Total variation as a multiplicative constraint for solving inverse problems (original) (raw)
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Contrast Source Inversion Method: State of Art - Abstract*
Journal of Electromagnetic Waves and Applications, 2001
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 L 2 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.
Contrast Source Inversion Method: State of Art
Progress in Electromagnetics Research-pier, 2001
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 L 2 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.
Multiplicative regularization for contrast profile inversion
Radio Science, 2003
In this paper we discuss a new type of regularization technique for the nonlinear inverse scattering problem, namely the multiplicative technique. The main advantage is that we do not have to determine the regularization parameter before the inversion process is started. We consider different norms of the total variation as regularization factor. Specifically, we investigate a weighted L2-norm, and by using an appropriate updating scheme we show that this multiplicative regularization factor does not increase the nonlinearity of the inversion problem. Numerical examples using synthetic and experimental data demonstrate the robustness of the presented method.
2004
We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods by using total variation regularization. We obtain rigorous convergence results and effective stopping criteria for the general procedure. The numerical results for denoising appear to give significant improvement over standard models, and preliminary results for deblurring/denoising are very encouraging.
An Alternating Direction Algorithm for Total Variation Reconstruction of Distributed Parameters
IEEE transactions on image processing : a publication of the IEEE Signal Processing Society, 2012
Augmented Lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems. The alternating direction method of multipliers (ADMM) is one of such formulations/ optimization methods. Very recently the number of applications of ADMM, or variants of it, to solve inverse problems in image and signal processing has increased at an exponential rate. The reason for this interest is that ADMM decomposes a difficult optimization problem into a sequence of much simpler problems. In this work we use ADMM to reconstruct piece-wise smooth distributed parameters of elliptic partial differential equations from noisy and linear (blurred) observations of the underlying field. The distributed parameters are estimated by solving an inverse problem with total variation regularization. The proposed instance of ADMM solves, in each iteration, an `2 and a decoupled `2 ¡ `1 optimization problems. An operator splitting is used to simplify ...
Probabilistic regularization in inverse optical imaging
Journal of the Optical Society of America A, 2000
The problem of object restoration in the case of spatially incoherent illumination is considered. A regularized solution to the inverse problem is obtained through a probabilistic approach, and a numerical algorithm based on the statistical analysis of the noisy data is presented. Particular emphasis is placed on the question of the positivity constraint, which is incorporated into the probabilistically regularized solution by means of a quadratic programming technique. Numerical examples illustrating the main steps of the algorithm are also given.
Special Issue on "Optimization Methods for Inverse Problems in Imaging": Guest Editorial
2013
The aim of this special issue is to focus on the growing interaction between inverse problems in imaging science and optimization, that in recent years has given rise to significant advances in both the areas: optimization-based tools have been developed to solve challenging image reconstruction problems while the experience with imaging problems has led to an improved and deeper understanding of certain optimization algorithms. The issue includes 10 peer reviewed papers whose contributions represent new advances in numerical optimization for inverse problems with significant impact in signal and image processing
Imaging of biomedical data using a multiplicative regularized contrast source inversion method
IEEE Transactions on Microwave Theory and Techniques, 2002
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.
Application of Multiplicative Regularization to the Finite-Element Contrast Source Inversion Method
Antennas and Propagation, IEEE …, 2011
Multiplicative regularization is applied to the finite-element contrast source inversion (FEM-CSI) algorithm recently developed for microwave tomography. It is described for the two-dimensional (2D) transverse-magnetic (TM) case and tested by inverting experimental data where the fields can be approximated as TM. The unknown contrast, which is to be reconstructed, is represented using nodal variables and first-order basis functions on triangular elements; the same first-order basis functions used in the FEM solution of the accompanying field problem. This approach is different from other MR-CSI implementations where the contrast variables are located on a uniform grid of rectangular cells and represented using pulse basis functions. The linear basis function representation of the contrast makes it difficult to apply the weighted -norm total variation multiplicative regularization which requires that gradient and divergence operators be applied to the predicted contrast at each iteration of the inversion algorithm; the use of finite-difference operators for this purpose becomes unwieldy. Thus, a new technique is introduced to perform these operators on the triangular mesh.