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Convex Generalizations of Total Variation based on the Structure Tensor with Applications to Inverse Problems
, 2013
"... We introduce a generic convex energy functional that is suitable for both grayscale and vector-valued images. Our functional is based on the eigenvalues of the structure tensor, therefore it penalizes image variation at every point by taking into account the information from its neighborhood. It g ..."
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Cited by 7 (3 self)
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We introduce a generic convex energy functional that is suitable for both grayscale and vector-valued images. Our functional is based on the eigenvalues of the structure tensor, therefore it penalizes image variation at every point by taking into account the information from its neighborhood. It generalizes several existing variational penalties, such as the Total Variation and vectorial extensions of it. By introducing the concept of patch-based Jacobian operator, we derive an equivalent formulation of the proposed regularizer that is based on the Schatten norm of this operator. Using this new formulation, we prove convexity and develop a dual definition for the proposed energy, which gives rise to an efficient and parallelizable minimization algorithm. Moreover, we establish a connection between the minimization of the proposed convex regularizer and a generic type of nonlinear anisotropic diffusion that is driven by a spatially regularized and adaptive diffusion tensor. Finally, we perform extensive experiments with image denoising and deblurring for grayscale and color images. The results show the effectiveness of the proposed approach as well as its improved performance compared to Total Variation and existing vectorial extensions of it.
Poisson Image Reconstruction with Hessian Schatten-norm Regularization
, 2013
"... Poisson inverse problems arise in many modern imaging applications, including biomedical and astronomical ones. The main challenge is to obtain an estimate of the underlying image from a set of measurements degraded by a linear operator and further corrupted by Poisson noise. In this paper, we prop ..."
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Cited by 2 (1 self)
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Poisson inverse problems arise in many modern imaging applications, including biomedical and astronomical ones. The main challenge is to obtain an estimate of the underlying image from a set of measurements degraded by a linear operator and further corrupted by Poisson noise. In this paper, we propose an efficient framework for Poisson image reconstruction, under a regularization approach, which depends on matrix-valued regularization operators. In particular, the employed regularizers involve the Hessian as the regularization operator and Schatten matrix norms as the potential functions. For the solution of the problem, we propose two optimization algorithms that are specifically tailored to the Poisson nature of the noise. These algorithms are based on an augmented-Lagrangian formulation of the problem and correspond to two variants of the alternating direction method of multipliers. Further, we derive a link that relates the proximal map of an ℓ p norm with the proximal map of a Schatten matrix norm of order p. This link plays a key role in the development of one of the proposed algorithms. Finally, we provide experimental results on natural and biological images for the task of Poisson image deblurring and demonstrate the practical relevance and effectiveness of the proposed framework.
Structure Tensor Total Variation
- SIAM J. IMAGING SCIENCES
, 2015
"... We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as Structure tensor Total Variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale ..."
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Cited by 1 (1 self)
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We introduce a novel generic energy functional that we employ to solve inverse imaging problems within a variational framework. The proposed regularization family, termed as Structure tensor Total Variation (STV), penalizes the eigenvalues of the structure tensor and is suitable for both grayscale and vector-valued images. It generalizes several existing variational penalties, including the Total Variation (TV) semi-norm and vectorial extensions of it. Meanwhile, thanks to the structure tensor’s ability of capturing first-order information around a local neighborhood, the STV functionals can provide more robust measures of image variation. Further, we prove that the STV regularizers are convex while they also satisfy several invariance properties w.r.t image transformations. These properties qualify them as ideal candidates for imaging applications. In addition, for the discrete version of the STV functionals we derive an equivalent definition that is based on the patch-based Jacobian operator, a novel linear operator which extends the Jacobian matrix. This alternative definition allow us to derive a dual problem formulation. The duality of the problem paves the way for employing robust tools from convex optimization and enables us to design an efficient and parallelizable optimization algorithm. Finally, we present extensive experiments on various inverse imaging problems, where we compare our regularizers with other competing regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually.
Recovery of Discontinuous Signals Using Group Sparse Higher Degree Total Variation
"... Abstract-We introduce a family of novel regularization penalties to enable the recovery of discrete discontinuous piecewise polynomial signals from undersampled or degraded linear measurements. The penalties promote the group sparsity of the signal analyzed under a nth order derivative. We introduc ..."
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Abstract-We introduce a family of novel regularization penalties to enable the recovery of discrete discontinuous piecewise polynomial signals from undersampled or degraded linear measurements. The penalties promote the group sparsity of the signal analyzed under a nth order derivative. We introduce an efficient alternating minimization algorithm to solve linear inverse problems regularized with the proposed penalties. Our experiments show that promoting group sparsity of derivatives enhances the compressed sensing recovery of discontinuous piecewise linear signals compared with an unstructured sparse prior. We also propose an extension to 2-D, which can be viewed as a group sparse version of higher degree total variation, and illustrate its effectiveness in denoising experiments.
Magnetic Resonance Spectroscopic Imaging at Superresolution: Overview
"... dx.doi.org/10.1016/j.jmr.2015.11.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof befor ..."
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dx.doi.org/10.1016/j.jmr.2015.11.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Non-local Structure Tensor Functionals for Image Regularization
, 2015
"... We present a non-local regularization framework that we apply to inverse imaging problems. As opposed to existing non-local regularization methods that rely on the graph gradient as the regularization operator, we introduce a family of non-local energy functionals that involves the standard image gr ..."
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We present a non-local regularization framework that we apply to inverse imaging problems. As opposed to existing non-local regularization methods that rely on the graph gradient as the regularization operator, we introduce a family of non-local energy functionals that involves the standard image gradient. Our motivation for designing these functionals is to exploit at the same time two important properties inherent in natural images, namely the local structural image regularity and the non-local image self-similarity. To this end, our regularizers employ as their regularization operator a novel non-local version of the structure tensor. This operator performs a non-local weighted average of the image gradients computed at every image location and, thus, is able to provide a robust measure of image variation. Further, we show a connection of the proposed regularizers to the Total Variation semi-norm and prove convexity. The convexity property allows us to employ powerful tools from convex opti-mization in order to design an efficient minimization algorithm. Our algorithm is based on a splitting variable strategy which leads to an augmented Lagrangian formulation. To solve the corresponding optimization problem we employ the alternating-direction methods of multipliers. Finally, we present extensive experiments on several inverse imaging problems, where we compare our regularizers with other competing local and non-local regularization approaches. Our results are shown to be systematically superior, both quantitatively and visually.
Improved Variational Denoising of Flow Fields with Application to Phase-Contrast MRI Data
"... Abstract—We propose a new variational framework for the problem of reconstructing flow fields from noisy measurements. The formalism is based on regularizers penalizing the singular values of the Jacobian of the field. Specifically, we rely on the nuclear norm. Our method is invariant with respect t ..."
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Abstract—We propose a new variational framework for the problem of reconstructing flow fields from noisy measurements. The formalism is based on regularizers penalizing the singular values of the Jacobian of the field. Specifically, we rely on the nuclear norm. Our method is invariant with respect to funda-mental transformations and can be efficiently solved. We conduct numerical experiments on several phantom data and report improved performance compared to existing vectorial extensions of total variation and curl-divergence regularizations. Finally, we apply our reconstruction method to an experimentally-acquired phase-contrast MRI recording for enhancing the data visualization. Index Terms—Denoising, flow fields, vector fields, Jacobian, regularization, Schatten norms, phase-constrast MRI, PCMRI, flow MRI, 4D MRI, vectorial total variation.
Isotropic inverse-problem approach for two-dimensional phase unwrapping
"... We propose a new technique for two-dimensional phase unwrapping. The unwrapped phase is found as the solution of an inverse problem that consists in the minimization of an energy functional. The latter includes a weighted data fidelity term that favors sparsity in the error between the true and wrap ..."
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We propose a new technique for two-dimensional phase unwrapping. The unwrapped phase is found as the solution of an inverse problem that consists in the minimization of an energy functional. The latter includes a weighted data fidelity term that favors sparsity in the error between the true and wrapped phase differences, as well as a regularizer based on higher-order total variation. One desirable feature of our method is its rotation invariance, which allows it to unwrap a much larger class of images compared to the state of the art. We dem-onstrate the effectiveness of our method through several experiments on simulated and real data obtained through the tomographic phase microscope. The proposed method can enhance the applicability and outreach of tech-niques that rely on quantitative phase evaluation. © 2015 Optical Society of America
3D POISSON MICROSCOPY DECONVOLUTION WITH HESSIAN SCHATTEN-NORM REGULARIZATION
"... Inverse problems with shot noise arise in many modern biomedical imaging applications. The main challenge is to obtain an estimate of the underlying specimen from measurements corrupted by Poisson noise. In this work, we propose an efficient framework for photon-limited image reconstruction, under a ..."
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Inverse problems with shot noise arise in many modern biomedical imaging applications. The main challenge is to obtain an estimate of the underlying specimen from measurements corrupted by Poisson noise. In this work, we propose an efficient framework for photon-limited image reconstruction, under a regularization approach that relies on matrix-valued operators. Our regularizers involve the Hes-sian operator and its eigenvalues. They are second-order regularizers that are well suited to biomedical images. For the solution of the aris-ing minimization problem, we propose an optimization algorithm based on an augmented-Lagrangian formulation and specifically tai-lored to the Poisson nature of the noise. To assess the quality of the reconstruction, we provide experimental results on 3D image stacks of biological images for microscopy deconvolution.
HIGH-PERFORMANCE 3D DECONVOLUTION OF FLUORESCENCE MICROGRAPHS
"... In this work, we describe our approach of combining the most effective ideas and tools developed during the past years to build a variational 3D deconvolution system that can be successfully em-ployed in fluorescence microscopy. In particular, the main compo-nents of our deconvolution system involve ..."
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In this work, we describe our approach of combining the most effective ideas and tools developed during the past years to build a variational 3D deconvolution system that can be successfully em-ployed in fluorescence microscopy. In particular, the main compo-nents of our deconvolution system involve proper handling of image boundaries, choice of a regularizer that is best suited to biological images, and use of an optimization algorithm that can be efficiently implemented on graphics processing units (GPUs) and fully bene-fit from their massive parallel computational capabilities. We show that our system leads to very competitive results and reduces the computational time by at least one order of magnitude compared to a CPU implementation. This makes the use of advanced deconvolu-tion techniques feasible in practice and attractive computationally. Index Terms — Graphics Processing Unit, image regularization, variational reconstruction, convex optimization.
