M´ ethodologies d’Estimation et de D´ etection Robuste en Conditions Non-Standards Pour le Traitement d’Antenne, l’Imagerie et le Radar Jean-Philippe Ovarlez 1 , 2 1 SONDRA, CentraleSup´ elec, France 2 French Aerospace Lab, ONERA DEMR/TSI, France Joint works with F. Pascal, P. Forster, G. Ginolhac, M. Mahot, J. Frontera-Pons, A. Breloy, G. Vasile, and many others eme ´ Ecole d’´ 12 ` Et´ e de Peyresq en Traitement du Signal et des Images 25 juin au 01 juillet 2017 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
Adaptive Robust Detection Schemes ... Other Refinements Contents Part A: Background on Radar, Array Processing, SAR and Hyperspectral Imaging Part B: Robust Detection and Estimation Schemes Part C: Applications and Results in Radar, STAP and Array Processing, SAR Imaging, Hyperspectral Imaging 1/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
Adaptive Robust Detection Schemes ... Other Refinements Part B Robust Detection and Estimation Schemes 2/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
Adaptive Robust Detection Schemes ... Other Refinements Part B: Contents 1 Adaptive Robust Detection Schemes in non-Gaussian Background CES distributions M -estimators and Tyler (FP) Estimator Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method 2 Other Refinements Exploiting Prior Information: Covariance Structure Low Rank Detectors Shrinkage of M -estimator RMT Theory and M -Estimator based Detectors 3/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Outline 1 Adaptive Robust Detection Schemes in non-Gaussian Background CES distributions M -estimators and Tyler (FP) Estimator Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method 2 Other Refinements Exploiting Prior Information: Covariance Structure Low Rank Detectors Shrinkage of M -estimator RMT Theory and M -Estimator based Detectors 4/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Modeling the background Let z be a complex circular random vector of length m . z has a Complex Elliptically Symmetric (CES) distribution ( CE ( µ , Σ, g . ) ) if its PDF is [Kelker, 1970, Frahm, 2004, Ollila et al., 2012]: g z ( z ) = π − m | Σ | − 1 h z (( z − µ ) H Σ − 1 ( z − µ )) , (1) where h z : [ 0 , ∞ ) → [ 0 , ∞ ) is the density generator, where µ is the statistical mean (generally � z z H � known or = 0 ) and Σ is the scatter matrix. In general, E = α Σ where α is known. Large class of distributions : Gaussian ( h z ( z ) = exp (− z ) , SIRV, MGGD ( h z ( z ) = exp (− z α ) ), etc. Closed under affine transformations (e.g. matched filter), z = d µ + R Au ( k ) , Stochastic representation theorem : where R ≥ 0, independent of u ( k ) and Σ = AA H is a factorization of Σ , where A ∈ C m × k with k = rank ( Σ ) . 5/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method SIRV: a CES subclass The m -vector z is a complex Spherically Invariant Random Vector [Yao, 1973, Jay, 2002] if its PDF can be put in the following form: � ( z − µ ) H Σ − 1 ( z − µ ) � � ∞ 1 1 g z ( z ) = p τ ( τ ) d τ , (2) τ m exp π m | Σ | τ 0 where p τ : [ 0 , ∞ ) → [ 0 , ∞ ) is the texture generator. Large class of distributions : Gaussian ( p τ ( τ ) = δ ( τ − 1 ) ), K-distribution ( p τ gamma), Weibull (no closed form), Student-t ( p τ inverse gamma), etc. Main Gaussian Kernel: closed under affine transformations, The texture random scalar is modeling the variation of the power of the Gaussian vector x along his support (e.g. heterogeneity of the noise along range bins, time, spatial domain, etc.), z = d µ + √ τ A x , where τ ≥ 0 is the texture, Stochastic representation theorem : independent of x and x ∼ CN ( 0 , Σ ) . 6/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Outline 1 Adaptive Robust Detection Schemes in non-Gaussian Background CES distributions M -estimators and Tyler (FP) Estimator Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method 2 Other Refinements Exploiting Prior Information: Covariance Structure Low Rank Detectors Shrinkage of M -estimator RMT Theory and M -Estimator based Detectors 7/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Estimating the covariance matrix: Conventional estimators Assuming n available SIRV secondary data z k = √ τ k x k where x k ∼ CN ( 0 , Σ ) and where τ k scalar random variable. The Sample Covariance Matrix (SCM) may be a poor estimate of the Elliptical/SIRV Scatter/Covariance Matrix because of the texture contamination: n n n S n = 1 � k = 1 � k � = 1 � ^ z k z H τ k x k x H x k x H k , n n n k = 1 k = 1 k = 1 The Normalized Sample Covariance Matrix (NSCM) may be a good candidate of the Elliptical SIRV Scatter/Covariance Matrix: n z k z H n x k x H � � Σ NSCM = 1 = 1 k k ^ , z H x H n k z k n k x k k = 1 k = 1 This estimate does not depend on the texture τ k but it is biased and share the same eigenvectors but have different eigenvalues, with the same ordering [Bausson et al., 2007]. 8/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Estimating the covariance matrix Let ( z 1 , ..., z n ) be a n -sample ∼ CE m ( 0 , Σ, g z ( . ) ) (Secondary data). PDF g z ( . ) specified: ML -estimator of Σ � � i � Σ − 1 z i z H n − g ′ � Σ = 1 z � � z i z H � i , n i � Σ − 1 z i z H g z i = 1 PDF g z ( . ) not specified: M -estimator of Σ n � � � Σ = 1 Σ − 1 z i � i � z H z i z H u i , n i = 1 [Maronna et al., 2006, Kent and Tyler, 1991, Pascal, 2006, Pascal et al., 2008a, Pascal et al., 2008b] Existence, Uniqueness, Convergence of the recursive algorithm, etc. 9/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Examples of M -estimators FPE (Tyler): SCM: Huber’s M -estimator: � K / e if r < = e u ( r ) = 1 u ( r ) = m u ( r ) = r K / r if r > e Huber = mix between SCM and FPE [Huber, 1964], FPE and SCM are “not” (theoretically) M -estimators, FPE is the most robust while SCM is the most efficient. 10/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Estimating the covariance matrix: Tyler’s M -estimators Let ( z 1 , ..., z n ) be a n -sample ∼ CE m ( 0 , Σ, g z ) (Secondary data). FP Estimate ([Tyler, 1987, Pascal et al., 2008a] n � z k z H Σ FPE = m � k . k � Σ − 1 n z H FPE z k k = 1 The FPE does not depend on the texture (SIRV or CES distributions), Existence, Uniqueness, Convergence of the recursive algorithm (identifiability condition: tr ( � Σ FPE ) = m ), True MLE under SIRV distributed noise with unknown deterministic texture { τ k } k ∈ [ 1 , n ] . 11/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
CES distributions Adaptive Robust Detection Schemes ... M -estimators and Tyler (FP) Estimator Other Refinements Robustness of M-estimators and ANMF MUltiple Signal Classification (MUSIC) method Some Weighting Functions of M -estimators Weighting functions for student -t distribution Weighting functions for K -distribution 10 1 10 1 n = 0.01 n = 0.1 n = 0.01 n = 0.5 n = 0.1 n = 0.5 n = 1 n = 1 n = 10 n = 10 10 0 m/t m/t 10 0 j ( t ) j ( t ) 10 –1 10 –1 10 –2 10 –2 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 t t √ ν K ν − m − 1 ( 4 ν t ) u ( t ) = ν + 2 m u ( t ) = K ν − m ( 4 ν t ) , ν + 2 t . t Σ = ^ ^ Σ = ^ ^ We have lim Σ FPE and lim Σ SCM . ν →∞ ν → 0 12/68 Jean-Philippe Ovarlez Sch´ emas de D´ etection Adaptative Robuste
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