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A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors Jorge Prendes 1 , 2 , Marie Chabert 1 , 3 , Fr´ eric Pascal 2 , ed´ Alain Giros 4 , Jean-Yves Tourneret 1 , 3 1 T´ eSA Laboratory, 2 Sup´ elec - SONDRA, 3 University of Toulouse, 4 CNES (French Space Agency) ICASSP 2014

Introduction Image Model Similarity Measure Results Conclusions Outline 1 Introduction 2 Image Model 3 Similarity Measure 4 Results 5 Conclusions J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 2 / 23

Introduction Image Model Similarity Measure Results Conclusions Introduction Motivation: Change detection on remote sensing images Monitor urban/rural area evolution Detect new constructions Track changes in agricultural areas Track urban growth Coordinate efforts after natural disasters Volcano eruptions Floodings Earthquakes Improve the analysis of remote sensing images Find new objects Different type of sensors: Optical, SAR, Hyperspectral, etc. Joint analysis of heterogeneous sensors! J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 3 / 23

Introduction Image Model Similarity Measure Results Conclusions Introduction Change Detection Framework Images Sliding Window: W Sliding window W Similarity measure on W W Opt W SAR Optical SAR Threshold Decision H 0 : Absence of change Statistical Similarity Measures Similarity Measure H 1 : Presence of change d = f ( W Opt , W SAR ) H 0 Dependency between pixel d ≷ τ H 1 intensities Correlation Coefficient Result Using several Linear dependency, Fails on homogeneous areas windows . . . Mutual Information Requires pdf estimation, Fails on homogeneous areas Objective: Similarity measure for homogeneous and heterogeneous sensors based on a statistical model J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 4 / 23

Introduction Image Model Similarity Measure Results Conclusions Image Model – Optical image for Homogeneous Regions Optical Sensor Affected by additive Gaussian noise I Opt = T Opt ( P ) + ν N (0 ,σ 2 ) 10 � T Opt ( P ) , σ 2 � I Opt | P ∼ N 5 where T Opt ( P ) is how an object with physical properties P would be ideally seen by an optical sensor 0 σ 2 is associated with the noise variance 0 1 I Opt Histogram of the normalized image J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 5 / 23

Introduction Image Model Similarity Measure Results Conclusions Image Model – SAR Image for Homogeneous Regions Radar Sensor Affected by multiplicative speckle noise (with gamma distribution) I SAR = T SAR ( P ) × ν Γ ( L , 1 L ) 4 � � L , T SAR ( P ) I SAR | P ∼ Γ L 2 where T SAR ( P ) is how an object with physical 0 properties P would be ideally seen by a SAR 0 1 I SAR sensor L is the number of looks of the SAR sensor Histogram of the normalized image J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 6 / 23

Introduction Image Model Similarity Measure Results Conclusions Image Model – Generic Image for Homogeneous Regions Generic Model: Sensor S I S | P = f S [ T S ( P ) , ν S ] SAR Image Optical Image I SAR = T SAR ( P ) × ν Γ ( L , 1 I Opt = T Opt ( P ) + ν N (0 ,σ 2 ) L ) T Opt ( P ) = µ P T SAR ( P ) = α P × θ P J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 7 / 23

Introduction Image Model Similarity Measure Results Conclusions Image Model – Joint Distribution for Homogeneous Regions Independence assumption for the sensor noises p( I S1 , I S2 | P ) = p( I S1 | P ) × p( I S2 | P ) 1 Conclusion I SAR Statistical dependency (CC, MI) is not always an 0 appropriate similarity 0 1 I Opt measure J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 8 / 23

Introduction Image Model Similarity Measure Results Conclusions Image Model – Heterogeneous Regions Sliding window W Usually includes a finite number of objects, K Different values of P for each object Pr( P = P k | W ) = w k 1 p( I S1 , I S2 | W ) = I SAR K � w k p( I S1 , I S2 | P k ) 0 0 1 I Opt k =1 Mixture distribution! J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 9 / 23

Introduction Image Model Similarity Measure Results Conclusions Image Model – Mixture Distribution Mixture Distribution K � p( I S1 , I S2 | W ) = w k p( I S1 , I S2 | P k ) k =1 Parameter Estimation Expectation Maximization Iteratively Algorithm Estimate class prob. π ( i ) n , k Maximize parameters θ ( i ) k Repeat Selection of the number of classes [1] [1] M. A. T. Figueiredo and A. K. Jain, ”Unsupervised learning of finite mixture models,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 24, no. 3, pp. 381–396, March 2002. J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 10 / 23

Introduction Image Model Similarity Measure Results Conclusions Similarity Measure – Introduction 1 Mixture distribution I SAR Parameter Estimates Related to P 0 Can be used to derive 0 I Opt 1 [ T S1 ( P ) , T S2 ( P ) , . . . ] for each object 1 Example: P 3 P 4 T SAR ( P ) P 2 T Opt ( P k ) = µ k T SAR ( P k ) = L × θ k P 1 0 0 1 T Opt ( P ) J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 11 / 23

Introduction Image Model Similarity Measure Results Conclusions Similarity Measure – Manifold Main assumption Several For each unchanged window, unchanged windows v ( P ) = [ T S1 ( P ) , T S2 ( P ) , . . . ] can be considered as a point on a manifold . . . Manifold 0 . 3 Describes the joint behavior of the T SAR ( P ) different images Belongs to a D -dimensional space 0 0 1 D : Number of combined channels T Opt ( P ) J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 12 / 23

Introduction Image Model Similarity Measure Results Conclusions Similarity Measure – Manifold Unchanged regions Changed regions Pixels belong to the same Pixels belong to different object objects P is the same for both P changes from one image images to another 0 . 3 0 . 3 T SAR ( P ) T SAR ( P ) 0 0 0 1 0 1 T Opt ( P ) T Opt ( P ) J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 13 / 23

Introduction Image Model Similarity Measure Results Conclusions Similarity Measure – Manifold Distance measure between H 0 : Absence of change Optical and SAR images H 1 : Presence of change PDF of v ( P ) Good distance measure K � H 0 w k � � p T ( � v W , k ) ≷ τ Learned using training data H 1 from unchanged images k =1 where Learning strategies w k is the estimated w k � Histogram v W , k is the estimated vector v for the k -th � Parzen windows component of the window W p T is the estimated density of v ( P ) � Mixture models τ is an application dependent threshold J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 14 / 23

Introduction Image Model Similarity Measure Results Conclusions Similarity Measure – Summary Mixture W Opt W SAR Sliding Window: W Using several windows Manifold Estimation . . . 0 . 3 � � θ 1 : θ 4 : . . . T SAR ( P ) 1 , � 4 , � σ 2 σ 2 � µ 1 , � k 1 , � α 1 � µ 4 , � k 4 , � α 4 0 . 3 T SAR ( P ) v P 1 : � � v P 4 : . . . � � � � 0 � T S 1 ( P 1 ) , � T S 2 ( P 1 ) T S 1 ( P 4 ) , � � T S 2 ( P 4 ) 0 1 T Opt ( P ) 0 0 1 T Opt ( P ) Manifold Samples 0 . 3 P 3 P 4 P 2 T S2 ( P ) P 1 0 0 1 T S1 ( P ) J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 15 / 23

Introduction Image Model Similarity Measure Results Conclusions Results – Synthetic Optical and SAR Images Mutual Correlation Proposed Method Information Coefficient Synthetic optical image Synthetic SAR image 1 PD Proposed Correlation Mutual Inf. 0 0 PFA 1 Performance – ROC Change mask J. Prendes T´ eSA – Sup´ elec-SONDRA – INP/ENSEEIHT – CNES A Multivariate Statistical Model for Multiple Images Acquired by Homogeneous or Heterogeneous Sensors 16 / 23