DOCSLIB.ORG

Gradient Descent in a Bregman Distance Framework † ‡ § ¶ Martin Benning , Marta M

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Gradient Descent in a Bregman Distance Framework † ‡ § ¶ Martin Benning , Marta M SIAM J. IMAGING SCIENCES © 2021 Society for Industrial and Applied Mathematics Vol. 14, No. 2, pp. 814{843 Choose Your Path Wisely: Gradient Descent in a Bregman Distance Framework∗ y z x { Martin Benning , Marta M. Betcke , Matthias J. Ehrhardt , and Carola-Bibiane Sch¨onlieb Abstract. We propose an extension of a special form of gradient descent|in the literature known as linearized Bregman iteration|to a larger class of nonconvex functions. We replace the classical (squared) two norm metric in the gradient descent setting with a generalized Bregman distance, based on a proper, convex, and lower semicontinuous function. The algorithm's global convergence is proven for functions that satisfy the Kurdyka{Lojasiewicz property. Examples illustrate that features of different scale are being introduced throughout the iteration, transitioning from coarse to fine. This coarse-to-fine approach with respect to scale allows us to recover solutions of nonconvex optimization problems that are superior to those obtained with conventional gradient descent, or even projected and proximal gradient descent. The effectiveness of the linearized Bregman iteration in combination with early stopping is illustrated for the applications of parallel magnetic resonance imaging, blind deconvolution, as well as image classification with neural networks. Key words. nonconvex optimization, nonsmooth optimization, gradient descent, Bregman iteration, linearized Bregman iteration, parallel MRI, blind deconvolution, deep learning AMS subject classifications. 49M37, 65K05, 65K10, 90C26, 90C30 DOI. 10.1137/20M1357500 1. Introduction. Nonconvex optimization methods are indispensable mathematical tools for a large variety of applications [62]. For differentiable objectives, first-order methods such as gradient descent have proven to be useful tools in all kinds of scenarios. Throughout the last decade, however, there has been an increasing interest in first-order methods for non- convex and nonsmooth objectives. These methods range from forward-backward, respectively, proximal-type, schemes [2,3,4, 18, 19], over linearized proximal schemes [80, 16, 81, 61], to inertial methods [63, 68], primal-dual algorithms [78, 52, 57, 12], scaled gradient projection methods [69], and nonsmooth Gauß–Newton extensions [35, 64]. In this paper, we follow a different approach of incorporating nonsmoothness into first- order methods for nonconvex problems. We present a direct generalization of gradient descent, ∗Received by the editors August 4, 2020; accepted for publication (in revised form) February 18, 2021; published electronically June 22, 2021. https://doi.org/10.1137/20M1357500 Funding: The work of the authors was supported by the Leverhulme Trust Early Career Fellowship \Learning from Mistakes: A Supervised Feedback-Loop for Imaging Applications," the Isaac Newton Trust, Engineering and Physical Sciences Research Council (EPSRC) grants EP/K009745/1, EP/M00483X/1, and EP/N014588/1, the Leverhulme Trust project \Breaking the Non-convexity Barrier," the Cantab Capital Institute for the Mathematics of Information, and the CHiPS Horizon 2020 RISE project grant. y School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK (m.benning@ qmul.ac.uk). zDepartment of Computer Science, University College London, London WC1V 6LJ, UK ([email protected]). xInstitute for Mathematical Innovation, University of Bath, Bath BA2 7JU, UK ([email protected]). {Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, Downloaded 08/04/21 to 193.60.238.99. Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms UK ([email protected]). 814 Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. CHOOSE YOUR PATH WISELY 815 first introduced in [10], where the usual squared two-norm metric that penalizes the gap of two subsequent iterates is being replaced by a potentially nonsmooth distance term. This distance term is given in form of a generalized Bregman distance [20, 22, 66], where the underlying function is proper, lower semicontinuous, and convex, but not necessarily smooth. If the underlying function is a Legendre function (see [73 , section 26] and [7]), the proposed gener- alization basically coincides with the recently proposed nonconvex extension of the Bregman proximal gradient method [17]. In the more general case, the proposed method is a gen- eralization of the so-called linearized Bregman iteration [33, 83, 25, 24] to nonconvex data fidelities. Motivated by inverse scale space methods (cf. [21, 22, 66]), the use of nonsmooth Bregman distances for the penalization of the iterates gap allows us to control the scale of features present in the individual iterates. Replacing the squared two-norm, for instance, with a squared two-norm plus the Bregman distance w.r.t. a one-norm leads to very sparse initial iterates, with iterates becoming more dense throughout the course of the iteration. This control of scale, i.e., the slow evolution from iterates with coarse structures to iterates with fine structures, can help to overcome unwanted minima of a nonconvex objective, as we are going to demonstrate with an example in section 2. This is in stark contrast to many of the nonsmooth, nonconvex first-order approaches mentioned above, where the methods are often initialized with random inputs that become more regular throughout the iteration. Our main contributions of this paper are the generalization of the linearized Bregman iteration to nonconvex functions, a detailed convergence analysis of the proposed method, as well as the presentation of numerical results that demonstrate that the use of coarse-to-fine scale space approaches in the context of nonconvex optimization can lead to superior solutions. The outline of the paper is as follows. Based on the nonconvex problem of blind decon- volution, we first give a motivation in section2 of why a coarse-to-fine approach in terms of scale can indeed lead to superior solutions of nonconvex optimization problems. Subse- quently, we define the extension of the linearized Bregman iteration for nonconvex functions in section3. Then, motivated by the informal convergence recipe of Bolte, Sabach, and Teboulle [16, section 3.2] we show a global convergence result in section4, which concludes the theoretical part. We conclude with the modeling of the applications of parallel mag- netic resonance imaging (MRI), blind deconvolution, and image classification in section5, followed by corresponding numerical results in section 6 as well as conclusions and outlook in section7. 2. Motivation. We want to motivate the use of the linearized Bregman iteration for non- convex optimization problems with the example of blind deconvolution. In blind (image) deconvolution the goal is to recover an unknown image u from a blurred and usually noisy image f. Assuming that the degradation is the same for each pixel, the problem of blind deconvolution can be modeled as the minimization of the energy 1 E (u; h) := ku ∗ h − fk2 +χ (h)(2.1) 1 2 2 C | {z } Downloaded 08/04/21 to 193.60.238.99. Redistribution subject SIAM license or copyright; see https://epubs.siam.org/page/terms =:F (u;h ) Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. 816 M. BENNING, M. M. BETCKE, M. J. EHRHARDT, AND C.-B. SCHONLIEB¨ n r with respect to the arguments u 2 R and h 2 R . Here ∗ denotes a discrete convolution operator, and χC is the characteristic function ( 0; h 2 C; χC (h) := 1; h 62 C; defined over the simplex constraint set 8 r 9 < r X = C := h 2 R hj = 1; hj ≥ 0; 8j 2 f1; : : : ; rg : : j=1 ; Even with data f in the range of the nonlinear convolution operator, i.e., f =u ^ ∗ h^ for some n u^ 2 R with h^ 2 C, it is usually still fairly challenging to recoveru ^ and h^ as solutions of n (2.1). A possible reason for this could be that (2.1 ) is an invex function on R × C, where every stationary point is already a global minimum. If we simply try to recoveru ^ and h^ via projected gradient descent, we usually require an initial point in the neighborhood of (^u; h^) in order to converge to that point. We want to illustrate this with a concrete example. Assume we are given an imageu ^ and a convolution kernel h^ as depicted in Figure 2.1, and f =u ^ ∗ h^ is as shown in Figure 2.1(b). Minimizing (2.1) via projected gradient descent leads to the following procedure: k+1 k k k k (2.2a) u = u − τ @u F (u ; h ) ; k+1 k k k k (2.2b) h = projC h − τ @h F (u ; h ) ; 0 T where projC denotes the projection onto the convex set C. If we initialize with u = (0;:::; 0) and h0 = (1;:::; 1)T =r, set τ 0 = 1, update τ k viabacktracking to ensure a monotonic decrease of the energy E1, and iterate (2.2) for 3500 iterations, we obtain the reconstructions visualized in Figure 2.1(c). Even without any noise present in the data f, the algorithm converges to a solution very different fromu ^ and h^. This is not necessarily surprising as we do not impose any regularity on the image. We can try to overcome this issue by modifying (2.1) as follows: E2(u; h) := F (u; h) + χC (h) + αTV(u)(2.3) = E1(u; h) + α TV(u) : Here TV denotes the discretized total variation, i.e., TV(u) := kjr ujk1 ; where r : n ! 2n is a (forward) finite difference discretization of the gradient operator, R R j · j the Euclidean vector norm, and k · k1 the one-norm, and α is a positive scalar. The minimization of (2.3) can easily be carried out by the proximal gradient descent method, also known as forward-backward splitting [54], which is a minor modification of the projected Downloaded 08/04/21 to 193.60.238.99.
Load more

Recommended publications
  • Total Variation Deconvolution Using Split Bregman
    Published in Image Processing On Line on 2012{07{30. Submitted on 2012{00{00, accepted on 2012{00{00. ISSN 2105{1232 c 2012 IPOL & the authors CC{BY{NC{SA This article is available online with supplementary materials, software, datasets and online demo at http://dx.doi.org/10.5201/ipol.2012.g-tvdc 2014/07/01 v0.5 IPOL article class Total Variation Deconvolution using Split Bregman Pascal Getreuer Yale University ([email protected]) Abstract Deblurring is the inverse problem of restoring an image that has been blurred and possibly corrupted with noise. Deconvolution refers to the case where the blur to be removed is linear and shift-invariant so it may be expressed as a convolution of the image with a point spread function. Convolution corresponds in the Fourier domain to multiplication, and deconvolution is essentially Fourier division. The challenge is that since the multipliers are often small for high frequencies, direct division is unstable and plagued by noise present in the input image. Effective deconvolution requires a balance between frequency recovery and noise suppression. Total variation (TV) regularization is a successful technique for achieving this balance in de- blurring problems. It was introduced to image denoising by Rudin, Osher, and Fatemi [4] and then applied to deconvolution by Rudin and Osher [5]. In this article, we discuss TV-regularized deconvolution with Gaussian noise and its efficient solution using the split Bregman algorithm of Goldstein and Osher [16]. We show a straightforward extension for Laplace or Poisson noise and develop empirical estimates for the optimal value of the regularization parameter λ.
    [Show full text]
  • Accelerated Split Bregman Method for Image Compressive Sensing Recovery Under Sparse Representation
    KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 10, NO. 6, Jun. 2016 2748 Copyright ⓒ2016 KSII Accelerated Split Bregman Method for Image Compressive Sensing Recovery under Sparse Representation Bin Gao1, Peng Lan2, Xiaoming Chen1, Li Zhang3 and Fenggang Sun2 1 College of Communications Engineering, PLA University of Science and Technology Nanjing, 210007, China [e-mail: [email protected], [email protected]] 2 College of Information Science and Engineering, Shandong Agricultural University Tai’an, 271018, China [e-mail: {lanpeng, sunfg}@sdau.edu.cn] 3 College of Optoelectronic Engineering, Nanjing University of Posts and Telecommunications Nanjing,, 210023, China [e-mail: [email protected]] *Corresponding author: Bin Gao Received December 14, 2015; revised March 13, 2016; accepted May 5, 2016; published June 30, 2016 Abstract Compared with traditional patch-based sparse representation, recent studies have concluded that group-based sparse representation (GSR) can simultaneously enforce the intrinsic local sparsity and nonlocal self-similarity of images within a unified framework. This article investigates an accelerated split Bregman method (SBM) that is based on GSR which exploits image compressive sensing (CS). The computational efficiency of accelerated SBM for the measurement matrix of a partial Fourier matrix can be further improved by the introduction of a fast Fourier transform (FFT) to derive the enhanced algorithm. In addition, we provide convergence analysis for the proposed method. Experimental results demonstrate that accelerated SBM is potentially faster than some existing image CS reconstruction methods. Keywords: Compressive sensing, sparse representation, split Bregman method, accelerated split Bregman method, image restoration http://dx.doi.org/10.3837/tiis.2016.06.016 ISSN : 1976-7277 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL.
    [Show full text]
  • Split Bregman Method for Large Scale Fused Lasso
    Split Bregman method for large scale fused Lasso Gui-Bo Ye and Xiaohui Xie Department of Computer Science, University of California, Irvine Institute for Genomics and Bioinformatics, University of California, Irvine e-mail: [email protected]; [email protected] Abstract: Ordering of regression or classification coefficients occurs in many real-world applications. Fused Lasso exploits this ordering by explicitly regularizing the differences between neighboring co- efficients through an ℓ1 norm regularizer. However, due to nonseparability and nonsmoothness of the regularization term, solving the fused Lasso problem is computationally demanding. Existing solvers can only deal with problems of small or medium size, or a special case of the fused Lasso problem in which the predictor matrix is identity matrix. In this paper, we propose an iterative algorithm based on split Bregman method to solve a class of large-scale fused Lasso problems, including a generalized fused Lasso and a fused Lasso support vector classifier. We derive our algorithm using augmented Lagrangian method and prove its convergence properties. The performance of our method is tested on both artificial data and real-world applications including proteomic data from mass spectrometry and genomic data from array CGH. We demonstrate that our method is many times faster than the existing solvers, and show that it is especially efficient for large p, small n problems. Keywords and phrases: Fused Lasso, Bregman iteration, ℓ1-norm, Mass spectrometry, Array CGH. 1. Introduction Regularization terms that encourage sparsity in coefficients are increasingly being used in regression and clas- sification procedures. One widely used example is the Lasso procedure for linear regression, which minimizes the usual sum of squared errors, but additionally penalizes the ℓ1 norm of the regression coefficients.
    [Show full text]
  • Fast Linearized Bregman Method for Compressed Sensing
    KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 7, NO. 9, Sep. 2013 2284 Copyright ⓒ 2013 KSII Fast Linearized Bregman Method for Compressed Sensing Zhenzhen Yang1,2 and Zhen Yang2 1 College of Communication & Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing 210003, China [e-mail: [email protected]] 2 Key Lab of “Broadband Wireless Communication and Sensor Network Technology” (Nanjing University of Posts and Telecommunications), Ministry of Education [e-mail: yangz@ njupt.edu.cn ] *Corresponding author: Zhenzhen Yang Received June 24, 2013; revised August 25, 2013; accepted September 21, 2013; published September 30, 2013 Abstract In this paper, a fast and efficient signal reconstruction algorithm for solving the basis pursuit (BP) problem in compressed sensing (CS) is proposed. This fast linearized Bregman method (FLBM), which is inspired by the fast method of Beck et al., is based on the fact that the linearized Bregman method (LBM) is equivalent to a gradient descent method when applied to a certain formulation. The LBM requires O1/ iterations to obtain an -optimal solution while the FLBM reduces this iteration complexity to O1/ and requiring almost the same computational effort on each iteration. Our experimental results show that the FLBM can be faster than some other existing signal reconstruction methods. Keywords: Fast linearized Bregman method, linearized Bregman method, compressed sensing, basis pursuit, signal reconstruction This work is supported by the National Basic Research Program of China (973 Program) (No.2011CB302903), the National Natural Science Foundation of China (No.60971129, 61070234, 61271335, 61271240), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions--Information and Communication Engineering, the Research and Innovation Project for College Graduates of Jiangsu Province (No.CXZZ12_0469), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China(No.13KJB510020 ).
    [Show full text]
  • Curvelet-Wavelet Regularized Split Bregman Iteration for Compressed Sensing
    Curvelet-Wavelet Regularized Split Bregman Iteration for Compressed Sensing Gerlind Plonka 1, Jianwei Ma 2,3 1 Department of Mathematics, University of Duisburg-Essen, Campus Duisburg, 47048 Duisburg, Germany 2 School of Aerospace, Tsinghua University, Beijing 100084, China 3 Centre of Geoscience, Ecole des Mines de Paris, 35 rue Saint-Honore 77305, Fontainebleau, France Abstract Compressed sensing is a new concept in signal processing. Assuming that a sig- nal can be represented or approximated by only a few suitably chosen terms in a frame expansion, compressed sensing allows to recover this signal from much fewer samples than the Shannon-Nyquist theory requires. Many images can be sparsely approximated in expansions of suitable frames as wavelets, curvelets, wave atoms and others. Generally, wavelets represent point-like features while curvelets repre- sent line-like features well. For a suitable recovery of images, we propose models that contain weighted sparsity constraints in two different frames. Given the in- K×N complete measurements f =Φu + with the measurement matrix Φ ∈ R , K<<N, we consider a jointly sparsity-constrained optimization problem of the form 1 2 argmin{ΛcΨcu1 + ΛwΨwu1 + 2 f − Φu2}.HereΨc and Ψw are the transform u matrices corresponding to the two frames, and the diagonal matrices Λc,Λw contain the weights for the frame coefficients. We present efficient iteration methods to solve the optimization problem, based on Alternating Split Bregman algorithms. The con- vergence of the proposed iteration schemes will be proved by showing that they can be understood as special cases of the Douglas-Rachford Split algorithm. Numerical experiments for compressed sensing based Fourier-domain random imaging show good performances of the proposed curvelet-wavelet regularized split Bregman (CWSpB) methods,whereweparticularlyuseacombination of wavelet and curvelet coefficients as sparsity constraints.
    [Show full text]
  • Total Variation Deconvolution Using Split Bregman
    Published in Image Processing On Line on 2012–07–30. Submitted on 2012–00–00, accepted on 2012–00–00. ISSN 2105–1232 c 2012 IPOL & the authors CC–BY–NC–SA This article is available online with supplementary materials, software, datasets and online demo at http://dx.doi.org/10.5201/ipol.2012.g-tvdc 2014/07/01 v0.5 IPOL article class Total Variation Deconvolution using Split Bregman Pascal Getreuer Yale University ([email protected]) Abstract Deblurring is the inverse problem of restoring an image that has been blurred and possibly corrupted with noise. Deconvolution refers to the case where the blur to be removed is linear and shift-invariant so it may be expressed as a convolution of the image with a point spread function. Convolution corresponds in the Fourier domain to multiplication, and deconvolution is essentially Fourier division. The challenge is that since the multipliers are often small for high frequencies, direct division is unstable and plagued by noise present in the input image. Effective deconvolution requires a balance between frequency recovery and noise suppression. Total variation (TV) regularization is a successful technique for achieving this balance in de- blurring problems. It was introduced to image denoising by Rudin, Osher, and Fatemi [4] and then applied to deconvolution by Rudin and Osher [5]. In this article, we discuss TV-regularized deconvolution with Gaussian noise and its efficient solution using the split Bregman algorithm of Goldstein and Osher [16]. We show a straightforward extension for Laplace or Poisson noise and develop empirical estimates for the optimal value of the regularization parameter λ.
    [Show full text]
  • Highly Undersampled Magnetic Resonance Image Reconstruction
    1290 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 32, NO. 7, JULY 2013 Highly Undersampled Magnetic Resonance Image Reconstruction Using Two-Level Bregman Method With Dictionary Updating Qiegen Liu*, Shanshan Wang, Kun Yang, Jianhua Luo, Yuemin Zhu, and Dong Liang Abstract—In recent years Bregman iterative method (or related I. INTRODUCTION augmented Lagrangian method) has shown to be an efficient opti- mization technique for various inverse problems. In this paper, we propose a two-level Bregman Method with dictionary updating for AGNETIC resonance imaging (MRI) is an essential highly undersampled magnetic resonance (MR) image reconstruc- M medical diagnostic tool which provides clinicians tion. The outer-level Bregman iterative procedure enforces the with important anatomical information in the absence of ion- sampled k-space data constraints, while the inner-level Bregman method devotes to updating dictionary and sparse representation izing radiation. However, despite its superiority in obtaining of small overlapping image patches, emphasizing local structure high-resolution images and excellent depiction of soft tissues, adaptively. Modified sparse coding stage and simple dictionary MRI still has its own limitations; specifically, one property updating stage applied in the inner minimization make the whole accompanying MRI is that its scanning time is linearly related algorithm converge in a relatively small number of iterations, to the amount of data it acquires. As reported in [1], increased and enable accurate MR image reconstruction from highly un- dersampled k-space data. Experimental results on both simulated scan duration may introduce some potential issues such as MR images and real MR data consistently demonstrate that the physiological motion artifacts and patient discomfort.
    [Show full text]
  • Split Bregman Method Based Level Set Formulations for Segmentation
    Magnetic Resonance Imaging 57 (2019) 50–67 Contents lists available at ScienceDirect Magnetic Resonance Imaging journal homepage: www.elsevier.com/locate/mri Original contribution Split Bregman method based level set formulations for segmentation and T correction with application to MR images and color images Yunyun Yanga,*, Dongcai Tiana, Wenjing Jiaa, Xiu Shua, Boying Wub a School of Science, Harbin Institute of Technology, Shenzhen, China b School of Science, Harbin Institute of Technology, Harbin, China ARTICLE INFO ABSTRACT Keywords: At present, magnetic resonance (MR) images have gradually become a major aid for clinical medicine, which has Image segmentation greatly improved the doctor's diagnosis rate. Accurate and fast segmentation of MR images plays an extremely Intensity inhomogeneity important role in medical research. However, due to the influence of external factors and the defects of imaging Split Bregman method devices, the MR images have severe intensity inhomogeneity, which poses a great challenge to accurately seg- Bias correction ment MR images. To deal with this problem, this paper presents an improved active contour model by combining Color images the level set evolution model (LSE) and the split Bregman method, and gives the two-phase, the multi-phase and MR images the vector-valued formulations of our model, respectively. The use of the split Bregman method accelerates the minimization process of our model by reducing the computation time and iterative times. A slowly varying bias field is added into the energy functional, which is the key to correct inhomogeneous images. By estimating the bias fields, not only can we get accurate image segmentation results, but also a homogeneous image aftercor- rection is provided.
    [Show full text]
  • An Alternating Direction Method for Total Variation Denoising
    An Alternating Direction Method for Total Variation Denoising Zhiwei (Tony) Qin ∗ Donald Goldfarb y Shiqian Ma z August 26, 2014 Abstract We consider the image denoising problem using total variation (TV) regulariza- tion. This problem can be computationally challenging to solve due to the non- differentiability and non-linearity of the regularization term. We propose an alter- nating direction augmented Lagrangian (ADAL) method, based on a new variable splitting approach that results in subproblems that can be solved efficiently and ex- actly. The global convergence of the new algorithm is established for the anisotropic TV model. For the isotropic TV model, by doing further variable splitting, we are able to derive an ADAL method that is globally convergent. We compare our meth- ods with the split Bregman method [16],which is closely related to it, and demon- strate their competitiveness in computational performance on a set of standard test images. 1 Introduction In signal processing, total variation (TV) regularization is a very popular and effective approach for noise reduction and has a wide array of applications in digital imaging. The total variation is the integral of the absolute gradient of the signal. Using TV regularization to remove noise from signals was originally proposed in [30] and is based on the observation that noisy signals have high total variation. By reducing the to- tal variation of a noisy signal while keeping the resulting signal close to the original one removes noise while preserving important details such as sharp edges. Other ex- isting denoising techniques include median filtering and Tikhonov-like regularization, P 2 2 kukTIK := i(rxu)i + (ryu)i of the desired solution u, where rx and ry are defined arXiv:1108.1587v4 [math.OC] 23 Aug 2014 ∗Department of Industrial Engineering and Operations Research, Columbia University, New York, NY 10027.
    [Show full text]
  • Image Restoration Using Total Variation with Overlapping Group
    1 Image Restoration using Total Variation with Overlapping Group Sparsity Jun Liu, Ting-Zhu Huang, Ivan W. Selesnick, Xiao-Guang Lv, Po-Yu Chen Abstract—Image restoration is one of the most fundamental stable and accurate solution. In other words, we seek to ap- issues in imaging science. Total variation (TV) regularization is proximately recover f by minimizing the following variational widely used in image restoration problems for its capability to problem: preserve edges. In the literature, however, it is also well known 1 2 for producing staircase-like artifacts. Usually, the high-order total min kg − Hfk2 + αϕ(f) , (2) f 2 variation (HTV) regularizer is an good option except its over- smoothing property. In this work, we study a minimization prob- where k·k2 denotes the Euclidean norm, ϕ is conventionally lem where the objective includes an usual l2 data-fidelity term and an overlapping group sparsity total variation regularizer called a regularization functional, and α> 0 is referred to as which can avoid staircase effect and allow edges preserving in the a regularization parameter which controls the balance between restored image. We also proposed a fast algorithm for solving the fidelity and regularization terms in (2). corresponding minimization problem and compare our method How to choose a good functional ϕ is an active area of with the state-of-the-art TV based methods and HTV based method. The numerical experiments illustrate the efficiency and research in imaging science. In the early 1960s, D. L. Phillips effectiveness of the proposed method in terms of PSNR, relative [3] and A.
    [Show full text]
  • Split Bregman Method for Minimization of Region-Scalable Fitting Energy for Image Segmentation
    Split Bregman Method for Minimization of Region-Scalable Fitting Energy for Image Segmentation Yunyun Yanga,b, Chunming Lic, Chiu-Yen Kaoa,d⋆, and Stanley Osher e a Department of Mathematics, The Ohio State University, OH 43202, U.S. b Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China c Department of Electrical Engineering, University of Pennsylvania, PA 19104, U.S. d Mathematical Biosciences Institute, The Ohio State University, OH 43210, U.S. e Department of Mathematics, University of California, Los Angeles, CA 90095, U.S. Abstract. In this paper, we incorporated the global convex segmenta- tion method and the Split Bregman technique into the region-scalable fitting energy model. The new proposed method based on the region- scalable model can draw upon intensity information in local regions at a controllable scale, so that it can segment images with intensity inho- mogeneity. Furthermore, with the application of the global convex seg- mentation method and the Split Bregman technique, the method is very robust and efficient. By using a non-negative edge detector function to the proposed method, the algorithm can detect the boundaries more eas- ily and achieve results that are very similar to those obtained through the classical geodesic active contour model. Experimental results for syn- thetic and real images have shown the robustness and efficiency of our method and also demonstrated the desirable advantages of the proposed method. Key words: Split Bregman, region-scalable model, image segmentation, intensity inhomogeneity 1 Introduction Image segmentation [1–4] is a fundamental and important task in image analysis and computer vision. Most of existing methods for image segmentation can be categorized into two classes: region-based methods [1,5–9] and edge-based meth- ods [3,4,10–13].
    [Show full text]
  • A Study of Total-Variation Based Noise-Reduction Algorithms for Low-Dose Cone-Beam Computed Tomography
    Sovanlal Mukherjee, Jonathan B. Farr & Weiguang Yao A Study of Total-Variation Based Noise-Reduction Algorithms For Low-Dose Cone-Beam Computed Tomography Sovanlal Mukherjee [email protected] Department of Radiation Oncology, St. Jude Children’s Research Hospital, 262 Danny Thomas Place, Memphis, TN 38105, USA Jonathan B. Farr [email protected] Department of Radiation Oncology, St. Jude Children’s Research Hospital, 262 Danny Thomas Place, Memphis, TN 38105, USA Weiguang Yao [email protected] Department of Radiation Oncology, St. Jude Children’s Research Hospital, 262 Danny Thomas Place, Memphis, TN 38105, USA Abstract In low-dose cone-beam computed tomography, the reconstructed image is contaminated with excessive quantum noise. In this work, we examined the performance of two popular noise- reduction algorithms—total-variation based on the split Bregman (TVSB) and total-variation based on Nesterov’s method (TVN)—on noisy imaging data from a computer-simulated Shepp–Logan phantom, a physical CATPHAN phantom and head-and-neck patient. Up to 15% Gaussian noise was added to the Shepp–Logan phantom. The CATPHAN phantom was scanned by a Varian OBI system with scanning parameters 100 kVp, 4 ms, and 20 mA. Images from the head-and-neck patient were generated by the same scanner, but with a 20-ms pulse time. The 4-ms low-dose image of the head-and-neck patient was simulated by adding Poisson noise to the 20-ms image. The performance of these two algorithms was quantitatively compared by computing the peak signal-to-noise ratio (PSNR), contrast-to-noise ratio (CNR) and the total computational time.
    [Show full text]