The Single Best Replacement algorithm

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Table of contents

1 Introduction generale 
1.1 Contexte applicatif et objectifs
1.1.1 Contexte general
1.1.2 Microscopie AFM
1.1.3 Objectifs du projet
1.2 Traitement de donnees et besoins algorithmiques
1.2.1 Traitement de courbes de force et d’images force-volume
1.2.2 Formulation du lissage d’un signal
1.2.3 Separation de sources parcimonieuses retardees
1.3 Contributions principales
1.3.1 Approximation parcimonieuse par minimisation d’un critere mixte `2-`0
1.3.2 Algorithme de continuation
1.3.3 Detection conjointe de discontinuites
1.3.4 Separation de sources parcimonieuses retardees
1.3.5 Segmentation d’une courbe de force et estimation de parametres physiques
1.4 Organisation des autres chapitres de la these
2 From Bernoulli-Gaussian deconvolution to sparse signal restoration 
2.1 Introduction
2.2 From Bernoulli-Gaussian signal modeling to sparse signal representation
2.2.1 Preliminary denitions and working assumptions
2.2.2 Bernoulli-Gaussian models
2.2.3 Bayesian formulation of sparse signal restoration
2.2.4 Mixed `2-`0 minimization as a limit case
2.3 Adaptation of SMLR to `0-penalized least-square optimization
2.3.1 Principle of SMLR and main notations
2.3.2 The Single Best Replacement algorithm (rst version)
2.3.3 Modied version of SBR (nal version)
2.3.4 Behavior and adaptations of SBR
2.4 Implementation issues
2.4.1 Basic implementation
2.4.2 Recursive implementation
2.4.3 Ecient strategy based on the Cholesky factorization
2.4.4 Memory requirements and computation burden
2.5 Deconvolution of a sparse signal with a Gaussian impulse response
2.5.1 Dictionary and simulated data
2.5.2 Separation of two close Gaussian features
2.5.3 Behavior of SBR for noisy data
2.6 Joint detection of discontinuities at dierent orders in a signal
2.6.1 Approximation of a spline of degree p
2.6.2 Approximation of a piecewise polynomial of maximum degree P
2.6.3 Adaptation of SBR
2.6.4 Numerical simulations
2.6.5 AFM data processing
2.6.6 Discussion
2.7 Conclusion
3 A continuation algorithm for `0-penalized least-square optimization 
3.1 Introduction
3.2 Mixed `2-`0 optimization
3.2.1 Denitions and working assumptions
3.2.2 Solution path
3.3 Continuation algorithm
3.3.1 Single Best Replacement
3.3.2 Recursive computation of
3.3.3 CSBR algorithm
3.4 Numerical simulations and real data processing
3.4.1 Discontinuity detection
3.4.2 Sparse spike train deconvolution
3.4.3 Sensitivity to the mutual coherence of the dictionary
3.5 Conclusion
4 Automated force volume image processing for biological samples 
4.1 Introduction
4.2 Theory and physical models
4.2.1 Physical models relevant for the approach curves
4.2.2 Physical models relevant for the retraction curves
4.3 Algorithms
4.3.1 Force curve segmentation
4.3.2 Fitting of the approach curves
4.3.3 Fitting of the retraction curves
4.3.4 Processing of a force-volume image
4.4 Results and Discussion
4.4.1 Bacterial culture and sample preparation
4.4.2 AFM measurements and preparation of experiments
4.4.3 Data processing for the approach force curve
4.4.4 Data processing for the retraction force curve
4.4.5 Discussion on the performance of the algorithm for force curve analysis
4.5 Conclusion
5 Separate sparse sources from a large number of shifted mixtures
5.1 Introduction and motivation
5.2 Sparse source separation problem
5.2.1 Dierent kinds of mixing models
5.2.2 Separation of instantaneous mixture of sparse sources
5.2.3 Degenerate Unmixing Estimation Technique (DUET)
5.2.4 Enlighting points of existing algorithms
5.3 Identiability
5.3.1 Modeling of sparse signals
5.3.2 Ambiguities of source separation problem
5.3.3 Denition of eigen interval
5.3.4 Sucient condition for identiability
5.3.5 Extension of sucient condition
5.4 Algorithm to separate sparse spike trains
5.4.1 Separate sparse spike trains in noise-free setting
5.4.2 Extension to noisy setting
5.5 Experiments
5.5.1 Eect of bin width of histogram
5.5.2 Eect of perturbation on locations of spikes
5.5.3 Eect of thresholding the histograms
5.5.4 AFM data processing
5.6 Conclusion
Conclusion