scm whispers 2015
TRANSCRIPT
A Spatial Compositional
Model for Linear Unmixing
Yuan Zhou, Anand Rangarajan and Paul D. GaderDepartment of CISE, University of Florida
Motivation
Hyperspectral image unmixing are important in many applications. Very little previous work on simultaneously:
Incorporating spatial information in the pixel likelihood. Modeling the uncertainty in the extracted endmembers.
Spatial compositional model (SCM) proposed for this purpose.
Linear mixing model (LMM) Spectral measurement ( , ) at wavelength , location in the 𝑔 𝒙 𝜆 𝜆 𝒙
image domain
𝑀 is the number of endmembers, is the spectral signature of the ith endmember, is the fractional abundance map of the ith endmember satisfying thepositivity and sum-to-one constraints,
( , ) is a noise field (usually modeled as Gaussian).𝑛 𝒙 𝜆
Linear mixing model: discretized version The discretized version of the ith pixel can be represented by
, , , . Combining this at all locations
,, , .
The uncertainty of endmembers One area of LMM with little previous work is to discover the model
uncertainty of the endmembers.
What are the uncertainties for the 3 endmember sets?
Endmember variability vs. endmember uncertainty
Endmember uncertainty: all the pixels are generated by the same endmember set.
Suppose B = 1. , , .
indicates that the density function of does not even exist.
Add some noise
The covariance matrix is not block diagonal.
Endmember variability: each pixel is generated by a different endmember set.
Let indicates
Assume pixel independence
The covariance matrix is block diagonal.
This is what we want to find
NCM
SCM
SCM: Advantages SCM’s features:
Estimating the uncertainty along with the endmembers and abundances by calculating the full likelihood.
A smoothness prior on abundances to help unmixing. A simple and efficient algorithm.
SCM: Density function assumptions We assume that
The probability function of the whole endmember set becomes
. Assume the noise follows
SCM: Spatial random variable transformations
The random variable transformation of indicates
If , it can be simplified as
is a diagonal matrix with diagonal elements . The estimation of full endmember covariance matrices will be
discussed later.
Not block diagonal
SCM: The prior on the abundances The prior probability for can be defined by
controls the spatial intimacy between node i and node j , is the symmetric positive semidefinite graph Laplacian matrix.
We can use
when node i and node j are neighbors and 0 otherwise.
SCM: The prior on the endmembers We use the following prior for parameters :
denotes the ith row of , denotes the kth column of . for all i and j. is 1 when and 0 otherwise.
Similar to , the prior can be written as
and are the corresponding Laplacian matrices.
SCM: Maximization of the posterior Using Bayes’ theorem,
Maximizing the posterior becomes minimizing
subject to
where and is a symmetric positive definite matrix. The unknowns are , , (), ().
The optimization is done by setting the derivatives w.r.t. the unknowns to 0 and alternately updating the solutions.
The extension to full endmember covariance matrices will be discussed later.
Pavia University: comparison of abundance maps
(a) SCM (b) NCMWithout the spatial prior in SCM
Which material does this abundance map correspond to?
Extension to full covariance matrices We extend the work to estimate the full endmember covariance
matrices, without assuming it is diagonal. The objective function turns out to be
subject to
where
The unknowns are .
Extension to full covariance matrices:Synthetic image uncertainty range
We use the full endmember covariance matrices here
Extension to full covariance matrices:Pavia University uncertainty range
We take the square root of the largest eigenvalue, , the corresponding eigenvector to calculate the uncertainty range .
Discussion
The features of SCM include estimating the uncertainty along with the endmembers and abundances.
The spatial prior with weighted smoothness leads to better unmixing. The objective function enables a simple and efficient algorithm, e.g. it
takes about 2 minutes to process the Pavia University dataset on a laptop.
The extended work on full endmember covariance matrices implies that the uncertainty estimation leads to error prediction.
We want to thank Jose Bioucas-Dias and Alina Zare for helpful conversations. We acknowledge support from NSF IIS 1065081.