DAGM 2011 Tutorial on Convex Optimization for Computer Vision

Transcription

1 DAGM 2011 Tuorial on Convex Opimizaion for Compuer Vision Par 3: Convex Soluions for Sereo and Opical Flow Daniel Cremers Compuer Vision Group Technical Universiy of Munich Graz Universiy of Technology Thomas Pock Insiue for Compuer Graphics and Vision Graz Universiy of Technology Frankfur, Augus 30, 2011

2 Graz Universiy of Technology Overview 1 Moion esimaion 2 Sereo esimaion Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 2 / 21

3 Graz Universiy of Technology Moion esimaion Moion esimaion (opical flow) is a cenral opic in compuer vision, Compues a 2D vecor field, describing he moion of pixel inensiies Applicaions: Tracking Video compression, video inerpolaion 3D reconsrucion Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 3 / 21

4 Graz Universiy of Technology Moion esimaion Moion esimaion (opical flow) is a cenral opic in compuer vision, Compues a 2D vecor field, describing he moion of pixel inensiies Applicaions: Tracking Video compression, video inerpolaion 3D reconsrucion Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 3 / 21

5 Graz Universiy of Technology Moion esimaion Moion esimaion (opical flow) is a cenral opic in compuer vision, Compues a 2D vecor field, describing he moion of pixel inensiies Applicaions: Tracking Video compression, video inerpolaion 3D reconsrucion Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 3 / 21

6 Graz Universiy of Technology Moion esimaion Moion esimaion (opical flow) is a cenral opic in compuer vision, Compues a 2D vecor field, describing he moion of pixel inensiies Applicaions: Tracking Video compression, video inerpolaion 3D reconsrucion Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 3 / 21

7 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

8 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

9 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

10 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

11 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

12 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

13 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

14 Graz Universiy of Technology Challenges Moion esimaion is sill a very difficul problem Aperure problem No informaion in unexured areas Illuminaion changes, shadows,... Large moion of small objecs, occlusions,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 4 / 21

15 Graz Universiy of Technology The correspondence problem Find corresponding poins in successive frames Brighness (color) consancy assumpion I 1 (x) I 2 (x + u(x)) 0 u(x) = (u 1 (x), u 2 (x)) is he displacemen vecor Ambiguiy: Many poins wih similar brighness (color)! Generalizaion: Consancy of image feaures (gradiens, NCC,...) Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 5 / 21

16 Graz Universiy of Technology Variaional moion esimaion Generic variaional model for moion esimaion min R(u) + I 1 (x) I 2 (x + u(x)) p dx u }{{} Ω }{{} Regularizaion erm Daa erm Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 6 / 21

17 Graz Universiy of Technology Variaional moion esimaion Generic variaional model for moion esimaion min R(u) + I 1 (x) I 2 (x + u(x)) p dx u }{{} Ω }{{} Regularizaion erm Daa erm Regularizaion erm: Should favor physically meaningful flow fields Popular convex regularizers: Quadraic, oal variaion,... Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 6 / 21

18 Graz Universiy of Technology Variaional moion esimaion Generic variaional model for moion esimaion min R(u) + I 1 (x) I 2 (x + u(x)) p dx u }{{} Ω }{{} Regularizaion erm Daa erm Regularizaion erm: Should favor physically meaningful flow fields Popular convex regularizers: Quadraic, oal variaion,... Daa erm: Highly non-convex hard o minimize Differen sraegies o deal wih he non-convexiy of he daa erm Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 6 / 21

19 Graz Universiy of Technology Variaional moion esimaion Generic variaional model for moion esimaion min R(u) + I 1 (x) I 2 (x + u(x)) p dx u }{{} Ω }{{} Regularizaion erm Daa erm Regularizaion erm: Should favor physically meaningful flow fields Popular convex regularizers: Quadraic, oal variaion,... Daa erm: Highly non-convex hard o minimize Differen sraegies o deal wih he non-convexiy of he daa erm Vas lieraure on moion esimaion: Window based opical flow: [Lucas, Kanade, 1981] Variaional opical flow: [Horn, Schunck, 1981] Disconinuiy preserving opical flow: [Shulman, Hervé 89] Robus opical flow: [Black, Anadan, 93] Highly accurae opical flow: [Brox, Bruhn, Papenberg, Weicker 04] Real-ime opical flow: [A. Bruhn, J. Weicker, T. Kohlberger, C. Schnörr 05] Primal-dual opimizaion on he GPU: [Zach, Pock, Bischof 07] Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 6 / 21

20 Graz Universiy of Technology Linearizaion of he image Perform a firs order Taylor expansion of he funcion I 2 (x + u(x)) a x + u 0 (x) [Horn, Schunck, 1981], [Lucas, Kanade, 1981] Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 7 / 21

21 Graz Universiy of Technology Linearizaion of he image Perform a firs order Taylor expansion of he funcion I 2 (x + u(x)) a x + u 0 (x) [Horn, Schunck, 1981], [Lucas, Kanade, 1981] I 2 (x + u(x)) I 2 (x + u 0 (x)) + I 2 (x + u 0 (x)), u(x) u 0 (x) Only valid close o u 0, i.e. u(x) u 0 (x) ε Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 7 / 21

22 Graz Universiy of Technology Linearizaion of he image Perform a firs order Taylor expansion of he funcion I 2 (x + u(x)) a x + u 0 (x) [Horn, Schunck, 1981], [Lucas, Kanade, 1981] I 2 (x + u(x)) I 2 (x + u 0 (x)) + I 2 (x + u 0 (x)), u(x) u 0 (x) Only valid close o u 0, i.e. u(x) u 0 (x) ε Leads o he classical opical flow consrain: ρ(u) = I 1 (x) I 2 (x + u 0 (x)) I 2 (x + u 0 (x)), u(x) u 0 (x) 0 Noe: ρ(u) is linear in u and hence ρ(u) is convex! Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 7 / 21

23 Graz Universiy of Technology TV-L 1 moion esimaion I urns ou ha oal variaion regularizaion in combinaion wih a L 1 daa erm performs well Toal variaion allows for moion disconinuiies L 1 daa erm allows for ouliers in he daa erm (occlusions, noise,...) min α Du + ρ(u) 1 u u 0 ε Ω Non-differeniable and hence difficul o solve Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 8 / 21

24 Graz Universiy of Technology TV-L 1 moion esimaion I urns ou ha oal variaion regularizaion in combinaion wih a L 1 daa erm performs well Toal variaion allows for moion disconinuiies L 1 daa erm allows for ouliers in he daa erm (occlusions, noise,...) min α Du + ρ(u) 1 u u 0 ε Ω Non-differeniable and hence difficul o solve Smoohing and fixed-poin ieraion: [Brox, Bruhn, Papenberg, Weicker 04] Primal-dual opimizaion: [Chambolle, Pock, 10] min max u div p dx + ρ(u) 1 u u 0 ε p α Ω Allows o compue he exac soluion Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 8 / 21

25 Graz Universiy of Technology Second-order approximaion of he daa erm Consider a more general non-convex daa erm of he form φ(x, u(x)) dx Ω Perform a second order Taylor expansion of he daa erm φ(x, u(x)) around u 0 (x) [Werlberger, Pock, Bischof 10] φ(x, u(x)) φ(x, u 0 (x)) + ( φ(x, u 0 (x))) T (u(x) u 0 (x)) + (u(x) u 0 (x)) T ( 2 φ(x, u 0 (x)) ) (u(x) u 0 (x)), To ensure convexiy he Hessian 2 φ(x, u 0 (x)) has o be posiive semidefinie We use he following diagonal approximaion of he Hessian [ 2 (φxx (x, u φ = 0 (x))) + ] 0 0 (φ yy (x, u 0 (x))) + Can be used wih arbirary daa erms: SAD, NCC,... Sill only valid in a small neighborhood around u 0 Minimizaion using primal-dual schemes Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 9 / 21

26 Graz Universiy of Technology Large displacemens How can we compue large displacemens? Inegrae he algorihm in a coarse-o fine / warping framework Similar o muligrid schemes, speeds up he minimizaion process Does no give any guaranees! Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 10 / 21

27 Graz Universiy of Technology Large displacemen opical flow wihou warping Consider he following equivalen generic formulaion [Seinbrücker, Pock, Cremers, 09] min R(u) + φ(u) dx min R(u) + φ(v) dx s.. u = v u Ω u,v Ω Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 11 / 21

28 Graz Universiy of Technology Large displacemen opical flow wihou warping Consider he following equivalen generic formulaion [Seinbrücker, Pock, Cremers, 09] min R(u) + φ(u) dx min R(u) + φ(v) dx s.. u = v u Ω u,v Ω Quadraic penaliy approach o obain a unconsrained formulaion min R(u) + 1 u,v 2θ u v 2 2 Ω + φ(v) dx Becomes equivalen o he consrained formulaion for θ 0 + Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 11 / 21

29 Graz Universiy of Technology Large displacemen opical flow wihou warping Consider he following equivalen generic formulaion [Seinbrücker, Pock, Cremers, 09] min R(u) + φ(u) dx min R(u) + φ(v) dx s.. u = v u Ω u,v Ω Quadraic penaliy approach o obain a unconsrained formulaion min R(u) + 1 u,v 2θ u v 2 2 Ω + φ(v) dx Becomes equivalen o he consrained formulaion for θ 0 + Observaions: Soluion wih respec o u reduces o an image denoising problem Soluion wih respec o v reduces o poinwise non-convex problems Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 11 / 21

30 Graz Universiy of Technology Large displacemen opical flow wihou warping Consider he following equivalen generic formulaion [Seinbrücker, Pock, Cremers, 09] min R(u) + φ(u) dx min R(u) + φ(v) dx s.. u = v u Ω u,v Ω Quadraic penaliy approach o obain a unconsrained formulaion min R(u) + 1 u,v 2θ u v 2 2 Ω + φ(v) dx Becomes equivalen o he consrained formulaion for θ 0 + Observaions: Soluion wih respec o u reduces o an image denoising problem Soluion wih respec o v reduces o poinwise non-convex problems Annealing-ype scheme: Alernaing minimizaion for a sequence of decreasing parameers θ i Advanages: No coarse-o-fine, no warping, arbirary daa erms Disadvanage: Resuls srongly depend on he sequence θ i Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 11 / 21

31 Graz Universiy of Technology Varying illuminaion Noe: opical flow moion esimaion! Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 12 / 21

32 Graz Universiy of Technology Varying illuminaion Noe: opical flow moion esimaion! Insead of (or in combinaion wih) he original images I 1,2 use gradien images I 1,2 [Brox, Bruhn, Papenberg, Weicker 04] Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 12 / 21

33 Graz Universiy of Technology Varying illuminaion Noe: opical flow moion esimaion! Insead of (or in combinaion wih) he original images I 1,2 use gradien images I 1,2 [Brox, Bruhn, Papenberg, Weicker 04] Phoomeric invarians: [Mileva, Bruhn, Weicker 07] Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 12 / 21

34 Graz Universiy of Technology Varying illuminaion Noe: opical flow moion esimaion! Insead of (or in combinaion wih) he original images I 1,2 use gradien images I 1,2 [Brox, Bruhn, Papenberg, Weicker 04] Phoomeric invarians: [Mileva, Bruhn, Weicker 07] Srucure-exure decomposiion: Illuminaion changes and shadows correspond o large image feaures [Wedel, Pock, Zach, Bischof, Cremers 08] Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 12 / 21

35 Graz Universiy of Technology Varying illuminaion Noe: opical flow moion esimaion! Insead of (or in combinaion wih) he original images I 1,2 use gradien images I 1,2 [Brox, Bruhn, Papenberg, Weicker 04] Phoomeric invarians: [Mileva, Bruhn, Weicker 07] Srucure-exure decomposiion: Illuminaion changes and shadows correspond o large image feaures [Wedel, Pock, Zach, Bischof, Cremers 08] Addiive decomposiion using he ROF model: I = S + T, S := arg min u TV(u) + λ 2 u I 2 2 (a) I (b) S (c) T Use exure componen T o compue he opical flow Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 12 / 21

36 Graz Universiy of Technology Modified opical flow consrain Recall he opical flow consrain ρ(u) = I1 (x) I2 (x + u0 (x)) h I2 (x + u0 (x)), u(x) u0 (x)i 0 Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 13 / 21

37 Graz Universiy of Technology Modified opical flow consrain Recall he opical flow consrain ρ(u) = I1 (x) I2 (x + u0 (x)) h I2 (x + u0 (x)), u(x) u0 (x)i 0 We can modify he consrain [Shulman, Herve 89] δ(u, v ) = I1 (x) I2 (x + u0 (x)) h I2 (x + u0 (x)), u(x) u0 (x)i v (x) 0 v (x) is a smooh funcion modeling illuminaion changes Noe ha δ(u, v ) is sill linear in u and v! Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 13 / 21

38 Graz Universiy of Technology Modified opical flow consrain Recall he opical flow consrain ρ(u) = I1 (x) I2 (x + u0 (x)) h I2 (x + u0 (x)), u(x) u0 (x)i 0 We can modify he consrain [Shulman, Herve 89] δ(u, v ) = I1 (x) I2 (x + u0 (x)) h I2 (x + u0 (x)), u(x) u0 (x)i v (x) 0 v (x) is a smooh funcion modeling illuminaion changes Noe ha δ(u, v ) is sill linear in u and v! Addiional regularizaion needed for v (x) Z Z α Du + β Dv + kδ(u(x), v (x))k1 min ku u0 k ε,v (a) Inpu Daniel Cremers and Thomas Pock Ω (b) Ground ruh Ω (c) Esimaed moion Frankfur, Augus 30, 2011 (d) Illuminaion Convex Opimizaion for Compuer Vision 13 / 21

39 Graz Universiy of Technology Overview 1 Moion esimaion 2 Sereo esimaion Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 14 / 21

40 Graz Universiy of Technology Sereo If I 1 and I 2 come from a sereo camera or a moving camera ha browses a saic scene, he displacemen can be resriced o 1D problems on he epipolar lines, [Slesareva, Bruhn, Weicker 05] Each sereo pair can be normalized such ha he displacemen is only horizonally The deph z can be compued from he displacemen u via z(x, y) = bf u(x, y) where b is he baseline and f is he focal lengh of he camera Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 15 / 21

41 Graz Universiy of Technology Sereo If I 1 and I 2 come from a sereo camera or a moving camera ha browses a saic scene, he displacemen can be resriced o 1D problems on he epipolar lines, [Slesareva, Bruhn, Weicker 05] Each sereo pair can be normalized such ha he displacemen is only horizonally The deph z can be compued from he displacemen u via z(x, y) = bf u(x, y) where b is he baseline and f is he focal lengh of he camera Opical flow consrain for sereo ˆρ(u) = I 1 I 2 (x + u 0 (x, y), y) x I 2 (x + u 0 (x, y), y)(u(x, y) u 0 (x, y)) 0 TV-L 1 based sereo min α Du + ˆρ(u(x, y)) 1 u u 0 ε Ω Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 15 / 21

42 Graz Universiy of Technology Sereo If I 1 and I 2 come from a sereo camera or a moving camera ha browses a saic scene, he displacemen can be resriced o 1D problems on he epipolar lines, [Slesareva, Bruhn, Weicker 05] Each sereo pair can be normalized such ha he displacemen is only horizonally The deph z can be compued from he displacemen u via z(x, y) = bf u(x, y) where b is he baseline and f is he focal lengh of he camera Opical flow consrain for sereo ˆρ(u) = I 1 I 2 (x + u 0 (x, y), y) x I 2 (x + u 0 (x, y), y)(u(x, y) u 0 (x, y)) 0 TV-L 1 based sereo min α Du + ˆρ(u(x, y)) 1 u u 0 ε Ω Advanages Highly accurae due o sub-pixel accuracy fas o compue (real-ime) Disadvanages Does no compue he globally opimal soluion (coarse-o-fine) Large baseline leads o more accurae resuls bu causes large displacemens Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 15 / 21

43 Disance along profile Graz Universiy of Technology Applicaion: range esimaion in a driving car (wih Daimler AG) Inpu images provided by a calibraed sereo rig (a) Lef image (b) Righ image Range image compued by he TV-L 1 based sereo algorihm (a) Range image (b) Profile of sree Toal variaion regularizaion leads o he saircasing effec! Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 16 / 21

44 Graz Universiy of Technology Toal generalized variaion The oal variaion can be wrien (via he convex conjugae) as { } TV α(u) = α Du = sup u div v dx v Cc(Ω, 1 R d ), v α, Ω Ω Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 17 / 21

45 Graz Universiy of Technology Toal generalized variaion The oal variaion can be wrien (via he convex conjugae) as { } TV α(u) = α Du = sup u div v dx v Cc(Ω, 1 R d ), v α, Ω Ω In [Bredies, Kunisch, Pock, SIIMS 10], we proposed a generalizaion of he oal variaion o higher order smoohness. { TGV k α(u) = sup u div k v dx Ω v C k c (Ω, Sym k (R d )), } div l v α l, l = 0,..., k 1, Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 17 / 21

46 Graz Universiy of Technology Toal generalized variaion The oal variaion can be wrien (via he convex conjugae) as { } TV α(u) = α Du = sup u div v dx v Cc(Ω, 1 R d ), v α, Ω Ω In [Bredies, Kunisch, Pock, SIIMS 10], we proposed a generalizaion of he oal variaion o higher order smoohness. { TGV k α(u) = sup u div k v dx Ω v C k c (Ω, Sym k (R d )), } div l v α l, l = 0,..., k 1, For k = 2 i can be wrien as TGV 2 α (u) = inf α 1 Du w + α 0 Dw w Ω Ω TGV 2 can be used o reconsruc piecewise affine funcions Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 17 / 21

47 Graz Universiy of Technology Image resoraion examples (a) Clean image (b) Noisy image (c) TV (d) TGV 2 Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 18 / 21

48 Graz Universiy of Technology Image resoraion examples (a) TV Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 18 / 21

49 Disance along profile Disance along profile Graz Universiy of Technology TGV based sereo Simply replace TV regularizaion by TGV regularizaion in he sereo model [Ranfl, Pock, Gehrig, Franke 11] min α 1 Du w + α 0 Dw + ˆρ(u(x, y)) 1 u u 0 ε,w Ω Ω Comparison on he sereo problem (a) TV (b) TGV Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 19 / 21

50 Graz Universiy of Technology Range esimaion from a driving car Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 20 / 21

51 Graz Universiy of Technology Summary and open quesions Inroduced he problem of moion esimaion in compuer vision Moion esimaion is sill a challenging problem, no near o be solved Highly non-convex daa erm leads o numerical difficulies A simple linearizaion approach works well in pracice Can be used for sereo esimaion TGV regularizaion avoids saircasing-arifacs Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 21 / 21

52 Graz Universiy of Technology Summary and open quesions Inroduced he problem of moion esimaion in compuer vision Moion esimaion is sill a challenging problem, no near o be solved Highly non-convex daa erm leads o numerical difficulies A simple linearizaion approach works well in pracice Can be used for sereo esimaion TGV regularizaion avoids saircasing-arifacs Global Soluions for Moion and Sereo? Daniel Cremers and Thomas Pock Frankfur, Augus 30, 2011 Convex Opimizaion for Compuer Vision 21 / 21