A Brief Survey of Image Processing Algorithms in Electrical Capacitance Tomography
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paper_title: Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems

paper_content:

This paper studies gradient-based schemes for image denoising and deblurring problems based on the discretized total variation (TV) minimization model with constraints. We derive a fast algorithm for the constrained TV-based image deburring problem. To achieve this task, we combine an acceleration of the well known dual approach to the denoising problem with a novel monotone version of a fast iterative shrinkage/thresholding algorithm (FISTA) we have recently introduced. The resulting gradient-based algorithm shares a remarkable simplicity together with a proven global rate of convergence which is significantly better than currently known gradient projections-based methods. Our results are applicable to both the anisotropic and isotropic discretized TV functionals. Initial numerical results demonstrate the viability and efficiency of the proposed algorithms on image deblurring problems with box constraints.
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paper_title: A Fast Wavelet-Based Reconstruction Method for Magnetic Resonance Imaging

paper_content:

In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary k-space trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k-space trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality.
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paper_title: Image reconstruction algorithms for electrical capacitance tomography

paper_content:

Electrical capacitance tomography (ECT) is used to image cross-sections of industrial processes containing dielectric material. This technique has been under development for more than a decade. The task of image reconstruction for ECT is to determine the permittivity distribution and hence material distribution over the cross-section from capacitance measurements. There are three principal difficulties with image reconstruction for ECT: (1) the relationship between the permittivity distribution and capacitance is non-linear and the electric field is distorted by the material present, the so-called 'soft-field' effect; (2) the number of independent measurements is limited, leading to an under-determined problem and (3) the inverse problem is ill posed and ill conditioned, making the solution sensitive to measurement errors and noise. Regularization methods are needed to treat this ill-posedness. This paper reviews existing image reconstruction algorithms for ECT, including linear back-projection, singular value decomposition, Tikhonov regularization, Newton–Raphson, iterative Tikhonov, the steepest descent method, Landweber iteration, the conjugate gradient method, algebraic reconstruction techniques, simultaneous iterative reconstruction techniques and model-based reconstruction. Some of these algorithms are examined by simulation and experiment for typical permittivity distributions. Future developments in image reconstruction for ECT are discussed.
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paper_title: Total variation image reconstruction for electrical capacitance tomography

paper_content:

A new iterative reconstruction technique for electrical capacitance tomography (ECT) is presented. This method applies iterative soft thresholding to penalize the total variation of the ECT image. This method is shown to allow for sharp changes between different regions of simulated ECT images, unlike conventional iterative reconstruction techniques such as ART, SIRT and the Landweber iteration. The technique is demonstrated on experimental results for a phantom with distinct regions of gas and solid. The technique is therefore appropriate for the imaging of multiphase systems.
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paper_title: A new alternating minimization algorithm for total variation image reconstruction

paper_content:

We propose, analyze, and test an alternating minimization algorithm for recovering images from blurry and noisy observations with total variation (TV) regularization. This algorithm arises from a new half-quadratic model applicable to not only the anisotropic but also the isotropic forms of TV discretizations. The per-iteration computational complexity of the algorithm is three fast Fourier transforms. We establish strong convergence properties for the algorithm including finite convergence for some variables and relatively fast exponential (or $q$-linear in optimization terminology) convergence for the others. Furthermore, we propose a continuation scheme to accelerate the practical convergence of the algorithm. Extensive numerical results show that our algorithm performs favorably in comparison to several state-of-the-art algorithms. In particular, it runs orders of magnitude faster than the lagged diffusivity algorithm for TV-based deblurring. Some extensions of our algorithm are also discussed.
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paper_title: Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems

paper_content:

This paper studies gradient-based schemes for image denoising and deblurring problems based on the discretized total variation (TV) minimization model with constraints. We derive a fast algorithm for the constrained TV-based image deburring problem. To achieve this task, we combine an acceleration of the well known dual approach to the denoising problem with a novel monotone version of a fast iterative shrinkage/thresholding algorithm (FISTA) we have recently introduced. The resulting gradient-based algorithm shares a remarkable simplicity together with a proven global rate of convergence which is significantly better than currently known gradient projections-based methods. Our results are applicable to both the anisotropic and isotropic discretized TV functionals. Initial numerical results demonstrate the viability and efficiency of the proposed algorithms on image deblurring problems with box constraints.
---
paper_title: Total variation image reconstruction for electrical capacitance tomography

paper_content:

A new iterative reconstruction technique for electrical capacitance tomography (ECT) is presented. This method applies iterative soft thresholding to penalize the total variation of the ECT image. This method is shown to allow for sharp changes between different regions of simulated ECT images, unlike conventional iterative reconstruction techniques such as ART, SIRT and the Landweber iteration. The technique is demonstrated on experimental results for a phantom with distinct regions of gas and solid. The technique is therefore appropriate for the imaging of multiphase systems.
---
paper_title: Nonlinear image reconstruction for electrical capacitance tomography using experimental data

paper_content:

Electrical capacitance tomography (ECT) attempts to image the permittivity distribution of an object by measuring the electrical capacitances between sets of electrodes placed around its periphery. Image reconstruction in ECT is a nonlinear and ill-posed inverse problem. Although reconstruction techniques based on a linear approximation are fast, they are not adequate for all cases. In this paper, we study the nonlinearity of the inverse permittivity problem of ECT. A regularized Gauss-Newton scheme has been implemented for nonlinear image reconstruction. The forward problem has been solved at each iteration using the finite element method and the Jacobian matrix is recalculated using an efficient adjoint field method. Regularization techniques are required to overcome the ill-posedness: smooth generalized Tikhonov regularization for the smoothly varying case, and total variation (TV) regularization when there is a sharp transition of the permittivity have been used. The reconstruction results for experimental ECT data demonstrate the advantage of TV regularization for jump changes, and show improvement of the image quality by using nonlinear reconstruction methods.
---
paper_title: A Fast Wavelet-Based Reconstruction Method for Magnetic Resonance Imaging

paper_content:

In this work, we exploit the fact that wavelets can represent magnetic resonance images well, with relatively few coefficients. We use this property to improve magnetic resonance imaging (MRI) reconstructions from undersampled data with arbitrary k-space trajectories. Reconstruction is posed as an optimization problem that could be solved with the iterative shrinkage/thresholding algorithm (ISTA) which, unfortunately, converges slowly. To make the approach more practical, we propose a variant that combines recent improvements in convex optimization and that can be tuned to a given specific k-space trajectory. We present a mathematical analysis that explains the performance of the algorithms. Using simulated and in vivo data, we show that our nonlinear method is fast, as it accelerates ISTA by almost two orders of magnitude. We also show that it remains competitive with TV regularization in terms of image quality.
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paper_title: Compressed sensing

paper_content:

Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces
---
paper_title: Image reconstruction algorithms for electrical capacitance tomography

paper_content:

Electrical capacitance tomography (ECT) is used to image cross-sections of industrial processes containing dielectric material. This technique has been under development for more than a decade. The task of image reconstruction for ECT is to determine the permittivity distribution and hence material distribution over the cross-section from capacitance measurements. There are three principal difficulties with image reconstruction for ECT: (1) the relationship between the permittivity distribution and capacitance is non-linear and the electric field is distorted by the material present, the so-called 'soft-field' effect; (2) the number of independent measurements is limited, leading to an under-determined problem and (3) the inverse problem is ill posed and ill conditioned, making the solution sensitive to measurement errors and noise. Regularization methods are needed to treat this ill-posedness. This paper reviews existing image reconstruction algorithms for ECT, including linear back-projection, singular value decomposition, Tikhonov regularization, Newton–Raphson, iterative Tikhonov, the steepest descent method, Landweber iteration, the conjugate gradient method, algebraic reconstruction techniques, simultaneous iterative reconstruction techniques and model-based reconstruction. Some of these algorithms are examined by simulation and experiment for typical permittivity distributions. Future developments in image reconstruction for ECT are discussed.
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paper_title: An Iterative Shrinkage Approach to Total-Variation Image Restoration

paper_content:

The problem of restoration of digital images from their degraded measurements plays a central role in a multitude of practically important applications. A particularly challenging instance of this problem occurs in the case when the degradation phenomenon is modeled by an ill-conditioned operator. In such a situation, the presence of noise makes it impossible to recover a valuable approximation of the image of interest without using some a priori information about its properties. Such a priori information--commonly referred to as simply priors--is essential for image restoration, rendering it stable and robust to noise. Moreover, using the priors makes the recovered images exhibit some plausible features of their original counterpart. Particularly, if the original image is known to be a piecewise smooth function, one of the standard priors used in this case is defined by the Rudin-Osher-Fatemi model, which results in total variation (TV) based image restoration. The current arsenal of algorithms for TV-based image restoration is vast. In this present paper, a different approach to the solution of the problem is proposed based upon the method of iterative shrinkage (aka iterated thresholding). In the proposed method, the TV-based image restoration is performed through a recursive application of two simple procedures, viz. linear filtering and soft thresholding. Therefore, the method can be identified as belonging to the group of first-order algorithms which are efficient in dealing with images of relatively large sizes. Another valuable feature of the proposed method consists in its working directly with the TV functional, rather then with its smoothed versions. Moreover, the method provides a single solution for both isotropic and anisotropic definitions of the TV functional, thereby establishing a useful connection between the two formulae. Finally, a number of standard examples of image deblurring are demonstrated, in which the proposed method can provide restoration results of superior quality as compared to the case of sparse-wavelet deconvolution.
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Title: A Brief Survey of Image Processing Algorithms in Electrical Capacitance Tomography

Section 1: Introduction
Description 1: Introduce the Electrical Capacitance Tomography (ECT) technique, its applications, characteristics, and provide an overview of the paper.

Section 2: Fundamentals of Electrical Capacitance Tomography
Description 2: Explain the underlying physical laws and principles of ECT, including Maxwell's equations, the relationship between capacitance and permittivity distribution, and the forward problem formulation.

Section 3: Reconstruction Algorithms
Description 3: Review various reconstruction algorithms used in ECT, categorizing them into direct and iterative algorithms.

Section 4: Linear Back Projection
Description 4: Describe the linear back projection technique, its basis on the sensitivity matrix model, and how it solves the inverse problem.

Section 5: Approach based on SVD
Description 5: Explain the Singular Value Decomposition (SVD) based approach, including least square solutions and the use of pseudoinverse for matrix computation.

Section 6: Tikhonov Regularization
Description 6: Discuss the Tikhonov regularization method to solve ill-posed inverse problems in ECT and its implementation details.

Section 7: Iterative Back Projection Algorithm
Description 7: Detail the iterative back projection algorithm, including its process of updating capacitances and iterating to minimize differences.

Section 8: Landweber Iteration and Steepest Descent Method
Description 8: Present the Landweber iteration and steepest descent methods for optimization in ECT, including the cost function minimization and gradient descent process.

Section 9: Iterative Soft Thresholding of Total Variation
Description 9: Review the iterative soft thresholding approach, combining least squares with total variation for image reconstruction.

Section 10: Other Algorithms
Description 10: Briefly mention other iterative algorithms like iterative Tikhonov methods, algebraic reconstruction technique, neural network-based algorithms, and those using compressive sensing theory.

Section 11: Conclusion and Future Work
Description 11: Summarize the survey, discuss recent advancements in 3D ECT and compressive sensing, and outline potential directions for future research.