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SIAM J. on Imaging Sciences

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2012

Volume 5, Issue 1 (partial)


Image Denoising Using Mean Curvature of Image Surface

Wei Zhu and Tony Chan

SIAM J. Imaging Sci. 5, pp. 1-32 (32 pages)

Online Publication Date: January 17, 2012

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We propose a new variational model for image denoising, which employs the $L^{1}$-norm of the mean curvature of the image surface $(x,f(x))$ of a given image $f:\Omega\rightarrow\mathbb{R}$. Besides eliminating noise and preserving edges of objects efficiently, our model can keep corners of objects and greyscale intensity contrasts of images and also remove the staircase effect. In this paper, we analytically study the proposed model and justify why our model can preserve object corners and image contrasts. We apply the proposed model to the denoising of curves and plane images, and also compare the results with those obtained by using the classical Rudin–Osher–Fatemi model [Phys. D, 60 (1992), pp. 259–268].

Dictionary Learning for Noisy and Incomplete Hyperspectral Images

Zhengming Xing, Mingyuan Zhou, Alexey Castrodad, Guillermo Sapiro, and Lawrence Carin

SIAM J. Imaging Sci. 5, pp. 33-56 (24 pages)

Online Publication Date: January 17, 2012

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We consider analysis of noisy and incomplete hyperspectral imagery, with the objective of removing the noise and inferring the missing data. The noise statistics may be wavelength dependent, and the fraction of data missing (at random) may be substantial, including potentially entire bands, offering the potential to significantly reduce the quantity of data that need be measured. To achieve this objective, the imagery is divided into contiguous three-dimensional (3D) spatio-spectral blocks of spatial dimension much less than the image dimension. It is assumed that each such 3D block may be represented as a linear combination of dictionary elements of the same dimension, plus noise, and the dictionary elements are learned in situ based on the observed data (no a priori training). The number of dictionary elements needed for representation of any particular block is typically small relative to the block dimensions, and all the image blocks are processed jointly (“collaboratively") to infer the underlying dictionary. We address dictionary learning from a Bayesian perspective, considering two distinct means of imposing sparse dictionary usage. These models allow inference of the number of dictionary elements needed as well as the underlying wavelength-dependent noise statistics. It is demonstrated that drawing the dictionary elements from a Gaussian process prior, imposing structure on the wavelength dependence of the dictionary elements, yields significant advantages, relative to the more conventional approach of using an independent and identically distributed Gaussian prior for the dictionary elements; this advantage is particularly evident in the presence of noise. The framework is demonstrated by processing hyperspectral imagery with a significant number of voxels missing uniformly at random, with imagery at specific wavelengths missing entirely, and in the presence of substantial additive noise.

Adaptive Compressed Image Sensing Using Dictionaries

Amir Averbuch, Shai Dekel, and Shay Deutsch

SIAM J. Imaging Sci. 5, pp. 57-89 (33 pages)

Online Publication Date: January 24, 2012

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In recent years, the theory of compressed sensing has emerged as an alternative for the Shannon sampling theorem, suggesting that compressible signals can be reconstructed from far fewer samples than required by the Shannon sampling theorem. In fact the theory advocates that nonadaptive, “random” functionals are in some sense optimal for this task. However, in practice, compressed sensing is very difficult to implement for large data sets, particularly because the recovery algorithms require significant computational resources. In this work, we present a new alternative method for simultaneous image acquisition and compression called adaptive compressed sampling. We exploit wavelet tree structures found in natural images to replace the “universal” acquisition of incoherent measurements with a direct and fast method for adaptive wavelet tree acquisition. The main advantages of this direct approach are that no complex recovery algorithm is in fact needed and that it allows more control over the compressed image quality, in particular, the sharpness of edges. Our experimental results show, by way of software simulations, that our adaptive algorithms perform better than existing nonadaptive methods in terms of image quality and speed.

Fast Algorithms for Image Reconstruction with Application to Partially Parallel MR Imaging

Yunmei Chen, William Hager, Feng Huang, Dzung Phan, Xiaojing Ye, and Wotao Yin

SIAM J. Imaging Sci. 5, pp. 90-118 (29 pages)

Online Publication Date: January 24, 2012

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This paper presents two fast algorithms for total variation–based image reconstruction in a magnetic resonance imaging technique known as partially parallel imaging (PPI), where the inversion matrix is large and ill-conditioned. These algorithms utilize variable splitting techniques to decouple the original problem into more easily solved subproblems. The first method reduces the image reconstruction problem to an unconstrained minimization problem, which is solved by an alternating proximal minimization algorithm. One phase of the algorithm solves a total variation (TV) denoising problem, and the second phase solves an ill-conditioned linear system. Linear and sublinear convergence results are given, and an implementation based on a primal-dual hybrid gradient (PDHG) scheme for the TV problem and on a Barzilai–Borwein scheme for the linear inversion is proposed. The second algorithm exploits the special structure of the PPI reconstruction problem by decomposing it into one subproblem involving Fourier transforms and another subproblem that can be treated by the PDHG scheme. Numerical results and comparisons with recently developed methods indicate the efficiency of the proposed algorithms.

Convergence Analysis of Primal-Dual Algorithms for a Saddle-Point Problem: From Contraction Perspective

Bingsheng He and Xiaoming Yuan

SIAM J. Imaging Sci. 5, pp. 119-149 (31 pages)

Online Publication Date: January 24, 2012

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Recently, some primal-dual algorithms have been proposed for solving a saddle-point problem, with particular applications in the area of total variation image restoration. This paper focuses on the convergence analysis of these primal-dual algorithms and shows that their involved parameters (including step sizes) can be significantly enlarged if some simple correction steps are supplemented. Some new primal-dual–based methods are thus proposed for solving the saddle-point problem. We show that these new methods are of the contraction type: the iterative sequences generated by these new methods are contractive with respect to the solution set of the saddle-point problem. The global convergence of these new methods thus can be obtained within the analytic framework of contraction-type methods. The novel study on these primal-dual algorithms from the perspective of contraction methods substantially simplifies existing convergence analysis. Finally, we show the efficiency of the new methods numerically.

A Variational Approach for Sharpening High Dimensional Images

Michael Möller, Todd Wittman, Andrea L. Bertozzi, and Martin Burger

SIAM J. Imaging Sci. 5, pp. 150-178 (29 pages)

Online Publication Date: January 24, 2012

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Earth-observing satellites usually not only take ordinary red-green-blue images but also provide several images including the near-infrared and infrared spectrum. These images are called multispectral, for about four to seven different bands, or hyperspectral, for higher dimensional images of up to 210 bands. The drawback of the additional spectral information is that each spectral band has rather low spatial resolution. In this paper we propose a new variational method for sharpening high dimensional spectral images with the help of a high resolution gray-scale image while preserving the spectral characteristics used for classification and identification tasks. We describe the application of split Bregman minimization to our energy, prove convergence speed, and compare the split Bregman method to a descent method based on the ideas of alternating directions minimization. Finally, we show results on Quickbird multispectral as well as on AVIRIS hyperspectral data.
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