Multigrid Acceleration of the Horn-Schunck Algorithm for the Optical Flow Problem
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1 Multigrid Acceleration of the Horn-Schunck Algorithm for the Optical Flow Problem El Mostafa Kalmoun Ulrich Ruede Institute of Computer Science 10 Friedrich Alexander University Germany
2 Overview Introduction and Related Works Optical Flow Constraints Regularization Horn Schunck Algorithm Multigrid Scheme A Simple Illustration Variational Multigrid Experimental Results Slide 2 of 23
3 Introduction and Related Works Frame at time t y x Slide 3 of 23
4 Introduction and Related Works Frame at time t Frame at time t + dt y dy dx y+dy x x+dx Slide 3 of 23
5 Introduction and Related Works Frame at time t Frame at time t + dt y dy dx y+dy x x+dx The optical flow at the pixel (x,y) is the 2D-velocity vector (u,v) = ( dx dt, dy dt ) Slide 3 of 23
6 Introduction and Related Works Frame at time t Frame at time t + dt y dy dx y+dy x x+dx The optical flow at the pixel (x,y) is the 2D-velocity vector (u,v) = ( dx dt, dy dt ) An approximation of the 2D-motion Slide 3 of 23
7 Introduction and Related Works Frame at time t Frame at time t + dt y dy dx y+dy x x+dx The optical flow at the pixel (x,y) is the 2D-velocity vector (u,v) = ( dx dt, dy dt ) An approximation of the 2D-motion Optical Flow Motion Slide 3 of 23
8 Introduction and Related Works Term originates with James Gibson in 1979 Quadratic smoothness Horn Schunck (1981) Registration technique with local constraints Lucas Kanade (1981) Oriented smoothness Nagel Enkelmann ( ) Multigrid relaxation Terzopolous (1986) Performance evaluation of popular algorithms Barron Fleet Beauchemin (1994) General anisotropic smoothness Weickert (1996) Optimal control framework Borzi Ito Kunisch (2002) Slide 4 of 23
9 Optical Flow Constraints I(x,y,t) : The image intensity of the pixel (x,y) at time t. I x,i y,i t : Spatial and temporal derivatives of I. Slide 5 of 23
10 Optical Flow Constraints I(x,y,t) : The image intensity of the pixel (x,y) at time t. I x,i y,i t : Spatial and temporal derivatives of I. Assumption : Objects keep the same intensity over time Slide 5 of 23
11 Optical Flow Constraints I(x,y,t) : The image intensity of the pixel (x,y) at time t. I x,i y,i t : Spatial and temporal derivatives of I. Assumption : Objects keep the same intensity over time I(x,y,t) = I(x + dx,y + dy,t + dt) Slide 5 of 23
12 Optical Flow Constraints I(x,y,t) : The image intensity of the pixel (x,y) at time t. I x,i y,i t : Spatial and temporal derivatives of I. Assumption : Objects keep the same intensity over time I(x,y,t) = I(x + dx,y + dy,t + dt) Taylor Expansion = I x u + I y v + I t = 0 The optical flow constraint equation (OFCE) Slide 5 of 23
13 Optical Flow Constraints An equivalent form of the (OFCE) : I. w = I t DI.( w,1) = 0 where I = (I x,i y ), DI = (I x,i y,i t ) and w = (u,v) Slide 6 of 23
14 Optical Flow Constraints An equivalent form of the (OFCE) : u I. w = I t DI.( w,1) = 0 where I = (I x,i y ), DI = (I x,i y,i t ) and w = (u,v) v Slide 6 of 23
15 Optical Flow Constraints An equivalent form of the (OFCE) : u I. w = I t DI.( w,1) = 0 where I = (I x,i y ), DI = (I x,i y,i t ) I and w = (u,v) v Slide 6 of 23
16 Optical Flow Constraints An equivalent form of the (OFCE) : I. w = I t DI.( w,1) = 0 I t / I u where I = (I x,i y ), DI = (I x,i y,i t ) I and w = (u,v) v Slide 6 of 23
17 Optical Flow Constraints An equivalent form of the (OFCE) : I. w = I t DI.( w,1) = 0 I t / I u where I = (I x,i y ), DI = (I x,i y,i t ) I and w = (u,v) v OFC line Slide 6 of 23
18 Optical Flow Constraints An equivalent form of the (OFCE) : I. w = I t DI.( w,1) = 0 I t / I u where I = (I x,i y ), DI = (I x,i y,i t ) w 1 I and w = (u,v) v OFC line Slide 6 of 23
19 Optical Flow Constraints An equivalent form of the (OFCE) : where I. w = I t DI.( w,1) = 0 I t / I w 2 u I = (I x,i y ), DI = (I x,i y,i t ) w 1 I and w = (u,v) v OFC line Slide 6 of 23
20 Optical Flow Constraints An equivalent form of the (OFCE) : where I. w = I t DI.( w,1) = 0 I t / I w 2 u I = (I x,i y ), DI = (I x,i y,i t ) w 1 I and w = (u,v) v OFC line We can only calculate the normal component of the velocity w and not the tangent flow Slide 6 of 23
21 Optical Flow Constraints An equivalent form of the (OFCE) : where I. w = I t DI.( w,1) = 0 I t / I w 2 u I = (I x,i y ), DI = (I x,i y,i t ) w 1 I and w = (u,v) v OFC line We can only calculate the normal component of the velocity w and not the tangent flow The aperture problem. Slide 6 of 23
22 Regularization The (OFCE) is replaced by min (u,v) x y E(u,v)dxdy (P) Slide 7 of 23
23 Regularization The (OFCE) is replaced by where min (u,v) x y E(u,v)dxdy (P) E = E d + αe r, E d (u,v) = (I x u + I y v + I t ) 2 (the data term) E r is the regularization term Slide 7 of 23
24 Regularization The (OFCE) is replaced by where min (u,v) x y E(u,v)dxdy (P) E = E d + αe r, E d (u,v) = (I x u + I y v + I t ) 2 (the data term) E r is the regularization term and α is a positive scalar for adjustment between E d and E r Slide 7 of 23
25 Regularization The (OFCE) is replaced by where min (u,v) x y E(u,v)dxdy (P) E = E d + αe r, E d (u,v) = (I x u + I y v + I t ) 2 (the data term) E r is the regularization term and α is a positive scalar for adjustment between E d and E r The method for solving (P) will depend on the choice of E r. Slide 7 of 23
26 Horn-Schunck Algorithm A standard choice of E r is the isotropic stabilizer : E r (u,v) = u 2 + v 2 Slide 8 of 23
27 Horn-Schunck Algorithm A standard choice of E r is the isotropic stabilizer : From calculus of variations, we get E r (u,v) = u 2 + v 2 α u I x (I x u + I y v + I t ) = 0 α v I y (I x u + I y v + I t ) = 0 Slide 8 of 23
28 Horn-Schunck Algorithm A standard choice of E r is the isotropic stabilizer : From calculus of variations, we get E r (u,v) = u 2 + v 2 α u I x (I x u + I y v + I t ) = 0 α v I y (I x u + I y v + I t ) = 0 We discretize the Laplacien by the standard 5 point stencil and use w = w w where w is the average of the neighbors. Slide 8 of 23
29 Horn-Schunck Algorithm Spatial and Temporal Image Derivatives Masks Slide 9 of 23
30 Horn-Schunck Algorithm Spatial and Temporal Image Derivatives Masks M x = 1 4 ( ) ( 1 1 M y = ) M t = 1 4 ( ) Slide 9 of 23
31 Horn-Schunck Algorithm Spatial and Temporal Image Derivatives Masks M x = 1 4 ( ) ( 1 1 M y = ) M t = 1 4 ( I x = M x (I 1 + I 2 ) I y = M y (I 1 + I 2 ) I t = M t (I 2 I 1 ) ) Coupled Gauss-Seidel relaxation Slide 9 of 23
32 Horn-Schunck Algorithm Spatial and Temporal Image Derivatives Masks M x = 1 4 ( ) ( 1 1 M y = ) M t = 1 4 ( I x = M x (I 1 + I 2 ) I y = M y (I 1 + I 2 ) I t = M t (I 2 I 1 ) ) Coupled Gauss-Seidel relaxation u k+1 = ū k I x I x ū k + I y v k + I t α + I 2 x + I 2 y v k+1 = v k I y I x ū k + I y v k + I t α + I 2 x + I 2 y Slide 9 of 23
33 Multigrid Scheme Multigrid Components Slide 10 of 23
34 Multigrid Scheme Multigrid Components Vertex-centered grid and standard coarsening Slide 10 of 23
35 Multigrid Scheme Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Slide 10 of 23
36 Multigrid Scheme Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Slide 10 of 23
37 Multigrid Scheme Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Discretization coarse grid approximation (DCA approach) Slide 10 of 23
38 A Simple Illustration Intensity value : I(x,y,t) = x + y +t Slide 11 of 23
39 A Simple Illustration Intensity value : I(x,y,t) = x + y +t Slide 11 of 23
40 A Simple Illustration Intensity value : I(x,y,t) = x + y +t Corresponding system of PDE s : α u = u + v + 1 α v = u + v + 1 Slide 11 of 23
41 A Simple Illustration Intensity value : I(x,y,t) = x + y +t Corresponding system of PDE s : α u = u + v + 1 α v = u + v + 1 Slide 11 of 23
42 A Simple Illustration Intensity value : I(x,y,t) = x + y +t Corresponding system of PDE s : α u = u + v + 1 α v = u + v + 1 Method α = 1 and u 0 v 0 Convergence rate Horn-Schunck V(1,0) V(1,1) V(2,1) V(3,3) Slide 11 of 23
43 Variational Multigrid Write the system of the two PDE s as ( u L v ) = ( f g ) Slide 12 of 23
44 Variational Multigrid Write the system of the two PDE s as ( u L v ) = ( f g ) where f = I x I t, g = I y I t ( α 0 L = L d + L r, L r = 0 α and L d is a 2x2 block diagonal matrix with entries ) ( I 2 x I x I y I x I y I 2 y ) Slide 12 of 23
45 Variational Multigrid Write the system of the two PDE s as ( u L v ) = ( f g ) where f = I x I t, g = I y I t ( α 0 L = L d + L r, L r = 0 α and L d is a 2x2 block diagonal matrix with entries ) ( I 2 x I x I y I x I y I 2 y ) L d (positive semi-definite) Slide 12 of 23
46 Variational Multigrid Write the system of the two PDE s as ( u L v ) = ( f g ) where f = I x I t, g = I y I t ( α 0 L = L d + L r, L r = 0 α and L d is a 2x2 block diagonal matrix with entries ) ( I 2 x I x I y I x I y I 2 y ) L d (positive semi-definite) + L r (positive definite) Slide 12 of 23
47 Variational Multigrid Write the system of the two PDE s as ( u L v ) = ( f g ) where f = I x I t, g = I y I t ( α 0 L = L d + L r, L r = 0 α and L d is a 2x2 block diagonal matrix with entries ) ( I 2 x I x I y I x I y I 2 y ) L d (positive semi-definite) + L r (positive definite) = L (positive definite). Slide 12 of 23
48 Variational Multigrid Variational Minimization Form Lw = F w = argmin a(z) Ω Slide 13 of 23
49 Variational Multigrid Variational Minimization Form Lw = F w = argmin a(z) Ω a(z) = 1 2 (Lz,z) (F,z) Slide 13 of 23
50 Variational Multigrid Variational Minimization Form Lw = F w = argmin a(z) Ω a(z) = 1 2 (Lz,z) (F,z) Galerkin Approach Let I h H : ΩH Ω h be a full rank linear mapping. Slide 13 of 23
51 Variational Multigrid Variational Minimization Form Lw = F w = argmin a(z) Ω a(z) = 1 2 (Lz,z) (F,z) Galerkin Approach Let I h H : ΩH Ω h be a full rank linear mapping. An optimal coarse grid correction I h H w H of w h is characterized by ((I h H) T L h I h H)w H = (I h H) T (F L h w h ) Slide 13 of 23
52 Variational Multigrid The CGO is chosen then as follows L H = I H h L h I h H and I H h = (I h H) T Slide 14 of 23
53 Variational Multigrid The CGO is chosen then as follows L H = I H h L h I h H and I H h = (I h H) T For our system, we get L H = = ( R 0 0 R )( L 1 h L 2 h L 2 h ( RL 1 h P RLh 2 P RLh 2 P RL3 h P ) L 3 h )( P 0 0 P ) Slide 14 of 23
54 Variational Multigrid Multigrid Components Slide 15 of 23
55 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Slide 15 of 23
56 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) Slide 15 of 23
57 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) The General Scheme Slide 15 of 23
58 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) The General Scheme READ FRAMES Slide 15 of 23
59 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) The General Scheme READ FRAMES COMPUTE SPATIAL and TEMPORAL FRAMES DERIVATIVES Slide 15 of 23
60 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) The General Scheme READ FRAMES COMPUTE SPATIAL and TEMPORAL FRAMES DERIVATIVES CALCULATE the FINEST GRID OPERATOR and the RHS Slide 15 of 23
61 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) The General Scheme READ FRAMES COMPUTE SPATIAL and TEMPORAL FRAMES DERIVATIVES CALCULATE the FINEST GRID OPERATOR and the RHS SET UP the COARSE GRID OPERATORS HIEARACHY with GALERKIN Slide 15 of 23
62 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) The General Scheme READ FRAMES COMPUTE SPATIAL and TEMPORAL FRAMES DERIVATIVES CALCULATE the FINEST GRID OPERATOR and the RHS SOLVE with MULTIGRID V-CYCLING SET UP the COARSE GRID OPERATORS HIEARACHY with GALERKIN Slide 15 of 23
63 Variational Multigrid Multigrid Components Vertex-centered grid and standard coarsening Coupled lexicographic point Gauss-Seidel smoother Full weighting and bilinear interpolation Galerkin coarse grid approximation (GCA approach) The General Scheme READ FRAMES COMPUTE SPATIAL and TEMPORAL FRAMES DERIVATIVES CALCULATE the FINEST GRID OPERATOR and the RHS WRITE the OPTICAL FLOW FIELD SOLVE with MULTIGRID V-CYCLING SET UP the COARSE GRID OPERATORS HIEARACHY with GALERKIN Slide 15 of 23
64 Experimental Results Slide 16 of 23
65 Experimental Results A rotating sphere (128x128) Slide 16 of 23
66 Experimental Results A rotating sphere (128x128) the flow field Slide 16 of 23
67 Experimental Results the flow field after scaling Slide 17 of 23
68 Experimental Results The marble sequence (512x512) Slide 18 of 23
69 Experimental Results The marble sequence (512x512) the flow field Slide 18 of 23
70 Experimental Results the flow field after scaling Slide 19 of 23
71 Experimental Results The Hamburg taxi sequence (256x190) Slide 20 of 23
72 Experimental Results The Hamburg taxi sequence (256x190) the flow field Slide 20 of 23
73 Experimental Results the flow field after scaling Slide 21 of 23
74 Experimental Results α = 5 and u 0 = v 0 = 0 on P4 2.4GHz Sphere Marble Taxi ρ CPU ρ CPU ρ CPU Horn-Schunck VMG V(2,1) Slide 22 of 23
75 Experimental Results α = 5 and u 0 = v 0 = 0 on P4 2.4GHz Sphere Marble Taxi ρ CPU ρ CPU ρ CPU Horn-Schunck VMG V(2,1) Robust (but not yet optimal convergence) Slide 22 of 23
76 Experimental Results α = 5 and u 0 = v 0 = 0 on P4 2.4GHz Sphere Marble Taxi ρ CPU ρ CPU ρ CPU Horn-Schunck VMG V(2,1) Robust (but not yet optimal convergence) * need matrix dependent transfer grid operators Slide 22 of 23
77 Experimental Results α = 5 and u 0 = v 0 = 0 on P4 2.4GHz Sphere Marble Taxi ρ CPU ρ CPU ρ CPU Horn-Schunck VMG V(2,1) Robust (but not yet optimal convergence) * need matrix dependent transfer grid operators Galerkin leads to high memory costs search for better representation of the CGOs Slide 22 of 23
78 Future work Application in medicine imaging Consider other regularization (e.g. by Nagel, Weickert...) Slide 23 of 23
79 Future work Application in medicine imaging Consider other regularization (e.g. by Nagel, Weickert...) PLEASE WAKE UP! Slide 23 of 23
80 Future work Application in medicine imaging Consider other regularization (e.g. by Nagel, Weickert...) PLEASE WAKE UP! THANKS FOR YOUR ATTENTION Slide 23 of 23
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