3,336 research outputs found
CS Decomposition Based Bayesian Subspace Estimation
In numerous applications, it is required to estimate the principal subspace of the data, possibly from a very limited number of samples. Additionally, it often occurs that some rough knowledge about this subspace is available and could be used to improve subspace estimation accuracy in this case. This is the problem we address herein and, in order to solve it, a Bayesian approach is proposed. The main idea consists of using the CS decomposition of the semi-orthogonal matrix whose columns span the subspace of interest. This parametrization is intuitively appealing and allows for non informative prior distributions of the matrices involved in the CS decomposition and very mild assumptions about the angles between the actual subspace and the prior subspace. The posterior distributions are derived and a Gibbs sampling scheme is presented to obtain the minimum mean-square distance estimator of the subspace of interest. Numerical simulations and an application to real hyperspectral data assess the validity and the performances of the estimator
Minimum mean square distance estimation of a subspace
We consider the problem of subspace estimation in a Bayesian setting. Since
we are operating in the Grassmann manifold, the usual approach which consists
of minimizing the mean square error (MSE) between the true subspace and its
estimate may not be adequate as the MSE is not the natural metric in
the Grassmann manifold. As an alternative, we propose to carry out subspace
estimation by minimizing the mean square distance (MSD) between and its
estimate, where the considered distance is a natural metric in the Grassmann
manifold, viz. the distance between the projection matrices. We show that the
resulting estimator is no longer the posterior mean of but entails
computing the principal eigenvectors of the posterior mean of .
Derivation of the MMSD estimator is carried out in a few illustrative examples
including a linear Gaussian model for the data and a Bingham or von Mises
Fisher prior distribution for . In all scenarios, posterior distributions
are derived and the MMSD estimator is obtained either analytically or
implemented via a Markov chain Monte Carlo simulation method. The method is
shown to provide accurate estimates even when the number of samples is lower
than the dimension of . An application to hyperspectral imagery is finally
investigated
Extrinsic Parameter Calibration for Line Scanning Cameras on Ground Vehicles with Navigation Systems Using a Calibration Pattern
Line scanning cameras, which capture only a single line of pixels, have been
increasingly used in ground based mobile or robotic platforms. In applications
where it is advantageous to directly georeference the camera data to world
coordinates, an accurate estimate of the camera's 6D pose is required. This
paper focuses on the common case where a mobile platform is equipped with a
rigidly mounted line scanning camera, whose pose is unknown, and a navigation
system providing vehicle body pose estimates. We propose a novel method that
estimates the camera's pose relative to the navigation system. The approach
involves imaging and manually labelling a calibration pattern with distinctly
identifiable points, triangulating these points from camera and navigation
system data and reprojecting them in order to compute a likelihood, which is
maximised to estimate the 6D camera pose. Additionally, a Markov Chain Monte
Carlo (MCMC) algorithm is used to estimate the uncertainty of the offset.
Tested on two different platforms, the method was able to estimate the pose to
within 0.06 m / 1.05 and 0.18 m / 2.39. We also propose
several approaches to displaying and interpreting the 6D results in a human
readable way.Comment: Published in MDPI Sensors, 30 October 201
Kernel Multivariate Analysis Framework for Supervised Subspace Learning: A Tutorial on Linear and Kernel Multivariate Methods
Feature extraction and dimensionality reduction are important tasks in many
fields of science dealing with signal processing and analysis. The relevance of
these techniques is increasing as current sensory devices are developed with
ever higher resolution, and problems involving multimodal data sources become
more common. A plethora of feature extraction methods are available in the
literature collectively grouped under the field of Multivariate Analysis (MVA).
This paper provides a uniform treatment of several methods: Principal Component
Analysis (PCA), Partial Least Squares (PLS), Canonical Correlation Analysis
(CCA) and Orthonormalized PLS (OPLS), as well as their non-linear extensions
derived by means of the theory of reproducing kernel Hilbert spaces. We also
review their connections to other methods for classification and statistical
dependence estimation, and introduce some recent developments to deal with the
extreme cases of large-scale and low-sized problems. To illustrate the wide
applicability of these methods in both classification and regression problems,
we analyze their performance in a benchmark of publicly available data sets,
and pay special attention to specific real applications involving audio
processing for music genre prediction and hyperspectral satellite images for
Earth and climate monitoring
Joint Bayesian endmember extraction and linear unmixing for hyperspectral imagery
This paper studies a fully Bayesian algorithm for endmember extraction and
abundance estimation for hyperspectral imagery. Each pixel of the hyperspectral
image is decomposed as a linear combination of pure endmember spectra following
the linear mixing model. The estimation of the unknown endmember spectra is
conducted in a unified manner by generating the posterior distribution of
abundances and endmember parameters under a hierarchical Bayesian model. This
model assumes conjugate prior distributions for these parameters, accounts for
non-negativity and full-additivity constraints, and exploits the fact that the
endmember proportions lie on a lower dimensional simplex. A Gibbs sampler is
proposed to overcome the complexity of evaluating the resulting posterior
distribution. This sampler generates samples distributed according to the
posterior distribution and estimates the unknown parameters using these
generated samples. The accuracy of the joint Bayesian estimator is illustrated
by simulations conducted on synthetic and real AVIRIS images
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
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