A survey on level set methods for inverse problems and optimal design

The aim of this paper is to provide a survey on the recent development in level set methods in inverse problems and optimal design. We give introductions on the general features of such problems involving geometries and on the general framework of the level set method. In subsequent parts we discuss shape sensitivity analysis and its relation to level set methods, various approaches on constructing optimization algorithms based on the level set approach, and special tools needed for the application of level set based optimization methods to ill-posed problems. Furthermore, we provide a review on numerical methods important in this context, and give an overview of applications treated with level set methods. Finally, we provide a discussion of the most challenging and interesting open problems in this field, that might be of interest for scientists who plan to start future research in this field.

[1]  Xiao Han,et al.  A Topology Preserving Level Set Method for Geometric Deformable Models , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[2]  Jean-Michel Morel,et al.  Level lines based disocclusion , 1998, Proceedings 1998 International Conference on Image Processing. ICIP98 (Cat. No.98CB36269).

[3]  Tony F. Chan,et al.  A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model , 2002, International Journal of Computer Vision.

[4]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[5]  M. Burger.  Levenberg–Marquardt level set methods for inverse obstacle problems , 2004 .

[6]  Hongkai Zhao,et al.  Imaging of location and geometry for extended targets using the response matrix , 2004 .

[7]  B. Engquist,et al.  Numerical approximations of singular source terms in differential equations , 2004 .

[8]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[9]  Otmar Scherzer,et al.  Regularization for Curve Representations: Uniform Convergence for Discontinuous Solutions of Ill-Posed Problems , 1998, SIAM J. Appl. Math..

[10]  M. Burger,et al.  Level set methods for geometric inverse problems in linear elasticity , 2004 .

[11]  U. Tautenhahn.  On the asymptotical regularization of nonlinear ill-posed problems , 1994 .

[12]  Björn Engquist,et al.  Regularization Techniques for Numerical Approximation of PDEs with Singularities , 2003, J. Sci. Comput..

[13]  Danping Peng,et al.  Weighted ENO Schemes for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[14]  O. Faugeras,et al.  The Inverse EEG and MEG Problems : The Adjoint State Approach I: The Continuous Case , 1999 .

[15]  Kazufumi Ito,et al.  Identification of contact regions in semiconductor transistors by level-set methods , 2003 .

[16]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[17]  Yoshikazu Giga,et al.  A LEVEL SET APPROACH TO SEMICONTINUOUS VISCOSITY SOLUTIONS FOR CAUCHY PROBLEMS , 1999 .

[18]  J. Sokolowski,et al.  The structure theorem for the Eulerian derivative of shape functionals defined in domains with cracks , 2000 .

[19]  E. Haber,et al.  Preconditioned all-at-once methods for large, sparse parameter estimation problems , 2001 .

[20]  F. Santosa,et al.  ENHANCED ELECTRICAL IMPEDANCE TOMOGRAPHY VIA THE MUMFORD{SHAH FUNCTIONAL , 2001 .

[21]  Michael Yu Wang,et al.  A level-set based variational method for design and optimization of heterogeneous objects , 2005, Comput. Aided Des..

[22]  Gerhard Dziuk,et al.  A fully discrete numerical scheme for weighted mean curvature flow , 2002, Numerische Mathematik.

[23]  Ted Belytschko,et al.  Modelling crack growth by level sets in the extended finite element method , 2001 .

[24]  Jianhong Shen,et al.  EULER'S ELASTICA AND CURVATURE BASED INPAINTINGS , 2002 .

[25]  J. Sethian,et al.  FRONTS PROPAGATING WITH CURVATURE DEPENDENT SPEED: ALGORITHMS BASED ON HAMILTON-JACOB1 FORMULATIONS , 2003 .

[26]  Chi-Wang Shu,et al.  A Discontinuous Galerkin Finite Element Method for Hamilton-Jacobi Equations , 1999, SIAM J. Sci. Comput..

[27]  J. Morel,et al.  Variational Methods in Image Segmentation: with seven image processing experiments , 1994 .

[28]  Martin Rumpf,et al.  Image Registration by a Regularized Gradient Flow. A Streaming Implementation in DX9 Graphics Hardware , 2004, Computing.

[29]  F. Santosa.  A Level-set Approach Inverse Problems Involving Obstacles , 1995 .

[30]  Steven J. Cox,et al.  Maximizing Band Gaps in Two-Dimensional Photonic Crystals , 1999, SIAM J. Appl. Math..

[31]  Zhilin Li,et al.  Convergence analysis of the immersed interface method , 1999 .

[32]  M. Burger.  A level set method for inverse problems , 2001 .

[33]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[34]  Morton E. Gurtin,et al.  The nature of configurational forces , 1995 .

[35]  Dominique Lesselier,et al.  CORRIGENDUM: Shape inversion from TM and TE real data by controlled evolution of level sets , 2001 .

[36]  Arian Novruzi,et al.  Structure of shape derivatives , 2002 .

[37]  J. Strain.  Fast Tree-Based Redistancing for Level Set Computations , 1999 .

[38]  Ted Belytschko,et al.  AN EXTENDED FINITE ELEMENT METHOD (X-FEM) FOR TWO- AND THREE-DIMENSIONAL CRACK MODELING , 2000 .

[39]  W. Eric L. Grimson,et al.  A shape-based approach to the segmentation of medical imagery using level sets , 2003, IEEE Transactions on Medical Imaging.

[40]  E. Miller,et al.  A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets , 2000 .

[41]  G. Barles,et al.  Front propagation and phase field theory , 1993 .

[42]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .

[43]  Zhilin Li.  A Fast Iterative Algorithm for Elliptic Interface Problems , 1998 .

[44]  K. Kunisch,et al.  Level-set function approach to an inverse interface problem , 2001 .

[45]  A. Kirsch,et al.  A simple method for solving inverse scattering problems in the resonance region , 1996 .

[46]  G. Allaire,et al.  A level-set method for shape optimization , 2002 .

[47]  O. Pironneau,et al.  Applied Shape Optimization for Fluids , 2001 .

[48]  Hend Ben Ameur,et al.  Identification of 2D cracks by elastic boundary measurements , 1999 .

[49]  O. Scherzer,et al.  LETTER TO THE EDITOR: On the relation between constraint regularization, level sets, and shape optimization , 2006 .

[50]  Xiaoming Wang,et al.  Structural shape and topology optimization in a level-set-based framework of region representation , 2004 .

[51]  E. Haber,et al.  On optimization techniques for solving nonlinear inverse problems , 2000 .

[52]  Yoshikazu Giga,et al.  A level set approach for computing discontinuous solutions of Hamilton-Jacobi equations , 2003, Math. Comput..

[53]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[54]  S. Osher,et al.  Motion of multiple junctions: a level set approach , 1994 .

[55]  Jan Sokolowski,et al.  On the Topological Derivative in Shape Optimization , 1999 .

[56]  Michael Hintermüller,et al.  An Inexact Newton-CG-Type Active Contour Approach for the Minimization of the Mumford-Shah Functional , 2004, Journal of Mathematical Imaging and Vision.

[57]  B. Engquist,et al.  Discretization of Dirac delta functions in level set methods , 2005 .

[58]  R. LeVeque,et al.  A comparison of the extended finite element method with the immersed interface method for elliptic equations with discontinuous coefficients and singular sources , 2006 .

[59]  Mark Sussman,et al.  An Efficient, Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow , 1999, SIAM J. Sci. Comput..

[60]  Morton E. Gurtin,et al.  Interface Evolution in Three Dimensions¶with Curvature-Dependent Energy¶and Surface Diffusion:¶Interface-Controlled Evolution, Phase Transitions, Epitaxial Growth of Elastic Films , 2002 .

[61]  P. Lions,et al.  Image recovery via total variation minimization and related problems , 1997 .

[62]  Irena Lasiecka,et al.  Analysis and Optimization of Differential Systems , 2002, IFIP — The International Federation for Information Processing.

[63]  Dominique Lesselier,et al.  Shape reconstruction of buried obstacles by controlled evolution of a level set: from a min-max formulation to numerical experimentation , 2001 .

[64]  Stefan Ritter,et al.  A linear sampling method for inverse scattering from an open arc Inverse Problems , 2000 .

[65]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[66]  M. Droske,et al.  A level set formulation for Willmore flow , 2004 .

[67]  Luca Rondi,et al.  Optimal Stability for the Inverse Problemof Multiple Cavities , 2001 .

[68]  M. Delfour,et al.  Shapes and Geometries: Analysis, Differential Calculus, and Optimization , 1987 .

[69]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[70]  P. Smereka,et al.  A Remark on Computing Distance Functions , 2000 .

[71]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[72]  S. Osher,et al.  Level Set Methods for Optimization Problems Involving Geometry and Constraints I. Frequencies of a T , 2001 .

[73]  N. Paragios.  A level set approach for shape-driven segmentation and tracking of the left ventricle , 2003, IEEE Transactions on Medical Imaging.

[74]  A. Chambolle,et al.  Design-dependent loads in topology optimization , 2003 .

[75]  Bert Jüttler,et al.  C1 Spline Implicitization of Planar Curves , 2002, Automated Deduction in Geometry.

[76]  P. Lions.  Generalized Solutions of Hamilton-Jacobi Equations , 1982 .

[77]  Stefan Kindermann,et al.  REGULARIZATION FOR SURFACE REPRESENTATIONS OF DISCONTINUOUS SOLUTIONS OF LINEAR ILL-POSED PROBLEMS , 2001 .

[78]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[79]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[80]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[81]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[82]  M. Burger,et al.  Incorporating topological derivatives into level set methods , 2004 .

[83]  Stanley Osher,et al.  Level set methods in image science , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[84]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .

[85]  Jan Sokolowski,et al.  Introduction to shape optimization , 1992 .

[86]  J. Sethian,et al.  Structural Boundary Design via Level Set and Immersed Interface Methods , 2000 .

[87]  Xiaoming Wang,et al.  Color level sets: a multi-phase method for structural topology optimization with multiple materials , 2004 .

[88]  Guillermo Sapiro,et al.  Variational Problems and Partial Differential Equations on Implicit Surfaces: Bye Bye Triangulated Surfaces? , 2003 .

[89]  F. Santosa,et al.  Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set , 1998 .

[90]  M. Burger.  A framework for the construction of level set methods for shape optimization and reconstruction , 2003 .

[91]  William Rundell,et al.  The determination of a discontinuity in a conductivity from a single boundary measurement , 1998 .

[92]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[93]  Michael Yu Wang,et al.  A Variational Method for Structural Topology Optimization , 2004 .

[94]  Eli Yablonovitch,et al.  Inverse Problem Techniques for the Design of Photonic Crystals (INVITED) , 2004 .

[95]  L. Ambrosio,et al.  A direct variational approach to a problem arising in image reconstruction , 2003 .

[96]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[97]  T. Chan,et al.  Level set and total variation regularization for elliptic inverse problems with discontinuous coefficients , 2004 .

[98]  S. Osher,et al.  Geometric Level Set Methods in Imaging, Vision, and Graphics , 2011, Springer New York.

[99]  Li-Tien Cheng,et al.  Variational Problems and Partial Differential Equations on Implicit Surfaces: The Framework and Exam , 2000 .

[100]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[101]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

[102]  Dominique Lesselier,et al.  On novel developments of controlled evolution of level sets in the field of inverse shape problems , 2002 .

[103]  E. Haber.  A multilevel, level-set method for optimizing eigenvalues in shape design problems , 2004 .

[104]  Tony F. Chan,et al.  Euler's Elastica and Curvature-Based Inpainting , 2003, SIAM J. Appl. Math..

[105]  Stefan Kindermann,et al.  Estimation of discontinuous parameters of elliptic partial differential equations by regularization for surface representations , 2001 .

[106]  Michael Hintermüller,et al.  A Second Order Shape Optimization Approach for Image Segmentation , 2004, SIAM J. Appl. Math..

[107]  Jan Sokołowski,et al.  Topological derivatives for elliptic problems , 1999 .

[108]  Michael Hintermüller,et al.  A level set approach for the solution of a state-constrained optimal control problem , 2004, Numerische Mathematik.

[109]  S. Osher,et al.  High-order essentially nonsocillatory schemes for Hamilton-Jacobi equations , 1990 .

[110]  F. Murat,et al.  Sur le controle par un domaine géométrique , 1976 .

[111]  T. O’Neil.  Geometric Measure Theory , 2002 .

[112]  Andrea Braides.  Gamma-Convergence for Beginners , 2002 .

[113]  James A. Sethian,et al.  Level Set Methods for Curvature Flow, Image Enchancement, and Shape Recovery in Medical Images , 1997, VisMath.

[114]  Y. Chen,et al.  Image registration via level-set motion: Applications to atlas-based segmentation , 2003, Medical Image Anal..

[115]  Eric T. Chung,et al.  Electrical impedance tomography using level set representation and total variational regularization , 2005 .

[116]  Martin Burger,et al.  Numerical Approximation of an SQP-Type Method for Parameter Identification , 2002, SIAM J. Numer. Anal..

[117]  Stefano Finzi Vita,et al.  Area-preserving curve-shortening flows: from phase separation to image processing , 2002 .

[118]  Antonino Morassi,et al.  Detecting an Inclusion in an Elastic Body by Boundary Measurements , 2002, SIAM Rev..

[119]  Steven J. Ruuth,et al.  A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods , 2002, SIAM J. Numer. Anal..