Generalized Morphological Component Analysis Research Articles

21-cm foregrounds and polarization leakage: cleaning and mitigation strategies

ABSTRACT The success of H i intensity mapping is largely dependent on how well 21-cm foreground contamination can be controlled. In order to progress our understanding further, we present a range of simulated foreground data from two different ∼3000 deg2 sky regions, with varying effects from polarization leakage. Combining these with cosmological H i simulations creates a range of intensity mapping test cases that require different foreground treatments. This allows us to conduct the most generalized study to date into 21-cm foregrounds and their cleaning techniques for the post-reionization era. We first provide a pedagogical review of the most commonly used blind foreground removal techniques [principal component analysis (PCA)/singular value decomposition (SVD), fast independent component analysis (FASTICA), and generalized morphological component analysis (GMCA)]. We also trial a non-blind parametric fitting technique and discuss potential hybridization of methods. We highlight the similarities and differences in these techniques finding that the blind methods produce near equivalent results, and we explain the fundamental reasons for this. Our results demonstrate that polarized foreground residuals should be generally subdominant to H i on small scales ($k\gtrsim 0.1\, h\, \text{Mpc}^{-1}$). However, on larger scales, results are more case dependent. In some cases, aggressive cleans severely damp H i power but still leave dominant foreground residuals. We find a changing polarization fraction has little impact on results within a realistic range (0.5–2 per cent); however, a higher level of Faraday rotation does require more aggressive cleaning. We also demonstrate the gain from cross-correlations with optical galaxy surveys, where extreme levels of residual foregrounds can be circumvented. However, these residuals still contribute to errors and we discuss the optimal balance between overcleaning and undercleaning.

The bispectrum and 21-cm foregrounds during the Epoch of Reionization

ABSTRACT Numerous studies have established the theoretical potential of the 21-cm bispectrum to boost our understanding of the Epoch of Reionization (EoR). We take a first look at the impact of foregrounds (FGs) and instrumental effects on the 21-cm bispectrum and our ability to measure it. Unlike the power spectrum for which (in the absence of instrumental effects) there is a window clear of smooth-spectrum FGs in which it may be detectable, there is no such ‘EoR window’ for the bispectrum. For the triangle configurations and scales we consider, the EoR structures are completely swamped by those of the FGs, and the EoR + FG bispectrum is entirely dominated by that of the FGs. By applying a rectangular window function on the sky combined with a Blackman–Nuttall filter along the frequency axis, we find that spectral, or in our case scale, leakage (caused by FFTing non-periodic data) suppresses the FG contribution so that cross-terms of the EoR and FGs dominate. While difficult to interpret, these findings motivate future studies to investigate whether filtering can be used to extract information about the EoR from the 21-cm bispectrum. We also find that there is potential for instrumental effects to seriously corrupt the bispectrum. FG removal using GMCA (generalized morphological component analysis) is found to recover the EoR bispectrum to a reasonable level of accuracy for many configurations. Further studies are necessary to understand the error and/or bias associated with FG removal before the 21-cm bispectrum can be practically applied in analysis of future data.

Recovery of 21-cm intensity maps with sparse component separation

ABSTRACT 21-cm intensity mapping has emerged as a promising technique to map the large-scale structure of the Universe. However, the presence of foregrounds with amplitudes orders of magnitude larger than the cosmological signal constitutes a critical challenge. Here, we test the sparsity-based algorithm generalized morphological component analysis (GMCA) as a blind component separation technique for this class of experiments. We test the GMCA performance against realistic full-sky mock temperature maps that include, besides astrophysical foregrounds, also a fraction of the polarized part of the signal leaked into the unpolarized one, a very troublesome foreground to subtract, usually referred to as polarization leakage. To our knowledge, this is the first time the removal of such component is performed with no prior assumption. We assess the success of the cleaning by comparing the true and recovered power spectra, in the angular and radial directions. In the best scenario looked at, GMCA is able to recover the input angular (radial) power spectrum with an average bias of ${\sim} 5{{\ \rm per\ cent}}$ for ℓ > 25 ($20\!-\!30 {{\ \rm per\ cent}}$ for $k_{\parallel } \gtrsim 0.02 \, h^{-1}$ Mpc), in the presence of polarization leakage. Our results are robust also when up to $40{{\ \rm per\ cent}}$ of channels are missing, mimicking a radio-frequency interference (RFI) flagging of the data. Having quantified the notable effect of polarization leakage on our results, in perspective we advocate the use of more realistic simulations when testing 21-cm intensity mapping capabilities.

Generalized Morphological Component Analysis for Hyperspectral Unmixing

Hyperspectral unmixing (HU) is an active research topic in the remote-sensing community. It aims at modeling mixed pixels using a collection of pure constituent materials ( endmembers ) weighted by their corresponding fractional abundances. Among existing unmixing schemes, nonnegative matrix factorization (NMF) has drawn significant attention due to its unsupervised nature, as well as its capacity to obtain both endmembers and fractional abundances simultaneously. In this article, we present a new blind unmixing method based on the generalized morphological component analysis (GMCA) framework, in which an additional constraint is introduced into the standard NMF model to represent the sparsity and morphological diversity of the abundance maps associated with each endmember. More specifically, we take into account the fact that different ground categories in a hyperspectral scene generally exhibit various spatial distributions and morphological characteristics. As a result, when providing a specific dictionary basis for these categories, their corresponding abundance maps (referred to as sources ) can be sparsely represented. In addition, due to the low correlation between different sources, their sparse representations will not share the same most significant coefficients. With this observation in mind, we can further promote source discrimination and separation in the unmixing process. Moreover, in order to obtain a stable solution of the involved optimization problem, we adopt an alternate iterative constrained algorithm with a threshold descent strategy. Our experiments, carried out on both synthetic and real hyperspectral scenes, reveal that our newly developed GMCA-based unmixing method obtains very promising results with fast convergence speed and requiring significantly less parameter tuning. This confirms the advantage of the proposed spatial morphological component approach for HU purposes.

A novel CMB component separation method: hierarchical generalized morphological component analysis

ABSTRACT We present a novel technique for cosmic microwave background (CMB) foreground subtraction based on the framework of blind source separation. Inspired by previous work incorporating local variation to generalized morphological component analysis (GMCA), we introduce hierarchical GMCA (HGMCA), a Bayesian hierarchical graphical model for source separation. We test our method on Nside = 256 simulated sky maps that include dust, synchrotron, free–free, and anomalous microwave emission, and show that HGMCA reduces foreground contamination by $25{{\ \rm per\ cent}}$ over GMCA in both the regions included and excluded by the Planck UT78 mask, decreases the error in the measurement of the CMB temperature power spectrum to the 0.02–0.03 per cent level at ℓ > 200 (and $\lt 0.26{{\ \rm per\ cent}}$ for all ℓ), and reduces correlation to all the foregrounds. We find equivalent or improved performance when compared to state-of-the-art internal linear combination type algorithms on these simulations, suggesting that HGMCA may be a competitive alternative to foreground separation techniques previously applied to observed CMB data. Additionally, we show that our performance does not suffer when we perturb model parameters or alter the CMB realization, which suggests that our algorithm generalizes well beyond our simplified simulations. Our results open a new avenue for constructing CMB maps through Bayesian hierarchical analysis.

Use of PALM for ℓ1 sparse matrix factorization: Difficulty and rationalization of a two-step approach

Blind Source Separation (BSS) is a key machine learning method, which has been successfully applied to analyze multichannel data in various domains ranging from medical imaging to astrophysics. Being an ill-posed matrix factorization problem, it is necessary to introduce extra regularizing priors on the sources. While using sparsity has led to improved factorization results, the quality of the separation process turns out to be dramatically dependent on the minimization strategy and the regularization parameters. In this scope, the Proximal Alternating Linearized Minimization (PALM) has recently attracted a lot of interest as a generic, fast and highly flexible algorithm. Using PALM for sparse BSS is theoretically well grounded, but getting good empirical results requires a fine tuning of the involved regularization parameters, which might be too computationally expensive with real-world large-scale data, therefore mandating automatic parameter choice strategies. In this article, we first investigate the empirical limitations of using the PALM algorithm to perform sparse BSS and we explain their origin. Based on this, we further study and justify an alternative two-step algorithmic framework combining PALM with a heuristic approach, namely the Generalized Morphological Component Analysis (GMCA). This method enables an automatic parameter choice for the PALM step. Numerical experiments with comparisons to standard algorithms are carried out on two realistic experiments in spectroscopy and astrophysics.

Kronecker least angle regression for unsupervised unmixing of hyperspectral imaging data

Spectral unmixing is an important data processing task that is commonly applied to hyperspectral imaging data. It uses a set of spectral pixels, i.e., multichannel data, to separate each spectral pixel, a multicomponent signal comprised of a linear mixture of pure spectral signatures, into its individual spectral signatures commonly known as endmembers. When no prior information about the required endmembers is available, the resulting unsupervised unmixing problem will be underdetermined, and additional constraints become necessary. A recent approach to solving this problem required that these endmembers be sparse in some dictionary. Sparse signal recovery is commonly solved using a basis pursuit optimization algorithm that requires specifying a data-dependent regularization parameter. Least angle regression (LARS) is a very efficient method to simultaneously solve the basis pursuit optimization problem for all relevant regularization parameter values. However, despite this efficiency of LARS, it has not been applied to the spectral unmixing problem before. This is likely because the application of LARS to large multichannel data could be very challenging in practice, due to the need for generation and storage of extremely large arrays (~ 1010 bytes in a relatively small area of spectral unmixing problem). In this paper, we extend the standard LARS algorithm, using Kronecker products, to make it suitable for practical and efficient recovery of sparse signals from large multichannel data, i.e., without the need to construct or process very large arrays, or the need for trial and error to determine the regularization parameter value. We then apply this new Kronecker LARS (K-LARS) algorithm to successfully achieve spectral unmixing of both synthetic and AVIRIS hyperspectral imaging data. We also compare our results to ones obtained using an earlier basis pursuit-based spectral unmixing algorithm, generalized morphological component analysis (GMCA). We show that these two results are similar, albeit our results were obtained without trial and error, or arbitrary choices, in specifying the regularization parameter. More important, our K-LARS algorithm could be a very valuable research tool to the signal processing community, where it could be used to solve sparse least squares problems involving large multichannel data.

Remote sensing image denoising application by generalized morphological component analysis

In this paper, we introduced a remote sensing image denoising method based on generalized morphological component analysis (GMCA). This novel algorithm is the further extension of morphological component analysis (MCA) algorithm to the blind source separation framework. The iterative thresholding strategy adopted by GMCA algorithm firstly works on the most significant features in the image, and then progressively incorporates smaller features to finely tune the parameters of whole model. Mathematical analysis of the computational complexity of GMCA algorithm is provided. Several comparison experiments with state-of-the-art denoising algorithms are reported. In order to make quantitative assessment of algorithms in experiments, Peak Signal to Noise Ratio (PSNR) index and Structural Similarity (SSIM) index are calculated to assess the denoising effect from the gray-level fidelity aspect and the structure-level fidelity aspect, respectively. Quantitative analysis on experiment results, which is consistent with the visual effect illustrated by denoised images, has proven that the introduced GMCA algorithm possesses a marvelous remote sensing image denoising effectiveness and ability. It is even hard to distinguish the original noiseless image from the recovered image by adopting GMCA algorithm through visual effect.

Speed-up generalized morphological component analysis technology used in remote sensing image inpainting application

In this paper, we introduced a remote sensing image inpainting method based on speed-up generalized morphological component analysis (SGMCA). Due to its capability to represent and separate the morphological diversities, generalized morphological component analysis (GMCA) algorithm is a state-of-the-art image inpainting method. SGMCA algorithm introduced in this paper can accelerate the iterative process of GMCA algorithm. By adding some more assumptions to GMCA algorithm, SGMCA algorithm is proven as a much faster algorithm which can handle very large-scale problems. Several experiments illustrate that SGMCA algorithm can recover the remote sensing images with different patterns of missing pixels. It is even hard to distinguish the original remote sensing image from the recovered image through visual effect. The peak signal to noise ratio and structural similarity indices explain why the salient visual effect is obtained, and confirm the marvelous inpainting capability of SGMCA algorithm. Quantitative analysis on time consumption proves that SGMCA algorithm can greatly improve the iterative speed of GMCA algorithm, indeed.

Reconstruction of the cosmic microwave background lensing forPlanck

<i>Aims. <i/>We prepare real-life cosmic microwave background (CMB) lensing extraction with the forthcoming <i>Planck<i/> satellite data by studying two systematic effects related to the foreground contamination: the impact of foreground residuals after a component separation on the lensed CMB map, and the impact of removing a large contaminated region of the sky.<i>Methods. <i/>We first use the generalized morphological component analysis (GMCA) method to perform a component separation within a simplified framework, which allows a high statistics Monte-Carlo study. For the second systematic, we apply a realistic mask on the temperature maps and then restore them with a recently developed inpainting technique on the sphere. We investigate the reconstruction of the CMB lensing from the resultant maps using a quadratic estimator in the flat sky limit and on the full sphere.<i>Results. <i/>We find that the foreground residuals from the GMCA method does not significantly alter the lensed signal, which is also true for the mask corrected with the inpainting method, even in the presence of point source residuals.

Hyperspectral BSS Using GMCA With Spatio-Spectral Sparsity Constraints

Generalized morphological component analysis (GMCA) is a recent algorithm for multichannel data analysis which was used successfully in a variety of applications including multichannel sparse decomposition, blind source separation (BSS), color image restoration and inpainting. Building on GMCA, the purpose of this contribution is to describe a new algorithm for BSS applications in hyperspectral data processing. It assumes the collected data is a mixture of components exhibiting sparse spectral signatures as well as sparse spatial morphologies, each in specified dictionaries of spectral and spatial waveforms. We report on numerical experiments with synthetic data and application to real observations which demonstrate the validity of the proposed method.

SZ and CMB reconstruction using generalized morphological component analysis

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.

Sparsity and Morphological Diversity in Blind Source Separation

Over the last few years, the development of multichannel 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-caIled 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 emergedas a novel and effective source of diversity for BSS. Here, we give some new and 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 introduces a new 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 BSS method. We present arguments and a discussion supporting the convergence of the GMCA algorithm. Numerical results in multivariate image and signal processing are given illustrating the good performance of GMCA and its robustness to noise.