Sparsity and adaptivity for the blind separation of partially correlated sources

Blind source separation (BSS) is a very popular technique to analyze multichannel data. In this context, the data are modeled as the linear combination of sources to be retrieved. For that purpose, standard BSS methods all rely on some discrimination principle, whether it is statistical independence or morphological diversity, to distinguish between the sources. However, dealing with real-world data reveals that such assumptions are rarely valid in practice: the signals of interest are more likely partially correlated, which generally hampers the performances of standard BSS methods. In this article, we introduce a novel sparsity-enforcing BSS method coined Adaptive Morphological Component Analysis (AMCA), which is designed to retrieve sparse and partially correlated sources. More precisely, it makes profit of an adaptive re-weighting scheme to favor/penalize samples based on their level of correlation. Extensive numerical experiments have been carried out which show that the proposed method is robust to the partial correlation of sources while standard BSS techniques fail. The AMCA algorithm is evaluated in the field of astrophysics for the separation of physical components from microwave data.Comment: submitted to IEEE Transactions on signal processin

A PCA-based automated finder for galaxy-scale strong lenses

We present an algorithm using Principal Component Analysis (PCA) to subtract galaxies from imaging data, and also two algorithms to find strong, galaxy-scale gravitational lenses in the resulting residual image. The combined method is optimized to find full or partial Einstein rings. Starting from a pre-selection of potential massive galaxies, we first perform a PCA to build a set of basis vectors. The galaxy images are reconstructed using the PCA basis and subtracted from the data. We then filter the residual image with two different methods. The first uses a curvelet (curved wavelets) filter of the residual images to enhance any curved/ring feature. The resulting image is transformed in polar coordinates, centered on the lens galaxy center. In these coordinates, a ring is turned into a line, allowing us to detect very faint rings by taking advantage of the integrated signal-to-noise in the ring (a line in polar coordinates). The second way of analysing the PCA-subtracted images identifies structures in the residual images and assesses whether they are lensed images according to their orientation, multiplicity and elongation. We apply the two methods to a sample of simulated Einstein rings, as they would be observed with the ESA Euclid satellite in the VIS band. The polar coordinates transform allows us to reach a completeness of 90% and a purity of 86%, as soon as the signal-to-noise integrated in the ring is higher than 30, and almost independent of the size of the Einstein ring. Finally, we show with real data that our PCA-based galaxy subtraction scheme performs better than traditional subtraction based on model fitting to the data. Our algorithm can be developed and improved further using machine learning and dictionary learning methods, which would extend the capabilities of the method to more complex and diverse galaxy shapes

Learning the Morphological Diversity

International audienceThis article proposes a new method for image separation into a linear combination of morphological components. Sparsity in global dictionaries is used to extract the cartoon and oscillating content of the image. Complicated texture patterns are extracted by learning adapted local dictionaries that sparsify patches in the image. These global and local sparsity priors together with the data fidelity define a non-convex energy and the separation is obtained as a stationary point of this energy. This variational optimization is extended to solve more general inverse problems such as inpainting. A new adaptive morphological component analysis algorithm is derived to find a stationary point of the energy. Using adapted dictionaries learned from data allows to circumvent some difficulties faced by fixed dictionaries. Numerical results demonstrate that this adaptivity is indeed crucial to capture complex texture patterns

Image decomposition and separation using sparse representations: an overview

International audienceThis paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by overviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck et al. [1], [2] proposed a novel decomposition method - Morphological Component Analysis (MCA) - based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called Morphological Components, that are morphologically distinct, e.g. sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will overview the Generalized MCA (GMCA) introduced by the authors in [3], [4] as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation