MCALab: Reproducible Research in Signal and Image Decomposition and Inpainting

International audienceMorphological Component Analysis (MCA) of signals and images is an ambitious and important goal in signal processing; successful methods for MCA have many far-reaching applications in science and technology. Because MCA is related to solving underdetermined systems of linear equations it might also be considered, by some, to be problematic or even intractable. Reproducible research is essential to to give such a concept a firm scientific foundation and broad base of trusted results. MCALab has been developed to demonstrate key concepts of MCA and make them available to interested researchers and technologists. MCALab is a library of Matlab routines that implement the decomposition and inpainting algorithms that we previously proposed in [1], [2], [3], [4]. The MCALab package provides the research community with open source tools for sparse decomposition and inpainting and is made available for anonymous download over the Internet. It contains a variety of scripts to reproduce the figures in our own articles, as well as other exploratory examples not included in the papers. One can also run the same experiments on one's own data or tune the parameters by simply modifying the scripts. The MCALab is under continuing development by the authors; who welcome feedback and suggestions for further enhancements, and any contributions by interested researchers

FASTLens (FAst STatistics for weak Lensing) : Fast method for Weak Lensing Statistics and map making

With increasingly large data sets, weak lensing measurements are able to measure cosmological parameters with ever greater precision. However this increased accuracy also places greater demands on the statistical tools used to extract the available information. To date, the majority of lensing analyses use the two point-statistics of the cosmic shear field. These can either be studied directly using the two-point correlation function, or in Fourier space, using the power spectrum. But analyzing weak lensing data inevitably involves the masking out of regions or example to remove bright stars from the field. Masking out the stars is common practice but the gaps in the data need proper handling. In this paper, we show how an inpainting technique allows us to properly fill in these gaps with only $N \log N$ operations, leading to a new image from which we can compute straight forwardly and with a very good accuracy both the pow er spectrum and the bispectrum. We propose then a new method to compute the bispectrum with a polar FFT algorithm, which has the main advantage of avoiding any interpolation in the Fourier domain. Finally we propose a new method for dark matter mass map reconstruction from shear observations which integrates this new inpainting concept. A range of examples based on 3D N-body simulations illustrates the results.Comment: Final version accepted by MNRAS. The FASTLens software is available from the following link : http://irfu.cea.fr/Ast/fastlens.software.ph

Morphological Diversity and Sparsity for Multichannel Data Restoration

International audienceOver the last decade, overcomplete dictionaries and the very sparse signal representations they make possible, have raised an intense interest from signal processing theory. In a wide range of signal processing problems, sparsity has been a crucial property leading to high performance. As multichannel data are of growing interest, it seems essential to devise sparsity-based tools accounting for such specific multichannel data. Sparsity has proved its efficiency in a wide range of inverse problems. Hereafter, we address some multichannel inverse problems issues such as multichannel morphological component separation and inpainting from the perspective of sparse representation. In this paper, we introduce a new sparsity-based multichannel analysis tool coined multichannel Morphological Component Analysis (mMCA). This new framework focuses on multichannel morphological diversity to better represent multichannel data. This paper presents conditions under which the mMCA converges and recovers the sparse multichannel representation. Several experiments are presented to demonstrate the applicability of our approach on a set of multichannel inverse problems such as morphological component decomposition and inpainting

Dictionary learning and low-rank sparse matrix decomposition for sparsity driven SAR image reconstruction

Synthetic aperture radar (SAR) is one of the most widely used remote sensing modalities, providing images for a variety of applications including those in defense, environmental science, and weather forecasting. However, conventionally formed SAR imagery from undersampled observed data, arising in several emerging applications and sensing scenarios, suffers from artifacts that might limit effective use of such imagery in remote sensing applications. Recently, sparsity-driven SAR imaging has emerged as an e ective framework to alleviate such problems. Sparsity-based methods for SAR imaging have employed overcomplete dictionaries to represent the magnitude of the complex-valued eld sparsely. Selection of an appropriate dictionary with respect to the features of the particular type of underlying scene plays an important role in these methods. In this thesis, we develop two new sparsity-driven SAR imaging methods that significantly expand the domain of applicability of sparsity-based methods in SAR imaging. Our first contribution involves the development of a new reconstruction method that is based on learning sparsifying dictionaries and using such learned dictionaries in the reconstruction process. Adaptive dictionaries learned from data can represent the magnitude of complex-valued field more effectively and hence have the potential to widen the applicability of sparsity-based radar imaging. Our framework allows the use of both adaptive dictionaries learned offline from a training set and those learned online from the undersampled data used in image formation. We demonstrate the effectiveness of the proposed dictionary learning-based SAR imaging approach as well as the improvements it provides, on both synthetic and real data. The second contribution of this thesis involves the development of a reconstruction method that decomposes the imaged field into a sparse and a low-rank component. Such a decomposition is of interest in image analysis tasks such as segmentation and background subtraction. Conventionally, such operations are performed after SAR image formation. We exploit recent work on sparse and low-rank decomposition of matrices and incorporate such a decomposition into the process of SAR image formation. The outcome is a method that jointly reconstructs a SAR image and decomposes the formed image into its low-rank background and spatially sparse components. We demonstrate the effectiveness of the proposed method on both synthetic and real SAR images

A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems

Non-Local Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the non-local variations, jointly for the different components, through various $\ell_{1,p}$ matrix norms with $p \ge 1$. To facilitate the choice of the hyper-parameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented thanks to the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for multispectral and hyperspectral images. The results demonstrate the interest of introducing a non-local structure tensor regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods

Decomposition methods for machine learning with small, incomplete or noisy datasets

In many machine learning applications, measurements are sometimes incomplete or noisy resulting in missing features. In other cases, and for different reasons, the datasets are originally small, and therefore, more data samples are required to derive useful supervised or unsupervised classification methods. Correct handling of incomplete, noisy or small datasets in machine learning is a fundamental and classic challenge. In this article, we provide a unified review of recently proposed methods based on signal decomposition for missing features imputation (data completion), classification of noisy samples and artificial generation of new data samples (data augmentation). We illustrate the application of these signal decomposition methods in diverse selected practical machine learning examples including: brain computer interface, epileptic intracranial electroencephalogram signals classification, face recognition/verification and water networks data analysis. We show that a signal decomposition approach can provide valuable tools to improve machine learning performance with low quality datasets.Fil: Caiafa, César Federico. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto Argentino de Radioastronomía. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto Argentino de Radioastronomía; ArgentinaFil: Sole Casals, Jordi. Center for Advanced Intelligence; JapónFil: Marti Puig, Pere. University of Catalonia; EspañaFil: Sun, Zhe. RIKEN; JapónFil: Tanaka,Toshihisa. Tokyo University of Agriculture and Technology; Japó

Robust Algorithms for Low-Rank and Sparse Matrix Models

Data in statistical signal processing problems is often inherently matrix-valued, and a natural first step in working with such data is to impose a model with structure that captures the distinctive features of the underlying data. Under the right model, one can design algorithms that can reliably tease weak signals out of highly corrupted data. In this thesis, we study two important classes of matrix structure: low-rankness and sparsity. In particular, we focus on robust principal component analysis (PCA) models that decompose data into the sum of low-rank and sparse (in an appropriate sense) components. Robust PCA models are popular because they are useful models for data in practice and because efficient algorithms exist for solving them. This thesis focuses on developing new robust PCA algorithms that advance the state-of-the-art in several key respects. First, we develop a theoretical understanding of the effect of outliers on PCA and the extent to which one can reliably reject outliers from corrupted data using thresholding schemes. We apply these insights and other recent results from low-rank matrix estimation to design robust PCA algorithms with improved low-rank models that are well-suited for processing highly corrupted data. On the sparse modeling front, we use sparse signal models like spatial continuity and dictionary learning to develop new methods with important adaptive representational capabilities. We also propose efficient algorithms for implementing our methods, including an extension of our dictionary learning algorithms to the online or sequential data setting. The underlying theme of our work is to combine ideas from low-rank and sparse modeling in novel ways to design robust algorithms that produce accurate reconstructions from highly undersampled or corrupted data. We consider a variety of application domains for our methods, including foreground-background separation, photometric stereo, and inverse problems such as video inpainting and dynamic magnetic resonance imaging.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/143925/1/brimoor_1.pd