Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

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Anisotropic diffusion in image processing

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.

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An Iterative Regularization Method for Total Variation-Based Image Restoration

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 ( $H^{*}H$ , where $H^{*}$ is the adjoint of the forward imaging operator, $H$ ) 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 $512\times 512$ image on the GPU.

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Deep Convolutional Neural Network for Inverse Problems in Imaging

Millimeter wave (mmWave) communications has been regarded as a key enabling technology for 5G networks, as it offers orders of magnitude greater spectrum than current cellular bands. In contrast to conventional multiple-input–multiple-output (MIMO) systems, precoding in mmWave MIMO cannot be performed entirely at baseband using digital precoders, as only a limited number of signal mixers and analog-to-digital converters can be supported considering their cost and power consumption. As a cost-effective alternative, a hybrid precoding transceiver architecture, combining a digital precoder and an analog precoder, has recently received considerable attention. However, the optimal design of such hybrid precoders has not been fully understood. In this paper, treating the hybrid precoder design as a matrix factorization problem, effective alternating minimization (AltMin) algorithms will be proposed for two different hybrid precoding structures, i.e., the fully-connected and partially-connected structures. In particular, for the fully-connected structure, an AltMin algorithm based on manifold optimization is proposed to approach the performance of the fully digital precoder, which, however, has a high complexity. Thus, a low-complexity AltMin algorithm is then proposed, by enforcing an orthogonal constraint on the digital precoder. Furthermore, for the partially-connected structure, an AltMin algorithm is also developed with the help of semidefinite relaxation. For practical implementation, the proposed AltMin algorithms are further extended to the broadband setting with orthogonal frequency division multiplexing modulation. Simulation results will demonstrate significant performance gains of the proposed AltMin algorithms over existing hybrid precoding algorithms. Moreover, based on the proposed algorithms, simulation comparisons between the two hybrid precoding structures will provide valuable design insights.

Alternating Minimization Algorithms for Hybrid Precoding in Millimeter Wave MIMO Systems

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.

Total variation blind deconvolution

Convergence of the alternating minimization algorithm for blind deconvolution

We consider applying the preconditioned conjugate gradient (PCG) method to solving linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators with Dirichlet boundary conditions Let (L + Σ)Σ−1(Lt + Σ) denote the block Cholesky factorization of A with lower block triangular matrix L and diagonal block matrix Σ We propose a preconditioner M = (Lˆ + Σˆ)Σˆ−1(Lˆt + Σˆ) with block diagonal matrix Σˆ and lower block triangular matrix Lˆ The diagonal blocks of Σˆ and the subdiagonal blocks of Lˆ are respectively the optimal sine transform approximations to the diagonal blocks of Σ and the subdiagonal blocks of L We show that for two-dimensional domains, the construction cost of M and the cost for each iteration of the PCG algorithm are of order O(n2 log n) Furthermore, for rectangular regions, we show that the condition number of the preconditioned system M−1A is of order O(1) In contrast, the system preconditioned by the MILU and MINV methods are of order O(n) We will also show that M can be obtained from A by taking the optimal sine transform approximations of each sub-block of A Thus, the construction of M is similar to that of Level-1 circulant preconditioners Our numerical results on two-dimensional square and L-shaped domains show that our method converges faster than the MILU and MINV methods Extension to higher-dimensional domains will also be discussed © 1997 John Wiley & Sons, Ltd

Sine transform based preconditioners for elliptic problems

The total variational (TV) regularization method was first proposed for gray scale images and was extended for vector valued images. In this work, we apply the TV regularization method to solve the multichannel image deconvolution problem. The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images. In this paper, a fast iterative method is developed to solve the deconvolution problem. Our method involves solving linear systems and the conjugate gradient method is applied in which Fourier transform type preconditioners are used to speed up the convergence rate. Numerical experiments demonstrate the effectiveness of the TV regularization method. In this paper, we present some preliminary results on multichannel blind deconvolution with TV regularization.

Chiu-Kwong Wong

Papers

Total variation blind deconvolution

Convergence of the alternating minimization algorithm for blind deconvolution

Sine transform based preconditioners for elliptic problems

Multichannel image deconvolution by total variation regularization