iee egold2010 presentazione_finale_veracini

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


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

…


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


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)


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


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”


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.


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


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


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


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


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


Experimental results: detection mapsExperimental results: detection maps

StMM-based AD

GMM-based AD

EM (6 components)EM (2 components)

Bayesian GMM learning


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

iee egold2010 presentazione_finale_veracini

Documents

hyperspectral data

mixture component

model parameters

finite mixture distributions

background continuous

detection threshold

mixture collapses

hyperspectral pixel