Error Analysis of Low-rank Three-Way Tensor Factorization Approach to Blind Source Separation

International audienceIn tensor factorization approaches to blind separation of multidimensional sources, two formulas for calculating the source tensor have emerged. In practice, it is observed that these two schemes exhibit different levels of robustness against perturbations of the factors involved in the tensor model. Motivated by both practical reasons and the will to better figure this out, we present error analyses in source tensor estimation performed by low-rank factorization of three-way tensors. To that aim, computer simulations as well as the analytical calculation of the theoretical error are carried out. The conclusions drawn from these numerical and analytical error analyses are supported by the results obtained thanks to tensor-based blind decomposition of an experimental multispectral image of a skin tumor

Bayesian Nonparametric Unmixing of Hyperspectral Images

Hyperspectral imaging is an important tool in remote sensing, allowing for accurate analysis of vast areas. Due to a low spatial resolution, a pixel of a hyperspectral image rarely represents a single material, but rather a mixture of different spectra. HSU aims at estimating the pure spectra present in the scene of interest, referred to as endmembers, and their fractions in each pixel, referred to as abundances. Today, many HSU algorithms have been proposed, based either on a geometrical or statistical model. While most methods assume that the number of endmembers present in the scene is known, there is only little work about estimating this number from the observed data. In this work, we propose a Bayesian nonparametric framework that jointly estimates the number of endmembers, the endmembers itself, and their abundances, by making use of the Indian Buffet Process as a prior for the endmembers. Simulation results and experiments on real data demonstrate the effectiveness of the proposed algorithm, yielding results comparable with state-of-the-art methods while being able to reliably infer the number of endmembers. In scenarios with strong noise, where other algorithms provide only poor results, the proposed approach tends to overestimate the number of endmembers slightly. The additional endmembers, however, often simply represent noisy replicas of present endmembers and could easily be merged in a post-processing step

HALS-based NMF with Flexible Constraints for Hyperspectral Unmixing

International audienceIn this article, the hyperspectral unmixing problem is solved with the nonnegative matrix factorization (NMF) algorithm. The regularized criterion is minimized with a hierarchical alternating least squares (HALS) scheme. Under the HALS framework, four constraints are introduced to improve the unmixing accuracy, including the sum-to-unity constraint, the constraints for minimum spectral dispersion and maximum spatial dispersion, and the minimum volume constraint. The derived algorithm is called F-NMF, for NMF with flexible constraints. We experimentally compare F-NMF with different constraints and combined ones. We test the sensitivity and robustness of F-NMF to many parameters such as the purity level of endmembers, the number of endmembers and pixels, the SNR, the sparsity level of abundances, and the overestimation of endmembers. The proposed algorithm improves the results estimated by vertex component analysis. A comparative analysis on real data is included. The unmixing results given by a geometrical method, the simplex identification via split augmented Lagrangian and the F-NMF algorithms with combined constraints are compared, which shows the relative stability of F-NMF

Robust unmixing of tumor states in array comparative genomic hybridization data

Motivation: Tumorigenesis is an evolutionary process by which tumor cells acquire sequences of mutations leading to increased growth, invasiveness and eventually metastasis. It is hoped that by identifying the common patterns of mutations underlying major cancer sub-types, we can better understand the molecular basis of tumor development and identify new diagnostics and therapeutic targets. This goal has motivated several attempts to apply evolutionary tree reconstruction methods to assays of tumor state. Inference of tumor evolution is in principle aided by the fact that tumors are heterogeneous, retaining remnant populations of different stages along their development along with contaminating healthy cell populations. In practice, though, this heterogeneity complicates interpretation of tumor data because distinct cell types are conflated by common methods for assaying the tumor state. We previously proposed a method to computationally infer cell populations from measures of tumor-wide gene expression through a geometric interpretation of mixture type separation, but this approach deals poorly with noisy and outlier data

Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Case

In blind hyperspectral unmixing (HU), the pure-pixel assumption is well-known to be powerful in enabling simple and effective blind HU solutions. However, the pure-pixel assumption is not always satisfied in an exact sense, especially for scenarios where pixels are heavily mixed. In the no pure-pixel case, a good blind HU approach to consider is the minimum volume enclosing simplex (MVES). Empirical experience has suggested that MVES algorithms can perform well without pure pixels, although it was not totally clear why this is true from a theoretical viewpoint. This paper aims to address the latter issue. We develop an analysis framework wherein the perfect endmember identifiability of MVES is studied under the noiseless case. We prove that MVES is indeed robust against lack of pure pixels, as long as the pixels do not get too heavily mixed and too asymmetrically spread. The theoretical results are verified by numerical simulations