Combining multiresolution analysis and non-smooth optimization for texture segmentation
|
|
- Dennis Farmer
- 5 years ago
- Views:
Transcription
1 Combining multiresolution analysis and non-smooth optimization for texture segmentation Nelly Pustelnik CNRS, Laboratoire de Physique de l ENS de Lyon
2 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Stochastic textures Geometric textures periodic Stochastic textures Conclusions
3 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Stochastic textures Geometric textures periodic Stochastic textures scale-free? Conclusions
4 Sinusoidal signal periodic Stochastic textures log power Time log frequency Sinusoidal signal + noise periodic log power Time log frequency Monofractal signal scale-free log power Time log frequency
5 Texture segmentation Ω 1 Ω 2 Mask Synthetic image Real texture Segmentation: Estimate the boundary between Ω 1 and Ω 2 - Contribution 1: Discrete Mumford-Shah, - Contribution 2: Chan-Vese model. Texture = local dependence = local regularity. - Contribution 3: Joint estimation and segmentation.
6 SIROCCO Projet (Start) Projet Jeunes Chercheur.e.s GdR ISIS Défi Imag In CNRS 2017 Joint work with : B. Pascal, M. Foare, P. Abry, V. Vidal, J.-C. Géminard (LPENSL), L. Condat (GIPSA-Lab), H. Wendt, N. Dobigeon (IRIT). Difficulties: large size data (> 2 million pixels), accurate transition, avoid irregular contour.
7 Summary 1. Basics: wavelets and proximal tools 2. Segmentation by means of proximal tools 3. Two-step texture segmentation relying on scale-free descriptor 4. Joint texture segmentation
8 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Wavelet transform and sparsity prox Wavelets: sparse representation of most natural signals. Dyadic wavelet transform, denoted F R Ω Ω filterbank implementation, orthonormal transform: FF = F F = I. g R Ω ζ = Fg
9 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Wavelet transform and sparsity prox g ζ = Fg softλ (F g) b = F softλ (F g) u
10 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Wavelet transform and sparsity prox g ζ = Fg softλ (F g) softλ (ζ) = max{ ζi λ, 0}sign(ζi ) i Ω X 1 νi = arg min kν ζk22 + λ ν 2 i {z } kνk1 1 b = arg min ku gk22 + λkf uk1 u u 2 b = F softλ (F g) u 10 8 Identity Soft-thresholding λ 0 λ αi
11 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Wavelet transform and sparsity prox g ζ = Fg softλ (F g) b = F softλ (F g) u 10 softλ (ζ) = max{ ζi λ, 0}sign(ζi ) i Ω 8 Identity Soft-thresholding 6 = proxλk k1 (ζ) 4 2 -λ 0 1 b = arg min ku gk22 + λkf uk1 u u 2 λ αi
12 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Wavelet transform and sparsity prox g ζ = Fg softλ (F g) b = F softλ (F g) u 10 softλ (ζ) = max{ ζi λ, 0}sign(ζi ) i Ω = proxλk k1 (ζ) 8 Identity Soft-thresholding λ 0 λ -2 b = proxλkf k1 (g) u αi
13 F s : linear operator, Non-smooth optimization û Argmin u R Ω f s : proper, convex, l.s.c functions. S f s (F s u) s=1 Since 2004, numerous proximal algorithms: [Bauschke-Combettes, 2017] - Forward-Backward S = 2, f 1 Lipschitz gradient and L 2 = Id - Douglas-Rachford S = 2 and F 1 = F 2 = Id - PPXA F 1 =... = F S = Id - ADMM Invert S i=1 F ifi - Primal-dual... Flexibility in the design of objective functions.
14 F s : linear operator, Non-smooth optimization û Argmin u R Ω f s : proper, convex, l.s.c functions. S f s (F s u) s=1 Handle with large size problems: Closed form expression of the proximity operators: Avoid splitting: prox s fs. prox fs u = arg min ν u 2 ν 2 + f s (ν). Exploit properties of f s (strong convexity) and of F s. Block-coordinate approach.
15 Summary 1. Basics: wavelets and proximal tools 2. Segmentation by means of proximal tools 3. Two-step texture segmentation relying on scale-free descriptor 4. Joint texture segmentation
16 minimize u,k Mumford-Shah 1 (u g) 2 dxdy + β 2 Ω }{{} fidelity Ω\K u 2 dxdy } {{ } smoothness [Mumford-Shah, 1989] Ω: image domain, g L (Ω): input (possibly noisy), u W 1,2 (Ω): piecewise smooth approximation of g, + λh 1 (K Ω) }{{} length W 1,2 (Ω) = { u L 2 (Ω) u L 2 (Ω) } where weak derivative operator K: set of discontinuities, H 1 : Hausdorff measure. g (û, K)
17 minimize u,k Mumford-Shah 1 (u g) 2 dxdy + β 2 Ω }{{} fidelity Ω\K u 2 dxdy } {{ } smoothness [Mumford-Shah, 1989] Ω: image domain, g L (Ω): input (possibly noisy), u W 1,2 (Ω): piecewise smooth approximation of g, + λh 1 (K Ω) }{{} length W 1,2 (Ω) = { u L 2 (Ω) u L 2 (Ω) } where weak derivative operator K: set of discontinuities, H 1 : Hausdorff measure. g (û, K)
18 Total variation model 1 minimize (u g) 2 dxdy + β u 2 dxdy + λh 1 (K Ω) u,k 2 Ω Ω\K Discrete piecewise constant relaxation minimize u 1 2 u g λtv(u) + Convex. + Fast implementation due to strong convexity. TV denotes some form of the 2-D discrete total variation, i.e., N 1 N 2 ( u R Ω ) TV(u)= u i1 +1,i 2 u i1,i u i1,i 2 +1 u i1,i 2 2 i 1 =1i 2 =1 = Du 2,1,
19 Total variation model g û TV with λ = 100 û TV with λ = 500
20 Proposed Discrete Mumford-Shah minimize u,e 1 2 u g β (1 e) Du 2 + λr(e), [Foare-Pustelnik-Condat, 2018] Ω = {1,..., N 1 } {1,..., N 2 } g R Ω : input (possibly noisy), u R Ω : piecewise smooth approximation of g, D R E Ω : models a finite difference operator, e R E : edges between nodes whose value is 1 when a contour change is detected and 0 otherwise, R: non-smooth to favor sparse solution (i.e. short K ).
21 Proposed Discrete Mumford-Shah minimize u,e [Foare-Pustelnik-Condat, 2018] Ω = {1,..., N 1 } {1,..., N 2 } g R Ω : input (possibly noisy), 1 2 u g β (1 e) Du 2 + λr(e), u R Ω : piecewise smooth approximation of g, D R E Ω : models a finite difference operator, e R E : edges between nodes whose value is 1 when a contour change is detected and 0 otherwise, R: non-smooth to favor sparse solution (i.e. short K ). Hybrid linearized proximal alternating minimization (alternative to [Bolte et al. 2013]
22 Segmentation methods: summary Total Variation Discrete MS Chan-Vese + Fast + Piecewise constant Not accurate contour + Extract contour + Identify smooth variations + Piecewise smooth piecewise constant Time consuming Tune parameters + Perform good segmentation results Time consuming Tune parameters: number of labels, mean value µ q [Pustelnik-Condat, 2017]
23 Segmentation methods: summary Total Variation Discrete MS Chan-Vese + Fast + Piecewise constant Not accurate contour + Extract contour + Identify smooth variations + Piecewise smooth piecewise constant Time consuming Tune parameters + Perform good segmentation results Time consuming Tune parameters: number of labels, mean value µ q [Pustelnik-Condat, 2017]
24 Summary 1. Basics: wavelets and proximal tools 2. Segmentation by means of proximal tools 3. Two-step texture segmentation relying on scale-free descriptor 4. Joint texture segmentation
25 Local regularity (1D)
26 Local regularity (1D) f α regular at y f (x) f (y) χ x y α Example: α = 1 10
27 Local regularity (1D) f α regular at y f (x) f (y) χ x y α Example: α = 1 2
28 Local regularity (1D) Definition ( y) h(y) = sup α such that f is α-regular at y. Compute h(y) at every point?
29 Pointwise regularity and wavelet transform modulus [Extracted from Mallat 1998] log 2 Wf (u, s) log 2 A + (α ) log 2 s. Extract α at each location compute the slope. Continuous wavelet transform not adapted to large size images.
30 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Local regularity and wavelet leaders Discrete wavelet coefficients: - Coefficients at scale j {1,..., J} and subband m = {1, 2, 3}: ζj,m = Hj,m g - Orthonormal transform: h i> > >,..., HJ,3 F = H1,1, L> where J,4 g N Hj,m R 4j ζ = Fg N N and LJ,4 R 4J N
31 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Local regularity and wavelet leaders Wavelet leader at scale j and location k - local supremum of all wavelet coefficients taken within a spatial neighborhood across all finer scales j 0 j ( λj,k = [k2j, (k + 1)2j ) S Lj,k = sup ζj 0,m,k where Λj,k = p { 1,0,1}2 λj,k+p m={1,2,3} λj 0,k 0 Λj,k
32 Multiresolution + nonlinearity local regularity Behavior through the scales [Jaffard, 2004] L j,k s n 2 jhn when 2 j 0 (where k = 2 j n) Linear regression across scales [Wendt et al., 2009] ĥ n = j w j,k log 2 L j,k
33 Multiresolution + nonlinearity local regularity Behavior through the scales [Jaffard, 2004] L j,k s n 2 jhn when 2 j 0 (where k = 2 j n) Linear regression across scales [Wendt et al., 2009] ĥ n = j w j,k log 2 L j,k Unbiased when { j w j,k 0 j jw j,k 1 Ω 1 Ω 2 Mask Original g Estimate ĥ
34 Multiresolution + nonlinearity + nonsmooth Total variation: piecewise constant estimate 1 ĥ TV = arg min u 2 u w j log 2 L j λ Du 1 j }{{} Nonlinear Linear transform transform Linear transform wavelet log 2 leaders linear regression ĥ Nonlinear transform l 1 minimisation
35 Multiresolution + nonlinearity + nonsmooth Ω 1 Ω 2 Mask Original g Estimate ĥ Estimate ĥtv
36 Summary 1. Basics: wavelets and proximal tools 2. Segmentation by means of proximal tools 3. Two-step texture segmentation relying on scale-free descriptor 4. Joint texture segmentation
37 Multiresolution + nonlinearity + nonsmooth Total variation: Joint estimation and segmentation [Pustelnik et al., 2016] (ĥtvw, ŵ) = arg min u,w 1 2 u w j log 2 L j λ Du 1 + d C (w) j }{{} Relax unbiased contraint: C = {w R J Ω ( k) j w j,k 0 and j jw j,k 1} d C (ŵ)) = w P C (w) 2 1 P C (ŵ)) = arg min ν C 2 ν w 2 2 ĥ
38 Multiresolution + nonlinearity + nonsmooth Ω 1 Ω 2 Mask Original g Estimate ĥ Estimate ĥtv Estimate ĥtvw
39 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Multiresolution + nonlinearity + nonsmooth Original g b Estimate h btv Estimate h btvw Estimate h Conclusions
40 Intro Basics Segmentation Two-step texture segmentation Joint Texture Segmentation Conclusions Multiresolution + nonlinearity + nonsmooth [Yuan et al. 2015] Original g [Arbelaez et al. 2011] b Estimate h btv Estimate h btvw Estimate h
41 Multiresolution + nonlinearity + nonsmooth 1 (ĥtvw, ŵ) = arg min (u,w) 2 u w j log 2 L j λ Du 1 + d C (ŵ) j }{{} ĥ + Good texture segmentation performance + Convex minimization formulation + Combined estimation and segmentation (contrary to ĥtv) Computational cost. Not adapted for large scale data.
42 Multiresolution + nonlinearity local regularity Behavior through the scales [Jaffard, 2004] L j,k s n 2 jhn as 2 j 0 (where k = 2 j n) log 2 L j,k log 2 s n + jh n as 2 j 0. PLOVER: Piecewise constant LOcal VariancE and Regularity estimation [Pascal et al., 2018] Find ( v, ĥ) Argmin j log 2 L j v jh η Dh 1 + ζ Dv 1 v,h with ŝ = 2 v + Strongly convex computationally efficient + Combine estimation and segmentation. + Joint estimation of the local variance and local regularity.
43 Multiresolution + nonlinearity + nonsmooth + fast (a) Synthetic texture x (b) s mask (c) h mask Linear regression Disjoint TV PLOVER Disjoint re-estimation SNR = SNR = SNR = SNR = PLOVER re-estimation SNR = Local variance SNR = SNR = SNR = SNR = SNR = Local regularity
44 Multiresolution + nonlinearity + nonsmooth + fast Image g R N Zoom of g PLOVER : ŝ PLOVER : ĥ [Arbelaez2011] [Yuan 2015] Disjoint TV PLOVER
45 Conclusions HL-PAM for fast discrete Mumford-Shah several applications going from image restoration to graph analysis. Proximity operator of a sum of two functions application to segmentation and depth map estimation. Scale-free descriptors in a variational framework large-scale texture segmentation procedure.
46 Perspectives TV denoising/ Chan-Vese/D-MS procedure allowing to propose to expert accurate estimation and segmentation. D-MS allows to go from piecewise smooth to piecewise constant. Both are of interest for the applications. HL-PAM and strong convexity? Quantify deadzone w.r.t. scale. Regularization parameter selection. Integrate anisotropy.
47 References N. Pustelnik, H. Wendt, P. Abry, N. Dobigeon, Combining local regularity estimation and total variation optimization for scale-free texture segmentation, IEEE Trans. on Computational Imaging, vol. 2, no. 4, pp , Dec N. Pustelnik, L. Condat, Proximity operator of a sum of functions; Application to depth map estimation, IEEE Signal Processing Letters, M. Foare, N. Pustelnik, L. Condat, A new proximal method for joint image restoration and edge detection with the Mumford-Shah model, accepted ICASSP B. Pascal, N. Pustelnik, P. Abry, M. Serres, V. Vidal, Joint estimation of local variance and local regularity for texture segmentation. Application to multiphase flow characterization, submitted IEEE ICIP J. Frecon, N. Pustelnik, N. Dobigeon, H. Wendt, and P. Abry, Bayesian selection for the regularization parameter in TVl0 denoising problems, IEEE Trans. on Signal Processing, 2017.
48 Proposed Discrete Mumford-Shah minimize u,e 1 2 u g β (1 e) Du 2 + λr(e) g û ê
49 Proposed Discrete Mumford-Shah minimize u,e 1 2 u g β (1 e) Du 2 + λr(e) R: favors binary, i.e. {0, 1} E and sparse solution (i.e. short K ) 1. Ambrosio-Tortorelli approximation: [Ambrosio-Tortorelli, 1990] [Foare-Lachaud-Talbot, 2016] R(e) = ε De ε e 2 2 with ε > 0 2. l 1 -norm: R(e) = e 1 3. Quadratic l 1 : [Foare-Pustelnik-Condat, 2017] R(e) = { } E i=1 max e i, e2 i. 4ε
50 minimize u,e Proposed Discrete Mumford-Shah Ψ(u, e) := 1 2 u g β (1 e) Du 2 +λr(e) }{{} S(e,Du) PALM [Bolte et al, 2014] Set u [0] R Ω and e [0] R E. For l N Set γ > 1 and c l = γχ(e [l] ) u [l+1] prox 1 g (u [l] c 2 c l u S ( e [l], Du [l])) l 2 Set δ > 1 and d k = δν(u [l+1] ) e [l+1] prox 1 λr (e [l] d l e S ( e [l], Du [l+1])) d l Under technical assumptions, the sequence (u [l], e [l] ) l N converges to a critical point (u, e ) of Ψ.
51 minimize u,e Proposed Discrete Mumford-Shah Ψ(u, e) := 1 2 u g β (1 e) Du 2 +λr(e), }{{} S(e,Du) Proposed HL-PAM [Foare-Pustelnik-Condat, 2017] Set u [0] R Ω and e [0] R E. For l N Set γ > 1 and c l = γχ(e [l] ). u [l+1] prox 1 g (u [l] c 2 c k u S ( e [l], Du [l])) k 2 Set d l > 0. ) e [l+1] prox 1 λr+s(,du d [l+1] ) (e [l] l Under technical assumptions, the sequence (u [l], e [l] ) l N converges to a critical point (u, e ) of Ψ.
52 Proposed Discrete Mumford-Shah Assumptions 1. The updating steps of u [l+1] and e [l+1] have closed form expressions; 2. e S is globally Lispchitz with moduli χ(e [l] ) for every l N and there exists χ, χ + > 0 such that χ χ(e [l] ) χ + ; 3. (d l ) l N is a positive sequence such that the stepsizes d l belongs to (d, d + ) for some positive d d +.
53 Proposed Discrete Mumford-Shah Proposition [Foare-Pustelnik-Condat, 2017] We assume that S is separable, i.e, ( e=(e i ) 1 i E ) R(e) = σ i (e i ), where σ i :R E ] ; + ] with a closed form proximity operator expression. Let d l > 0, then prox 1 d l λr+s(,du [l+1] ) (e[l] ) = ( prox λσ i 2β(Du [l] ) 2 i +d l E i=1 [l] i β(du[l+1] ) 2 i + d le 2 β(du [l+1] ) 2 i + d l 2 ) i E
54 Proposed Discrete Mumford-Shah Proposition [Foare-Pustelnik-Condat, 2017] For every η R and τ, ɛ > 0 { [ ( prox. τ max{., 2 }(η) = sign(η) max 0, min η τ, max 4ɛ, 4ɛ η )]} τ 2ɛ + 1
55 Proposed Discrete Mumford-Shah Convergence PALM versus HL-PALM: Ψ(u [l], e [l] ) w.r.t. iterations l PALM, d l = 0.5/β HL-PAM, d l = 0.5/β HL-PAM, d l = 5/β HL-PAM, d l = 50/β HL-PAM, d l = 500/β
56 Data g TV [Strekalovskiy-Cremers, 2014] [Foare-Lachaud-Talbot, 2016] l 1 quadratic-l 1
57 Proposed Discrete Mumford-Shah Data g TV [Strekalovskiy-Cremers, 2014] [Foare-Lachaud-Talbot, 2016] l 1 quadratic-l 1
58 Proposed Discrete Mumford-Shah Data g TV [Strekalovskiy-Cremers, 2014] [Foare-Lachaud-Talbot, 2016] l 1 quadratic-l 1
59 Proposed Discrete Mumford-Shah Data g TV [Strekalovskiy-Cremers, 2014] [Foare-Lachaud-Talbot, 2016] l 1 quadratic-l 1
60 Proposed Discrete Mumford-Shah Convergence speed: Ψ(u [l+1], e [l+1] ) Ψ(u [l], e [l] ) < 10 4 TV [Foare-Lachaud-Talbot, 2016] l 1 quadratic l 1 dots ( Ω = ) dots ( Ω = ) dots ( Ω = ) ellipse ( Ω = ) ellipse ( Ω = ) ellipse ( Ω = ) peppers ( Ω = ) peppers ( Ω = ) peppers ( Ω = )
61 minimize u,k Chan-Vese model 1 (u g) 2 dxdy + β u 2 dxdy + λh 1 (K Ω) 2 Ω Ω\K Discrete piecewise constant relaxation with fixed label number [Chan-Vese, 2001] Q Q minimize θ (q 1) θ (q), (µ q g) 2 + λ TV(θ (q 1) θ (q) ) (θ (q) ) 1 q Q 1 q=1 q=1 s.t. 1 θ (0) θ (1)... θ (Q 1) θ (Q) 0, Ω 3 Ω 3 Ω 2 Ω 1 Ω 1 Ω Ω Ω g
62 minimize u,k Chan-Vese model 1 (u g) 2 dxdy + β u 2 dxdy + λh 1 (K Ω) 2 Ω Ω\K Discrete piecewise constant relaxation with fixed label number [Chan-Vese, 2001] Q Q minimize θ (q 1) θ (q), (µ q g) 2 + λ TV(θ (q 1) θ (q) ) (θ (q) ) 1 q Q 1 q=1 q=1 s.t. 1 θ (0) θ (1)... θ (Q 1) θ (Q) 0, Ω 3 Ω 2 Ω 1 g θ (0) θ (1) θ (2) θ (3)
63 minimize u,k Chan-Vese model 1 (u g) 2 dxdy + β u 2 dxdy + λh 1 (K Ω) 2 Ω Ω\K Discrete piecewise constant relaxation with fixed label number [Chan-Vese, 2001] Q Q minimize θ (q 1) θ (q), (µ q g) 2 + λ TV(θ (q 1) θ (q) ) (θ (q) ) 1 q Q 1 q=1 q=1 s.t. 1 θ (0) θ (1)... θ (Q 1) θ (Q) 0, Ω 3 Ω 3 Ω 2 Ω Ω Ω 1 1 Ω Ω g θ (0) θ (1) θ (1) θ (2) θ (2) θ (3)
64 Chan-Vese model minimize Θ=(θ (q) ) 1 q Q 1 Q 1 Q β (q), θ (q) + λ DH q Θ 2,1 + ι [0,1] Q Ω (Θ) + ι E (Θ) q=1 q=1 β (q) = (µ q+1 g) 2 (µ q g) 2, H q : R Q Ω R Ω : Θ θ (q 1) θ (q), E = {Θ R Q Ω : θ (1)... θ (Q 1) }. Use of splitting proximal algorithms to deal with a sum of convex but non-smooth functions.
65 Chan-Vese model Three-term splitting : minimize Θ Q 1 q=1 β(q), θ (q) + λ Q q=1 DH qθ 2,1 + ι [0,1] Q Ω (Θ) + ι E (Θ) Two-term splitting : minimize Θ Q 1 q=1 β(q), θ (q) + λ Q q=1 DH qθ 2,1 + ι [0,1] Q Ω (Θ) + ι E (Θ) Question: When is it possible to compute the proximity operator of a sum of functions rather splitting. Would it be more efficient?
66 Chan-Vese model Proposition [Pustelnik, Condat, 2017] (i) For some function h 0 Γ 0(R), h is separable, with ( ) x = (xi ) i Ω h(x) = h 0(x i ). i Ω (ii) g has the following form: ( x = (xi ) i Ω ) g(x) = (m,m ) Υ Ω 2 σ Cm,m (x m x m), where σ Cm,m : t R sup {tp, p C m,m } is the support function of a closed real interval C m,m, such that inf C m,m = a m,m and sup C m,m = b m,m, for some a m,m R { } and b m,m R {+ }, with a m,m b m,m. a m,m t if t < 0, ( t R) σ Cm,m (t) = 0 if t = 0, b m,m t if t > 0, Under assumptions (i) and (ii), prox g+h = prox h prox g.
67 Chan-Vese model Particular cases: Fused Lasso: Ω = {1,..., N} and Υ = {(1, 2), (2, 3),..., (N 1, N)}, b n,n+1 = a n,n+1 = ω n 0, h 0 = λ, g(x) = N 1 n=1 ω n x n+1 x n Chan-Vese: Ω = {1,..., Q} and Υ = {(1, 2), (2, 3),..., (Q 1, Q)}, a n,n+1 = 0 b n,n+1 = +, h 0 = ι [0,1], g(x) = ι E Compute P E with Pool Adjacent Violators Algorithm (PAVA) [Ayer et al., 1995]
68 Chan-Vese model g λ = 10 3 λ = 10 4
69 Chan-Vese model Minimal splitting (proposed method) Intermediate splitting 10 8 Full splitting
A Parallel Block-Coordinate Approach for Primal-Dual Splitting with Arbitrary Random Block Selection
EUSIPCO 2015 1/19 A Parallel Block-Coordinate Approach for Primal-Dual Splitting with Arbitrary Random Block Selection Jean-Christophe Pesquet Laboratoire d Informatique Gaspard Monge - CNRS Univ. Paris-Est
More informationIn collaboration with J.-C. Pesquet A. Repetti EC (UPE) IFPEN 16 Dec / 29
A Random block-coordinate primal-dual proximal algorithm with application to 3D mesh denoising Emilie CHOUZENOUX Laboratoire d Informatique Gaspard Monge - CNRS Univ. Paris-Est, France Horizon Maths 2014
More informationProximal tools for image reconstruction in dynamic Positron Emission Tomography
Proximal tools for image reconstruction in dynamic Positron Emission Tomography Nelly Pustelnik 1 joint work with Caroline Chaux 2, Jean-Christophe Pesquet 3, and Claude Comtat 4 1 Laboratoire de Physique,
More informationA Primal-dual Three-operator Splitting Scheme
Noname manuscript No. (will be inserted by the editor) A Primal-dual Three-operator Splitting Scheme Ming Yan Received: date / Accepted: date Abstract In this paper, we propose a new primal-dual algorithm
More informationAbout Split Proximal Algorithms for the Q-Lasso
Thai Journal of Mathematics Volume 5 (207) Number : 7 http://thaijmath.in.cmu.ac.th ISSN 686-0209 About Split Proximal Algorithms for the Q-Lasso Abdellatif Moudafi Aix Marseille Université, CNRS-L.S.I.S
More informationInverse problem and optimization
Inverse problem and optimization Laurent Condat, Nelly Pustelnik CNRS, Gipsa-lab CNRS, Laboratoire de Physique de l ENS de Lyon Decembre, 15th 2016 Inverse problem and optimization 2/36 Plan 1. Examples
More informationA memory gradient algorithm for l 2 -l 0 regularization with applications to image restoration
A memory gradient algorithm for l 2 -l 0 regularization with applications to image restoration E. Chouzenoux, A. Jezierska, J.-C. Pesquet and H. Talbot Université Paris-Est Lab. d Informatique Gaspard
More informationAdaptive Primal Dual Optimization for Image Processing and Learning
Adaptive Primal Dual Optimization for Image Processing and Learning Tom Goldstein Rice University tag7@rice.edu Ernie Esser University of British Columbia eesser@eos.ubc.ca Richard Baraniuk Rice University
More informationOn a multiscale representation of images as hierarchy of edges. Eitan Tadmor. University of Maryland
On a multiscale representation of images as hierarchy of edges Eitan Tadmor Center for Scientific Computation and Mathematical Modeling (CSCAMM) Department of Mathematics and Institute for Physical Science
More informationVariational Image Restoration
Variational Image Restoration Yuling Jiao yljiaostatistics@znufe.edu.cn School of and Statistics and Mathematics ZNUFE Dec 30, 2014 Outline 1 1 Classical Variational Restoration Models and Algorithms 1.1
More informationPrimal-dual algorithms for the sum of two and three functions 1
Primal-dual algorithms for the sum of two and three functions 1 Ming Yan Michigan State University, CMSE/Mathematics 1 This works is partially supported by NSF. optimization problems for primal-dual algorithms
More informationShiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 9. Alternating Direction Method of Multipliers
Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 9 Alternating Direction Method of Multipliers Shiqian Ma, MAT-258A: Numerical Optimization 2 Separable convex optimization a special case is min f(x)
More informationSplitting Techniques in the Face of Huge Problem Sizes: Block-Coordinate and Block-Iterative Approaches
Splitting Techniques in the Face of Huge Problem Sizes: Block-Coordinate and Block-Iterative Approaches Patrick L. Combettes joint work with J.-C. Pesquet) Laboratoire Jacques-Louis Lions Faculté de Mathématiques
More informationSparse linear models
Sparse linear models Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda 2/22/2016 Introduction Linear transforms Frequency representation Short-time
More informationMaster 2 MathBigData. 3 novembre CMAP - Ecole Polytechnique
Master 2 MathBigData S. Gaïffas 1 3 novembre 2014 1 CMAP - Ecole Polytechnique 1 Supervised learning recap Introduction Loss functions, linearity 2 Penalization Introduction Ridge Sparsity Lasso 3 Some
More informationA General Framework for a Class of Primal-Dual Algorithms for TV Minimization
A General Framework for a Class of Primal-Dual Algorithms for TV Minimization Ernie Esser UCLA 1 Outline A Model Convex Minimization Problem Main Idea Behind the Primal Dual Hybrid Gradient (PDHG) Method
More informationInverse problems Total Variation Regularization Mark van Kraaij Casa seminar 23 May 2007 Technische Universiteit Eindh ove n University of Technology
Inverse problems Total Variation Regularization Mark van Kraaij Casa seminar 23 May 27 Introduction Fredholm first kind integral equation of convolution type in one space dimension: g(x) = 1 k(x x )f(x
More information1 Sparsity and l 1 relaxation
6.883 Learning with Combinatorial Structure Note for Lecture 2 Author: Chiyuan Zhang Sparsity and l relaxation Last time we talked about sparsity and characterized when an l relaxation could recover the
More informationINVERSE PROBLEM FORMULATION FOR REGULARITY ESTIMATION IN IMAGES
INVERSE PROBLEM FORMULATION FOR REGULARITY ESTIMATION IN IMAGES Nelly Pustelnik, Patrice Abry, Herwig Wendt 2 and Nicolas Dobigeon 2 Physics Dept. - ENSL, UMR CNRS 5672, F-69364 Lyon, France, firstname.lastname@ens-lyon.fr
More informationA Majorize-Minimize subspace approach for l 2 -l 0 regularization with applications to image processing
A Majorize-Minimize subspace approach for l 2 -l 0 regularization with applications to image processing Emilie Chouzenoux emilie.chouzenoux@univ-mlv.fr Université Paris-Est Lab. d Informatique Gaspard
More informationSemi-Linearized Proximal Alternating Minimization for a Discrete Mumford Shah Model
SemiLinearized Proximal Alternating Minimization for a Discrete Mumford Shah Model Marion Foare, Nelly Pustelnik, Laurent Condat To cite this version: Marion Foare, Nelly Pustelnik, Laurent Condat. SemiLinearized
More informationA GENERAL FRAMEWORK FOR A CLASS OF FIRST ORDER PRIMAL-DUAL ALGORITHMS FOR CONVEX OPTIMIZATION IN IMAGING SCIENCE
A GENERAL FRAMEWORK FOR A CLASS OF FIRST ORDER PRIMAL-DUAL ALGORITHMS FOR CONVEX OPTIMIZATION IN IMAGING SCIENCE ERNIE ESSER XIAOQUN ZHANG TONY CHAN Abstract. We generalize the primal-dual hybrid gradient
More informationA GENERAL FRAMEWORK FOR A CLASS OF FIRST ORDER PRIMAL-DUAL ALGORITHMS FOR TV MINIMIZATION
A GENERAL FRAMEWORK FOR A CLASS OF FIRST ORDER PRIMAL-DUAL ALGORITHMS FOR TV MINIMIZATION ERNIE ESSER XIAOQUN ZHANG TONY CHAN Abstract. We generalize the primal-dual hybrid gradient (PDHG) algorithm proposed
More informationGeneralized greedy algorithms.
Generalized greedy algorithms. François-Xavier Dupé & Sandrine Anthoine LIF & I2M Aix-Marseille Université - CNRS - Ecole Centrale Marseille, Marseille ANR Greta Séminaire Parisien des Mathématiques Appliquées
More informationSparse Regularization via Convex Analysis
Sparse Regularization via Convex Analysis Ivan Selesnick Electrical and Computer Engineering Tandon School of Engineering New York University Brooklyn, New York, USA 29 / 66 Convex or non-convex: Which
More informationVariable Metric Forward-Backward Algorithm
Variable Metric Forward-Backward Algorithm 1/37 Variable Metric Forward-Backward Algorithm for minimizing the sum of a differentiable function and a convex function E. Chouzenoux in collaboration with
More informationStochastic Proximal Gradient Algorithm
Stochastic Institut Mines-Télécom / Telecom ParisTech / Laboratoire Traitement et Communication de l Information Joint work with: Y. Atchade, Ann Arbor, USA, G. Fort LTCI/Télécom Paristech and the kind
More informationA posteriori error control for the binary Mumford Shah model
A posteriori error control for the binary Mumford Shah model Benjamin Berkels 1, Alexander Effland 2, Martin Rumpf 2 1 AICES Graduate School, RWTH Aachen University, Germany 2 Institute for Numerical Simulation,
More informationRecent developments on sparse representation
Recent developments on sparse representation Zeng Tieyong Department of Mathematics, Hong Kong Baptist University Email: zeng@hkbu.edu.hk Hong Kong Baptist University Dec. 8, 2008 First Previous Next Last
More informationAccelerated Dual Gradient-Based Methods for Total Variation Image Denoising/Deblurring Problems (and other Inverse Problems)
Accelerated Dual Gradient-Based Methods for Total Variation Image Denoising/Deblurring Problems (and other Inverse Problems) Donghwan Kim and Jeffrey A. Fessler EECS Department, University of Michigan
More informationMarkov Random Fields
Markov Random Fields Umamahesh Srinivas ipal Group Meeting February 25, 2011 Outline 1 Basic graph-theoretic concepts 2 Markov chain 3 Markov random field (MRF) 4 Gauss-Markov random field (GMRF), and
More informationProximal splitting methods on convex problems with a quadratic term: Relax!
Proximal splitting methods on convex problems with a quadratic term: Relax! The slides I presented with added comments Laurent Condat GIPSA-lab, Univ. Grenoble Alpes, France Workshop BASP Frontiers, Jan.
More informationA First Order Primal-Dual Algorithm for Nonconvex T V q Regularization
A First Order Primal-Dual Algorithm for Nonconvex T V q Regularization Thomas Möllenhoff, Evgeny Strekalovskiy, and Daniel Cremers TU Munich, Germany Abstract. We propose an efficient first order primal-dual
More informationECE G: Special Topics in Signal Processing: Sparsity, Structure, and Inference
ECE 18-898G: Special Topics in Signal Processing: Sparsity, Structure, and Inference Sparse Recovery using L1 minimization - algorithms Yuejie Chi Department of Electrical and Computer Engineering Spring
More informationconsistent learning by composite proximal thresholding
consistent learning by composite proximal thresholding Saverio Salzo Università degli Studi di Genova Optimization in Machine learning, vision and image processing Université Paul Sabatier, Toulouse 6-7
More informationSolving DC Programs that Promote Group 1-Sparsity
Solving DC Programs that Promote Group 1-Sparsity Ernie Esser Contains joint work with Xiaoqun Zhang, Yifei Lou and Jack Xin SIAM Conference on Imaging Science Hong Kong Baptist University May 14 2014
More informationGradient Sliding for Composite Optimization
Noname manuscript No. (will be inserted by the editor) Gradient Sliding for Composite Optimization Guanghui Lan the date of receipt and acceptance should be inserted later Abstract We consider in this
More informationI P IANO : I NERTIAL P ROXIMAL A LGORITHM FOR N ON -C ONVEX O PTIMIZATION
I P IANO : I NERTIAL P ROXIMAL A LGORITHM FOR N ON -C ONVEX O PTIMIZATION Peter Ochs University of Freiburg Germany 17.01.2017 joint work with: Thomas Brox and Thomas Pock c 2017 Peter Ochs ipiano c 1
More informationSignal Processing and Networks Optimization Part VI: Duality
Signal Processing and Networks Optimization Part VI: Duality Pierre Borgnat 1, Jean-Christophe Pesquet 2, Nelly Pustelnik 1 1 ENS Lyon Laboratoire de Physique CNRS UMR 5672 pierre.borgnat@ens-lyon.fr,
More informationProximal Methods for Optimization with Spasity-inducing Norms
Proximal Methods for Optimization with Spasity-inducing Norms Group Learning Presentation Xiaowei Zhou Department of Electronic and Computer Engineering The Hong Kong University of Science and Technology
More informationVariational methods for restoration of phase or orientation data
Variational methods for restoration of phase or orientation data Martin Storath joint works with Laurent Demaret, Michael Unser, Andreas Weinmann Image Analysis and Learning Group Universität Heidelberg
More informationarxiv: v2 [math.oc] 21 Nov 2017
Unifying abstract inexact convergence theorems and block coordinate variable metric ipiano arxiv:1602.07283v2 [math.oc] 21 Nov 2017 Peter Ochs Mathematical Optimization Group Saarland University Germany
More informationAdaptive discretization and first-order methods for nonsmooth inverse problems for PDEs
Adaptive discretization and first-order methods for nonsmooth inverse problems for PDEs Christian Clason Faculty of Mathematics, Universität Duisburg-Essen joint work with Barbara Kaltenbacher, Tuomo Valkonen,
More informationarxiv: v4 [math.oc] 29 Jan 2018
Noname manuscript No. (will be inserted by the editor A new primal-dual algorithm for minimizing the sum of three functions with a linear operator Ming Yan arxiv:1611.09805v4 [math.oc] 29 Jan 2018 Received:
More informationPrimal-dual coordinate descent A Coordinate Descent Primal-Dual Algorithm with Large Step Size and Possibly Non-Separable Functions
Primal-dual coordinate descent A Coordinate Descent Primal-Dual Algorithm with Large Step Size and Possibly Non-Separable Functions Olivier Fercoq and Pascal Bianchi Problem Minimize the convex function
More informationOWL to the rescue of LASSO
OWL to the rescue of LASSO IISc IBM day 2018 Joint Work R. Sankaran and Francis Bach AISTATS 17 Chiranjib Bhattacharyya Professor, Department of Computer Science and Automation Indian Institute of Science,
More informationArtifact-free Wavelet Denoising: Non-convex Sparse Regularization, Convex Optimization
IEEE SIGNAL PROCESSING LETTERS. 22(9):164-168, SEPTEMBER 215. (PREPRINT) 1 Artifact-free Wavelet Denoising: Non-convex Sparse Regularization, Convex Optimization Yin Ding and Ivan W. Selesnick Abstract
More informationARock: an algorithmic framework for asynchronous parallel coordinate updates
ARock: an algorithmic framework for asynchronous parallel coordinate updates Zhimin Peng, Yangyang Xu, Ming Yan, Wotao Yin ( UCLA Math, U.Waterloo DCO) UCLA CAM Report 15-37 ShanghaiTech SSDS 15 June 25,
More informationPrimal-dual coordinate descent
Primal-dual coordinate descent Olivier Fercoq Joint work with P. Bianchi & W. Hachem 15 July 2015 1/28 Minimize the convex function f, g, h convex f is differentiable Problem min f (x) + g(x) + h(mx) x
More informationConvex Hodge Decomposition and Regularization of Image Flows
Convex Hodge Decomposition and Regularization of Image Flows Jing Yuan, Christoph Schnörr, Gabriele Steidl April 14, 2008 Abstract The total variation (TV) measure is a key concept in the field of variational
More informationInexact Alternating Direction Method of Multipliers for Separable Convex Optimization
Inexact Alternating Direction Method of Multipliers for Separable Convex Optimization Hongchao Zhang hozhang@math.lsu.edu Department of Mathematics Center for Computation and Technology Louisiana State
More informationOptimisation in imaging
Optimisation in imaging Hugues Talbot ICIP 2014 Tutorial Optimisation on Hierarchies H. Talbot : Optimisation 1/59 Outline of the lecture 1 Concepts in optimization Cost function Constraints Duality 2
More informationConvex Hodge Decomposition of Image Flows
Convex Hodge Decomposition of Image Flows Jing Yuan 1, Gabriele Steidl 2, Christoph Schnörr 1 1 Image and Pattern Analysis Group, Heidelberg Collaboratory for Image Processing, University of Heidelberg,
More informationCoordinate Update Algorithm Short Course Operator Splitting
Coordinate Update Algorithm Short Course Operator Splitting Instructor: Wotao Yin (UCLA Math) Summer 2016 1 / 25 Operator splitting pipeline 1. Formulate a problem as 0 A(x) + B(x) with monotone operators
More informationAdaptive Corrected Procedure for TVL1 Image Deblurring under Impulsive Noise
Adaptive Corrected Procedure for TVL1 Image Deblurring under Impulsive Noise Minru Bai(x T) College of Mathematics and Econometrics Hunan University Joint work with Xiongjun Zhang, Qianqian Shao June 30,
More informationInvestigating the Influence of Box-Constraints on the Solution of a Total Variation Model via an Efficient Primal-Dual Method
Article Investigating the Influence of Box-Constraints on the Solution of a Total Variation Model via an Efficient Primal-Dual Method Andreas Langer Department of Mathematics, University of Stuttgart,
More informationConvex Optimization. (EE227A: UC Berkeley) Lecture 15. Suvrit Sra. (Gradient methods III) 12 March, 2013
Convex Optimization (EE227A: UC Berkeley) Lecture 15 (Gradient methods III) 12 March, 2013 Suvrit Sra Optimal gradient methods 2 / 27 Optimal gradient methods We saw following efficiency estimates for
More information2 Regularized Image Reconstruction for Compressive Imaging and Beyond
EE 367 / CS 448I Computational Imaging and Display Notes: Compressive Imaging and Regularized Image Reconstruction (lecture ) Gordon Wetzstein gordon.wetzstein@stanford.edu This document serves as a supplement
More informationCombining local regularity estimation and total variation optimization for scale-free texture segmentation
arxiv:154.5776v3 [cs.cv] 24 Jun 216 Combining local regularity estimation and total variation optimization for scale-free texture segmentation N. Pustelnik H. Wendt P. Abry N. Dobigeon June 27, 216 Abstract
More informationAccelerated primal-dual methods for linearly constrained convex problems
Accelerated primal-dual methods for linearly constrained convex problems Yangyang Xu SIAM Conference on Optimization May 24, 2017 1 / 23 Accelerated proximal gradient For convex composite problem: minimize
More informationarxiv: v1 [math.oc] 13 Dec 2018
A NEW HOMOTOPY PROXIMAL VARIABLE-METRIC FRAMEWORK FOR COMPOSITE CONVEX MINIMIZATION QUOC TRAN-DINH, LIANG LING, AND KIM-CHUAN TOH arxiv:8205243v [mathoc] 3 Dec 208 Abstract This paper suggests two novel
More informationMMSE Denoising of 2-D Signals Using Consistent Cycle Spinning Algorithm
Denoising of 2-D Signals Using Consistent Cycle Spinning Algorithm Bodduluri Asha, B. Leela kumari Abstract: It is well known that in a real world signals do not exist without noise, which may be negligible
More informationADMM and Fast Gradient Methods for Distributed Optimization
ADMM and Fast Gradient Methods for Distributed Optimization João Xavier Instituto Sistemas e Robótica (ISR), Instituto Superior Técnico (IST) European Control Conference, ECC 13 July 16, 013 Joint work
More informationGauge optimization and duality
1 / 54 Gauge optimization and duality Junfeng Yang Department of Mathematics Nanjing University Joint with Shiqian Ma, CUHK September, 2015 2 / 54 Outline Introduction Duality Lagrange duality Fenchel
More informationA New Look at First Order Methods Lifting the Lipschitz Gradient Continuity Restriction
A New Look at First Order Methods Lifting the Lipschitz Gradient Continuity Restriction Marc Teboulle School of Mathematical Sciences Tel Aviv University Joint work with H. Bauschke and J. Bolte Optimization
More informationMonotone Operator Splitting Methods in Signal and Image Recovery
Monotone Operator Splitting Methods in Signal and Image Recovery P.L. Combettes 1, J.-C. Pesquet 2, and N. Pustelnik 3 2 Univ. Pierre et Marie Curie, Paris 6 LJLL CNRS UMR 7598 2 Univ. Paris-Est LIGM CNRS
More informationProximal methods. S. Villa. October 7, 2014
Proximal methods S. Villa October 7, 2014 1 Review of the basics Often machine learning problems require the solution of minimization problems. For instance, the ERM algorithm requires to solve a problem
More informationSEMI-SMOOTH SECOND-ORDER TYPE METHODS FOR COMPOSITE CONVEX PROGRAMS
SEMI-SMOOTH SECOND-ORDER TYPE METHODS FOR COMPOSITE CONVEX PROGRAMS XIANTAO XIAO, YONGFENG LI, ZAIWEN WEN, AND LIWEI ZHANG Abstract. The goal of this paper is to study approaches to bridge the gap between
More informationϕ ( ( u) i 2 ; T, a), (1.1)
CONVEX NON-CONVEX IMAGE SEGMENTATION RAYMOND CHAN, ALESSANDRO LANZA, SERENA MORIGI, AND FIORELLA SGALLARI Abstract. A convex non-convex variational model is proposed for multiphase image segmentation.
More informationEE 367 / CS 448I Computational Imaging and Display Notes: Image Deconvolution (lecture 6)
EE 367 / CS 448I Computational Imaging and Display Notes: Image Deconvolution (lecture 6) Gordon Wetzstein gordon.wetzstein@stanford.edu This document serves as a supplement to the material discussed in
More informationDenoising of NIRS Measured Biomedical Signals
Denoising of NIRS Measured Biomedical Signals Y V Rami reddy 1, Dr.D.VishnuVardhan 2 1 M. Tech, Dept of ECE, JNTUA College of Engineering, Pulivendula, A.P, India 2 Assistant Professor, Dept of ECE, JNTUA
More informationConvex relaxation for Combinatorial Penalties
Convex relaxation for Combinatorial Penalties Guillaume Obozinski Equipe Imagine Laboratoire d Informatique Gaspard Monge Ecole des Ponts - ParisTech Joint work with Francis Bach Fête Parisienne in Computation,
More informationConvergence of Fixed-Point Iterations
Convergence of Fixed-Point Iterations Instructor: Wotao Yin (UCLA Math) July 2016 1 / 30 Why study fixed-point iterations? Abstract many existing algorithms in optimization, numerical linear algebra, and
More informationDual methods for the minimization of the total variation
1 / 30 Dual methods for the minimization of the total variation Rémy Abergel supervisor Lionel Moisan MAP5 - CNRS UMR 8145 Different Learning Seminar, LTCI Thursday 21st April 2016 2 / 30 Plan 1 Introduction
More information2D HILBERT-HUANG TRANSFORM. Jérémy Schmitt, Nelly Pustelnik, Pierre Borgnat, Patrick Flandrin
2D HILBERT-HUANG TRANSFORM Jérémy Schmitt, Nelly Pustelnik, Pierre Borgnat, Patrick Flandrin Laboratoire de Physique de l Ecole Normale Suprieure de Lyon, CNRS and Université de Lyon, France first.last@ens-lyon.fr
More informationStochastic and online algorithms
Stochastic and online algorithms stochastic gradient method online optimization and dual averaging method minimizing finite average Stochastic and online optimization 6 1 Stochastic optimization problem
More informationMath 273a: Optimization Overview of First-Order Optimization Algorithms
Math 273a: Optimization Overview of First-Order Optimization Algorithms Wotao Yin Department of Mathematics, UCLA online discussions on piazza.com 1 / 9 Typical flow of numerical optimization Optimization
More informationRegularization Methods for Prediction in Dynamic Graphs and e-marketing Applications
Regularization Methods for Prediction in Dynamic Graphs and e-marketing Applications Emile Richard CMLA-ENS Cachan 1000mercis PhD defense Advisors: Th. Evgeniou (INSEAD), N. Vayatis (CMLA-ENS Cachan) November
More informationOn the equivalence of the primal-dual hybrid gradient method and Douglas Rachford splitting
Mathematical Programming manuscript No. (will be inserted by the editor) On the equivalence of the primal-dual hybrid gradient method and Douglas Rachford splitting Daniel O Connor Lieven Vandenberghe
More informationSolving Corrupted Quadratic Equations, Provably
Solving Corrupted Quadratic Equations, Provably Yuejie Chi London Workshop on Sparse Signal Processing September 206 Acknowledgement Joint work with Yuanxin Li (OSU), Huishuai Zhuang (Syracuse) and Yingbin
More informationOptimization methods
Optimization methods Optimization-Based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_spring16 Carlos Fernandez-Granda /8/016 Introduction Aim: Overview of optimization methods that Tend to
More informationA Linearly Convergent First-order Algorithm for Total Variation Minimization in Image Processing
A Linearly Convergent First-order Algorithm for Total Variation Minimization in Image Processing Cong D. Dang Kaiyu Dai Guanghui Lan October 9, 0 Abstract We introduce a new formulation for total variation
More informationA Level Set Based. Finite Element Algorithm. for Image Segmentation
A Level Set Based Finite Element Algorithm for Image Segmentation Michael Fried, IAM Universität Freiburg, Germany Image Segmentation Ω IR n rectangular domain, n = (1), 2, 3 u 0 : Ω [0, C max ] intensity
More information2D Wavelets. Hints on advanced Concepts
2D Wavelets Hints on advanced Concepts 1 Advanced concepts Wavelet packets Laplacian pyramid Overcomplete bases Discrete wavelet frames (DWF) Algorithme à trous Discrete dyadic wavelet frames (DDWF) Overview
More informationMinimizing Isotropic Total Variation without Subiterations
MITSUBISHI ELECTRIC RESEARCH LABORATORIES http://www.merl.com Minimizing Isotropic Total Variation without Subiterations Kamilov, U. S. TR206-09 August 206 Abstract Total variation (TV) is one of the most
More informationOslo Class 6 Sparsity based regularization
RegML2017@SIMULA Oslo Class 6 Sparsity based regularization Lorenzo Rosasco UNIGE-MIT-IIT May 4, 2017 Learning from data Possible only under assumptions regularization min Ê(w) + λr(w) w Smoothness Sparsity
More informationSmoothing Proximal Gradient Method. General Structured Sparse Regression
for General Structured Sparse Regression Xi Chen, Qihang Lin, Seyoung Kim, Jaime G. Carbonell, Eric P. Xing (Annals of Applied Statistics, 2012) Gatsby Unit, Tea Talk October 25, 2013 Outline Motivation:
More informationRelaxed linearized algorithms for faster X-ray CT image reconstruction
Relaxed linearized algorithms for faster X-ray CT image reconstruction Hung Nien and Jeffrey A. Fessler University of Michigan, Ann Arbor The 13th Fully 3D Meeting June 2, 2015 1/20 Statistical image reconstruction
More informationNonnegative Tensor Factorization using a proximal algorithm: application to 3D fluorescence spectroscopy
Nonnegative Tensor Factorization using a proximal algorithm: application to 3D fluorescence spectroscopy Caroline Chaux Joint work with X. Vu, N. Thirion-Moreau and S. Maire (LSIS, Toulon) Aix-Marseille
More informationSparse Optimization Lecture: Dual Methods, Part I
Sparse Optimization Lecture: Dual Methods, Part I Instructor: Wotao Yin July 2013 online discussions on piazza.com Those who complete this lecture will know dual (sub)gradient iteration augmented l 1 iteration
More informationarxiv: v1 [math.na] 3 Jan 2019
arxiv manuscript No. (will be inserted by the editor) A Finite Element Nonoverlapping Domain Decomposition Method with Lagrange Multipliers for the Dual Total Variation Minimizations Chang-Ock Lee Jongho
More informationSélection adaptative des paramètres pour le débruitage des images
Journées SIERRA 2014, Saint-Etienne, France, 25 mars, 2014 Sélection adaptative des paramètres pour le débruitage des images Adaptive selection of parameters for image denoising Charles Deledalle 1 Joint
More informationNetwork Newton. Aryan Mokhtari, Qing Ling and Alejandro Ribeiro. University of Pennsylvania, University of Science and Technology (China)
Network Newton Aryan Mokhtari, Qing Ling and Alejandro Ribeiro University of Pennsylvania, University of Science and Technology (China) aryanm@seas.upenn.edu, qingling@mail.ustc.edu.cn, aribeiro@seas.upenn.edu
More informationAccelerated Proximal Gradient Methods for Convex Optimization
Accelerated Proximal Gradient Methods for Convex Optimization Paul Tseng Mathematics, University of Washington Seattle MOPTA, University of Guelph August 18, 2008 ACCELERATED PROXIMAL GRADIENT METHODS
More informationLasso: Algorithms and Extensions
ELE 538B: Sparsity, Structure and Inference Lasso: Algorithms and Extensions Yuxin Chen Princeton University, Spring 2017 Outline Proximal operators Proximal gradient methods for lasso and its extensions
More informationSparse linear models and denoising
Lecture notes 4 February 22, 2016 Sparse linear models and denoising 1 Introduction 1.1 Definition and motivation Finding representations of signals that allow to process them more effectively is a central
More informationENERGY METHODS IN IMAGE PROCESSING WITH EDGE ENHANCEMENT
ENERGY METHODS IN IMAGE PROCESSING WITH EDGE ENHANCEMENT PRASHANT ATHAVALE Abstract. Digital images are can be realized as L 2 (R 2 objects. Noise is introduced in a digital image due to various reasons.
More informationAn Introduction to Wavelets and some Applications
An Introduction to Wavelets and some Applications Milan, May 2003 Anestis Antoniadis Laboratoire IMAG-LMC University Joseph Fourier Grenoble, France An Introduction to Wavelets and some Applications p.1/54
More informationError Analysis for H 1 Based Wavelet Interpolations
Error Analysis for H 1 Based Wavelet Interpolations Tony F. Chan Hao-Min Zhou Tie Zhou Abstract We rigorously study the error bound for the H 1 wavelet interpolation problem, which aims to recover missing
More informationCoordinate Update Algorithm Short Course Proximal Operators and Algorithms
Coordinate Update Algorithm Short Course Proximal Operators and Algorithms Instructor: Wotao Yin (UCLA Math) Summer 2016 1 / 36 Why proximal? Newton s method: for C 2 -smooth, unconstrained problems allow
More information