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Remote Sensing Remote Sensing & &
Image Processing GroupImage Processing Group
A Novel Anomaly Detection Scheme A Novel Anomaly Detection Scheme for Hyperspectral Images for Hyperspectral Images
Based on a Non-Gaussian Mixture ModelBased on a Non-Gaussian Mixture Model
Tiziana VeraciniTiziana Veracini, Marco Diani, Giovanni Corsini
Dipartimento di Ingegneria dell’Informazione, Università di Pisavia G. Caruso 16, 56122 Pisa, Italy
2010 IEEE GOLD REMOTE SENSING CONFERENCE2010 IEEE GOLD REMOTE SENSING CONFERENCE29, 30 april 201029, 30 april 2010
Accademia Navale, Livorno, ItalyAccademia Navale, Livorno, Italy
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group2
OutlineOutline
Mixture of Student’s t distributionsMixture of Student’s t distributions. Student’s t distribution Mixture Model (StMM) PDF
is assumed and its estimation is carried out within a Bayesian model selection.
Background Background distributiondistribution
Experimental results and conclusions.Experimental results and conclusions. Data set description. Design of the experiments and results.
Experimental Experimental resultsresults
Anomaly detection strategy: GLRT.Anomaly detection strategy: GLRT. The AD scheme adopts a background Probability
Density Function (PDF) estimation in conjunction with the Generalized Likelihood Ratio Test (GLRT).
Anomaly Anomaly detectiondetection
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group3
Hyperspectral remote sensingHyperspectral remote sensing
The high spectral resolution of hyperspectral sensors has made it possible the distinction among different types of materials on the scene.
Spatial dimension(across the flight
line swath)
Spectraldimension
Image data cube
Imaging spectrometer
Wavelength [μm]
0.40
2.5
1
SpatialDimension
(along the flight line)
Refle
ctan
ceHyperspectral remote sensing applications
Earth observation
- Terrain classification- Land use management- Environmental monitoring- Wide-area surveillance
…
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group4
Anomaly Detection (AD) strategyAnomaly Detection (AD) strategy
Anomaly detection Anomaly detection algorithms explore the image to detect pixels whose spectral content is significantly different from that of background ones, without any previous knowledge of the objects of interest.
The statistical approach to anomaly detection is to solve a binary hypothesis hypothesis testing problemtesting problem:
Where x is a realization of the random vector X used for modeling the pixel under test, whereas H0 and H1 denote the target absent and the target present hypothesis, respectively.
The Generalized Likelihood Ratio Test Generalized Likelihood Ratio Test (GLRT) creates a decision rule that detects anomalies as image pixel that do not fit properly that given model.
pixelbackgroundnoais:H
pixelbackgroundais:HH
1
0
x
xx
The decision criterion is given by:
where η is the detection threshold.
ηHfH
H
HGLRT
0
1
0 0| |log
xx X
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group5
The Student’s t distribution Mixture Model (StMM) assumes the data originate from a weighted sum of several multivariate Student’s t distribution sources. It assumes:
Finite mixture distributions are able to approximate any continuous PDF, provided the model has a sufficient number of components and the model parameters are chosen correctly, as:
Mixture of Student’s t distributionsMixture of Student’s t distributions
Mixing proportion
Multivariate PDF of x given the component distribution j controlled by the parameters vector θj
J
jjjgπf
1
;θxx XX
{πk}
{μk}
{Tk}
fX(x)
{νk}
νdνανStνg
t μxΛμxΛΛμxΛμxX 1,,,,;,,;
Mean vector Scale
matrixNumber of degrees of freedom
Normalization factor (it depends on the number of spectral channels d)
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group6
Expectation Maximization (EM) algorithmExpectation Maximization (EM) algorithm
The well- known EM algorithm aims to maximize the likelihood function with respect to the parameters.
Several limitations of EM approach can be highlighted:
Number of components
•It should be chosen a priori by the user.
Convergence•It is ensured to a local maximum of the likelihood function.
•It depends on initial conditions.
Covariance matrix
•It can happen that the PDF of one or more component of the mixture collapses onto a specific data point.
•The likelihood function may be unlimited due to singular covariance matrices.
Problems of the EM algorithm
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group7
How to solve the problems of EM algorithm?How to solve the problems of EM algorithm?
It treats parameters of the distribution to be fitted as random variables with a given prior probability distribution.
The functions with respect to which the optimization is performed assume a specific form.
Bayesian approach
Factorized distributions
Prarameters calculation by “Variational
Bayesian Approximation”
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group8
Learning approach Learning approach [1][1]
Gauss-Wishart prior distribution governs both the mean vector and the precision matrix of each component.
The analysis is carried out by using PRIOR DISTRIBUTIONS CONJUGATE TO THE LIKELIHOOD PRIOR DISTRIBUTIONS CONJUGATE TO THE LIKELIHOOD
FUNCTIONFUNCTION.
Dirichlet prior distribution governs the mixing proportions.
The number of degrees of freedom of each component is obtained by maximizing the expected log-likelihood function, evaluated taking into account the observed data and the parameters considered as random variables, and imposing νj>2.
No prior distribution is imposed on the number of degrees of freedom of each mixture component.
[1] C. Archambeau, and M. Verleysen, “Robust Bayesian Clustering”, Neural Networks, vol.20, pp. 129-138, 2007.
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group9
Anomaly detection strategy: StMM GLRT ADAnomaly detection strategy: StMM GLRT AD
The image pixels are grouped into clusters.The clustering strategy is conducted by assuming a StMM PDF for each hyperspectral pixel.
1.Clustering
step
The background continuous PDF is approximated as a linear combination of Student’s t distributions based on the statistics estimated for each cluster.
2.PDF
estimation
The anomaly detector decision criterion is given by:
where η is the detection threshold.
ηνπH
H
J
jjjjjGLRTStMM
0
1
1
,,;Stlog
Λμxx
Once the PDF has been estimated, the Generalized Likelihood Ratio Test (GLRT) can be applied.
3.GLRT
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group10
Data set descriptionData set description
Sensor HYPER/SIM-GA
Type Push-broom
Spectral Range 400-1000 nm (VNIR)
Spectral Sampling ≈1.2 nm
# Spectral pixels 512
# Spatial pixels 1024
IFOV 0.7 mrad
GSD @ 1000 m 0.7m
Swath @ 1000 m 715 m
FOV ±20°
Focal length 17mm
F# (min. value) 2.0
Quantization accuracy 12 bits
Platform airborne
Flight altitude 1725 m
Main technical specifications of SIM-GA sensor.
Representation of the hyperspectral data employed
For testing and validating the proposed method on real hyperspectral data an acquired at-sensor radiance image was utilized. A spectral binning, aimed at increase Signal to Noise
Ratio (SNR), was performed. Besides this, water-vapor absorption and noisy bands were discarded.
Panel 11x1 m2
Panel 22x2 m2
Panel 44x4 m2
Panel 32x2 m2
Panel 54x4 m2
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group11
Experimental comparisonExperimental comparison
Performance was compared to that obtained by employing a Gaussian Mixture Model (GMM). The GMM assumes:
μxCμx
CCμxCμxX
1
2/12/ 2
1exp
2
1,;,; t
dπNg
Mean vector
Covariancematrix
[1] McLachlan, G., and D. Peel, Finite Mixture Models, John Wiley & Sons, New York, 2000.[2] C. Constantinopoulos, and A. Likas; “Unsupervised Learning of Gaussian Mixtures Based on Variational Component Splitting”, IEEE Trans. on Neural Networks, vol. 18, pp. 745 – 755, 2007.
Both the conventional EM algorithm[1] and a Bayesian approach[2] were taken into account for learning the GMM.
Clustering step
The background continuous PDF was approximated as a linear combination of distributions based on the statistics estimated for each cluster.
PDF estimation
The GLRT decision rule was applied.
GLRT
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group12
Experimental results: cluster mapsExperimental results: cluster maps
EM (2 components) EM (6 components)
Bayesian GMM learning
Clustering strategy conducted by assuming a StMM PDF for each pixel
Clustering strategies conducted by assuming a GMM PDF for each pixel
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group13
Exceedance plotsExceedance plots
Mahalanobis distance Mt of multivariate Student’s t distributed data :
dνMνL
ν
νLM
t
t ,F~2
μxΛμx
Probability of exceedanceProbability of exceedance of the Mahalanobis distances between each pixel of the cluster and the cluster itself.
Mahalanobis distance M of multivariate normal data:
dχM t 21 ~μxCμx
ν
StMM learning: cluster 1
GMM learning
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Experimental results: detection mapsExperimental results: detection maps
StMM-based AD
GMM-based AD
EM (6 components)EM (2 components)
Bayesian GMM learning
Remote Sensing & Image Processing GroupRemote Sensing & Image Processing Group15
ConclusionsConclusions
Only the pure target pixels were assumed as targets to detect, whereas evident undesired anomalies were neglected.
AD philosophy
The proposed strategy combines a StMM PDF for modeling each hyperspectral pixel along with the GLRT decision rule.
The StMM learning is based on a Bayesian approach that automatically estimates the mixture parameters during the learning procedure.
AD: StMM vs GMM
The experimental analysis has shown the BStMM ability to reliably estimating the background PDF, and its effectiveness in detecting rare anomalous objects within the image employed.
The conducted analysis has highlighted how the discrete search over the number of components in a mixture distribution conducted by adopting the classical EM learning can be avoided by adopting a Bayesian philosophy within AD schemes.
Remote Sensing Remote Sensing & &
Image Processing GroupImage Processing Group
Thanks for your attentionThanks for your attention