Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Robust target detection for Hyperspectral Imaging Joana Frontera-Pons SONDRA PhD defense Under the supervision of Frédéric Pascal & Jean-Philippe Ovarlez (PhD director) December 10, 2013 Defense 1/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Hyperspectral Imaging (HSI) � ANOMALY DETECTION IN HYPERSPECTRAL IMAGES To detect all that is “different " from the background (Mahalanobis distance) - No information about the targets of interest available. � DETECTION OF TARGETS IN HYPERSPECTRAL IMAGES To detect targets characterized by a given spectral signature p - Regulation of False Alarm. 0 . 4 0 . 3 Reflectance 0 . 2 0 . 1 0 20 40 60 80 100 Wavelength Defense 2/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly � Many methodologies for detection and classification in hyperspectral images can be found in radar detection community. We can retrieve all the detectors family commonly used in radar detection (AMF (intensity detector), ACE (angle detector), sub-spaces detectors, ...). � Almost all the conventional techniques for anomaly detection and targets detection are based on Gaussian assumption and on spatial homogeneity in hyperspectral images. All these techniques need to estimate the data covariance matrix Σ (whitening process) and the mean vector µ . Defense 3/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Outline 1 Preliminary Notions 2 Target Detection in Gaussian background 3 Target Detection in non-Gaussian background 4 Anomaly Detection 5 Conclusions Defense 4/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Outline 1 Preliminary Notions 2 Target Detection in Gaussian background 3 Target Detection in non-Gaussian background 4 Anomaly Detection 5 Conclusions Defense 5/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Problem Statement � In a m -dimensional observation vector x , the problem of detecting a complex known signal s = α p ( p is the steering vector and α the target amplitude), corrupted by an additive noise b , can be stated as the following binary hypothesis test : � Hypothesis H 0 : x = b x i = b i i = 1 , . . . , N Hypothesis H 1 : x = s + b x i = b i i = 1 , . . . , N where the x i ’s are N "signal-free" independent observations (secondary data) used to estimate the background parameters . ⇒ Neyman-Pearson criterion : Mazimize the probability of detection for a fixed probability of false alarm. Defense 6/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Problem Statement � Detection test: comparison between the Likelihood Ratio (LR) Λ ( x ) and a detection threshold λ : Λ ( x ) = p ( x | H 1 ) H 1 ≷ η . p ( x | H 0 ) H 0 λ is determined for a fixed value of PFA (set by the user): � Probability of False Alarm (type-I error): PFA = P ( Λ ( x ; H 0 ) > λ ) � Probability of Detection (to evaluate the performance): PD = P ( Λ ( x ; H 1 ) > λ ) for different Signal-to-Noise Ration (SNR). Defense 7/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Gaussian distribution A m -dimensional vector x has a complex Gaussian distribution denoted CN ( µ , Σ ) . If the probability density function exists, it is of the form: f x ( x ) = π − m | Σ | − 1 exp { −( x − µ ) H Σ − 1 ( x − µ ) } . Maximum Likelihood Estimators: Let x 1 , . . . , x N be an IID N -sample, where x i ∼ CN ( µ , Σ ) . Thus, the SMV and the SCM can be written as: N N µ SMV = 1 � Σ SCM = 1 � µ ) H . ^ ^ x i , ( x i − ^ µ )( x i − ^ N N i = 1 i = 1 � Simplicity of analysis and well-known statistical properties: consistent, unbiased and efficient, � ^ Σ SCM is Wishart distributed. Defense 8/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Outline 1 Preliminary Notions 2 Target Detection in Gaussian background 3 Target Detection in non-Gaussian background 4 Anomaly Detection 5 Conclusions Defense 9/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Matched Filter The Matched Filter is the optimal filter for maximizing the SNR under Gaussian background assumption: Λ MF = | p H Σ − 1 ( x − µ ) | 2 H 1 ≷ λ ( p H Σ − 1 p ) H 0 PFA-threshold relationship PFA MF = exp (− λ ) 10 0 10 − 1 log 10 ( PFA ) 10 − 2 MF theo. MF Monte-Carlo 10 − 3 10 − 2 10 − 1 10 0 10 1 10 2 Defense 10/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging Threshold λ
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Adaptive Matched Filter Unknown Covariance matrix: − 1 ( x − µ ) | 2 Σ = | p H ^ Σ H 1 Λ ( N ) ≷ λ − 1 p ) AMF ^ ( p H ^ Σ H 0 PFA-threshold relationship � � N − m + 1 , N − m + 2 ; N + 1 ; − λ PFA AMF ^ Σ = 2 F 1 N 10 0 N = 6 10 − 1 log 10 ( PFA ) N = 10 10 − 2 MF theo. N = 20 MF Monte-Carlo µ known theo. µ known MC 10 − 3 10 − 2 10 − 1 10 0 10 1 10 2 Threshold λ Defense 11/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Adaptive Matched Filter Unknown Covariance matrix and Mean Vector: − 1 ( x − ^ µ = | p H ^ µ ) | 2 Σ H 1 Λ ( N ) ≷ λ − 1 p ) AMF ^ Σ , ^ ( p H ^ Σ H 0 PFA-threshold relationship λ ′ � � PFA AMF ^ µ = 2 F 1 N − m , N − m + 1 ; N ; − Σ , ^ N − 1 10 0 N = 6 10 − 1 log 10 ( PFA ) N = 10 MF theo. 10 − 2 MF Monte-Carlo µ known theo. N = 20 µ known MC Eq.( ?? ) µ unknown MC 10 − 3 10 − 2 10 − 1 10 0 10 1 10 2 Threshold λ Defense 12/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Kelly detection test The Kelly detector is based on the Generalized Likelihood Ratio Test assuming Gaussian distribution and unknown covariance matrix Σ : | p H ^ − 1 SCM ( x − µ ) | 2 Σ H 1 Λ ( N ) Σ = ≷ λ Kelly ^ � − 1 � � − 1 � p H ^ N + ( x − µ ) H ^ Σ SCM p Σ SCM ( x − µ ) H 0 PFA-threshold relationship PFA Kelly = ( 1 − λ ) N − m + 1 0 − 0 . 5 − 1 N = 6 − 1 . 5 log 10 ( PFA ) − 2 − 2 . 5 N = 10 − 3 − 3 . 5 µ known theo. N = 20 µ known MC − 4 0 1 2 3 4 5 6 7 8 Defense 13/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging Threshold η
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly Kelly “Plug-in" detection test Unknown Covariance matrix and Mean Vector: | p H ^ − 1 µ SMV ) | 2 SCM ( x − ^ Σ H 1 Λ ( N ) µ = ≷ λ Kelly ^ Σ , ^ � − 1 � � − 1 � p H ^ µ SMV ) H ^ H 0 Σ SCM p N + ( x − ^ Σ SCM ( x − ^ µ SMV ) PFA-threshold relationship � 1 �� m − N Γ ( N ) � λ � u u N − m ( 1 − u ) m − 2 du µ = 1 + 1 − PFA Kelly ^ Σ , ^ Γ ( N − m + 1 ) Γ ( m − 1 ) 1 − λ N + 1 0 0 − 0 . 5 − 1 N = 6 − 1 . 5 log 10 ( PFA ) − 2 − 2 . 5 N = 10 − 3 µ known theo. µ known MC − 3 . 5 Eq.( ?? ) N = 20 µ unknown MC − 4 0 1 2 3 4 5 6 7 8 Defense 14/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging Threshold η
Preliminary Notions Target Detection in Gaussian background Target Detection in non-Gaussian background Anomaly New Kelly detection test Unknown Covariance matrix and Mean Vector: Generalized Kelly detector � p H ^ � 2 � S − 1 � β ( N ) 0 ( x − ^ µ 0 ) H 1 Λ = ≷ λ ( p H ^ µ 0 ) H ^ S − 1 � S − 1 � 0 p ) 1 + ( x − ^ ( x − ^ µ 0 ) H 0 0 � � N N � 1 � µ 0 ) H , and ^ where ^ S 0 = ( x i − ^ µ 0 )( x i − ^ µ 0 = x + x i . N + 1 i = 1 i = 1 � New detector derived when both the mean vector and the covariance matrix are unknown, Generalized Likelihood Ratio Test, � The covariance matrix ^ S 0 and the mean vector ^ µ 0 estimates depend on the vector under test x , � ^ µ 0 are not independent and ^ S 0 and x − ^ S 0 is NOT Wishart distributed, � The distribution of the detector is unknown. Defense 15/ 51 J.Frontera-Pons Robust target detection for Hyperspectral Imaging
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