International audienceOptical flow motion estimation from two images is limited by the aperture problem. A method to deal with this problem is to use regularization techniques. Usually, one adds a regularization term with appriopriate weighting parameter to the optical flow cost funtion. Here, we suggest a new approach to regularization for optical flow motion estimation. In this approach, all the regularization informations are used in the definition of an appropriate norm for the cost function via a trust function to be defined, one does not ever need weighting parameter. A simple derivation of such a trust function from images is proposed and a comparison with usual approaches is presented. These results show the superiority of such approach over usual ones

Kamgnia, Emmanuel

Le Dimet, François-Xavier

Souopgui, Innocent

Vidard, Arthur

Hal - Université Grenoble Alpes

A new approach for regularization of inverse problems in images processing

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Inverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionA new approach for regularization of inverseproblems in image processingI. Souopgui1,2, E. Kamgnia2, F.-X. Le Dimet1, A. Vidard1(1) INRIA / LJK Grenoble(2) University of Yaounde I10th African Conference on Research in Computer Science and AppliedMathematics - CARI 2010October 18 - 21, 2010, Yamoussoukro, Côte d’IvoireF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusion1 Inverse problems : variational formulationDefinitionA priori knowledgesRegularization : Vector fields2 From classical regularization to Generalized DiffusionRegularization as smoothing operatorsCase of gradient penalization3 Application to geophysical fluid motion estimationMotion estimation problemFrom regularization to pseudo covariance operatorExperimental result4 ConclusionF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionDefinitionA priori knowledgesRegularizationIngredientsPhysical systemM : V → Yv 7→ y =M(v)y ∈ Y, the system statev ∈ V, the control variableM, model mapping V to YObservationyoyo ∈ O, observed stateObservation systemH : Y → Oy 7→ H(y)H observation operatormapping Y to OF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionDefinitionA priori knowledgesRegularizationDefinitionGiving observed state yo,Inverse problem (unconstrained)Find v∗ = MinArg(J(v)), v ∈ V whereJ(v) = Jo(v) =12‖H(M(v))− yo‖2O (1)under adequate conditions, the solution v∗ is given by theEuler-Lagrange Equation ∇J(v∗) = 0Problemsill-posedness ⇒ use a priori knowledges;ill-conditionning ⇒ use preconditioning.F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionDefinitionA priori knowledgesRegularizationA priori knowledgesFor a priori knowledge A, set J = Jo + JAwhere JA is defined to force the solution to satisfy AUse of a priori informationsBackground vb and background error covariance BJb =12αb‖v − vb‖2B−1 (2)Regularity of the solution : Φ-smooth (minimum gradient)Jr =12αr‖Φ(v)‖2 (3)Φ function of the derivatives of vF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionDefinitionA priori knowledgesRegularizationVector fields regularizationfirst order regularization : first order derivatives of vΦ(1)(∂vi∂xj)1≤i ,j≤n,gradient penalization : J∇(v) =12α∇∫Ωn∑i=1‖∇vi‖2dxsecond order regularization : second order derivatives of vΦ(2)(∂2vi∂xj∂xk)1≤i,j,k≤n,Suter regularization :Jsuter (v) =12∫Ωα∇div‖∇div(v)‖2 + α∇curl‖∇curl(v)‖2dx⇒ difficult to defined optimal weighting parameter(s)F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionRegularization as smoothing operatorsCase of gradient penalizationNotations and definitionLet :v(x) be an incomplete/inconsistent control variable, withx ∈ Ω the physical spaceΦ(v) regularization operator as defined previouslyϕ(x) a scalar positive trust function given the quality of v at x{small value meaning bad/lack/inconsistent control variablelarge value for good quality control variablewe define restored control variable u∗ = MinArg(ε(u)),u ∈ Vε(u) =12∫Ω‖2Φ(u(x))‖2 + ϕ(x)‖u(x)− v(x)‖2dx (4)F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionRegularization as smoothing operatorsCase of gradient penalizationε(u) =12∫Ω‖Φ(u(x))‖2 + ϕ(x)‖2u(x)− v(x)‖dxε 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 theEuler-Lagrange condition∇uε(u) = 0 (5)Gateaux derivatives development leads to∇uε(u) = Φ∗ ◦ Φ(u(x)) + ϕ(x)(u(x)− v(x)) (6)F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionRegularization as smoothing operatorsCase of gradient penalizationGradient penalization : mathematical expressionJ∇(v) =12α∇∫Ωn∑i=1‖∇vi‖2dxApplied as smoothing operator, we getΦ∗∇ ◦ Φ∇ = −∆,with boundary conditions : ∇ui ⊥ ν on ∂Ω⇒ ∇ε∇(ui ) = −∆ui (x) + ϕ(x)(ui (x)− vi (x)), 1 ≤ i ≤ n (7)F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionRegularization as smoothing operatorsCase of gradient penalizationNumerical implementationGeneralized diffusion implementationClassical implementation : given ∇ε, use descent-typealgorithms.Problem : solve the Euler-Lagrange equation∆ui − ϕ(x)(ui (x)− vi (x)) = 0, 1 ≤ i ≤ n (8)considers ui as a function of time and solve the equivalent problem∂∂tui (x, t) = ∆ui (x, t)− ϕ(x)(ui (x, t)− vi (x))), 1 ≤ i ≤ n (9)known as the generalized diffusion equations.As diffusion operator, it can directly be used in backgroundcovariance [see Weaver et al.]F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultoptical flow of Horn and Shunck : luminance conservationdfdt= 0 (10)f (x, t) noted f is the luminance function.For geophysical fluid images, the mass conservation equation ismore adequate [Fitzpatrick 1985]dfdt+ f (∇ · v) = 0 (11)v(x) is the velocity at xgiven the luminance function f (x, 0) = f 0(x) at time 0, solution toequations (10) or (11) defines f (x, t) as function of the staticvelocity field v(x)M : V → Fv 7→ f =M(v)F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultmotion estimation is reduced to the inverse problem :δv∗ = MinArg(J(δv))withJ(δv) =12‖M(vb + δv)− fo‖2F +12‖δv‖2V (12)Aperture problem :only motion along the normal to iso-contours can be inferred ⇒use regularizationF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultTrust function for motion estimationProposition: define trust function ϕto have large values on discontinuities (contours) for motioncomponent along the normal to the contour, and small values inhomogeneous areas.Example : set ϕ to be the contours map c1 or c2 defined asc1(x, f ) = ‖∇xf (x)‖2c2(x, f ) = ‖∇x(Gσ(x) ∗ f (x))‖2F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultTwin experimentsDirect Image sequences assimilation [Titaud et al 2009] ⇒ true initialstate (velocity fields)Images from [J.-B. Flór (LEGI) and I. Eames, 2002]F.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultError analysis : Tikhonov regularizationEvolution of diagnostic functions with respect to the weighting parameterαmean values max valuesF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultError analysis : gradient penalizationEvolution of diagnostic functions with respect to the weighting parameterα∇mean values max valuesF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultError analysis : Generalised diffusionEvolution of diagnostic functions with respect to the weighting parameterαGDmean values max valuesF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultError analysis : Comparison - cost functionEvolution of the observation cost function with minimization iterationsmean valuesF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultError analysis : Comparison - velocity errorEvolution of the velocity error with minimization iterationsmean values max valuesF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultError analysis : Comparison - vorticity errorEvolution of the vorticity error with minimization iterationsmean values max valuesF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultError analysis : Comparison - angular errorEvolution of the angular error with minimization iterationsmean values max valuesF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultAnalysis : vector fieldtrue Tikhonov regularizationF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultAnalysis : vector fieldtrue gradient penalizationF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionMotion estimation problemFrom regularization to pseudo covariance operatorExperimental resultAnalysis : vector fieldtrue Generalised DiffusionF.-X. Le Dimet regularization for inverse problemInverse problems : variational formulationGeneralized Diffusion regularizationApplicationConclusionConclusionConclusionInverse problems :ill-posed ⇒ use regularizationill-conditioned ⇒ use preconditionerproposed : promising approach for regularization of inverseproblems.F.-X. Le Dimet regularization for inverse problem

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A new approach for regularization of inverse problems in images processing

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