In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described by sparse linear combinations of these atoms. Applications that use sparse representation are many and include compression, regularization in inverse problems, feature extraction, and more. Recent activity in this field has concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a given dictionary. Designing dictionaries to better fit the above model can be done by either selecting one from a prespecified set of linear transforms or adapting the dictionary to a set of training signals. Both of these techniques have been considered, but this topic is largely still open. In this paper we propose a novel algorithm for adapting dictionaries in order to achieve sparse signal representations. Given a set of training signals, we seek the dictionary that leads to the best representation for each member in this set, under strict sparsity constraints. We present a new method-the K-SVD algorithm-generalizing the K-means clustering process. K-SVD is an iterative method that alternates between sparse coding of the examples based on the current dictionary and a process of updating the dictionary atoms to better fit the data. The update of the dictionary columns is combined with an update of the sparse representations, thereby accelerating convergence. The K-SVD algorithm is flexible and can work with any pursuit method (e.g., basis pursuit, FOCUSS, or matching pursuit). We analyze this algorithm and demonstrate its results both on synthetic tests and in applications on real image data

https://users.ece.utexas.edu/~sanghavi/courses/papers/KSVD.pdf

$rm K$ -SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation

We address the image denoising problem, where zero-mean white and homogeneous Gaussian additive noise is to be removed from a given image. The approach taken is based on sparse and redundant representations over trained dictionaries. Using the K-SVD algorithm, we obtain a dictionary that describes the image content effectively. Two training options are considered: using the corrupted image itself, or training on a corpus of high-quality image database. Since the K-SVD is limited in handling small image patches, we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image. We show how such Bayesian treatment leads to a simple and effective denoising algorithm. This leads to a state-of-the-art denoising performance, equivalent and sometimes surpassing recently published leading alternative denoising methods

Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries

1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

https://genoplivning.dk/wp-content/uploads/2013/07/european-resuscitation-council-guidelines-for-resuscitation-2010-section-4-adult-advanced-life-support.pdf

European Resuscitation Council Guidelines for Resuscitation 2010 Section 4 Adult advanced life support

A full-rank matrix ${\bf A}\in \mathbb{R}^{n\times m}$ with $n<m$ generates an underdetermined system of linear equations ${\bf Ax} = {\bf b}$ having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries. Can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena, in particular the existence of easily verifiable conditions under which optimally sparse solutions can be found by concrete, effective computational methods.

Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable, but there is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems have energized research on such signal and image processing problems—to striking effect.

In this paper we review the theoretical results on sparse solutions of linear systems, empirical results on sparse modeling of signals and images, and recent applications in inverse problems and compression in image processing. This work lies at the intersection of signal processing and applied mathematics, and arose initially from the wavelets and harmonic analysis research communities. The aim of this paper is to introduce a few key notions and applications connected to sparsity, targeting newcomers interested in either the mathematical aspects of this area or its applications.

/pdf/from-sparse-solutions-of-systems-of-equations-to-sparse-1y2kv866zu.pdf

From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images

A frame design technique for use with vector selection algorithms, for example matching pursuits (MP), is presented. The design algorithm is iterative and requires a training set of signal vectors. The algorithm, called method of optimal directions (MOD), is an improvement of the algorithm presented by Engan, Aase and Husoy see (Proc. ICASSP '98, Seattle, USA, p.1817-20, 1998). The MOD is applied to speech and electrocardiogram (ECG) signals, and the designed frames are tested on signals outside the training sets. Experiments demonstrate that the approximation capabilities, in terms of mean squared error (MSE), of the optimized frames are significantly better than those obtained using frames designed by the algorithm of Engan et. al. Experiments show typical reduction in MSE by 20-50%.

Method of optimal directions for frame design

Multi-frame compression: theory and design

Background—In 156 patients with out-of-hospital cardiac arrest of cardiac cause, we analyzed the ability of 4 spectral features of ventricular fibrillation before a total of 868 shocks to discriminate or not between segments that correspond to return of spontaneous circulation (ROSC). Methods and Results—Centroid frequency, peak power frequency, spectral flatness, and energy were studied. A second decorrelated feature set was generated with the coefficients of the principal component analysis transformation of the original feature set. Each feature set was split into training and testing sets for improved reliability in the evaluation of nonparametric classifiers for each possible feature combination. The combination of centroid frequency and peak power frequency achieved a mean±SD sensitivity of 92±2% and specificity of 27±2% in testing. The highest performing classifier corresponded to the combination of the 2 dominant decorrelated spectral features with sensitivity and specificity equal to 92±2% and 42...

Predicting Outcome of Defibrillation by Spectral Characterization and Nonparametric Classification of Ventricular Fibrillation in Patients With Out-of-Hospital Cardiac Arrest

The method of optimal directions (MOD) is an iterative method for designing frames for sparse representation purposes using a training set. In this paper we use frames designed by MOD in a multiframe compression (MFC) scheme. Both the MOD and the MFC need a vector selection algorithm, and orthogonal matching pursuit (OMP) is used in this paper. In the MFC scheme several different frames are used, each optimized for a fixed number of selected frame vectors in each approximation. We apply the MOD and the MFC scheme to ECG signals, and do experiments with both fixed size and variable size on the different frames used in the MFC scheme. Compared to traditional transform based compression, the experiments demonstrate improved rate-distortion performance by 1-4 dB, and that variable sized frames perform better than fixed sized frames.

Frame based signal compression using method of optimal directions (MOD)

Chest compression is a vital part of cardiopulmonary resuscitation (CPR). This paper demonstrates how the compression depth can be estimated using the principles of inertia navigation. The proposed method uses accelerometer sensors, one placed on the patient's chest, the other beside the patient. The acceleration-to-position conversion is performed using discrete-time digital signal processing (DSP). Instability problems due to integration are combated using a set of boundary conditions. The proposed algorithm is tested on a mannequin in harsh environments, where the patient is exposed to external forces as in a boat or car, as well as improper sensor/patient alignment. The overall performance is an estimation depth error of 4.3 mm in these environments, which is reduced to 1.6 mm in a regular, flat-floor controlled environment.

S.O. Aase

Papers

Method of optimal directions for frame design

Multi-frame compression: theory and design

Predicting Outcome of Defibrillation by Spectral Characterization and Nonparametric Classification of Ventricular Fibrillation in Patients With Out-of-Hospital Cardiac Arrest

Frame based signal compression using method of optimal directions (MOD)

Compression depth estimation for CPR quality assessment using DSP on accelerometer signals