269 research outputs found
Fast and Robust Recursive Algorithms for Separable Nonnegative Matrix Factorization
In this paper, we study the nonnegative matrix factorization problem under
the separability assumption (that is, there exists a cone spanned by a small
subset of the columns of the input nonnegative data matrix containing all
columns), which is equivalent to the hyperspectral unmixing problem under the
linear mixing model and the pure-pixel assumption. We present a family of fast
recursive algorithms, and prove they are robust under any small perturbations
of the input data matrix. This family generalizes several existing
hyperspectral unmixing algorithms and hence provides for the first time a
theoretical justification of their better practical performance.Comment: 30 pages, 2 figures, 7 tables. Main change: Improvement of the bound
of the main theorem (Th. 3), replacing r with sqrt(r
Hyperspectral Unmixing Overview: Geometrical, Statistical, and Sparse Regression-Based Approaches
Imaging spectrometers measure electromagnetic energy scattered in their
instantaneous field view in hundreds or thousands of spectral channels with
higher spectral resolution than multispectral cameras. Imaging spectrometers
are therefore often referred to as hyperspectral cameras (HSCs). Higher
spectral resolution enables material identification via spectroscopic analysis,
which facilitates countless applications that require identifying materials in
scenarios unsuitable for classical spectroscopic analysis. Due to low spatial
resolution of HSCs, microscopic material mixing, and multiple scattering,
spectra measured by HSCs are mixtures of spectra of materials in a scene. Thus,
accurate estimation requires unmixing. Pixels are assumed to be mixtures of a
few materials, called endmembers. Unmixing involves estimating all or some of:
the number of endmembers, their spectral signatures, and their abundances at
each pixel. Unmixing is a challenging, ill-posed inverse problem because of
model inaccuracies, observation noise, environmental conditions, endmember
variability, and data set size. Researchers have devised and investigated many
models searching for robust, stable, tractable, and accurate unmixing
algorithms. This paper presents an overview of unmixing methods from the time
of Keshava and Mustard's unmixing tutorial [1] to the present. Mixing models
are first discussed. Signal-subspace, geometrical, statistical, sparsity-based,
and spatial-contextual unmixing algorithms are described. Mathematical problems
and potential solutions are described. Algorithm characteristics are
illustrated experimentally.Comment: This work has been accepted for publication in IEEE Journal of
Selected Topics in Applied Earth Observations and Remote Sensin
Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search are Related
This paper considers a recently emerged hyperspectral unmixing formulation
based on sparse regression of a self-dictionary multiple measurement vector
(SD-MMV) model, wherein the measured hyperspectral pixels are used as the
dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is
special in that it allows simultaneous identification of the endmember spectral
signatures and the number of endmembers. Previous SD-MMV studies mainly focus
on convex relaxations. In this study, we explore the alternative of greedy
pursuit, which generally provides efficient and simple algorithms. In
particular, we design a greedy SD-MMV algorithm using simultaneous orthogonal
matching pursuit. Intriguingly, the proposed greedy algorithm is shown to be
closely related to some existing pure pixel search algorithms, especially, the
successive projection algorithm (SPA). Thus, a link between SD-MMV and pure
pixel search is revealed. We then perform exact recovery analyses, and prove
that the proposed greedy algorithm is robust to noise---including its
identification of the (unknown) number of endmembers---under a sufficiently low
noise level. The identification performance of the proposed greedy algorithm is
demonstrated through both synthetic and real-data experiments
Non-convex regularization in remote sensing
In this paper, we study the effect of different regularizers and their
implications in high dimensional image classification and sparse linear
unmixing. Although kernelization or sparse methods are globally accepted
solutions for processing data in high dimensions, we present here a study on
the impact of the form of regularization used and its parametrization. We
consider regularization via traditional squared (2) and sparsity-promoting (1)
norms, as well as more unconventional nonconvex regularizers (p and Log Sum
Penalty). We compare their properties and advantages on several classification
and linear unmixing tasks and provide advices on the choice of the best
regularizer for the problem at hand. Finally, we also provide a fully
functional toolbox for the community.Comment: 11 pages, 11 figure
Dictionary-based Tensor Canonical Polyadic Decomposition
To ensure interpretability of extracted sources in tensor decomposition, we
introduce in this paper a dictionary-based tensor canonical polyadic
decomposition which enforces one factor to belong exactly to a known
dictionary. A new formulation of sparse coding is proposed which enables high
dimensional tensors dictionary-based canonical polyadic decomposition. The
benefits of using a dictionary in tensor decomposition models are explored both
in terms of parameter identifiability and estimation accuracy. Performances of
the proposed algorithms are evaluated on the decomposition of simulated data
and the unmixing of hyperspectral images
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