Polynomial Smoothing of Time Series With Additive Step Discontinuities

This paper addresses the problem of estimating simultaneously a local polynomial signal and an approximately piecewise constant signal from a noisy additive mixture. The approach developed in this paper synthesizes the total variation filter and least-square polynomial signal smoothing into a unified problem formulation. The method is based on formulating an l1-norm regularized inverse problem. A computationally efficient algorithm, based on variable splitting and the alternating direction method of multipliers (ADMM), is presented. Algorithms are derived for both unconstrained and constrained formulations. The method is illustrated on experimental data involving the detection of nano-particles with applications to real-time virus detection using a whispering-gallery mode detector.

[1]  S. Arnold,et al.  Shift of whispering-gallery modes in microspheres by protein adsorption. , 2003, Optics letters.

[2]  Ashraf A. Kassim,et al.  Signal and Image Approximation Using Interval Wavelet Transform , 2007, IEEE Transactions on Image Processing.

[3]  Robert D. Nowak,et al.  Wavelet-based statistical signal processing using hidden Markov models , 1998, IEEE Trans. Signal Process..

[4]  Jaakko Astola,et al.  Local Approximation Techniques in Signal and Image Processing (SPIE Press Monograph Vol. PM157) , 2006 .

[5]  R. Schafer,et al.  What Is a Savitzky-Golay Filter? , 2022 .

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

[7]  Michael Elad,et al.  Submitted to Ieee Transactions on Image Processing Image Decomposition via the Combination of Sparse Representations and a Variational Approach , 2022 .

[8]  Simon Foucart,et al.  Hard Thresholding Pursuit: An Algorithm for Compressive Sensing , 2011, SIAM J. Numer. Anal..

[9]  Mike E. Davies,et al.  Normalized Iterative Hard Thresholding: Guaranteed Stability and Performance , 2010, IEEE Journal of Selected Topics in Signal Processing.

[10]  Junfeng Yang,et al.  A New Alternating Minimization Algorithm for Total Variation Image Reconstruction , 2008, SIAM J. Imaging Sci..

[11]  Patrick L. Combettes,et al.  Proximal Splitting Methods in Signal Processing , 2009, Fixed-Point Algorithms for Inverse Problems in Science and Engineering.

[12]  S. Arnold,et al.  Single virus detection from the reactive shift of a whispering-gallery mode , 2008, Proceedings of the National Academy of Sciences.

[13]  Martin Vetterli,et al.  Wavelet footprints: theory, algorithms, and applications , 2003, IEEE Trans. Signal Process..

[14]  José M. Bioucas-Dias,et al.  Fast Image Recovery Using Variable Splitting and Constrained Optimization , 2009, IEEE Transactions on Image Processing.

[15]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[16]  Antonin Chambolle,et al.  Image Decomposition into a Bounded Variation Component and an Oscillating Component , 2005, Journal of Mathematical Imaging and Vision.

[17]  M. Nikolova.  An Algorithm for Total Variation Minimization and Applications , 2004 .

[18]  Peter Steffen,et al.  Some advanced topics in filter design , 1987 .

[19]  C. Sidney Burrus,et al.  Design of FIR digital filters , 1997 .

[20]  Jacques Froment,et al.  Reconstruction of Wavelet Coefficients Using Total Variation Minimization , 2002, SIAM J. Sci. Comput..

[21]  Laurent Condat,et al.  A Direct Algorithm for 1-D Total Variation Denoising , 2013, IEEE Signal Processing Letters.

[22]  Justin K. Romberg,et al.  Wavelet-domain approximation and compression of piecewise smooth images , 2006, IEEE Transactions on Image Processing.

[23]  I. Kozlov,et al.  Sparse Solutions of Underdetermined Linear Systems , 2015 .

[24]  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..

[25]  Tony F. Chan,et al.  ENO-Wavelet Transforms for Piecewise Smooth Functions , 2002, SIAM J. Numer. Anal..

[26]  Ronald W. Schafer,et al.  What Is a Savitzky-Golay Filter? [Lecture Notes] , 2011, IEEE Signal Processing Magazine.

[27]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

[28]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[29]  A. Savitzky,et al.  Smoothing and Differentiation of Data by Simplified Least Squares Procedures. , 1964 .

[30]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

[31]  Benedetto Piccoli,et al.  A fast computation method for time scale signal denoising , 2009, Signal Image Video Process..

[32]  José M. Bioucas-Dias,et al.  An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems , 2009, IEEE Transactions on Image Processing.

[33]  Nick G. Kingsbury,et al.  Redundant representation with complex wavelets: how to achieve sparsity , 2003, Proceedings 2003 International Conference on Image Processing (Cat. No.03CH37429).

[34]  Liedewij Laan,et al.  Assembly dynamics of microtubules at molecular resolution , 2006, Nature.

[35]  Robert D. Nowak,et al.  Majorization–Minimization Algorithms for Wavelet-Based Image Restoration , 2007, IEEE Transactions on Image Processing.

[36]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

[37]  Tony F. Chan,et al.  The digital TV filter and nonlinear denoising , 2001, IEEE Trans. Image Process..

[38]  Ivan W. Selesnick,et al.  Total variation filtering , 2009 .

[39]  Javier Portilla,et al.  L0-based sparse approximation: two alternative methods and some applications , 2007, SPIE Optical Engineering + Applications.

[40]  Stanley Osher,et al.  Image Denoising and Decomposition with Total Variation Minimization and Oscillatory Functions , 2004, Journal of Mathematical Imaging and Vision.

[41]  Tony F. Chan,et al.  Total Variation Wavelet Thresholding , 2007, J. Sci. Comput..

[42]  V. R. Dantham,et al.  Taking whispering gallery-mode single virus detection and sizing to the limit , 2012 .