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# Mesh Processing Course : Mesh Parameterization

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### Mesh Processing Course : Mesh Parameterization

1. 1. Parameterization and Flattening www.numerical-tours.comGabriel PeyréCEREMADE, Université Paris-Dauphine
2. 2. Mesh Parameterization - Overview tex ma ture pp ing M R3 parameterization 2
3. 3. Mesh Parameterization - Overview tex ma ture pp ing R3 re- M s am 1 pli parameterization ng zoom D R2 2
4. 4. Overview • Dirichlet Energy on Meshes • Harmonic Parameterization • Spectral Flattening • Barycentric Coordinates for Warping • Geodesic Flattening • High Dimensional Data Analysis 3
5. 5. Local Averaging >0 if j Vi ,Local operator: W = (wij )i,j V where wij = 0 otherwise. (W f )i = wij fj . (i,j) E Examples: for i j, wij = 1 wij = 1 ||xj xi ||2 wij = cot( ij ) + cot(⇥ij ) combinatorial distance conformal (explanations later) ˜ wijLocal averaging operator W = (wij )i,j ˜ V : ⇥ (i, j) E, wij = ˜ . (i,j) E wij ˜ W =D 1 W with D = diagi (di ) where di = wij . (i,j)⇥E ˜Averaging: W 1 = 1. 4
6. 6. Voronoi and Dual MeshDeﬁnition for a planar triangulation M of a mesh M R2 .Voronoi for vertices: ⇧ i ⇤ V, Ei = {x ⇤ M ⇧ j ⌅= i, ||x xi || ⇥ ||x xj ||}Voronoi for edges: ⌅ e = (i, j) ⇥ E, Ee = {x ⇥ M ⌅ e ⇤= e, d(x, e) d(x, e )}Partition of the mesh: M = Ei = Ee . i V e E Ai i i cf A(ij) cf j j Dual mesh 1:3 subdivided mesh 5
7. 7. Approximating Integrals on MeshesApproximation of integrals on vertices and edges: ⇥ f (x)dx Ai f (xi ) Ae f ([xi , xj ]). M i V e=(i,j) E iT heorem : ⇥ e = (i, j) E, A(ij) cf 1 Ae = Area(Ee ) = ||xi xj ||2 (cot( ij ) + cot(⇥ij )) j 2
8. 8. Approximating Integrals on MeshesApproximation of integrals on vertices and edges: ⇥ f (x)dx Ai f (xi ) Ae f ([xi , xj ]). M i V e=(i,j) E iT heorem : ⇥ e = (i, j) E, A(ij) cf 1 Ae = Area(Ee ) = ||xi xj ||2 (cot( ij ) + cot(⇥ij )) j 2 AProof: + + = 2 ||AB||A(ABO) = ||AB|| h = ||AB|| tan( ) 2 O h 2 ||AB|| ⇤A(ABO) = tan ( + ⇥) C 2 2 B
9. 9. Cotangent WeightsSobolev norm (Dirichlet energy): J(f ) = || f (x)||2 dx M
10. 10. Cotangent Weights ⇧ (i, j) ⇤ E, wij = 1. Distance weights: they depends both on the geometry and the to require faces information,Sobolev norm (Dirichlet energy): J(f ) = 2 1 || f (x)|| dx ⇧ (i, j) ⇤ E, wij = ||xj xi ||2 . M Conformal weights: they depends on the full geometrical realiza require the face informationApproximation of Dirichelet energy: ⇧ (i, j) ⇤ E, wij = cot( ij ) + cot(⇥ ⇥ |f (xj ) f (xi )|2 Figure 1.2 shows the geometrical meaning of the angles ij and ||⇤x f || dx ⇥ 2 Ae |(Gf )e | = 2 Ae ij = ⇥(xi , xj , xk1 ) ||xj xand ⇥ij = ⇥(xi 2 M i || e E (i,j) E where (i, j, k1 ) ⇤ F and (i, j, k2 ) ⇤ F are the two faces adjacent = wij |fnext ) f (xiexplanation of these celebrated cotangent in the (xj section the )|2 (i,j) E where wij = cot( ij ) + cot(⇥ij ). xi ij xk1 ij xk2 xj 7
11. 11. Cotangent Weights ⇧ (i, j) ⇤ E, wij = 1. Distance weights: they depends both on the geometry and the to require faces information,Sobolev norm (Dirichlet energy): J(f ) = 2 1 || f (x)|| dx ⇧ (i, j) ⇤ E, wij = ||xj xi ||2 . M Conformal weights: they depends on the full geometrical realiza require the face informationApproximation of Dirichelet energy: ⇧ (i, j) ⇤ E, wij = cot( ij ) + cot(⇥ ⇥ |f (xj ) f (xi )|2 Figure 1.2 shows the geometrical meaning of the angles ij and ||⇤x f || dx ⇥ 2 Ae |(Gf )e | = 2 Ae ij = ⇥(xi , xj , xk1 ) ||xj xand ⇥ij = ⇥(xi 2 M i || e E (i,j) E where (i, j, k1 ) ⇤ F and (i, j, k2 ) ⇤ F are the two faces adjacent = wij |fnext ) f (xiexplanation of these celebrated cotangent in the (xj section the )|2 (i,j) E where wij = cot( ij ) + cot(⇥ij ). xi xk1T heorem : wij > 0 ⇥ ij + ⇥ij < ⇤ ij ij xk2 xj 7
12. 12. Overview • Dirichlet Energy on Meshes • Harmonic Parameterization • Spectral Flattening • Barycentric Coordinates for Warping • Geodesic Flattening • High Dimensional Data Analysis 8
13. 13. Mesh ParameterizationParameterization: bijection : M ⇤ D ⇥ R2 .Hypothesis: =( 1, 2) is smooth, minimizes min0 wi,j (| 1 (i) 1 (j)| 2 +| 2 (i) 2 (j)| ) 2 (i,j) EWith boundary conditions 0 : ⇥ xi ⇥M, (i) = 0 (i) ⇥D. 3D space (x,y,z) 2D parameter domain (u,v) boundary boundary 9
14. 14. Mesh ParameterizationParameterization: bijection : M ⇤ D ⇥ R2 .Hypothesis: =( 1, 2) is smooth, minimizes min0 wi,j (| 1 (i) 1 (j)| 2 +| 2 (i) 2 (j)| ) 2 (i,j) EWith boundary conditions 0 : ⇥ xi ⇥M, (i) = 0 (i) ⇥D. ⇥ i / ⇥M, (L 1 )(i) = (L 2 )(i) =0Optimality conditions: ⇥ i ⇥M, (i) = 0 (i) ⇥D. ⇥ sparse linear system to solve. 3D space (x,y,z) 2D parameter domain (u,v) boundary boundary 9
15. 15. Mesh ParameterizationParameterization: bijection : M ⇤ D ⇥ R2 .Hypothesis: =( 1, 2) is smooth, minimizes min0 wi,j (| 1 (i) 1 (j)| 2 +| 2 (i) 2 (j)| ) 2 (i,j) EWith boundary conditions 0 : ⇥ xi ⇥M, (i) = 0 (i) ⇥D. ⇥ i / ⇥M, (L 1 )(i) = (L 2 )(i) =0Optimality conditions: ⇥ i ⇥M, (i) = 0 (i) ⇥D. ⇥ sparse linear system to solve. 3D space (x,y,z) 2D parameter domain (u,v) Remark: each point is the average of its neighbors: ⇥ i / ⇥M, (i) = wi,j (j). ˜ (i,j) ETheorem: (Tutte) if i, j, wij > 0, then is a bijection. boundary boundary 9
16. 16. Examples of Parameterization Combinatorial Conformal Mesh 10
17. 17. Examples of Parameterization Combinatorial Conformal Mesh 11
18. 18. Application to Remeshing parameterization re-sampling 1 zoom P. Alliez et al., Isotropic Surface Remeshing, 2003. 12
19. 19. Application to Texture Mapping texture g(u) pa ra m et er iza tio n color g( (x)) 13
20. 20. Mesh Parameterization #1 : S0 ⇥ S ⇥x S0 ⇥S0 , =0 S0 ⇥ S S S0 14
21. 21. Mesh Parameterization #1 : S0 ⇥ S ⇥x S0 ⇥S0 , =0 S0 ⇥ S S Discretization of : W = make_sparse(n,n); for i=1:3 i1 = mod(i-1,3)+1; i2 = mod(i ,3)+1; i3 = mod(i+1,3)+1; pp = vertex(:,faces(i2,:)) - vertex(:,faces(i1,:)); qq = vertex(:,faces(i3,:)) - vertex(:,faces(i1,:)); % normalize the vectors pp = pp ./ repmat( sqrt(sum(pp.^2,1)), [3 1] ); qq = qq ./ repmat( sqrt(sum(qq.^2,1)), [3 1] ); % compute angles a = 1 ./ tan( acos(sum(pp.*qq,1)) ); a = max(a, 1e-2); % avoid degeneracy W = W + make_sparse(faces(i2,:),faces(i3,:), a, n, n ); W = W + make_sparse(faces(i3,:),faces(i2,:), a, n, n ); end S0 14
22. 22. Mesh Parameterization #1 : S0 ⇥ S ⇥x S0 ⇥S0 , =0 S0 ⇥ S S Discretization of : W = make_sparse(n,n); for i=1:3 i1 = mod(i-1,3)+1; i2 = mod(i ,3)+1; i3 = mod(i+1,3)+1; pp = vertex(:,faces(i2,:)) - vertex(:,faces(i1,:)); qq = vertex(:,faces(i3,:)) - vertex(:,faces(i1,:)); % normalize the vectors pp = pp ./ repmat( sqrt(sum(pp.^2,1)), [3 1] ); qq = qq ./ repmat( sqrt(sum(qq.^2,1)), [3 1] ); % compute angles a = 1 ./ tan( acos(sum(pp.*qq,1)) ); a = max(a, 1e-2); % avoid degeneracy W = W + make_sparse(faces(i2,:),faces(i3,:), a, n, n ); W = W + make_sparse(faces(i3,:),faces(i2,:), a, n, n ); end S0Formation of the linear system: D = spdiags(full( sum(W,1) ), 0, n,n); L = D - W; L1 = L; L1(boundary,:) = 0; L1(boundary + (boundary-1)*n) = 1; 14
23. 23. Mesh Parameterization #1 : S0 ⇥ S ⇥x S0 ⇥S0 , =0 S0 ⇥ S S Discretization of : W = make_sparse(n,n); for i=1:3 i1 = mod(i-1,3)+1; i2 = mod(i ,3)+1; i3 = mod(i+1,3)+1; pp = vertex(:,faces(i2,:)) - vertex(:,faces(i1,:)); qq = vertex(:,faces(i3,:)) - vertex(:,faces(i1,:)); % normalize the vectors pp = pp ./ repmat( sqrt(sum(pp.^2,1)), [3 1] ); qq = qq ./ repmat( sqrt(sum(qq.^2,1)), [3 1] ); % compute angles a = 1 ./ tan( acos(sum(pp.*qq,1)) ); a = max(a, 1e-2); % avoid degeneracy W = W + make_sparse(faces(i2,:),faces(i3,:), a, n, n ); W = W + make_sparse(faces(i3,:),faces(i2,:), a, n, n ); end S0Formation of the linear system: D = spdiags(full( sum(W,1) ), 0, n,n); L = D - W; L1 = L; L1(boundary,:) = 0; L1(boundary + (boundary-1)*n) = 1;Formation of the RHS and resolution: Rx = zeros(n,1); Rx(boundary) = x0; Ry = zeros(n,1); Ry(boundary) = y0; x = L1 Rx; y = L1 Ry; 14
24. 24. Mesh Parameterization #2Geometry image: re-sample X/Y/Z coordinates of on a grid. store the surface as a color (R/G/B) image.Exercise: perform the linear interpolation of the parameterization.Exercise: display the geometry image using a checkboard texture. 15
25. 25. Mesh Parameterization #3Exercise: Locate the position of the eyes / the mouth in the texture and on the mesh.Exercise: Compute an a ne transformation to re-align the texture. 16
26. 26. Mesh DeformationsInitial position: xi R3 .Displacement of anchors: i I, xi xi = xi + i R3 I xiLinear deformation: xi xi = xi + (i) i I, (i) = i i / I, (i) = 0 17
27. 27. Mesh DeformationsInitial position: xi R3 . % modify Laplacian L1 = L; L1(I,:) = 0;Displacement of anchors: L1(I + (I-1)*n) = 1; % displace vertices vertex = vertex + ( L1 Delta0 ); i I, xi xi = xi + i R 3 I xiLinear deformation: xi xi = xi + (i) i I, (i) = i i / I, (i) = 0 17
28. 28. Mesh DeformationsInitial position: xi R3 . % modify Laplacian L1 = L; L1(I,:) = 0;Displacement of anchors: L1(I + (I-1)*n) = 1; % displace vertices vertex = vertex + ( L1 Delta0 ); i I, xi xi = xi + i R 3 I xiLinear deformation: xi xi = xi + (i) i I, (i) = i i / I, (i) = 0 ˜ ni xiNon-linear deformation: xi = xi + i ˜ ˜ xi coarse details scale Linear deformation: xi ˜ ˜ xi Extrusion along normals: xi = xi + i, ˜ n i ni 17
29. 29. Overview • Dirichlet Energy on Meshes • Harmonic Parameterization • Spectral Flattening • Barycentric Coordinates for Warping • Geodesic Flattening • High Dimensional Data Analysis 18
30. 30. Mesh FlatteningNo boundary condition, minimize: ⇥ || 1 ||= 1, minn ||G 1 || = 2 wi,j | 1 (i) 1 (j)| 2 with 1 ⇥R i j ⇥ 1 , 1⇤ = 0. ⇧ ⌅ || ⇤ 2 ||= 1, minn ||G 2 || 2 = wi,j | 2 (i) 2 (j)| 2 with ⇥ 2 , 1 ⇤ = 0, 2 ⇥R ⌅ ⇥ 2 , 1⇤ = 0. i j ⇥ 19
31. 31. Mesh FlatteningNo boundary condition, minimize: ⇥ || 1 ||= 1, minn ||G 1 || = 2 wi,j | 1 (i) 1 (j)| 2 with 1 ⇥R i j ⇥ 1 , 1⇤ = 0. ⇧ ⌅ || ⇤ 2 ||= 1, minn ||G 2 || 2 = wi,j | 2 (i) 2 (j)| 2 with ⇥ 2 , 1 ⇤ = 0, 2 ⇥R ⌅ ⇥ 2 , 1⇤ = 0. i j ⇥Theorem: ⇥i = i+1 L⇥i , where 0 =0 1 2 ... n 1 are eigenvalues of L = G⇥ G. ( 1 (i), 2 (i)) R2 combinatorial conformal 19
32. 32. ProofSpectral decomposition: L=G G=D W =U U = diag( i ) where 0 = 1 < 2 ... n U = (ui )n orthonormal basis of Rn . i=1 u1 = 1 20
33. 33. Proof Spectral decomposition: L=G G=D W =U U = diag( i ) where 0 = 1 < 2 ... n U = (ui )n orthonormal basis of Rn . i=1 u1 = 1 nE(⇥) = ||G⇥||2 = i | ⇥, ui ⇥|2 i=1 nIf , 1 = 0, then E(⇥) = i ai where ai = | ⇥, ui ⇥|2 i=2 20
34. 34. Proof Spectral decomposition: L=G G=D W =U U = diag( i ) where 0 = 1 < 2 ... n U = (ui )n orthonormal basis of Rn . i=1 u1 = 1 nE(⇥) = ||G⇥||2 = i | ⇥, ui ⇥|2 i=1 nIf , 1 = 0, then E(⇥) = i ai where ai = | ⇥, ui ⇥|2 i=2 nConstrained minimization: Pnmin i ai i=2 ai =1 i=2 linear program: minimum reached at a = i. ±u2 = argmin E( ) , 1⇥=0,|| || 20
35. 35. Flattening ExamplesMain issue: No guarantee of being valid (bijective). combinatorial conformal 21
36. 36. Overview • Dirichlet Energy on Meshes • Harmonic Parameterization • Spectral Flattening • Barycentric Coordinates for Warping • Geodesic Flattening • High Dimensional Data Analysis 22
37. 37. Barycentric Coordinates x1 x x3x2 ˜ i (x) = A(x, xi+1 , xi+2 ) A(x1 , x2 , x3 ) 23
38. 38. Barycentric Coordinates x x1 1 x2 x x x3 x2 x3 ˜ i (x) = A(x, xi+1 , xi+2 ) A(x1 , x2 , x3 ) i (x)Barycentric coordinates: { i (x)}i I Normalized: ˜ i (x) = j j (x) Positivity: i (x) 0. ˜ Interpolation: ⇥i (xj ) = i,j Reproduction of a ne functions: ˜ i (x)xi = x i I 23
39. 39. Barycentric Coordinates x x1 1 x2 x x x3 x2 x3 ˜ i (x) = A(x, xi+1 , xi+2 ) A(x1 , x2 , x3 ) i (x)Barycentric coordinates: { i (x)}i I Normalized: ˜ i (x) = j j (x) Positivity: i (x) 0. ˜ Interpolation: ⇥i (xj ) = i,j Reproduction of a ne functions: ˜ i (x)xi = x i IApplication: interpolation of data {fi }i f (x) = ˜ i (x)fi I i I 23
40. 40. Barycentric Coordinates x x1 1 x2 xi x x x3 x2 x3 xj ˜ i (x) = A(x, xi+1 , xi+2 ) A(x1 , x2 , x3 ) i (x)Barycentric coordinates: { i (x)}i I Normalized: ˜ i (x) = j j (x) Positivity: i (x) 0. ˜ Interpolation: ⇥i (xj ) = i,j Reproduction of a ne functions: ˜ i (x)xi = x i IApplication: interpolation of data {fi }i f (x) = ˜ i (x)fi I i IApplication: mesh parameterization: wi,j = i (xj ) 23
41. 41. Mean Value CoordinatesConformal Laplacian weights: ⇥i (x) = cotan( i (x)) + cotan( ˜i (x)) xi+1 not necessarily positive. xi i iMean-value coordinates: x ˜i tan( i (x)/2) + tan(˜ i (x)/2) ˜i ⇥i (x) = ||x xi || valid coordinates. xi extend to non-convex coordinates (oriented angles). ˜ 1 (x) ˜ 2 (x) 24
42. 42. Barycentric Coordinates for Warping Cage C: polygon with vertices {xi }i I . Data points: {yj }j J C. Example: textured grid, 3D model, etc. Initialization: data anchoring, compute j J, i I, i,j = i (yj ). Satisﬁes: yi = i,j xi i I Cage warping: xi xi Data warping: yj yj = i,j xi i I xi y xi yi i 25
43. 43. Harmonic CoordinatesMean value coordinates: “non-physical” behavior, passes “through” the cage.Harmonic mapping: ⇥ x C, i (x) = 0.Boundary conditions: ⇥x ⇥C, i (x) = i (x). 0 Mean value Harmonic 26
44. 44. Warping Comparison Initial shape Mean value Harmonic 27
45. 45. Overview • Dirichlet Energy on Meshes • Harmonic Parameterization • Spectral Flattening • Barycentric Coordinates for Warping • Geodesic Flattening • High Dimensional Data Analysis 28
46. 46. Geodesic Distances 1 def.Length of a curve (t) M: L( ) = W ( (t))|| (t)||dt. 0Geodesic distance: dM (x, y) = min L( ) (0)=x, (1)=yGeodesic curve : dM (x, y) = L( ) Euclidean Shape Isotropic W = 1 Surface 29
47. 47. Computation of Geodesic DistancesDistance map to a point: Ux0 (x) = dM (x0 , x). Ux0Non-linear PDE: || Ux0 (x)|| = W (x) (viscosity) Ux0 (x0 ) = 0, x0Upwind ﬁnite di erences approximation.Fast Marching: front propagation in O(N log(N )) operations. 30
48. 48. Manifold FlatteningInput manifold M, dM geodesic distance on M. ˜Input geodesic distance matrix: D = (dM (˜i , xj )2 )i,j , for xi x ˜ ˜ M.Flattening: ﬁnd X = (xi )p ⇤ Rn i=1 p such that ||xi xj || ⇥ dM (˜i , xj ). x ˜ ˜ x1 x1 x1 ˜ x1 x2 x2 ˜ x2 ˜ x2 M R3 M R2 Surface parameterization Bending invariant 31
49. 49. Stress MinimizationGeodesic stress: S(X) = |||xi xj || di,j |2 , di,j = dM (˜i , xj ) x ˜ i,j 1 ()SMACOF algorithm: X ( +1) = X B(X ( ) ) N di,j where B(X)i,j = ||xi xj ||Non-convex functional : X ( ) X local minimizer of S. 32
50. 50. Projection on Distance Matrices D(X)i,j = ||xi xj ||2min |||xi xj ||2 d2 |2 = ||D(X) i,j D||2 , Di,j = d2X1=0 i,j i,j ||xi xj ||2 = ||xi ||2 + ||xj ||2 2⇥xi , xj ⇤ =⇥ D(X) = d1T + 1d 2X T X where d = (||xi ||2 )i ⇤ Rn 33
51. 51. Projection on Distance Matrices D(X)i,j = ||xi xj ||2min |||xi xj ||2 d2 |2 = ||D(X) i,j D||2 , Di,j = d2X1=0 i,j i,j ||xi xj ||2 = ||xi ||2 + ||xj ||2 2⇥xi , xj ⇤ =⇥ D(X) = d1T + 1d 2X T X where d = (||xi ||2 )i ⇤ RnCentering matrix: J = Idn 11T /N 1 JX = XFor centered points: JD(X)J = X T X 2 J1 = 0 33
52. 52. Projection on Distance Matrices D(X)i,j = ||xi xj ||2min |||xi xj ||2 d2 |2 = ||D(X) i,j D||2 , Di,j = d2X1=0 i,j i,j ||xi xj ||2 = ||xi ||2 + ||xj ||2 2⇥xi , xj ⇤ =⇥ D(X) = d1T + 1d 2X T X where d = (||xi ||2 )i ⇤ RnCentering matrix: J = Idn 11T /N 1 JX = XFor centered points: JD(X)J = X T X 2 J1 = 0 Replace ||D(X) D|| by min || J(D(X) D)J/2|| = ||X T X + JDJ/2|| XExplicit solution: diagonalize 1 2 JDJ = U UT i i 1 = diag( 0 , . . . , k 1 ), k X = k Uk T Uk = (u0 , . . . , uk 1 ) , 33
53. 53. Isomap vs. Laplacian Flattening: f = (f1 , f2 ) R2 . Laplacian: local smoothness: fi = argmin ||Gf || subj. to ||f || = 1. ⇥ (f1 , f2 ) eigenvectors (#2,#3) of L = GT G. Isomap: global constraints: ||f (x) f (y)|| ⇥ dM (x, y). ⇥ (f1 , f2 ) eigenvectors (#1,#2) of J(dM (xi , xj )2 )ij J.BijectiveNot bijective Mesh Lapl. combin. Lapl. conformal Isomap 34
54. 54. Bending Invariants of SurfacesBending invariants: [Elad, Kimmel, 2003].Surface M, Isomap dimension reduction: x ⇥ M ⇤ IM (x) ⇥ R3 . MIM [Elad, Kimmel, 2003]. 35
55. 55. Bending Invariants of SurfacesBending invariants: [Elad, Kimmel, 2003].Surface M, Isomap dimension reduction: x ⇥ M ⇤ IM (x) ⇥ R3 .Geodesic isometry :M M : dM (x, y) = dM ( (x), (y)).Theorem: up to rigid motion, IM is invariant to geodesic isometries: IM (x) = v + U IM ( (x)) where U O(3) and v R3 . MIM [Elad, Kimmel, 2003]. 35
56. 56. Bending Invariants of SurfacesBending invariants: [Elad, Kimmel, 2003].Surface M, Isomap dimension reduction: x ⇥ M ⇤ IM (x) ⇥ R3 .Geodesic isometry :M M : dM (x, y) = dM ( (x), (y)).Theorem: up to rigid motion, IM is invariant to geodesic isometries: IM (x) = v + U IM ( (x)) where U O(3) and v R3 . MIM [Elad, Kimmel, 2003]. [Bronstein et al., 2005]. 35
57. 57. Bending Invariants M IM 36
58. 58. Face Recognition Rigid similarity Non-rigid similarity Alex A. M. Bronstein et al., IJCV, 2005 37
59. 59. Overview • Dirichlet Energy on Meshes • Harmonic Parameterization • Spectral Flattening • Barycentric Coordinates for Warping • Geodesic Flattening • High Dimensional Data Analysis 38
60. 60. High Dimensional Data Sets 39
61. 61. Graph and Geodesics 40
62. 62. Isomap Dimension Reduction 41
63. 63. Isomap vs PCA Flattening 42
64. 64. Laplacian Spectral Dimension Reduction 43
65. 65. Parameterization of Image Datasets 44
66. 66. Library of Images 45
67. 67. When Does it Works? 46
68. 68. Local patches in images 47