Iee egold2010 presentazione_finale_veracini
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  • terrain classification, environmental monitoring, agricultural monitoring, geological exploration, and surveillance. (stein,2002)
  • Dire che la matrice di precisione è legata a quella di covarianza solo se ne>2!!!
  • (comprising the means and covariances of the components and the mixing coefficients).
  • Approccio bayesiano: cioè suppongo di avere delle conoscenze a priori sui parametri che devo stimare
  • Dire il perchè dell’uso delle distribuzioni coniugate!!
  • The GLRT assumes that PDFs have a parametric form dependent on unknown parameters, and replace the unknown parameters by their estimates.
  • The red rectangle represents the portion of the scene containing the targets of interest.
  • - Cluster map produced by the BGMMS algorithm. - Color-scale detection statistical map obtained by using the AD algorithm based on VBGMMS for mixture learning.
  • - Cluster map produced by the BGMMS algorithm. - Color-scale detection statistical map obtained by using the AD algorithm based on VBGMMS for mixture learning.
  • Solo FULL PIXEL!!!

Iee egold2010 presentazione_finale_veracini Iee egold2010 presentazione_finale_veracini Presentation Transcript

  • A Novel Anomaly Detection Scheme for Hyperspectral Images Based on a Non-Gaussian Mixture Model Tiziana Veracini , Marco Diani, Giovanni Corsini   Dipartimento di Ingegneria dell’Informazione, Università di Pisa via G. Caruso 16, 56122 Pisa, Italy 2010 IEEE GOLD REMOTE SENSING CONFERENCE 29, 30 april 2010 Accademia Navale, Livorno, Italy
  • Outline
      • Mixture 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 distribution
      • Experimental results and conclusions.
        • Data set description.
        • Design of the experiments and results.
    Experimental results
      • 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 detection
  • Hyperspectral 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) Spectral dimension Image data cube Imaging spectrometer Wavelength [ μ m ] 0.4 0 2.5 1 Spatial Dimension (along the flight line) Reflectance
      • Hyperspectral remote sensing applications
    Earth observation - Terrain classification - Land use management - Environmental monitoring - Wide-area surveillance …
  • Anomaly Detection (AD) strategy
    • 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 testing problem :
      • Where x is a realization of the random vector X used for modeling the pixel under test, whereas H 0 and H 1 denote the target absent and the target present hypothesis, respectively.
    • The Generalized Likelihood Ratio Test (GLRT) creates a decision rule that detects anomalies as image pixel that do not fit properly that given model.
    The decision criterion is given by: where η is the detection threshold.
    • 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 distributions
    • 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:
    Mixing proportion Multivariate PDF of x given the component distribution j controlled by the parameters vector θ j { π k } { μ k } { T k } f X ( x ) { ν k } Mean vector Scale matrix Number of degrees of freedom Normalization factor (it depends on the number of spectral channels d )
  • Expectation 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?
      • 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 [1] [1] C. Archambeau, and M. Verleysen, “Robust Bayesian Clustering”, Neural Networks, vol.20, pp. 129-138, 2007. 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 FUNCTION . 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 ν j >2. No prior distribution is imposed on the number of degrees of freedom of each mixture component.
  • Anomaly detection strategy: StMM GLRT AD 1. Clustering step 2. PDF estimation 3. GLRT The image pixels are grouped into clusters. The clustering strategy is conducted by assuming a StMM PDF for each hyperspectral pixel. The background continuous PDF is approximated as a linear combination of Student’s t distributions based on the statistics estimated for each cluster. The anomaly detector decision criterion is given by: where η is the detection threshold. Once the PDF has been estimated, the Generalized Likelihood Ratio Test (GLRT) can be applied.
  • Data set description 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.
    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 Panel 1 1x1 m 2 Panel 2 2x2 m 2 Panel 4 4x4 m 2 Panel 3 2x2 m 2 Panel 5 4x4 m 2
  • Experimental comparison
    • Performance was compared to that obtained by employing a Gaussian Mixture Model (GMM). The GMM assumes:
    [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. Mean vector Covariance matrix 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 maps
    • In the EM algorithm the number of components is a user-specified parameter.
      • Several configurations for this parameter were tested and the best performance was selected.
    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 Color-scale detection statistical maps
  • Exceedance plots
    • ν
    StMM learning: cluster 1 GMM learning Mahalanobis distance M t of multivariate Student’s t distributed data : Probability of exceedance of the Mahalanobis distances between each pixel of the cluster and the cluster itself. Mahalanobis distance M of multivariate normal data:
  • Experimental results: detection maps
    • In the EM algorithm the number of components is a user-specified parameter.
      • Several configurations for this parameter were tested and the best performance was selected.
    StMM-based AD GMM-based AD EM (6 components) EM (2 components) Bayesian GMM learning Color-scale detection statistical maps
  • Conclusions 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. Only the pure target pixels were assumed as targets to detect, whereas evident undesired anomalies were neglected. AD: StMM vs GMM The experimental analysis has shown the BStMM ability to reliably estimate 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. 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.
  • Thanks for your attention