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- 1. Geodesic Sampling and Meshing http://www.ceremade.dauphine.fr/~peyre/Gabriel PeyréCEREMADE, Université Paris Dauphine
- 2. Overview•Riemannian Metrics.• Riemannian Voronoi and Delaunay.• Farthest Point Sampling.• Anisotropic Triangulations.• Anisotropic Delaunay Reﬁnement. 2
- 3. Riemannian ManifoldRiemaniann manifold: abstract parametric space M Rn .Metric: M equiped x ⇥ M ⇤ H(x) ⇥ Rn n positive deﬁnite. 1⇥ def. T Length of a curve (t) M: L( ) = (t) H( (t)) (t)dt.Examples: 0 Euclidean space: M = Rn and H(x) = Idn . 2D shape: M R2 and H(x) = Id2 . Parametric surface: H(x) = Ix ﬁrst fundamental form. Isotropic metric: H(x) = W (x)Idn , W (x) > 0 weight function. Image processing: I : [0, 1]2 ⇥ R, W (x) = ( + ||⇤x I||) 1 . DTI imaging: M = [0, 1]3 , H(x) =di usion tensor.W (x) 3
- 4. Geodesic Distances Geodesic distance metric over M Rn def. ⇥ (x, y) M2 , dM (x, y) = min L( ) T >0, PT (x,y) def. where PT (x, y) = { (0) = x and (T ) = y} . Geodesic curve: (t) such that L( ) = dM (x, y). def. Distance map to a starting1057 x0 2 ECCV-08 submission ID point M: Ux0 (x) = dM (x0 , x).metricgeodesics Euclidean Shape Isotropic Anisotropic Surface 4
- 5. Overview• Riemannian Metrics.• Riemannian Voronoi and Delaunay.• Farthest Point Sampling.• Anisotropic Triangulations.• Anisotropic Delaunay Reﬁnement. 5
- 6. Euclidean Delaunay Triangulation Delaunay graph DS : (xi , xj ) ⇤ DS ⇥ Ci ⌃ Cj ⌅= ⇧. 6
- 7. Euclidean Voronoi Segmentation 7
- 8. Euclidean Delaunay Triangulation Delaunay graph DS : (xi , xj ) ⇤ DS ⇥ Ci ⌃ Cj ⌅= ⇧. 8
- 9. Geodesic Voronoi and DelaunayVoronoi segmentation: Outer cell: C0 = Closure( c ). Delaunay graph DS : (xi , xj ) ⇤ DS ⇥ Ci ⌃ Cj ⌅= ⇧. 9
- 10. Voronoi Diagrams on Surfaces 10
- 11. Double and Triple PointsS large enough = DS is a planar triangulation.Double point: wi,j = argmin d(x, xi ). x Ci ⇥CjTripple point: si,j,k Ci ⇥ Cj ⇥ Ck . si,j,k xk xi xj Fast computation in O(n log(n)) with Fast Marching. wi,jGeometric realization of DS : Union of two geodesics starting from wi,j . Special case: boundary curves if xi and xj . 11
- 12. Anisotropic Voronoi Segmentation ECCV-08 submission ID1057 7 Distances Voronoi f = .95 = .1 = .2 = .7 = .5 = .5 =1 = 0Fig. 4. Examples of anisotropic distances (top row) and Voronoi diagrams (bottomrow) with an decreasing anisotropy . The metric tensor is computed using the structuretensor, equation (8). 12
- 13. Overview• Riemannian Metrics.• Riemannian Voronoi and Delaunay.•Farthest Point Sampling.• Anisotropic Triangulations.• Anisotropic Delaunay Reﬁnement. 13
- 14. Coverings and Packings 14
- 15. Geodesic Delaunay and Voronoi xi,j,k xi xk xi,j x xj xi,j, 15
- 16. Farthest Point SamplingSurface sampling {x1 , . . . , xn } M. ⇤ ⇤Parametric domain sampling: : [0, 1] ⇥ 2 M, xi,j = (i/ n, j/ n). def. -covering: i B (xi ) = M, where B (x) = {y dM (x, y) }. -separated: min(dM (xi , xj )) . 1. Initialization: x1 random, d(x) dM (x0 , x), set i = 1.Farthest point 2. Select point: xi+1 = argmaxx d(x), = d(xi+1 ). 3. Local update of the distance: d(x) min(d(x), dM (xi+1 , x)). 4. Stop: If i < n or > 0 , set i i + 1 and go back to 2.Theorem: The sampling {x1 , . . . , xn } is an -covering that is -separated for = max min dM (xi , xj ). i=1,...,n j=1,...,n 16
- 17. Farthest Point SamplingW (x)W (x)W (x) Metric # samples W (x) small = front moves slowly, = denser sampling. 17
- 18. Farthest Point Triangulation # samples 18
- 19. Meshing a Planar ShapeUniformAdaptive 19
- 20. Uniform Remeshing 20
- 21. Remeshing of the David Original: Remeshed: 100k vert. 10k vert. 21
- 22. Adaptive Remeshing # samples 22
- 23. Density Given by a TextureParameterized surface: :D [0, 1]2 ⇥ S R3 .Texture: I = [0, 1]2 R. ⇥p S, W (p) = ( + ||⇤ 1 (p) I||) 1 I( 1 (p)) || 1 (p) I|| W =1 large small 23
- 24. Curvature of a Height ﬁeldHeight ﬁeld: x ⇥ [0, 1]2 (x, f (x)) ⇥ R3 . f (x + h) = f (x) + ⇤x f, h⇥ + Hx (f )h, h⇥ + O(||h||2 ) Hx = µ1 e1 e1 T + µ2 e2 e2 T with |µ1 | > |µ2 |. f (x + h) (f (x) + ⇥⌅x f, h⇤) = µ1 ⇥h, e1 ⇤2 + µ2 ⇥h, e2 ⇤2 + O(h2 ). x e1 (x) e2 (x)Gaussian curvature: µ1 (x)µ2 (x) = det(Hx (f )). IntrinsicMean curvature: (µ1 (x) + µ2 (x))/2 = tr(Hx (f ))/2.Total curvature: |µ1 (x)| + |µ2 (x)|. Extrinsic 24
- 25. Curvature of a SurfaceParametric surface: x ⇥ R2 ⇤ (x) ⇥ R3 . ⇥ ⇥ ⇥ ⇥Normal: n = ⇥x1 ⇥x2 / ⇥x1 ⇥x2 . ⇥ 2 (x) ⇥Second fundamental form: Hx = , n(x)⇥ ⇥xi ⇥xj i,j=1,2Numerical estimation: • polynomial ﬁt. • local covariance analysis. |µ1 (x)| + |µ2 (x)|. 25
- 26. Edge-aware RemeshingCurvature-driven metric: W (x) = ( + |µ1 (x)| + |µ2 (x)|) 1 Original mesh W =1 Small 26
- 27. Overview• Riemannian Metrics.• Riemannian Voronoi and Delaunay.• Farthest Point Sampling.•Anisotropic Triangulations.• Anisotropic Delaunay Reﬁnement. 27
- 28. Anisotropy is Important 8 Jonathan Richard Shewchuk Better respect of features. Isotropic Anisotropic, Anisotropic, Figure 2: A visual illustration of how large angles, but not small angles, can cause the error f− g to bad shape good shape explode. In each triangulation, 200 triangles are used to render a paraboloid. Better approximation of 20 40 20 functions. 40 40 e1f (x + h) = f (x) + ⇤x f, h⇥ + Hx (f )h, h⇥ + O(||h|| ) 2 35 65 50 20 40 x Hx = µ1 e1 e1 + µ2 e2 e2 T T with |µ1 | > |µ2 |. Figure 3: As the large angle of the triangle approaches 180◦ , or the sliver tetrahedron becomes arbitrarily ⇥ ﬂat, the magnitude of the vertical component of g becomes arbitrarily large. ⇥ µ1 µ2 sphere, then perturbing one of the vertices just off the equator so that the sliver has some (but not much) volume.f (x + h) (f (x) + ⇥⌅x f, h⇤) = µ1 ⇥h, ge1and2not+−µ2,⇥h, cangenerators+ O(h2 ). elements. Section 6.1 presents , ⇤ which ebe ⇤ Because of this sensitivity, mesh f g ∞ 2 usually choose the shapes of elements to control f − 2 reduced simply by using smaller ∞ quality measures that judge the shape of elements based on their ﬁtness for interpolation. = Local optimal shape: width/length = “weaker butf2−|/|µ1 |a triangle.3c r ﬁrstisupperasbound is almost tight, to within a factor of two. The |µ simpler upper bound” of The not good an indicator as Table 2 gives two upper bounds on g over ∞ t circ 28 the stronger upper bound, but it has the advantages of being smooth almost everywhere (and therefore more
- 29. TheFor a geometricallyinterpolation ff2 that is C (V, T ) is C edges, one cani )however ibuilt an⇥ is piecewise linear regular image ⇥ , of f on outside such that fM (x = f (x ) and fM 2 2 ⇥ Approximation of Images with Edges ||f fM⇥ j 2 CM . M basis. The following theorem sketch the construction of such a triangulation. (9.2) linear on each triangular face t|| . This is a two-dimensional extension performance of a wavelet adapted triangulation that enhances signiﬁcantly the approximation of the spline approximations. We give here only a sketch ofThis9.4. Geometric Image Approximationdepends on the position of the vertices studied in Section ??. the proof. [todo linear approximationtubes] Near an edges, piecewise : show an image of the ngths of thethe connectivity T of the M 1 and their widths should be of order M 2 . We V and triangles should be of order triangulation. In order to e⇥ciently approximate a given functione a thinwith Moftriangles, 2 around all the ﬁnd the optimal shape Since the edge curve are f band width M one needs to edge curves, see ﬁgure ??. of the triangles. f f∗hhis band can be sub-divided in elongated tubes of length proportional to M 1 ,2each of witch is For a geometrically regular image f , that isisC outside L is one can length 2 C edges, one can however built anmposed in two elongated triangles. The number of such triangle proof of theorem ??,the totaldesign triangles of width and legnth M As already seen in the 2LM wheree edges in the image. A special care should be taken near edge corner approximation but weof M , whicha wavelet p adapted triangulation that enhances so that the error in this area is also of the performance of concludes the the tube signiﬁcantly the 2 or edge crossings ordere these technicalities here. Over these tubes, the function is theorem shows that triangulation. is smoothed by an unknow basis. The following theorem sketch the construction andsuch thewhen an image The proof of this bounded of thus a approximation e ||f f ||2 2 ( ) in the tubes is of the order of⇤, the triangulation should . In the on ⇤ in order to get a fast decay of the app 2 2 2 L width area( )||f ||⇤ = LM ||f ||⇤ depend complementarye tubes c , one packs M large equilateral triangles of edge lengths approximativelythe⇤ M 1/2 . error. To reach an error decay of O(M 2 ), in neighborhood of a contour smoothed 2 f f∗h e over ⇤ 1/4 M 1/2 and a width of order ⇤ f is C outside the edge curve, theof width ⇤, the triangle should have a length offorder such approximation error of a linear interpolation e||⇤( c ||f ||C 2 and thus the error satisﬁes??. The scale ⇤ is most of the time unknown and one thus needs an as shown on ﬁgure Sharp edgesgles is ||f f Blurred edges Figure 9.2: Approximation with ﬁnite devise 2the on aof the triangles.for a function without and with algorithm to elements size triangulation e ||f f ||2 = O(LM 2 ||f ||2 + ||f ||C M 2 ). ⇤ additional blurring. M −2 s 3/ 4M −1/ 2 Theorem 2. If f is C outside a set of C2 contours, then it exists a constant C such that for 2 M −α 4the1/ 2 any M , it exists an adapted triangulation (V, T ) over which M piecewise-linear interpolation fM s1/ satisﬁes M −1 triangles 1/ 4 M −12 Figure 9.2: Approximation with ﬁnite elements|| on a triangulation for a s ||f fM CM . 2 M −1/ 2 function without (9.2) with and additional blurring. for the Sharp edges Figure 9.3: Finite We give here only approximation around[todo : show an image of the tubes] Near an edges, elements a singularity curve. Blurred edges Proof. a sketch of the proof. 1 2 the lengths of the triangles should be 9.4:order M ratio of triangles for the approximationM a. blurred contou Figure of Aspect and their widths should be of order of We 2 deﬁne a thin band of width M around all the edge curves, see ﬁgure ??. Since the edge curve aretheorem shows that an adapted triangulation should balance the approximation error 1 2 Theorem, 2. If f curves. outside a set locatedcontours, from these singularities C this band is Cbe singularity can of C2 far away then it exists a constant C such that for 2 sub-divided in elongated tubes of length proportional to M , each of witch isd outside thedecomposed in two elongated triangles.that locally, an such triangle is 2LM where L is the total length The triangles The number of optimal triangle should be aligned with the direction µ This shows ⇥ any M , it existswhileadapted triangulation (V, T ) over which the piecewise-linear interpolation fMe large and isotropic, an triangles that cover the edges should be stretched along the function is thecare should be taken near edge corner or edge crossings but wewell descr of the edges in the image. A special most regular. The local behavior of a smooth C2 image is satisﬁes ignore these technicalities here. Over these tubes, the function is bounded and thus y curves. This construction can second order by a quadratic approximation higher order the approximation be generalized by replacing triangles by ⇥ 2 C , 2.c primitives whose||f f ||2 2 ( ) in the tubes is of ||f order||of area(M as2 shown on ||f ||2 rightthe complementary (9.2) error boundaries are polynomial curves M degree )||f ||⇥ = LM 2 the . In e the f of L ⇥ f (x + h) = f (x) + ⌅⌥xhigher order )h, h⇧ + O(||h||2 ) f, h⇧ + ⌅Hx (fgure ??. Theof the tubes , one packs using polynomials deﬁned on edge lengths approximatively ⌃ ⇥ M 1/2 . adapted approximation M large equilateral triangles of M such c Proof. We give 2here only a sketch of the proof. [todo : show an image of the tubes] e Near an edges, 29 Since f is C outside the edge curve, the approximation error of a linear interpolation f over such 2⇥2
- 30. Metric Design for Anisotropic Meshing Idea: design the metric H(x). x e1 (x) Local orientation: e1 (x). Heuristic Algorithms for 2 (x) (x) Local shape: 1 (x)/ 2 (x). Generating Anisotropic Meshes 1 Bossen−Heckbert [1996] George−Borouchaki [1998] 30
- 31. Image Approximation Garland & Heckbert, Approximation of Terrains 19 Figure 16: Mandrill original, a 200×200 raster image. Figure 17: Mandrill approximated with Gouraud shaded triangles created by subsampling on a uniform 20×20 grid (400 vertices). Original image Figure 16: Mandrill original, a 200×200 raster image. Isotropic mesh (400Gouraud Figure 17: Mandrill approximated with vert) shaded triangles created by subsampling on a uniform 20×20 grid (400 vertices). Anisotropic meshFigure 19: Mesh for the image to the left. (400 vert) Figure 18: Mandrill approximated with Gouraud 31
- 32. Image Compression 16 L. Demaret, N. Dyn, M.S. Floater, A. Iske 16 L. Demaret, N. Dyn, M.S. Floater, A. Iske Original image (a) Wavelets(b) @.25bpp (a) (b) (c) (d) (c) Triangulation @.25bpp (d) Fig. 8. Reflex. (a) Original image of size 128 × 128. Compression at 0.251 bpp and reconstruction by(a) Original imagePSNR 30.42 db, (d) AT∗ with PSNR 41.73 and Fig. 8. Reflex. (b) SPIHT with of size 128 × 128. Compression at 0.251 bpp db. 2 32
- 33. Overview• Riemannian Metrics.• Riemannian Voronoi and Delaunay.• Farthest Point Sampling.• Anisotropic Triangulations.• Anisotropic Delaunay Reﬁnement. 33
- 34. 105 This section, shows how several tools from computational geometry extend Planar Domains 106 to the setting of a Riemannian metric. 107 Starting from a set of points S = {xi }m , one can deﬁne graphs and trian- i=1 ECCV-08 submission ID1057 108 gulations that 11reﬂect the geometry of the Riemannian manifold. These pointsECCV and the corresponding graphs are the basic building blocks of the algorithms for 2D Riemannian manifold: [0, 1] equipped with H(x) ⇥ R 2 2 2 . 109 perceptual grouping and planar domain meshing. e to the boundary. A boundary sub-curve i,j is 245 245 if a point xk is located too close to the boundary. 110S if it exists a triple point wi,k,0 i,jInsee ﬁgure 246 the boundary ⇧ encroached by xk , the following, 246 said to be of is S if it exists a triplennot be part of the Delaunay triangulation, anda 247 of closedSuch ancurves. x1 assumed to be set 247 7. smooth encroached edge cannot be part of the orithm by inserting a mid point. Similarly, triple 248 of S is automatically split by the algorithm by inserting At least one point 248 is located on each x2 Boundaries: curve, and these boundary points segmentcroach any boundary sub-curve (the sub-curve is 249 249 points are not added if they encroach any boundar ⇥ ⇧ as a set of sub-curves i P(x , x ) i,j250 250 subdivided instead). j ⇥ = with Ω i,j ⇥ S = {x , x }. 111 ⇥ Another) x3he Delaunay graph DS of S is not ⇧ = ⇥i,j i,j251 251 ⇥i,j ⇤ P(xi , xjj di⇥culty is that the Delaunay graph necessarily awith i i,j ⇥i,j ↵ S = {xi , xj }. ng S is not dense enough, see [7]. This is because 252 252 valid triangulation if the sampling S is not dense en i,j connected to only one other point of S in DS . 253 253 of some isolated point, that is connected to only Requirement: S. (one can have xi = xj if there is only one Dpoint on a curve). Ωcdd points on the Voronoi cell boundary of such a 254 254 The algorithm automatically4add points on the Vo x 255 255 point. 112 2.1 Delaunay and Voronoi Graphs 113 The segmentation of the domain in Riemannian Voronoi cells is ⇥ x3 = C0 Ci where Ci = {x ⇤ ⇧ j ⌅= i, d(xi , x) d(xj , x)} . (7) xi S x2 x1 x2 x1 x2 x1 114 The outer Voronoi cell is deﬁned as C0 = Closure( c ). x3 The Delaunay graph DS of S is a graph where two points are connected if their respective Voronoi cells are adjacent (xi , xj ) ⇤ DS ⇥ Ci ↵ Cj ⌅= ⌃. To each Delaunay (x Bad:corresponds a double point 1,2 ) D x3 encroaches wi,j Good situationvertexpoint toedgeandi,xxjj7.⇤theS common Voronoi cell boundary , which isaches the boundary curve ⇥ . Right:the closest x 1,2 the xi Fig. on Left: the vertex x encroaches the boundary 3 3e (x1 , x2 ) is a Delaunay edge. does not encroach anymore because (x1 , x2 ) is a Delau wi,j = argmin d(x, xi ). x Ci ⌅Cj 34
- 35. Face and Edge Splitting Voronoi edge Missing boundary x3 x4 Delaunay edgeBad quality triangle (x1 , x2 , x3 ) 35
- 36. It extends the isotropic farthest point seeding strategy of [?] w 242Riemannian Delaunay ReﬁnementFig. 7. Left: the vertex 243 encroaches and domains with arbitrary boundaries. vertex x3 x3 metrics the boundary curve ⇥1,2 . Right: thedoes not encroach anymore becauseOur ,anisotropic meshing algorithm proceeds by iteratively i (x1 x2 ) is a Delaunay edge. xpoint si,j,k ⇥ S to an already computed set of points S. In or 1 an anisotropic mesh with triangles of high quality with resp x2 x2 metric, one inserts si,j,k circumradius triangle (xi , xj , xk ) w for a Delaunay = shortest edge Table 2 details this algorithm. A boundshortest edge ratio the reﬁnement to x1 s1,2,3 circumradius to ⇥⇥ on ⇥ enforcesreach some quality criterion, while a bound U ⇥ enforces a uniform reﬁnement to x3 x3 s1,2,3 d(si,j,k , xi )match some desired triangle density. ⇥(si,j,k ) = , min(d(xi , xj ), d(xj , xk ), d(xk , xi )) 244 which is a quantity computed for each triple point in parallel to 1. Initialization: set S with at ing propagation. In the Euclidean of , compute US(xi , xj , xk ) 245 least one point on each curve domain, a triangle with a Fast Marching. 246 of ⇥(si,j,k ) is badly shaped since its smallest angle is close to 0 . 2. Boundary enforcement: while it exists ⇥i,j extends encroached by some xk ⇤ S, angl 247 [?], this property ⌃ to an anisotropic metric H(x) if subdivide: S ⇥ S ⌃ argmax using the inner product deﬁned by Fast Marching. 248 U (x). Update U with a local H(x). S S x⇤⇤i,j 3. Triangulation enforcement: while it exists (xi , xj ) ⇤ DS with xi or xj isolated, insert w⇥ = argmax d(xi , w). w⇤Ci ⇧Cj ⇧ 4. Select point: s ⇥ argmin ⇤(s). s⇤ S ⌃⇥ ⇧ ⇧ – If in S ⌃ {s }, s encroaches some ⇥i,j ⌃ , subdivide: S ⇥ S ⌃ argmax US (x). x⇤⇤i,j ⇧ – Otherwise, add it: S ⇥ S ⌃ s . Update US with a local Fast Marching. 5. Stop: while ⇤(s⇧ ) > ⇤⇧ or US (s⇧ ) > U ⇧ , go back to 2. 36
- 37. Isotropic vs. Anisotropic MeshingIsotropicAnisotropic 37
- 38. Examples of Anisotropic Meshingetric dual of an anisotropic Voronoi di- Given polygonal domain and metric tensor field M, angulation. We describe conditions in guaranteed to be entirely visible from Main Resultction 5. For the special case of two di- ons also guarantee that the planar dual with no inverted triangles. M is smooth with bounded If metric tensor nisotropic mesh. generates high- has angle < 20° as ible an algorithm that no triangle derivatives, isotropic meshes by reﬁ ning anpoint in the triangle. measured by any aniso-nforce the conditions that guarantee thatndgenerate any poor-quality elements to remove anisotropic mesh. 38
- 39. Bonus • Centroidal Tesselations 39
- 40. Centroidal TesselationA ⇥ Rn , Euclidean center of gravity: g(A) = argmin A ||x y||2 dy. xA M, geodesic center of gravity: gM (A) = argmin A dM (x, y)2 dy xSampling {xi }m , Voronoi tessellation VoronoiM ({xi }i ) = {Vi }m . i=1 i=1Centroidal tessellation i, xi = gM (Vi ), local minimizer of m ⇥ E({xi }, {Vi }i ) = dM (xi , y)2 dy. i=1 Vi ⇥ EA (x) = dM (x, y) dy 2 =⇤ ⌅x EA = dM (x, y)nx (y)dy A A ⇥ nx (y) 1. Initialize: {xi }i random.Lloyd-Max x 2. Update regions: ⇥ i, {Vi }i VoronoiM ({xi }i ). y 3. Update centers: ⇥ i, xi gM (Vi ). 4. Stop: while not converged, go back to 2. 40
- 41. Centroidal Tesselations of the Planeuniform #iterationsadaptive = Delaunay = Voronoi 41
- 42. Centroidal Tesselations of Surfaces #iterations 42

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