Smoothing for signals with discontinuities using higher order Mumford–Shah models

Minimizing the Mumford-Shah functional is frequently used for smoothing signals or time series with discontinuities. A significant limitation of the standard Mumford-Shah model is that linear trends -- and in general polynomial trends -- in the data are not well preserved. This can be improved by building on splines of higher order which leads to higher order Mumford-Shah models. In this work, we study these models in the univariate situation: we discuss important differences to the first order Mumford-Shah model, and we obtain uniqueness results for their solutions. As a main contribution, we derive fast minimization algorithms for Mumford-Shah models of arbitrary orders. We show that the worst case complexity of all proposed schemes is quadratic in the length of the signal. Remarkably, they thus achieve the worst case complexity of the fastest solver for the piecewise constant Mumford-Shah model (which is the simplest model of the class). Further, we obtain stability results for the proposed algorithms. We complement these results with a numerical study. Our reference implementation processes signals with more than 10,000 elements in less than one second.

[1]  G. Winkler,et al.  Smoothers for Discontinuous Signals , 2002 .

[3]  Andreas D. Baxevanis,et al.  The Molecular Biology Database Collection: 2003 update , 2003, Nucleic Acids Res..

[4]  William M. Rand,et al.  Objective Criteria for the Evaluation of Clustering Methods , 1971 .

[5]  Nira Dyn,et al.  Interpolation and Approximation of Piecewise Smooth Functions , 2005, SIAM J. Numer. Anal..

[6]  Martin Storath,et al.  Iterative Potts and Blake–Zisserman minimization for the recovery of functions with discontinuities from indirect measurements , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[7]  Olaf Wittich,et al.  Don't shed tears over breaks , 2005 .

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

[9]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[10]  Axel Munk,et al.  Idealizing Ion Channel Recordings by a Jump Segmentation Multiresolution Filter , 2013, IEEE Transactions on NanoBioscience.

[11]  Lorenzo Bruzzone,et al.  Piecewise Linear Approximation of Vector-Valued Images and Curves via Second-Order Variational Model , 2017, IEEE Transactions on Image Processing.

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

[13]  Andreas Weinmann,et al.  Fast Partitioning of Vector-Valued Images , 2014, SIAM J. Imaging Sci..

[14]  Gaofeng Meng,et al.  Level set evolution with locally linear classification for image segmentation , 2011, 2011 18th IEEE International Conference on Image Processing.

[15]  C. Joo,et al.  Advances in single-molecule fluorescence methods for molecular biology. , 2008, Annual review of biochemistry.

[16]  Andreas Weinmann,et al.  Iterative Potts Minimization for the Recovery of Signals with Discontinuities from Indirect Measurements: The Multivariate Case , 2018, Found. Comput. Math..

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

[18]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.

[19]  Gerhard Winkler,et al.  Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction , 2002 .

[20]  R. Bellman,et al.  Curve Fitting by Segmented Straight Lines , 1969 .

[21]  Sergio Amat,et al.  On an New Algorithm for Function Approximation with Full Accuracy in the Presence of Discontinuities Based on the Immersed Interface Method , 2018, J. Sci. Comput..

[22]  G. Winkler,et al.  Complexity Penalized M-Estimation , 2008 .

[23]  Camille Roth,et al.  Natural Scales in Geographical Patterns , 2017, Scientific Reports.

[24]  A. Munk,et al.  Multiscale change point inference , 2013, 1301.7212.

[25]  Grace Wahba,et al.  Spline Models for Observational Data , 1990 .

[26]  Olaf Wittich,et al.  Complexity penalized least squares estimators: Analytical results , 2008 .

[27]  R. Ramlau,et al.  Regularization of ill-posed Mumford–Shah models with perimeter penalization , 2010 .

[28]  Milan Sonka,et al.  Hedgehog Shape Priors for Multi-Object Segmentation , 2016, 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[29]  Edmund Taylor Whittaker.  On a New Method of Graduation , 1922, Proceedings of the Edinburgh Mathematical Society.

[30]  Jiaolong Yang,et al.  Dense, accurate optical flow estimation with piecewise parametric model , 2015, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[31]  I E Auger,et al.  Algorithms for the optimal identification of segment neighborhoods. , 1989, Bulletin of mathematical biology.

[32]  Max A. Little,et al.  Generalized methods and solvers for noise removal from piecewise constant signals. I. Background theory , 2011, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[33]  Zhizhou Wang,et al.  DTI segmentation using an information theoretic tensor dissimilarity measure , 2005, IEEE Transactions on Medical Imaging.

[34]  M. Jiang,et al.  Regularizing properties of the Mumford–Shah functional for imaging applications , 2014 .

[35]  Daniel Cremers,et al.  Real-Time Minimization of the Piecewise Smooth Mumford-Shah Functional , 2014, ECCV.

[36]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[37]  Daniel Lemire,et al.  A Better Alternative to Piecewise Linear Time Series Segmentation , 2006, SDM.

[38]  M. Carriero,et al.  A Survey on the Blake–Zisserman Functional , 2015 .

[39]  Daniel Cremers,et al.  A convex representation for the vectorial Mumford-Shah functional , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.

[40]  Massimo Fornasier,et al.  Iterative Thresholding Meets Free-Discontinuity Problems , 2009, Found. Comput. Math..

[41]  Ronny Ramlau,et al.  A Mumford-Shah level-set approach for the inversion and segmentation of X-ray tomography data , 2007, J. Comput. Phys..

[42]  M. Unser,et al.  Jump-Penalized Least Absolute Values Estimation of Scalar or Circle-Valued Signals , 2017 .

[43]  Martin Storath,et al.  An algorithmic framework for Mumford–Shah regularization of inverse problems in imaging , 2015 .

[44]  I. Johnstone,et al.  Ideal spatial adaptation by wavelet shrinkage , 1994 .

[45]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

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

[47]  R. Zabih,et al.  Efficient Graph-Based Energy Minimization Methods in Computer Vision , 1999 .

[48]  Michael Unser,et al.  Fast Piecewise-Affine Motion Estimation Without Segmentation , 2018, IEEE Transactions on Image Processing.

[49]  Michio Homma,et al.  Direct observation of steps in rotation of the bacterial flagellar motor , 2005, Nature.

[50]  Max A. Little,et al.  Generalized methods and solvers for noise removal from piecewise constant signals. II. New methods , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[51]  Andreas Weinmann,et al.  The L1-Potts Functional for Robust Jump-Sparse Reconstruction , 2012, SIAM J. Numer. Anal..

[52]  M. Fornasier,et al.  Existence of minimizers of the Mumford-Shah functional with singular operators and unbounded data , 2013 .

[53]  Vladimir Kolmogorov,et al.  Total Variation on a Tree , 2015, SIAM J. Imaging Sci..

[54]  Massimo Fornasier,et al.  Linearly Constrained Nonsmooth and Nonconvex Minimization , 2012, SIAM J. Optim..

[55]  Olga Veksler,et al.  Fast Approximate Energy Minimization via Graph Cuts , 2001, IEEE Trans. Pattern Anal. Mach. Intell..

[56]  P. Fearnhead,et al.  Optimal detection of changepoints with a linear computational cost , 2011, 1101.1438.

[57]  W. E. Gentleman.  Least Squares Computations by Givens Transformations Without Square Roots , 1973 .

[58]  A. Harten.  Multiresolution representation of data: a general framework , 1996 .

[59]  Leszek Wojnar,et al.  Image Analysis , 1998 .

[60]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[61]  Volkmar Liebscher,et al.  Specificity assessment from fractionation experiments (SAFE): a novel method to evaluate microarray probe specificity based on hybridisation stringencies. , 2003, Nucleic acids research.

[62]  Guoliang Xu,et al.  A general framework of piecewise-polynomial Mumford–Shah model for image segmentation , 2017, Int. J. Comput. Math..

[63]  Valeria Ruggiero,et al.  Numerical minimization of a second-order functional for image segmentation , 2016, Commun. Nonlinear Sci. Numer. Simul..

[64]  Ajay N. Jain,et al.  Assembly of microarrays for genome-wide measurement of DNA copy number , 2001, Nature Genetics.

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

[66]  Anthony J. Yezzi,et al.  Curve evolution implementation of the Mumford-Shah functional for image segmentation, denoising, interpolation, and magnification , 2001, IEEE Trans. Image Process..

[67]  L. Ambrosio,et al.  Approximation of functional depending on jumps by elliptic functional via t-convergence , 1990 .

[68]  Andrew Blake,et al.  Comparison of the Efficiency of Deterministic and Stochastic Algorithms for Visual Reconstruction , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[69]  T. Stephenson.  Image analysis , 1992, Nature.

[70]  Adam N. Letchford,et al.  Detecting Changes in Slope With an L0 Penalty , 2017, Journal of Computational and Graphical Statistics.

[71]  ROGER C. WOOD,et al.  On optimum quantization , 1969, IEEE Trans. Inf. Theory.

[72]  Stochastic Relaxation , 2014, Computer Vision, A Reference Guide.

[73]  Jeffrey D. Scargle,et al.  An algorithm for optimal partitioning of data on an interval , 2003, IEEE Signal Processing Letters.

[74]  Emmanuel Barillot,et al.  Analysis of array CGH data: from signal ratio to gain and loss of DNA regions , 2004, Bioinform..

[75]  Nahum Kiryati,et al.  Variational Pairing of Image Segmentation and Blind Restoration , 2004, ECCV.

[76]  A. Chambolle.  FINITE-DIFFERENCES DISCRETIZATIONS OF THE MUMFORD-SHAH FUNCTIONAL , 1999 .

[77]  R. B. Potts.  Some generalized order-disorder transformations , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[78]  Andreas Weinmann,et al.  Mumford–Shah and Potts Regularization for Manifold-Valued Data , 2014, Journal of Mathematical Imaging and Vision.

[79]  Éva Tardos,et al.  Algorithm design , 2005 .

[80]  Antonin Chambolle,et al.  Image Segmentation by Variational Methods: Mumford and Shah Functional and the Discrete Approximations , 1995, SIAM J. Appl. Math..

[81]  Karl Kunisch,et al.  Total Generalized Variation , 2010, SIAM J. Imaging Sci..

[82]  L. Rondi.  On the regularization of the inverse conductivity problem with discontinuous conductivities , 2008 .

[83]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .