Morphological Component Analysis

In power quality monitoring systems, a large volume of data is generated. A compression method should be used to perform the storage of data more efficiently. The present study proposes a compression method of power quality disturbances based on independent component analysis, Fast Fourier Transform and an adaptive threshold obtained using mathematical morphology. The independent component analysis algorithm returns three statistically independent components among themselves. Considering the noise and disturbances are statistically independent of the fundamental component, the independent component analysis algorithm isolates the noise and disturbances in the third component, where a threshold is applied, while the first and second components are presented to the Fast Fourier Transform in order to characterize the fundamental component. The results were compared with a Wavelet-based method in which the proposed method achieved compression rates slightly higher than the Wavelet-based method.

The research context of this work is dynamic texture analysis and characterization. Many dynamic textures can be modeled as a large scale propagating wave and local oscillating phenomena. The Morphological Component Analysis algorithm (MCA) is used to retrieve these components using a well chosen dictionary. We define a new strategy for adaptive thresholding in the MCA framework, which greatly reduces the computation time when applied on videos. Tests on synthetic and real image sequences illustrate the efficiency of the proposed method and future prospects are finally exposed.

The research context of this work is dynamic texture analysis and characterization. Many dynamic textures can be modeled as large scale propagating wavefronts and local oscillating phenomena. After introducing a formal model for dynamic textures, the Morphological Component Analysis (MCA) approach with a well chosen dictionary is used to retrieve the components of dynamic textures. We define two new strategies for adaptive thresholding in the MCA framework, which greatly reduce the computation time when applied on videos. Tests on real image sequences illustrate the efficiency of the proposed method. An application to global motion estimation is proposed and future prospects are finally exposed.

This paper contains the study on a hybrid algorithm combining Matching Pursuit (MP) and wavelet shrinkage. In this algorithm, we propose to shrink the scalar product of the element that best correlates with the residue, before modifying the residue. The study concerns for a rather broad family of shrinkage functions. Using simple properties of these shrinkage functions, we show that the algorithm converges towards the orthogonal projection of the data on the linear space generated by the dictionary, modulo a precision characterized by the shrinkage function. Finally, under a mild assumption on the shrinkage function (for instance, the hard shrinkage satisfies this assumption), this algorithm converges in a finite time which can be estimated from the properties of the shrinkage function. Experimental results demonstrate that in the presence of noise, our new algorithm outperforms the original MP algorithm when the the correlation of dictionary with the noise is not negligible.

This article proposes a new method for image separation into a linear combination of morphological components. This method is applied to decompose an image into meaningful cartoon and textural layers and is used to solve more general inverse problems such as image inpainting. For each of these components, a dictionary is learned from a set of exemplar images. Each layer is characterized by a sparse expansion in the corresponding dictionary. The separation inverse problem is formalized within a variational framework as the optimization of an energy functional. The morphological component analysis algorithm allows to solve iteratively this optimization problem under sparsity-promoting penalties. 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.

The Morphological Component Analysis (MCA) is a a new method which allows us to separate features contained in an image when these features present different morphological aspects. We show that MCA can be very useful for decomposing images into texture and piecewise smooth (cartoon) parts or for inpainting applications. We extend MCA to a multichannel MCA (MMCA) for analyzing multispectral data and present a range of examples which illustrates the results.

This paper describes a new method for blind source separation, adapted to the case of sources having different morphologies. We show that such morphological diversity leads to a new and very efficient separation method, even in the presence of noise. The algorithm, coined MMCA (Multichannel Morphological Component Analysis), is an extension of the Morphological Component Analysis method (MCA). The latter takes advantage of the sparse representation of structured data in large overcomplete dictionaries to separate features in the data based on their morphology. MCA has been shown to be an efficient technique in such problems as separating an image into texture and piecewise smooth parts or for inpainting applications. The proposed extension, MMCA, extends the above for multichannel data, achieving a better source separation in those circumstances. Furthermore, the new algorithm can efficiently achieve good separation in a noisy context where standard ICA methods fail. The efficiency of the proposed scheme is confirmed in numerical experiments.

In the last decade, the study of cosmic microwave background (CMB) data has become one of the most powerful tools for studying and understanding the Universe. More precisely, measuring the CMB power spectrum leads to the estimation of most cosmological parameters. Nevertheless, accessing such precious physical information requires extracting several different astrophysical components from the data. Recovering those astrophysical sources (CMB, Sunyaev-Zel'dovich clusters, galactic dust) thus amounts to a component separation problem which has already led to an intensive activity in the field of CMB studies. In this paper, we introduce a new sparsity-based component separation method coined Generalized Morphological Component Analysis (GMCA). The GMCA approach is formulated in a Bayesian maximum a posteriori (MAP) framework. Numerical results show that this new source recovery technique performs well compared to state-of-the-art component separation methods already applied to CMB data.

Recently morphological diversity and sparsity have emerged as new and effective sources of diversity for Blind Source Separation. Based on these new concepts, novel methods such as Generalized Morphological Component Analysis have been put forward. The latter takes advantage of the very sparse representation of structured data in large overcomplete dictionaries, to separate sources based on their morphology. Building on GMCA, the purpose of this contribution is to describe a new algorithm for hyperspectral data processing. Large-scale hyperspectral data refers to collected data that exhibit sparse spectral signatures in addition to sparse spatial morphologies, in specified dictionaries of spectral and spatial waveforms. Numerical experiments are reported which demonstrate the validity of the proposed extension for solving source separation problems involving hyperspectral data.

Over the last few years, the development of multi-channel sensors motivated interest in methods for the coherent processing of multivariate data. Some specific issues have already been addressed as testified by the wide literature on the so-called blind source separation (BSS) problem. In this context, as clearly emphasized by previous work, it is fundamental that the sources to be retrieved present some quantitatively measurable diversity. Recently, sparsity and morphological diversity have emerged as a novel and effective source of diversity for BSS. We give here some essential insights into the use of sparsity in source separation and we outline the essential role of morphological diversity as being a source of diversity or contrast between the sources. This paper overviews a sparsity-based BSS method coined Generalized Morphological Component Analysis (GMCA) that takes advantages of both morphological diversity and sparsity, using recent sparse overcomplete or redundant signal representations. GMCA is a fast and efficient blind source separation method. In remote sensing applications, the specificity of hyperspectral data should be accounted for. We extend the proposed GMCA framework to deal with hyperspectral data. In a general framework, GMCA provides a basis for multivariate data analysis in the scope of a wide range of classical multivariate data restorate. Numerical results are given in color image denoising and inpainting. Finally, GMCA is applied to the simulated ESA/Planck data. It is shown to give effective astrophysical component separation.

This 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.

Over 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.

We outline digital implementations of two newly developed multiscale representation systems, namely, the ridgelet and curvelet transforms. We apply these digital transforms to the problem of restoring an image from noisy data and compare our results with those obtained via well established methods based on the thresholding of wavelet coefficients. We show that the curvelet transform allows us also to well enhance elongated features contained in the data. Finally, we describe the Morphological Component Analysis, which consists in separating features in an image which do not present the same morphological characteristics. A range of examples illustrates the results.

To remove artifacts from multi-channel Electroencephalography (EEG) data, we propose the use of Generalized Morphological Component Analysis (GMCA). GMCA separates the EEG signals into sources that have different morphological characteristics. Each source is sparse in an overcomplete dictionary, which is constructed using discrete cosine transform, Daubechies wavelet basis and Dirac basis. The sources related to artifacts are then removed. Semi-simulated EEG signals of movement-related potentials trials contaminated by eye-blink and muscle artifacts are used to evaluate the algorithm's performance. The performance of GMCA is compared with those of two other blind source separation algorithms, AMUSE and EFICA. The results demonstrate that GMCA successfully removes artifacts from EEG signals and the resulting distortions in both time and frequency domains are significantly lower than those of the other algorithms.

To reduce the effects of artifacts in electroencephalography (EEG), we propose the use of Morphological Component Analysis (MCA). Taking advantage of the sparse representation of data in overcomplete dictionaries, MCA decomposes EEG signals into parts that have different morphological characteristics. For denoising purpose, the parts related to artifacts are removed. An overcomplete dictionary is constructed using the discrete cosine transform, Daubechies wavelet basis, and Dirac basis. Movement-related potentials (MRP) and EEG signals contaminated by spikes, eye-blinks, and muscle artifacts caused by eyebrow raising are used to evaluate the performance of the method. The results demonstrate that MCA can be used to decompose the single-channel EEG signals into artifacts and MRP components. The correlation coefficient between the denoised MRP and the original MRP using MCA is significantly higher than that obtained using stationary wavelet transform.