Multiscale Segmentation via Bregman Distances and Nonlinear Spectral Analysis

In biomedical imaging reliable segmentation of objects (e.g. from small cells up to large organs) is of fundamental importance for automated medical diagnosis. New approaches for multi-scale segmentation can considerably improve performance in case of natural variations in intensity, size and shape. This paper aims at segmenting objects of interest based on shape contours and automatically finding multiple objects with different scales. The overall strategy of this work is to combine nonlinear segmentation with scales spaces and spectral decompositions recently introduced in literature. For this we generalize a variational segmentation model based on total variation using Bregman distances to construct an inverse scale space. This offers the new model to be accomplished by a scale analysis approach based on a spectral decomposition of the total variation. As a result we obtain a very efficient, (nearly) parameter-free multiscale segmentation method that comes with an adaptive regularization parameter choice. The added benefit of our method is demonstrated by systematic synthetic tests and its usage in a new biomedical toolbox for identifying and classifying circulating tumor cells. Due to the nature of nonlinear diffusion underlying, the mathematical concepts in this work offer promising extensions to nonlocal classification problems.

[1]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[2]  Daniel Cremers,et al.  An algorithm for minimizing the Mumford-Shah functional , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[3]  Arjan Kuijper,et al.  Scale Space and Variational Methods in Computer Vision , 2013, Lecture Notes in Computer Science.

[4]  Tom Goldstein,et al.  The Split Bregman Method for L1-Regularized Problems , 2009, SIAM J. Imaging Sci..

[5]  Jahn Müller,et al.  Higher-Order TV Methods—Enhancement via Bregman Iteration , 2012, Journal of Scientific Computing.

[6]  A.D. Hoover,et al.  Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response , 2000, IEEE Transactions on Medical Imaging.

[7]  W. M. Carey,et al.  Digital spectral analysis: with applications , 1986 .

[8]  Otmar Scherzer,et al.  Inverse Total Variation Flow , 2007, Multiscale Model. Simul..

[9]  Stephen M. Pizer,et al.  A Multiresolution Hierarchical Approach to Image Segmentation Based on Intensity Extrema , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[10]  Max A. Viergever,et al.  Probabilistic Multiscale Image Segmentation , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Thomas Brox,et al.  On the Equivalence of Soft Wavelet Shrinkage, Total Variation Diffusion, Total Variation Regularization, and SIDEs , 2004, SIAM J. Numer. Anal..

[12]  M. Novaga,et al.  The Total Variation Flow in RN , 2002 .

[13]  S. Osher,et al.  Decomposition of images by the anisotropic Rudin‐Osher‐Fatemi model , 2004 .

[14]  Yiqiu Dong,et al.  Exact Relaxation for Classes of Minimization Problems with Binary Constraints , 2012, 1210.7507.

[15]  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.

[16]  Xavier Bresson,et al.  Completely Convex Formulation of the Chan-Vese Image Segmentation Model , 2012, International Journal of Computer Vision.

[17]  Andrew P. Witkin,et al.  Scale-space filtering: A new approach to multi-scale description , 1984, ICASSP.

[18]  Tony F. Chan,et al.  Active Contours without Edges for Vector-Valued Images , 2000, J. Vis. Commun. Image Represent..

[19]  Luc Florack,et al.  The Topological Structure of Scale-Space Images , 2000, Journal of Mathematical Imaging and Vision.

[20]  Max A. Viergever,et al.  Multiscale Segmentation of Three-Dimensional MR Brain Images , 1999, International Journal of Computer Vision.

[21]  Martin Burger,et al.  Primal and Dual Bregman Methods with Application to Optical Nanoscopy , 2011, International Journal of Computer Vision.

[22]  Mads Nielsen,et al.  Non-linear Diffusion for Interactive Multi-scale Watershed Segmentation , 2000, MICCAI.

[23]  Gene H. Golub,et al.  A Nonlinear Primal-Dual Method for Total Variation-Based Image Restoration , 1999, SIAM J. Sci. Comput..

[24]  Max A. Viergever,et al.  Nonlinear Multiscale Representations for Image Segmentation , 1997, Comput. Vis. Image Underst..

[25]  John W. Park,et al.  Circulating Tumor Cells , 2017, Methods in Molecular Biology.

[26]  Jonathan W. Uhr,et al.  Tumor Cells Circulate in the Peripheral Blood of All Major Carcinomas but not in Healthy Subjects or Patients With Nonmalignant Diseases , 2004, Clinical Cancer Research.

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

[28]  Wotao Yin,et al.  An Iterative Regularization Method for Total Variation-Based Image Restoration , 2005, Multiscale Model. Simul..

[29]  Matthew Peet,et al.  Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares , 2014, 1408.5119.

[30]  K. Pienta,et al.  Circulating Tumor Cells Predict Survival Benefit from Treatment in Metastatic Castration-Resistant Prostate Cancer , 2008, Clinical Cancer Research.

[31]  Xue-Cheng Tai,et al.  Efficient Global Minimization for the Multiphase Chan-Vese Model of Image Segmentation , 2009, EMMCVPR.

[32]  Wotao Yin,et al.  Bregman Iterative Algorithms for (cid:2) 1 -Minimization with Applications to Compressed Sensing ∗ , 2008 .

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

[34]  Narendra Ahuja,et al.  Multiscale image segmentation by integrated edge and region detection , 1997, IEEE Trans. Image Process..

[35]  Maurizio Paolini,et al.  Characterization of facet-breaking for nonsmooth mean curvature flow in the convex case , 2001 .

[36]  V. Caselles,et al.  Minimizing total variation flow , 2000, Differential and Integral Equations.

[37]  Permalink IMAGE SEGMENTATION WITH DYNAMIC ARTIFACTS DETECTION AND BIAS CORRECTION , 2015 .

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

[39]  Guy Gilboa,et al.  A Total Variation Spectral Framework for Scale and Texture Analysis , 2014, SIAM J. Imaging Sci..

[40]  Guillermo Sapiro,et al.  Geodesic Active Contours , 1995, International Journal of Computer Vision.

[41]  Mila Nikolova,et al.  Algorithms for Finding Global Minimizers of Image Segmentation and Denoising Models , 2006, SIAM J. Appl. Math..

[42]  Stanley Osher,et al.  A Unified Primal-Dual Algorithm Framework Based on Bregman Iteration , 2010, J. Sci. Comput..

[43]  Guy Gilboa,et al.  A Spectral Approach to Total Variation , 2013, SSVM.

[44]  Petre Stoica,et al.  Spectral Analysis of Signals , 2009 .

[45]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[46]  Xavier Bresson,et al.  Fast Global Minimization of the Active Contour/Snake Model , 2007, Journal of Mathematical Imaging and Vision.

[47]  Wiro J Niessen,et al.  Segmentation of tumors in magnetic resonance brain images using an interactive multiscale watershed algorithm. , 2004, Academic radiology.

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

[49]  Otmar Scherzer,et al.  Exact solutions of one-dimensional total generalized variation , 2015 .

[50]  Alison Stopeck,et al.  Circulating tumor cells, disease progression, and survival in metastatic breast cancer. , 2004, The New England journal of medicine.

[51]  Michael Möller,et al.  Spectral Decompositions Using One-Homogeneous Functionals , 2016, SIAM J. Imaging Sci..

[52]  Tony Lindeberg,et al.  Feature Detection with Automatic Scale Selection , 1998, International Journal of Computer Vision.

[53]  Jesús Ildefonso Díaz Díaz,et al.  Some qualitative properties for the total variation flow , 2002 .

[54]  Michael Möller,et al.  Spectral Representations of One-Homogeneous Functionals , 2015, SSVM.

[55]  Alfred M. Bruckstein,et al.  Scale Space and Variational Methods in Computer Vision , 2011, Lecture Notes in Computer Science.

[56]  K. Bredies,et al.  A study of the one dimensional total generalised variation regularisation problem , 2013, 1309.5900.

[57]  Guy Gilboa,et al.  Nonlinear Inverse Scale Space Methods for Image Restoration , 2005, VLSM.

[58]  Martin Burger,et al.  Ground States and Singular Vectors of Convex Variational Regularization Methods , 2012, 1211.2057.

[59]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[60]  Demetri Terzopoulos,et al.  Snakes: Active contour models , 2004, International Journal of Computer Vision.

[61]  Mingqiang Zhu,et al.  An Efficient Primal-Dual Hybrid Gradient Algorithm For Total Variation Image Restoration , 2008 .

[62]  Anthony J. Yezzi,et al.  Gradient flows and geometric active contour models , 1995, Proceedings of IEEE International Conference on Computer Vision.

[63]  J. Bono,et al.  All circulating EpCAM+CK+CD45- objects predict overall survival in castration-resistant prostate cancer. , 2010, Annals of oncology : official journal of the European Society for Medical Oncology.

[64]  Xavier Bresson,et al.  Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction , 2010, J. Sci. Comput..

[65]  Adel Said Elmaghraby,et al.  Graph cut optimization for the Mumford-Shah model , 2007 .

[66]  Michael Möller,et al.  Nonlinear Spectral Analysis via One-Homogeneous Functionals: Overview and Future Prospects , 2015, Journal of Mathematical Imaging and Vision.

[67]  Tony F. Chan,et al.  A General Framework for a Class of First Order Primal-Dual Algorithms for Convex Optimization in Imaging Science , 2010, SIAM J. Imaging Sci..

[68]  Lin He,et al.  Solving the Chan-Vese Model by a Multiphase Level Set Algorithm Based on the Topological Derivative , 2007, SSVM.

[69]  J. Koenderink The structure of images , 2004, Biological Cybernetics.