STATISTICAL MODELING OF HYPERSPECTRAL DATA USING MARKOV RANDOM FIELDS YAHYA M. MASALMAH , Prof. MIGUEL VELEZ-REYES University of Puerto Rico Mayaguez Campus GOALS Hyperspectral imagery provides high spectral and spatial resolution that can be used to discriminate between object and natural clutter in environmental monitoring applications such as coastal and coral reef monitoring. In this research, we are looking at developing algorithms for parameter estimation for coastal monitoring application using CenSSIS physics-based approach. Here we are studying the integration of statistical models for 3D random variables such as Gaussian Markov Random Fields (GMRF) and radiative transfer models of light in water for developing improved environmental parameter estimation algorithms.
INTRODUCTION Hyperspectral sensors collect hundreds of narrow and contiguously spaced spectral bands of data organized in the so called hyperspectral cube, Fig. 1. Hyperspectral imagery provides fully registered spatial and high resolution spectral information that is invaluable in discriminating between objects and natural clutter backgrounds, since the objects and the clutter have unique spectral signatures that are captured by the data. Our focus is on using hyperspectral sensor data for the estimation of parameters of objects in natural clutter backgrounds. In this research we are looking into the development of parameter estimation algorithm and estimate the desired spectral signature or parameters of the observed objects using spatially–spectrally correlated data models and physical models relating parameters of interest and the recovered signature.
Underlying Assumptions
Spatial –spectral algorithms Those algorithms that use both spatial and spectral features from the hyperspectral imagery. In [6], an adaptive spatial/spectral detection method is presented in which it is assumed that the clutter is fully spatially and spectrally correlated. The method uses a separate spatial matched filter for each spectral band to adapt to the clutter before using maximum likelihood (ML) ratio to perform the detection. The algorithm requires taking the inverse of a spectral covariance matrix which has dimensions equal to the number of spectral bands used for processing. Another approach which employs Gauss-Markov random fields [7] was introduced as an extension of non-causal Gauss Markov Fields [8] to hyperspectral imagery. An adaptive anomaly detection algorithm that is computationally efficient and exhibits a low false alarm rate with high detection probability was designed. In this work , a 3-D GMRF was employed to construct a covariance matrix with simple parameterization.
APPROACH Each pixel in the hyperspectral image can be represented by two spatial and one spectral indices. In a stochastic representation, the hyperspectral image is considered to be a sample function of an array of random variables (RVs) called a random field(RF) [1].
Location (i,j,k) N
MOTIVATION Many of the existing estimation/detection algorithms in hyperspectral imagery were developed for multispectral imagery which has few spectral bands and low spectral resolution. Hyperspectral imagery (HSI) has hundreds of high spectral resolution bands collected over a given spectral region. The high spectral resolution available from hyperspectral sensors motivate the idea of looking into statistical models that can take advantage of the spatial and spectral information in HSI and combining it with forward models to implement the physics-based signal processing philosophy underlying CENSSIS approach for estimation of parameters of interest in coastal monitoring and other environmental and biomedical applications considering the use of HSI.
STATE OF THE ART In the literature, algorithms to process hyperspectral imagery can be classified in three categories: only spectral, spectral initially and then spatial for post processing purposes, and spatial-spectral algorithms.
Spectral - Only algorithms In this type of algorithms , only spectral information from hyperspectral data is being used in the attempt of detecting targets of interest. They can be either linear mixture model methods [2] or statistical measurement approaches, like the spectral matched filter (SMF) [3], the spectral angle mapper (SAM) [4], and principal component analysis (PCA)[5] . SMF and SAM algorithms depend on knowing a priori, the spectral signature of the target of interest. SMF, basically , processes the data by correlating the known signature with the signature at every pixel in the data set, while SAM approach measures the angle between the signature of interest and the signature at each pixel. Those locations for which the signature is most highly correlated, or has the smallest angle, are considered targets. Both algorithms are limited because they ignore the spatial information , and they don’t account for variations in the target signature due to atmospheric and illumination effects. PCA is used prior to another detection algorithm for purposes of reducing the dimensionality of the data set.
A Gauss-Markov random field (GMRF) model have been proposed in [7] using the nearest spectral and spatial neighbors that results in a covariance matrix parameterized by four parameters. Unlike previous algorithms which are limited to certain number of spectral band , this model works for any number of spectral bands since no inversion of large matrices is required. To simplify our initial work we are considering the problem of estimating the spectral response of an object obscured by additive clutter. Estimating the spectral response requires removing the clutter effect from the each pixel spectral value. The problem of interest can be formulated as a hyperspectral image with two different regions: clutter region (ie. sand or turbid water), and unknown region (ie. coral). The pixel value in the second region is a mixture of both target and clutter. The clutter needs to be modelled statistically and both spatial and spectral interaction should be considered. The processing window should be separated into two regions: Clutter region and an interior unknown region of dimension Nt x Nt see figure 4 below. The algorithm is summarized in the diagram shown in figure 3 below. Data
I Clutter Mean Estimation& removal
Wij(k+1) Wi(j+1)k
W(i+1)jk
Fig. 6 neighboring system
Correlated Clutter Model w = βv (w +w )+ ijk (i−1)jk (i+1)jk
β (w +w ) h (i−1)jk (i+1)jk
+ βs (w +w ) + ε } ⇒ Aw = ε ij(k−1) ij(k+1) ijk βh , βv , βs are the predictor coefficients, or potentials, A is the potential matrix, W is the clutter vector, Є is the modeling error
Model Structure
Clutter Vector
Potential Matrix A1 A 2 A= 0 0 0
A2
0
0
A1 A2 0 0
A2 . . 0
0 . . A2
=
0 0 0 A2 A 1
• A is highly structured mean Gaussian colored noise
,
W is NiNjNkx1 є is NiNjNk x1 and is a sample from a zeroprocesses with ∑w=б2 A.
A 1 = INi ⊗ B + H1Ni ⊗ C A 2 = INi ⊗ D − =
D = −βsINj
The inverse of the clutter covariance matrix is simply a scaled version of A
Clutter Area : Z q = Yq - mx = W
1≤ q ≤ P
Target Area : Z q = Yq - mx = W + Rq where Yq is a column vector obtained by operating vec on a Markov window in the unknown area, P is the number of Markov windows in the unknown region, and mx is the clutter mean obtained by averaging all the veced Markov windows in the clutter region. Parameter Estimation Techniques
βh , βv , βs , and б2 can be estimated using one of the following techniques: • Maximum - Likelihood (ML) •Least Squares (LS) •Approximate Maximum – Likelihood (AML)
CURRENT STATUS
“unknown”
Nt J
Parameter Estim.β
Spectral Signature Estimation R=F(λ,α)
Nt
Wijk
Currently we are working on testing the algorithm with synthetic images generated using the autoregressive model defined above using different sigma square values(0.01,0.1,1.0) . Some images are shown below classified by Euclidean distance, maximum likelihood, mahalanobis distance.
“Clutter”
Clutter Modeling
Wij(k-1)
and stationary within processing windows
C = −βvINj
Fig.2 3-D Lattice
Imagery- Nasa Web site
•2nd order statistics of clutter are unknown
B = −βhH1Nj + INj
K
Wi(j-1)k
•Modeled by a non-causal 3-D GMRF
A = I Nk ⊗ A1 + H1Nk ⊗ A2
M
Fig 1. Hyperspectral
W(i-1)jk
•Clutter is spatially – spectrally correlated
Fig. 4 processing window
Object Detection Fig. 7 synthetic images generated using the autoregressive model with betas(0.2,0.1,0.2)
Physical parameter Estimation, αml=f-1(R)ml
and sigma squared (.01,0.1,1) and classified by different classifiers
Future Plans The model will be validated using: synthetic data generated with Radiative Transfer code and experimental data from Sea Bed.
Fig. 3 Algorithm block diagram
References 1.Anil K. Jain. Fundamentals of Digital Image Processing. Prentice Hall, 1989. Chapter 2: Two Dimensional Systems and Mathematical Preliminaries. 2.Chein-I Chang, Xiao-Li Zhao, Mark L. G. Althouse, and Jeng Jong Pan. Least squares subspace projection approach to mixed pixel classification for hyperspectral images. IEEE Transactions on Geoscience and Remote Sensing , 36(3):898 – 912, May 1998. 3.E. Crist , C. Schwartz, and A. Stocker. Pairwise adaptive linear matched filter algorithm , January 1999. I
4.Hanna T. Hasket and Arun K. Sood. Adaptive real-time endmember selection algorithm for sub-pixel target detection using hyperspectral data. Proceedings of the 1997 IRIS Specialty Group on Camouflag, Concealment and Deception, October 1997. 5.Jim Breckenridge and Rob Stribl. Direct principal components algorithm, January 1999. 6.Charles F. Ferrara. Adaptive spatial/spectral detection of subpixel targets with unknown spectral characteristics. SPIE: Signal and Data Processing of small targets, 2235:82- 93,1994. 7.A. M. Schweizer and Jose M. F. Moura. Efficient detection in hyperspectral imagery. IEEE Transactions on image processing , 10(4) :584-597 April 2001. 8.Nikhil Balram and Jose M. F. Moura. Noncausal Gauss-Markov random field and Transactions on Information Theory, 39(4):1333-1355, July 1993.
parameter structure and estimation. IEEE
9.S. M. Schweizer , The GMRF detector for hyperspectral imagery : An efficient fully – adaptive maximum likelihood detector. Ph.D. dissertation, Carnegie Mellon Univ. , Pittsburgh, PA, Dec. 1999.
Acknowledgements This work was supported by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the Engineering Research Centers Program of the National Science Foundation (Award Number EEC-9986821).
Fig. 5 Processing Window divided into independent Markov Windows
CONTACT : Yahya M. Masalmah,
[email protected] 