Image processing based on partial differential equations: proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005 PDF
Preview Image processing based on partial differential equations: proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8-12, 2005
Mathematics and Visualization SeriesEditors GeraldFarin Hans-ChristianHege DavidHoffman ChristopherR.Johnson KonradPolthier MartinRumpf Xue-Cheng Tai Knut-Andreas Lie Tony F. Chan Stanley Osher Editors Image Processing Based on Partial Differential Equations Proceedings of the International Conference on PDE-Based Image Processing and Related Inverse Problems, CMA, Oslo, August 8–1 2, 2005 With 174 Figures, 22 in Color and 18 Tables ABC Xue-Cheng Tai Knut-Andreas Lie Professor of Mathematics SINTEF ICT, Dept. Applied Math. Department of Mathematics PO Box 124 Blindern University of Bergen, N-0314 Oslo, Norway Johannes Brunsgate 12, [email protected] Bergen, N-5008, Norway [email protected] Tony F. Chan Stanley Osher Assistant Director for Math & Physical Department of Mathematics Sciences Directorate Math Science Building The National Science Foundation University of California at Los Angeles 4201 Wilson Boulevard 520 Portola Plaza Arlington, Virginia 22230 Los Angeles, CA 90095, USA USA [email protected] [email protected] LibraryofCongressControlNumber:2006935256 MathematicsSubjectClassification (2000):35-06, 49-06 (49L25, 49M15, 49M30, 49N45), 65-06 (65K10, 65M06, 65M32, 65M55), 76-06 (76S05) ISBN-10 3-540-33266-9SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-33266-4SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsandSPiusingaSpringerLATEXmacropackage Coverdesign:design&productionWMXDesign GmbH,Heidelberg Printedonacid-freepaper SPIN:11693628 46/SPi/3100 543210 Preface Thebookcontainstwenty-twooriginalscientificresearcharticlesthataddress the state-of-the-art in using partial differential equations for image and signal processing. The articles arose from presentations given at the interna- tional conference on PDE-Based Image Processing and Related Inverse Prob- lems, held at the Centre of Mathematics for Applications, University of Oslo, Norway, August 8-12, 2005. The purpose of the conference was to bring together international re- searchers to present various aspects of new developments in using numerical techniques for partial differential equations to analyse and process digital im- ages.Variousaspectsofnewtrendsandtechniquesinthisfieldwerediscussed in the conference, covering the following topics: • Level set methods and applications • Total variation regularization and other nonlinear filters • Noise analysis and removal • Image inpainting • Image dejittering • Optical flow estimation • Image segmentation • Image registration • Analysis and processing of MR images and brain mapping • Image construction techniques • Level set methods for inverse problems Inverse problems for partial differential equations have large areas of appli- cations. Although image analysis and PDE inverse problems seem to be un- related at a first glance, there are many techniques used in one of these two areasthatareusefulfortheother.Onegoaloftheconferencewastohighlight some of the recent efforts in merging some of the techniques for these two research areas. Wehavearrangedthetwenty-tworesearcharticlesofthebookinsixparts VI Preface Part I Digital Image Inpainting, Image Dejittering, and Optical Flow Esti- mation Part II Denoising and Total Variation Methods Part III Image Segmentation Part IV Fast Numerical Methods Part V Image Registration Part VI Inverse Problems The book collects new developments in these topics and points to the newest literature results. As such, it should be a good resource for people working on related problems, as well as for people who are new in the field. The book should also be suitable for readers working with computer vision and visual- ization,imageandsignalprocessing,aswellasmedicalimaging.Moreover,the partial differential equations usedfor different problems discussedherein pro- vide some rich research topics for people working with mathematical analysis and numerical simulation. To ensure the scientific quality of the contributions to this book, each contributedpaperwascarefullyreviewed.Specialthanksgotoallcontributors and referees, without whom making this book would not have been possible. Finally,wewishtothankthosewhosupportedandhelpedtoorganizethe conference.Firstandforemostitisapleasuretoacknowledgethegenerousfi- nancialsupportfromtheCentreofMathematicsforApplications(CMA)and in particular the great help offered by Helge Galdal who has contributed to the practical work in organising the conference. In addition, partial financial support was given by Centre of Integrated Petroleum Research (University of Bergen), Simula Research Laboratory, and the Research Council of Norway (grant number 169281/V30). Moreover, we would like to thank the organis- ing committee: Helge Galdal, Knut–Andreas Lie, Arvid Lundervold, Marius Lysaker, Hans Munthe–Kaas, Xue-Cheng Tai, Ragnar Winther, and Sigurd Aanonsen, for valuable contributions for making the conference a successful one. The participants of the conference deserve special thanks for making the conference a memorable event. Last but not least, the friendly and effective collaboration with Springer-Verlag through Martin Peters and Ute McCrory is kindly appreciated. Bergen/Oslo/Los Angeles, Xue-Cheng Tai August 2006 Knut–Andreas Lie Tony F. Chan Stanley Osher Contents Part I Digital Image Inpainting, Image Dejittering, and Optical Flow Estimation Image Inpainting Using a TV-Stokes Equation Xue-Cheng Tai, Stanley Osher, Randi Holm ......................... 3 Error Analysis for H1 Based Wavelet Interpolations Tony F. Chan, Hao-Min Zhou, Tie Zhou............................ 23 Image Dejittering Based on Slicing Moments Sung Ha Kang, Jianhong (Jackie) Shen............................. 35 CLG Method for Optical Flow Estimation Based on Gradient Constancy Assumption Adam Rabcewicz ................................................. 57 Part II Denoising and Total Variation Methods On Multigrids for Solving a Class of Improved Total Variation Based Staircasing Reduction Models Joseph Savage, Ke Chen .......................................... 69 A Method for Total Variation-based Reconstruction of Noisy and Blurred Images Qianshun Chang, Weicheng Wang, Jing Xu ......................... 95 Minimization of an Edge-Preserving Regularization Functional by Conjugate Gradient Type Methods Jian-Feng Cai, Raymond H. Chan, Benedetta Morini.................109 A Newton-type Total Variation Diminishing Flow Wolfgang Ring...................................................123 VIII Contents Chromaticity Denoising using Solution to the Skorokhod Problem Dariusz Borkowski ...............................................149 Improved 3D Reconstruction of Interphase Chromosomes Based on Nonlinear Diffusion Filtering Jan Huben´y, Pavel Matula, Petr Matula, Michal Kozubek .............163 Part III Image Segmentation Some Recent Developments in Variational Image Segmentation Tony Chan, Mark Moelich, Berta Sandberg..........................175 Application of Non-Convex BV Regularization for Image Segmentation Klaus Frick, Otmar Scherzer ......................................211 Region-Based Variational Problems and Normal Alignment – Geometric Interpretation of Descent PDEs Jan Erik Solem, Niels Chr. Overgaard .............................229 Fast PCLSM with Newton Updating Algorithm Xue-Cheng Tai, Chang-Hui Yao ...................................249 Part IV Fast Numerical Methods Nonlinear Multilevel Schemes for Solving the Total Variation Image Minimization Problem Tony F. Chan, Ke Chen, Xue-Cheng Tai............................265 Fast Implementation of Piecewise Constant Level Set Methods Oddvar Christiansen, Xue-Cheng Tai...............................289 The Multigrid Image Transform Paul M. de Zeeuw................................................309 Minimally Stochastic Schemes for Singular Diffusion Equations Bernhard Burgeth, Joachim Weickert, Sibel Tari .....................325 Contents IX Part V Image Registration Total Variation Based Image Registration Claudia Frohn-Schauf, Stefan Henn, Lars Ho¨mke, Kristian Witsch .....343 Variational Image Registration Allowing for Discontinuities in the Displacement Field Sven Kabus, Astrid Franz, Bernd Fischer ...........................363 Part VI Inverse Problems Shape Reconstruction from Two-Phase Incompressible Flow Data using Level Sets Rossmary Villegas, Oliver Dorn, Miguel Moscoso, Manuel Kindelan ....381 Reservoir Description Using a Binary Level Set Approach with Additional Prior Information About the Reservoir Model Lars Kristian Nielsen, Xue-Cheng Tai, Sigurd Ivar Aanonsen, Magne S. Espedal.......................................................403 Color Figures..................................................427 Part I Digital Image Inpainting, Image Dejittering, and Optical Flow Estimation Image Inpainting Using a TV-Stokes Equation Xue-Cheng Tai1, Stanley Osher2, and Randi Holm1 1 Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5007 Bergen, Norway. E-mail: [email protected], url: http://www.mi.uib.no/˜tai. 2 Department of Mathematics, UCLA, California, USA. E-mail: [email protected] Summary. Based on some geometrical considerations, we propose a two-step method to do digital image inpainting. In the first step, we try to propagate the isophotedirectionsintotheinpaintingdomain.Anenergyminimizationmodelcom- bined with the zero divergence condition is used to get a nonlinear Stokes equa- tion. Once the isophote directions are constructed, an image is restored to fit the constructed directions. Both steps reduce to the solving of some nonlinear partial differential equations. Details about the discretization and implementation are ex- plained. The algorithms have been intensively tested on synthetic and real images. The advantages of the proposed methods are demonstrated by these experiments. 1 Introduction For a digital image, inpainting refers to the process of filling-in missing data. It ranges from removing objects from an image to repairing damaged images and photographs. The term of “digital inpainting” seems to have been introduced into im- age processing by Bertalmio, Sapiro, Caselles and Ballester [2]. In the past fewyears,severaldifferentapproacheshavebeenproposedtotacklethiscom- plicated image processing task. The basic idea for most of the inpainting techniquesistodoasmoothpropagationoftheinformationintheregionsur- rounding the inpainting area and interpolating level curves in a proper way [2, 21, 6]. However, there are different strategies to achieve these goals. In [2], the authors proposed to minimize an energy to compute the restored image and this results in the solving of coupled nonlinear differential equations. In a related work [4], this idea was further extended to guarantee that the level curvesarepropagatedintotheinpaintingdomain.In[3],aconnectionbetween the isophote direction of the image and the Navier-Stokes equation was ob- servedandtheyproposedtosolvetransportequationstofillintheinpainting domain. This is related to our method. Another related work is [11] where a minimization of the divergence is done to construct optical flow functions.
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