Sampling and Reconstruction of Sparse Signals on Circulant Graphs - An Introduction to Graph-FRI

With the objective of employing graphs toward a more generalized theory of signal processing, we present a novel sampling framework for (wavelet-)sparse signals defined on circulant graphs which extends basic properties of Finite Rate of Innovation (FRI) theory to the graph domain, and can be applied to arbitrary graphs via suitable approximation schemes. At its core, the introduced Graph-FRI-framework states that any K-sparse signal on the vertices of a circulant graph can be perfectly reconstructed from its dimensionality-reduced representation in the graph spectral domain, the Graph Fourier Transform (GFT), of minimum size 2K. By leveraging the recently developed theory of e-splines and e-spline wavelets on graphs, one can decompose this graph spectral transformation into the multiresolution low-pass filtering operation with a graph e-spline filter, and subsequent transformation to the spectral graph domain; this allows to infer a distinct sampling pattern, and, ultimately, the structure of an associated coarsened graph, which preserves essential properties of the original, including circularity and, where applicable, the graph generating set.Comment: To appear in Appl. Comput. Harmon. Anal. (2017

Sub-Nyquist Sampling: Bridging Theory and Practice

Sampling theory encompasses all aspects related to the conversion of continuous-time signals to discrete streams of numbers. The famous Shannon-Nyquist theorem has become a landmark in the development of digital signal processing. In modern applications, an increasingly number of functions is being pushed forward to sophisticated software algorithms, leaving only those delicate finely-tuned tasks for the circuit level. In this paper, we review sampling strategies which target reduction of the ADC rate below Nyquist. Our survey covers classic works from the early 50's of the previous century through recent publications from the past several years. The prime focus is bridging theory and practice, that is to pinpoint the potential of sub-Nyquist strategies to emerge from the math to the hardware. In that spirit, we integrate contemporary theoretical viewpoints, which study signal modeling in a union of subspaces, together with a taste of practical aspects, namely how the avant-garde modalities boil down to concrete signal processing systems. Our hope is that this presentation style will attract the interest of both researchers and engineers in the hope of promoting the sub-Nyquist premise into practical applications, and encouraging further research into this exciting new frontier.Comment: 48 pages, 18 figures, to appear in IEEE Signal Processing Magazin

Compressed sensing and finite rate of innovation for efficient data acquisition of quantitative acoustic microscopy images

La microscopie acoustique quantitative (MAQ) est une modalité d'imagerie bien établie qui donne accès à des cartes paramétriques 2D représentatives des propriétés mécaniques des tissus à une échelle microscopique. Dans la plupart des études sur MAQ, l'échantillons est scanné ligne par ligne (avec un pas de 2µm) à l'aide d'un transducteur à 250 MHz. Ce type d'acquisition permet d'obtenir un cube de données RF 3D, avec deux dimensions spatiales et une dimension temporelle. Chaque signal RF correspondant à une position spatiale dans l'échantillon permet d'estimer des paramètres acoustiques comme par exemple la vitesse du son ou l'impédance. Le temps d'acquisition en MAQ est directement proportionnel à la taille de l'échantillon et peut aller de quelques minutes à quelques dizaines de minutes. Afin d'assurer des conditions d'acquisition stables et étant donnée la sensibilité des échantillons à ces conditions, diminuer le temps d'acquisition est un des grand défis en MAQ. Afin de relever ce défi, ce travail de thèse propose plusieurs solutions basées sur l'échantillonnage compressé (EC) et la théories des signaux ayant un faible nombre de degré de liberté (finite rate of innovation - FRI, en anglais). Le principe de l'EC repose sur la parcimonie des données, sur l'échantillonnage incohérent de celles-ci et sur les algorithmes d'optimisation numérique. Dans cette thèse, les phénomènes physiques derrière la MAQ sont exploités afin de créer des modèles adaptés aux contraintes de l'EC et de la FRI. Plus particulièrement, ce travail propose plusieurs pistes d'application de l'EC en MAQ : un schéma d'acquisition spatiale innovant, un algorithme de reconstruction d'images exploitant les statistiques des coefficients en ondelettes des images paramétriques, un modèle FRI adapté aux signaux RF et un schéma d'acquisition compressée dans le domaine temporel.Quantitative acoustic microscopy (QAM) is a well-accepted modality for forming 2D parameter maps making use of mechanical properties of soft tissues at microscopic scales. In leading edge QAM studies, the sample is raster-scanned (spatial step size of 2µm) using a 250 MHz transducer resulting in a 3D RF data cube, and each RF signal for each spatial location is processed to obtain acoustic parameters, e.g., speed of sound or acoustic impedance. The scanning time directly depends on the sample size and can range from few minutes to tens of minutes. In order to maintain constant experimental conditions for the sensitive thin sectioned samples, the scanning time is an important practical issue. To deal with the current challenge, we propose the novel approach inspired by compressed sensing (CS) and finite rate of innovation (FRI). The success of CS relies on the sparsity of data under consideration, incoherent measurement and optimization technique. On the other hand, the idea behind FRI is supported by a signal model fully characterized as a limited number of parameters. From this perspective, taking into account the physics leading to data acquisition of QAM system, the QAM data can be regarded as an adequate application amenable to the state of the art technologies aforementioned. However, when it comes to the mechanical structure of QAM system which does not support canonical CS measurement manners on the one hand, and the compositions of the RF signal model unsuitable to existing FRI schemes on the other hand, the advanced frameworks are still not perfect methods to resolve the problems that we are facing. In this thesis, to overcome the limitations, a novel sensing framework for CS is presented in spatial domain: a recently proposed approximate message passing (AMP) algorithm is adapted to account for the underlying data statistics of samples sparsely collected by proposed scanning patterns. In time domain, as an approach for achieving an accurate recovery from a small set of samples of QAM RF signals, we employ sum of sincs (SoS) sampling kernel and autoregressive (AR) model estimator. The spiral scanning manner, introduced as an applicable sensing technique to QAM system, contributed to the significant reduction of the number of spatial samples when reconstructing speed of sound images of a human lymph node. Furthermore, the scanning time was also hugely saved due to the merit of the mechanical movement of the proposed sensing pattern. Together with the achievement in spatial domain, the introduction of SoS kernel and AR estimator responsible for an innovation rate sampling and a parameter estimation respectively led to dramatic reduction of the required number of samples per RF signal compared to a conventional approach. Finally, we showed that both data acquisition frameworks based on the CS and FRI can be combined into a single spatio-temporal solution to maximize the benefits stated above

Efficient network-guided multi-locus association mapping with graph cuts

As an increasing number of genome-wide association studies reveal the limitations of attempting to explain phenotypic heritability by single genetic loci, there is growing interest for associating complex phenotypes with sets of genetic loci. While several methods for multi-locus mapping have been proposed, it is often unclear how to relate the detected loci to the growing knowledge about gene pathways and networks. The few methods that take biological pathways or networks into account are either restricted to investigating a limited number of predetermined sets of loci, or do not scale to genome-wide settings. We present SConES, a new efficient method to discover sets of genetic loci that are maximally associated with a phenotype, while being connected in an underlying network. Our approach is based on a minimum cut reformulation of the problem of selecting features under sparsity and connectivity constraints that can be solved exactly and rapidly. SConES outperforms state-of-the-art competitors in terms of runtime, scales to hundreds of thousands of genetic loci, and exhibits higher power in detecting causal SNPs in simulation studies than existing methods. On flowering time phenotypes and genotypes from Arabidopsis thaliana, SConES detects loci that enable accurate phenotype prediction and that are supported by the literature. Matlab code for SConES is available at http://webdav.tuebingen.mpg.de/u/karsten/Forschung/scones/Comment: 20 pages, 6 figures, accepted at ISMB (International Conference on Intelligent Systems for Molecular Biology) 201