inversion scheme is very robust, handling noisy as well as limited data very well, without the necessity of artificial regularization parameters. 190 Van den Berg and Abubakar
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TL;DR: This review attempts to illuminate the state of the art of FWI by building accurate starting models with automatic procedures and/or recording low frequencies, and improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.
Abstract: Full-waveform inversion FWI is a challenging data-fitting procedure based on full-wavefield modeling to extract quantitative information from seismograms. High-resolution imaging at half the propagated wavelength is expected. Recent advances in high-performance computing and multifold/multicomponent wide-aperture and wide-azimuth acquisitions make 3D acoustic FWI feasible today. Key ingredients of FWI are an efficient forward-modeling engine and a local differential approach, in which the gradient and the Hessian operators are efficiently estimated. Local optimization does not, however, prevent convergence of the misfit function toward local minima because of the limited accuracy of the starting model, the lack of low frequencies, the presence of noise, and the approximate modeling of the wave-physics complexity. Different hierarchical multiscale strategiesaredesignedtomitigatethenonlinearityandill-posedness of FWI by incorporating progressively shorter wavelengths in the parameter space. Synthetic and real-data case studies address reconstructing various parameters, from VP and VS velocities to density, anisotropy, and attenuation. This review attempts to illuminate the state of the art of FWI. Crucial jumps, however, remain necessary to make it as popular as migration techniques. The challenges can be categorized as 1 building accurate starting models with automatic procedures and/or recording low frequencies, 2 defining new minimization criteria to mitigate the sensitivity of FWI to amplitude errors and increasing the robustness of FWI when multiple parameter classes are estimated, and 3 improving computational efficiency by data-compression techniquestomake3DelasticFWIfeasible.
2,981 citations
Cites background from "Contrast Source Inversion Method: S..."
...Alternatively, van den Berg and Abubakar 2001 implement TV regularization as a multiplicative constraint in he original misfit function....
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TL;DR: In this paper, a general framework for the inversion of electromagnetic measurements in cases where parametrization of the unknown configuration is possible is developed, which can advantageously be used over gradient-based approaches.
Abstract: In this paper, we developed a general framework for the inversion of electromagnetic measurements in cases where parametrization of the unknown configuration is possible. Due to the ill-posed nature of this nonlinear inverse scattering problem, this parametrization approach is needed when the available measurement data are limited and measurements are only carried out from limited transmitter-receiver positions (limited data diversity). By carrying out this parametrization, the number of unknown model parameters that need to be inverted is manageable. Hence the Newton based approach can advantageously be used over gradient-based approaches. In order to guarantee an error reduction of the optimization process, the iterative step is adjusted using a line search algorithm. Further unlike the most available Newton-based approaches available in the literature, we enhanced the Newton based approaches presented in this paper by constraining the inverted model parameters with nonlinear transformation. This constrain forces the reconstruction of the unknown model parameters to lie within their physical bounds. In order to deal with cases where the measurements are redundant or lacking sensitivity to certain model parameters causing non-uniqueness of solution, the cost function to be minimized is regularized by adding a penalty term. One of the crucial aspects of this approach is how to determine the regularization parameter determining the relative importance of the misfit between the measured and predicted data and the penalty term. We reviewed different approaches to determine this parameter and proposed a robust and simple way of choosing this regularization parameter with aid of recently developed multiplicative regularization analysis. By combining all the techniques mentioned above we arrive at an effective and robust parametric algorithm. As numerical examples we present results of electromagnetic inversion at induction frequency in the deviated borehole configuration.
329 citations
Cites methods from "Contrast Source Inversion Method: S..."
...This new approach of choosing the regularization parameter is inspired by the work of [12, 13] on the multiplicative regularization for gradient-type algorithm....
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...Recently [12] introduced an automatic way to choose the regularization parameter....
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TL;DR: In this paper, the recently developed multiplicative regularized contrast source inversion method is applied to microwave biomedical applications, which is fully iterative and avoids solving any forward problem in each iterative step.
Abstract: In this paper, the recently developed multiplicative regularized contrast source inversion method is applied to microwave biomedical applications. The inversion method is fully iterative and avoids solving any forward problem in each iterative step. In this way, the inverse scattering problem can efficiently be solved. Moreover, the recently developed multiplicative regularizer allows us to apply the method blindly to experimental data. We demonstrate inversion from experimental data collected by a 2.33-GHz circular microwave scanner using a two-dimensional (2-D) TM polarization measurement setup. Further some results of a feasibility study of the present inversion method to the 2-D TE polarization and the full-vectorial three-dimensional measurement will be presented as well.
329 citations
TL;DR: A review of the state-of-the-art and most recent advances of compressive sensing and related methods as applied to electromagnetics can be found in this article, where a wide set of applicative scenarios comprising the diagnosis and synthesis of antenna arrays, the estimation of directions of arrival, and the solution of inverse scattering and radar imaging problems are reviewed.
Abstract: Several problems arising in electromagnetics can be directly formulated or suitably recast for an effective solution within the compressive sensing (CS) framework. This has motivated a great interest in developing and applying CS methodologies to several conventional and innovative electromagnetic scenarios. This work is aimed at presenting, to the best of the authors knowledge, a review of the state-of-the-art and most recent advances of CS formulations and related methods as applied to electromagnetics. Toward this end, a wide set of applicative scenarios comprising the diagnosis and synthesis of antenna arrays, the estimation of directions of arrival, and the solution of inverse scattering and radar imaging problems are reviewed. Current challenges and trends in the application of CS to the solution of traditional and new electromagnetic problems are also discussed.
318 citations
Cites background or methods from "Contrast Source Inversion Method: S..."
...Although the standard framework of an imaging problem cannot be directly tackled with CS strategies because of its intrinsic nonlinearity [74], several alternative formulations have been proposed either within the “fully-nonlinear” framework...
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...The inversion procedure is then aimed at solving (40) by looking for the unknown contrast source J kðrÞ starting from the knowledge of the incident electric field and the samples of the scattered electric field at each illumination (the dielectric profile ðrÞ is then retrieved by means of the State equation [74])....
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..., without approximations), the so-called contrast source formulation [74] has been usually used [25, 75]....
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TL;DR: In this article, a 2.5D fast and rigorous forward and inversion algorithm for deep electromagnetic (EM) applications that include crosswell and controlled-source EM measurements is presented.
Abstract: We present 2.5D fast and rigorous forward and inversion algorithms for deep electromagnetic (EM) applications that include crosswell and controlled-source EM measurements. The forward algorithm is based on a finite-difference approach in which a multifrontal LU decomposition algorithm simulates multisource experiments at nearly the cost of simulating one single-source experiment for each frequency of operation. When the size of the linear system of equations is large, the use of this noniterative solver is impractical. Hence, we use the optimal grid technique to limit the number of unknowns in the forward problem. The inversion algorithm employs a regularized Gauss-Newton minimization approach with a multiplicative cost function. By using this multiplicative cost function, we do not need a priori data to determine the so-called regularization parameter in the optimization process, making the algorithm fully automated. The algorithm is equipped with two regularization cost functions that allow us to reconstruct either a smooth or a sharp conductivity image. To increase the robustness of the algorithm, we also constrain the minimization and use a line-search approach to guarantee the reduction of the cost function after each iteration. To demonstrate the pros and cons of the algorithm, we present synthetic and field data inversion results for crosswell and controlled-source EM measurements.
280 citations
References
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TL;DR: In this article, a constrained optimization type of numerical algorithm for removing noise from images is presented, where the total variation of the image is minimized subject to constraints involving the statistics of the noise.
15,225 citations
Book•
01 Jan 1992
TL;DR: Inverse Medium Problem (IMP) as discussed by the authors is a generalization of the Helmholtz Equation for direct acoustical obstacle scattering in an Inhomogeneous Medium (IMM).
Abstract: Introduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- III-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle Scattering.- Acoustic Waves in an Inhomogeneous Medium.- Electromagnetic Waves in an Inhomogeneous Medium.- The Inverse Medium Problem.-References.- Index
5,126 citations
TL;DR: The main purpose of this paper is to advocate the use of the graph associated with Tikhonov regularization in the numerical treatment of discrete ill-posed problems, and to demonstrate several important relations between regularized solutions and the graph.
Abstract: When discrete ill-posed problems are analyzed and solved by various numerical regularization techniques, a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector. In particular, the graph associated with Tikhonov regularization plays a central role. The main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill-posed problems. The graph is characterized quantitatively, and several important relations between regularized solutions and the graph are derived. It is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic L-shaped “corner” of the graph.
3,585 citations
"Contrast Source Inversion Method: S..." refers background in this paper
...The addition of the total variation to the cost functional has a very positive effect on the quality of the reconstructions for both ‘blocky’ and smooth profiles, but a drawback is the presence of an artificial weighting parameter in the cost functional, which can only be determined through considerable numerical experimentation [15] and a priori information of the desired reconstruction....
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TL;DR: This paper proposes a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable, which leads to the definition of an original reconstruction algorithm, called ARTUR, which can be applied in a large number of applications in image processing.
Abstract: Many image processing problems are ill-posed and must be regularized. Usually, a roughness penalty is imposed on the solution. The difficulty is to avoid the smoothing of edges, which are very important attributes of the image. In this paper, we first give conditions for the design of such an edge-preserving regularization. Under these conditions, we show that it is possible to introduce an auxiliary variable whose role is twofold. First, it marks the discontinuities and ensures their preservation from smoothing. Second, it makes the criterion half-quadratic. The optimization is then easier. We propose a deterministic strategy, based on alternate minimizations on the image and the auxiliary variable. This leads to the definition of an original reconstruction algorithm, called ARTUR. Some theoretical properties of ARTUR are discussed. Experimental results illustrate the behavior of the algorithm. These results are shown in the field of 2D single photon emission tomography, but this method can be applied in a large number of applications in image processing.
1,360 citations
TL;DR: A blind deconvolution algorithm based on the total variational (TV) minimization method proposed is presented, and it is remarked that psf's without sharp edges, e.g., Gaussian blur, can also be identified through the TV approach.
Abstract: We present a blind deconvolution algorithm based on the total variational (TV) minimization method proposed by Acar and Vogel (1994). The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images as well as some blurring functions, e.g., motion blur and out-of-focus blur. An alternating minimization (AM) implicit iterative scheme is devised to recover the image and simultaneously identify the point spread function (PSF). Numerical results indicate that the iterative scheme is quite robust, converges very fast (especially for discontinuous blur), and both the image and the PSF can be recovered under the presence of high noise level. Finally, we remark that PSFs without sharp edges, e.g., Gaussian blur, can also be identified through the TV approach.
1,220 citations
"Contrast Source Inversion Method: S..." refers methods in this paper
...Hence, the noise will, at all times, be suppressed in the reconstruction process and we automatically fulfill the need of a larger TV-regularization when the data contains noise as suggested by Chan and Wong [6] and Rudin et al....
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...Hence, the noise will, at all times, be suppressed in the reconstruction process and we automatically fulfill the need of a larger TV-regularization when the data contains noise as suggested by Chan and Wong [6] and Rudin et al. [25]....
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