TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Variational denoising for manifold-valued data Andreas Weinmann Helmholtz Center Munich & TU M¨ unchen Paris, le 21 novembre 2014 1 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Overview An algorithm for TV minimization for manifold-valued data (joint work with L. Demaret and M.Storath) Second order TV type functionals for S 1 -valued data (joint work with R. Bergmann, F. Laus, G. Steidl) Potts and Blake-Zisserman functionals for manifold-valued signals with a few jumps (joint work with L. Demaret and M.Storath) 2 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Manifold-valued data in DTI • In diffusion tensor imaging (DTI) (Basser et al. ’94) the data are positive(-definite) matrices. • It is reasonable (cf. Pennec et al. ’2004) to equip Pos n with the Riemannian metric g P ( A , B ) = trace ( P − 1 2 AP − 1 BP − 1 2 ) , P positive and A , B symmetric. Positive Matrices visualized as ellipsoids. 3 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Manifold-valued data in DTI • In diffusion tensor imaging (DTI) (Basser et al. ’94) the data are positive(-definite) matrices. • It is reasonable (cf. Pennec et al. ’2004) to equip Pos n with the Riemannian metric g P ( A , B ) = trace ( P − 1 2 AP − 1 BP − 1 2 ) , P positive and A , B symmetric. • Pos n with the metric g P is a Cartan Hadamard manifold (complete, nonpositive sectional curvature, Positive Matrices visualized as ellipsoids. simply connected). • log and exp can be computed explicitly by 1 2 log ( P − 1 2 QP − 1 1 1 2 exp ( P − 1 2 AP − 1 1 2 ) P 2 , 2 ) P 2 . log P Q = P exp P A = P 3 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals TV functionals for manifold-valued data We consider the variational denoising problem given by the (discrete, anisotropic, bivariate) functionals dist ( u i , f i ) p + α dist ( u ij , u i − 1 , j ) q + α � � � dist ( u ij , u i , j − 1 ) q , F α ( u ) = i , j i , j i , j with data f and p , q ≥ 1 . • Choosing q=1 corresponds to (anisotropic) TV minimization/ROF model in Lagrange form (Rudin, Osher, Fatemi ’90) . 4 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals TV functionals for manifold-valued data We consider the variational denoising problem given by the (discrete, anisotropic, bivariate) functionals dist ( u i , f i ) p + α dist ( u ij , u i − 1 , j ) q + α � � � dist ( u ij , u i , j − 1 ) q , F α ( u ) = i , j i , j i , j with data f and p , q ≥ 1 . • Choosing q=1 corresponds to (anisotropic) TV minimization/ROF model in Lagrange form (Rudin, Osher, Fatemi ’90) . • Choose the Riemannian distance dist to obtain the corresponding functionals for manifold-valued data. 4 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals TV functionals for manifold-valued data We consider the variational denoising problem given by the (discrete, anisotropic, bivariate) functionals dist ( u i , f i ) p + α dist ( u ij , u i − 1 , j ) q + α � � � dist ( u ij , u i , j − 1 ) q , F α ( u ) = i , j i , j i , j with data f and p , q ≥ 1 . • Choosing q=1 corresponds to (anisotropic) TV minimization/ROF model in Lagrange form (Rudin, Osher, Fatemi ’90) . • Choose the Riemannian distance dist to obtain the corresponding functionals for manifold-valued data. • Increase anisotropy by additionally considering diagonals, knight moves, . . . 4 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals TV denoising on real DTI data (Camino project, Cook et. al. ’06) Our method for ℓ 2 − TV Real data α = 0 . 11. 5 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Minimization algorithms - TV problem • Idea: Write (for simplicity univariate, multivarite analogous): i dist ( u i , u i − 1 ) q + j dist ( u j , f j ) p = � � � F ( u ) = γ i F i ( u ) + G ( u ) , � F i ( u ) = γ dist ( u i , u i − 1 ) q , j dist ( u j , f j ) p . where G = 6 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Minimization algorithms - TV problem • Idea: Write (for simplicity univariate, multivarite analogous): i dist ( u i , u i − 1 ) q + j dist ( u j , f j ) p = � � � F ( u ) = γ i F i ( u ) + G ( u ) , � F i ( u ) = γ dist ( u i , u i − 1 ) q , j dist ( u j , f j ) p . where G = • Apply the cyclic proximal point algorithm (Bacak, Bertsekas) : Iterate the proximal mappings (Moreau) of G and F i , i = l , . . . , r , 2 dist ( u , v ) 2 + λ F i ( v ) . 1 prox λ F i ( u ) = arg min v • Central Point: The proximal mappings of F i , G can be computed explicitely (next slide). 6 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Minimization algorithms - TV problem Minimize F ( u ) = � i F i ( u ) + G ( u ) , � F i ( u ) = γ dist ( u i , u i − 1 ) q , j dist ( u j , f j ) p . G = • The proximal mapping of G is explicitly given by 2 λ ( 1 + 2 λ ) dist ( u i , f i ) for p= 2 , prox λ G ( u ) i = [ u i , f i ] t , t = min ( λ, dist ( u i , f i )) for p= 1 . (“Soft thresholding” for p = 1.) • The proximal mapping of F i is explicitly given by (Demaret, Storath, W.) u j if j � i , i − 1 , prox λ F i ( u ) j = [ u i , u i − 1 ] t if j = i , [ u i − 1 , u i ] t if j = i − 1 , γλ ( 2 + 2 γλ ) dist ( u i , u i − 1 ) for q =2 , t = min ( λγ, 1 t = 2 dist ( u i , u i − 1 )) for q =1 . 7 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Minimization algorithms - TV problem Minimize F ( u ) = � i F i ( u ) + � i G i ( u ) , F i ( u ) = γ dist ( u i , u i − 1 ) q , G i = dist ( u i , f i ) p . • A parallel proximal point algorithm : Calculate the proximal mappings of F i , G i at u ( k ) u ( k + 1 ) u ( k + 1 ) = prox λ F i ( u ( k ) ) , = prox λ G i ( u ( k ) ) , i n + i and then average them using intrinsic means (Cartan, Frechet, Karcher, . . . ) u ( k + 1 ) = arg min � i dist ( u , u ( k + 1 ) ) 2 . i u 8 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Minimization algorithms - TV problem Minimize F ( u ) = � i F i ( u ) + � i G i ( u ) , F i ( u ) = γ dist ( u i , u i − 1 ) q , G i = dist ( u i , f i ) p . • A parallel proximal point algorithm : Calculate the proximal mappings of F i , G i at u ( k ) u ( k + 1 ) u ( k + 1 ) = prox λ F i ( u ( k ) ) , = prox λ G i ( u ( k ) ) , i n + i and then average them using intrinsic means (Cartan, Frechet, Karcher, . . . ) v = u ( k + 1 ) = arg min � i dist ( u , u ( k + 1 ) ) 2 . i u • To compute the minimizer, we use the gradient descent (Karcher) � N i = 1 log v old u ( k + 1 ) v new = exp v old ( 1 ) . N i 8 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Minimization algorithms - TV problem Minimize F ( u ) = � i F i ( u ) + � i G i ( u ) , F i ( u ) = γ dist ( u i , u i − 1 ) q , G i = dist ( u i , f i ) p . • A parallel proximal point algorithm : Calculate the proximal mappings of F i , G i at u ( k ) u ( k + 1 ) u ( k + 1 ) = prox λ F i ( u ( k ) ) , = prox λ G i ( u ( k ) ) , i n + i and then average them using intrinsic means (Cartan, Frechet, Karcher, . . . ) v = u ( k + 1 ) = arg min � i dist ( u , u ( k + 1 ) ) 2 . i u • To compute the minimizer, we use the gradient descent (Karcher) � N i = 1 log v old u ( k + 1 ) v new = exp v old ( 1 ) . N i • Fast Variant: Approximate the mean by iterated geodesic averages. 8 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Synthetic DTI example Synthetic DT image Rician noise, σ = 90. ℓ 2 -TV (our cyclic PPA) ℓ 2 -TV (our parallel PPA) 9 / 31
TV minimization for manifold data Second order TV type functionals Potts and Blake-Zisserman functionals Analytic Results Theorem (Demaret, Storath, W.) In a Cartan-Hadamard manifold (complete, simply connected, nonpositive sectional curvature) the proposed algorithms (cyclic, parallel and the parallel variant with approximative mean computation) for L p -TV minimization converge towards a global minimizer. 10 / 31
Recommend
More recommend
