Inverse problems : variational formulation Generalized Diffusion regularization Application Conclusion A new approach for regularization of inverse problems in image processing I. Souopgui 1 , 2 , E. Kamgnia 2 , F.-X. Le Dimet 1 , A. Vidard 1 (1) INRIA / LJK Grenoble (2) University of Yaounde I 10 th African Conference on Research in Computer Science and Applied Mathematics - CARI 2010 October 18 - 21, 2010, Yamoussoukro, Cˆ ote d’Ivoire F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Generalized Diffusion regularization Application Conclusion Inverse problems : variational formulation 1 Definition A priori knowledges Regularization : Vector fields From classical regularization to Generalized Diffusion 2 Regularization as smoothing operators Case of gradient penalization Application to geophysical fluid motion estimation 3 Motion estimation problem From regularization to pseudo covariance operator Experimental result Conclusion 4 F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Definition Generalized Diffusion regularization A priori knowledges Application Regularization Conclusion Ingredients Physical system y ∈ Y , the system state v ∈ V , the control variable M : V → Y M , model mapping V to Y �→ y = M ( v ) v Observation y o ∈ O , observed state y o Observation system H observation operator H : Y → O mapping Y to O y �→ H ( y ) F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Definition Generalized Diffusion regularization A priori knowledges Application Regularization Conclusion Definition Giving observed state y o , Inverse problem (unconstrained) Find v ∗ = MinArg( J ( v )), v ∈ V where J ( v ) = J o ( v ) = 1 2 �H ( M ( v )) − y o � 2 (1) O under adequate conditions, the solution v ∗ is given by the Euler-Lagrange Equation ∇ J ( v ∗ ) = 0 Problems ill-posedness ⇒ use a priori knowledges; ill-conditionning ⇒ use preconditioning. F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Definition Generalized Diffusion regularization A priori knowledges Application Regularization Conclusion A priori knowledges For a priori knowledge A , set J = J o + J A where J A is defined to force the solution to satisfy A Use of a priori informations Background v b and background error covariance B J b = 1 2 α b � v − v b � 2 (2) B − 1 Regularity of the solution : Φ-smooth (minimum gradient) J r = 1 2 α r � Φ( v ) � 2 (3) Φ function of the derivatives of v F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Definition Generalized Diffusion regularization A priori knowledges Application Regularization Conclusion Vector fields regularization first order regularization : first order derivatives of v � ∂ v i � Φ (1) ∂ x j 1 ≤ i , j ≤ n , n gradient penalization : J ∇ ( v ) = 1 � � �∇ v i � 2 d x 2 α ∇ Ω i =1 second order regularization : second order derivatives of v � ∂ 2 v i � Φ (2) ∂ x j ∂ x k 1 ≤ i , j , k ≤ n , Suter regularization : J suter ( v ) = 1 � α ∇ div �∇ div ( v ) � 2 + α ∇ curl �∇ curl ( v ) � 2 d x 2 Ω ⇒ difficult to defined optimal weighting parameter(s) F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Generalized Diffusion regularization Regularization as smoothing operators Application Case of gradient penalization Conclusion Notations and definition Let : v ( x ) be an incomplete/inconsistent control variable, with x ∈ Ω the physical space Φ( v ) regularization operator as defined previously ϕ ( x ) a scalar positive trust function given the quality of v at x � small value meaning bad/lack/inconsistent control variable large value for good quality control variable we define restored control variable u ∗ = MinArg( ε ( u )) , u ∈ V ε ( u ) = 1 � � 2 Φ( u ( x )) � 2 + ϕ ( x ) � u ( x ) − v ( x ) � 2 d x (4) 2 Ω F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Generalized Diffusion regularization Regularization as smoothing operators Application Case of gradient penalization Conclusion ε ( u ) = 1 � � Φ( u ( x )) � 2 + ϕ ( x ) � 2 u ( x ) − v ( x ) � d x 2 Ω ε is minimized by setting u to be : close to v when ϕ is large ( v has adequate properties) Φ − regular when ϕ is small (otherwise) Under adequate conditions MinArg( ε ) is given by the Euler-Lagrange condition ∇ u ε ( u ) = 0 (5) Gateaux derivatives development leads to ∇ u ε ( u ) = Φ ∗ ◦ Φ( u ( x )) + ϕ ( x )( u ( x ) − v ( x )) (6) F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Generalized Diffusion regularization Regularization as smoothing operators Application Case of gradient penalization Conclusion Gradient penalization : mathematical expression n 1 � � �∇ v i � 2 d x J ∇ ( v ) = 2 α ∇ Ω i =1 Applied as smoothing operator, we get Φ ∗ ∇ ◦ Φ ∇ = − ∆ , with boundary conditions : ∇ u i ⊥ ν on ∂ Ω ⇒ ∇ ε ∇ ( u i ) = − ∆ u i ( x ) + ϕ ( x )( u i ( x ) − v i ( x )) , 1 ≤ i ≤ n (7) F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Generalized Diffusion regularization Regularization as smoothing operators Application Case of gradient penalization Conclusion Numerical implementation Generalized diffusion implementation Classical implementation : given ∇ ε , use descent-type algorithms. Problem : solve the Euler-Lagrange equation ∆ u i − ϕ ( x )( u i ( x ) − v i ( x )) = 0 , 1 ≤ i ≤ n (8) considers u i as a function of time and solve the equivalent problem ∂ ∂ t u i ( x , t ) = ∆ u i ( x , t ) − ϕ ( x )( u i ( x , t ) − v i ( x ))) , 1 ≤ i ≤ n (9) known as the generalized diffusion equations. As diffusion operator, it can directly be used in background covariance [see Weaver et al.] F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Motion estimation problem Generalized Diffusion regularization From regularization to pseudo covariance operator Application Experimental result Conclusion optical flow of Horn and Shunck : luminance conservation df dt = 0 (10) f ( x , t ) noted f is the luminance function. For geophysical fluid images, the mass conservation equation is more adequate [Fitzpatrick 1985] df dt + f ( ∇ · v ) = 0 (11) v ( x ) is the velocity at x given the luminance function f ( x , 0) = f 0 ( x ) at time 0, solution to equations (10) or (11) defines f ( x , t ) as function of the static velocity field v ( x ) M : V → F �→ f = M ( v ) v F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Motion estimation problem Generalized Diffusion regularization From regularization to pseudo covariance operator Application Experimental result Conclusion motion estimation is reduced to the inverse problem : δ v ∗ = MinArg( J ( δ v )) with J ( δ v ) = 1 F + 1 2 �M ( v b + δ v ) − f o � 2 2 � δ v � 2 (12) V Aperture problem : only motion along the normal to iso-contours can be inferred ⇒ use regularization F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Motion estimation problem Generalized Diffusion regularization From regularization to pseudo covariance operator Application Experimental result Conclusion Trust function for motion estimation Proposition: define trust function ϕ to have large values on discontinuities (contours) for motion component along the normal to the contour, and small values in homogeneous areas. Example : set ϕ to be the contours map c 1 or c 2 defined as c 1 ( x , f ) �∇ x f ( x ) � 2 = c 2 ( x , f ) �∇ x ( G σ ( x ) ∗ f ( x )) � 2 = F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Motion estimation problem Generalized Diffusion regularization From regularization to pseudo covariance operator Application Experimental result Conclusion Twin experiments Direct Image sequences assimilation [Titaud et al 2009] ⇒ true initial state (velocity fields) Images from [J.-B. Fl´ or (LEGI) and I. Eames, 2002] F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Motion estimation problem Generalized Diffusion regularization From regularization to pseudo covariance operator Application Experimental result Conclusion Error analysis : Tikhonov regularization Evolution of diagnostic functions with respect to the weighting parameter α mean values max values F.-X. Le Dimet regularization for inverse problem
Inverse problems : variational formulation Motion estimation problem Generalized Diffusion regularization From regularization to pseudo covariance operator Application Experimental result Conclusion Error analysis : gradient penalization Evolution of diagnostic functions with respect to the weighting parameter α ∇ mean values max values F.-X. Le Dimet regularization for inverse problem
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