Introduction to Computational Manifolds and Applications

Transcription

1 IMPA - Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil Introduction to Computational Manifolds and Applications Part 1 - Constructions Prof. Marcelo Ferreira Siqueira mfsiqueira@dimap.ufrn.br Departmento de Informática e Matemática Aplicada Uniersidade Federal do Rio Grande do Norte Natal, RN, Brazil

2 The User Perspectie While interacting with a subdiision surface or a B-spline surface for the purposes of editing, rendering, or physical simulation, one uses the "control mesh" as an interface. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 2

3 The User Perspectie More specifically, the user assumes that there is a one-to-one correspondence between the control mesh and the surface. So, each point in the control mesh corresponds to a point in the surface, and thus the control mesh is iewed as a parameter space. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 3

4 The User Perspectie In reality, each point on the surface is the image of a (parameter) point in a triangle in E 2. E 2 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 4

5 The User Perspectie But, since we can relate the triangle in E 2 with a triangle of the control mesh (ia a barycentric mapping), the user "illusion" works fine. We will use a similar mechanism here. E 2 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 5

6 The User Perspectie Here, the control mesh is the underlying surface, K, of the input simplicial surface, K. Let σ be a triangle in K and let p be any point in σ. σ K p Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 6

7 The User Perspectie Map the point p to a point q belonging to the equilateral triangle, E 2, with ertices (0, 0), (1, 0), and (1/2, 3/2). 12, 3 2 w p u q (0, 0) (1, 0) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 7

8 The User Perspectie We can do that by using the barycentric map that takes the ertices, u, and w of σ to the ertices (0, 0), (1, 0), and (1/2, 3/2) of, respectiely. So, if p = λ + µ u + ν w, where λ, µ, ν R, λ, µ, ν 0, and λ + µ + ν = 1, the coordinates of q are gien by q = λ (0, 0)+µ (1, 0)+ν (1/2, 3/2). 12, 3 2 w p u q (0, 0) (1, 0) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 8

9 The User Perspectie Now, we can map q to the p-domain Ω using the map ru 1 g 1 wheneer the distance from q to (0, 0) is smaller than cos(π/6), which is the radius of the circle corresponding to g (Ω {(0, 0)}). 12, 3 2 w p u q (0, 0) (1, 0) g (Ω {(0, 0)}) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 9

10 The User Perspectie If this is not the case, then we compute the distance from q to (1, 0). If this distance is smaller than cos(π/6), we map q to Ω u. Otherwise, the distance from q to (1/2, 3/2) has to be smaller than cos(π/6) why is that so?, and thus we map q to Ω w. 12, 3 2 w p u q (0, 0) (1, 0) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 10

11 To map q to Ω u, we use ru 1 gu 1 Ω w. The User Perspectie h. In turn, we use r 1 w g 1 w h r π 3 to map q to 12, 3 2 w p u q (0, 0) (1, 0) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 11

12 The User Perspectie Regardless of the p-domain where q will be mapped to, the important point is that q will be mapped to some p-domain, establishing a correspondence between K and the PPS. 12, 3 2 w p u q (0, 0) (1, 0) Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 12

13 The User Perspectie K w w p u r,σ r,σ 12, r w gw 1 h r π 3 ru 1 g 1 q ru 1 gu 1 h (0, 0) (1, 0) K K u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 13

14 The User Perspectie KEEP IN MIND: the user does not hae to know about the existence of gluing data and parametrizations. All the user needs to interact with the PPS is a triangle, σ, in K and a point p in σ. Once σ and p are gien, we can take p to some p-domain and then to the image, S, of the PPS using the approach we just saw. So, all the user sees is still a "control mesh" (i.e., K ). Let us now demo the code and discuss the implementation... Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 14

15 Now, we assume that we are gien a quadrilateral surface mesh, K. A quadrilateral mesh, K, in E 3 is a set consisting of a finite number of (conex or nonconex) quadrilaterals, along with their edges and ertices, such that if σ 1 and σ 2 are any two quadrilaterals in K, then σ 1 σ 2 is either empty or a ertex or edge of both σ 1 and σ 2. Here, we assume that the quadrilaterals of K are planar objects. This allows us to define the underlying space, K, of K just like we did for simplicial complexes: the union K = σ. σ K Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 15

16 The notions of star and link of a ertex in K can be defined for quadrilateral meshes as well. If σ is a ertex in K, then the star, st(σ, K), of σ in K is the set of quadrilaterals that contain σ as a ertex, along with their edges and ertices. In turn, the link, lk(σ, K), of σ in K is the set of edges in st(σ, K) that do not contain σ as a ertex, along with their ertices. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 16

17 If eery edge of K is incident with exactly two quadrilaterals of K, and if the underlying space of lk(σ, K) is homeomorphic to the unit circle S 1 = {x E 2 x = 1}, for eery ertex σ in K, then we say that K is a quadrilateral surface mesh (without boundary). The underlying space, K, of a quadrilateral surface mesh, K, is a topological surface in E 3. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 17

18 We can now discuss the construction deised by Ying and Zorin (SIGGRAPH, 2003). We start by the definition of the gluing data, G = (Ω i ) i I, (Ω ij ) (i,j) I I, (ϕ ji ) (i,j) K. We let I be the set of ertices of K, as we assign a (distinct) p-domain to each ertex of K. Let be any ertex of K, and let n be the number of ertices of lk(, K). Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 18

19 Recall that z n = r n (cos (n θ) + i sin (n θ)), is the polar form of the complex number, z = x + iy, where x = r cos(θ) and y = r sin(θ). Let L be the square with ertices (0, 0), ( 2/2, 2/2), ( 2, 0), and ( 2/2, 2/2). 2 2, 2 2 (0, 0) 2, 0 2 2, 2 2 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 19

20 If we apply the map f (z) =z 4 n n = 6): to L we obtain a "cured" quadrilateral (e.g., for f (z) =z 4 6 As usual, the parameter n is the degree of the ertex in K. For a quadrilateral surface mesh, the alue of n is half the number of ertices in the link, lk(, K), of in K. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 20

21 We define the p-domain, Ω, as the interior of the union set n 1 k=0 r (2k+1) π n f (L), where r (2k+1) π n is a counterclockwise rotation around the origin by the angle (2k+1) π n. n = 3 n = 4 n = 6 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 21

22 Fix a counterclockwise enumeration, 0, 1,..., 2 n 1, of the ertices in lk(, K). We can then identify with the point =(0, 0) of Ω Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 22

23 For each i = 0, 1,..., 2 n 1, we can also identify the point i with the point i r 2 2 = (i+1) π f n 2, 2 if i is een; otherwise, we identify i with the point i r = (i+1) π f ( 2, 0). n Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 23 5

24 The aforementioned identification can be represented by a function, which we name f. So, = f (), 0 = f ( 0 ), 1 = f ( 1 ), and so on so forth Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 24 5

25 Suppose that and u, with = u, are the ertices of an edge, [, u], of K. Then, the sets st(, K) and st(u, K) share exactly two quadrilaterals, which in turn share an edge. t w u x y Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 25

26 Let, x, y, u, w, and t be the ertices of these two adjacent quadrilaterals. We define Ω u as the subset of Ω corresponding to the interior of the union of the two "cured" quadrilaterals defined by the ertices f (), f (x), f (y), f (u), f (w), and f (t). t w u x y Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 26

27 f (t) f (w) f () f (u) f (x) f (y) t w u x y Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 27

28 We define Ω u in a similar fashion. t w u x y Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 28

29 f (t) f (w) f u (y) f u (x) f () f (u) f u (u) f u () f (x) f (y) t w f u (w) f u (t) u x y Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 29

30 Suppose that and w, with = w, are the ertices of a quadrilateral of K, but not ertices of an edge of K. Then, the sets st(, K) and st(w, K) share exactly one quadrilateral. t w u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 30

31 Let, u, t, and w be the ertices of the common quadrilateral. We define the gluing domain, Ω w, as the subset of Ω corresponding to the interior of the "cured" quadrilateral defined by the ertices f (), f (u), f (w), and f (t). t w u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 31

32 f (t) f (w) f () f (u) t w u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 32

33 We define Ω w in a similar fashion. t w u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 33

34 f (t) f (w) f w (u) f w () f () f (u) f w (w) f w (t) t w u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 34

35 So, there are two types of gluing domains (the single and the double cured quadrilateral). We notice that a point in a p-domain can be identified either with no other p-domain (this is the case for the point located at (0, 0) only), with two p-domains, or with four p-domains. one p-domain four p-domains two p-domains Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 35

36 How can we define the transition functions? t w u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 36

37 Ω w Ω t t w u Ω Ω u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 37

38 Consider the transition function ϕ u : Ω u Ω u and let p be a point in Ω u. Ω p Ω u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 38

39 Since [, u] is an edge of K, we rotate Ω by an angle, θ, that makes [ f (), f (u)] coincide with the x axis. If f (u) = i, for some i = 0, 1,..., 2 n 1, with i een, then θ = i π n. i p r i π n Ω Ω p Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 39

40 We then take the set r i π n two squares: (Ω u ) onto a canonical set, say Q, which is the union of Q l =[(0, 0), (0, 1), (1, 1), (1, 0)] and Q u =[(0, 0), (1, 0), (1, 1), (0, 1)]. Q u Ω p Q l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 40

41 This is done in a piecewise manner. More specifically, the upper cured quadrilateral of r i π (Ω n u ) is mapped onto Q u using the composition map r π z n 4 r 4 π. So, we n get r π z n 4. 4 r (i+1) π n Similarly, the lower cured quadrilateral of Ω u is mapped onto Q l using the composition r π z n 4. 4 r (i 1) π n r π n z n 4 r π 4 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 41

42 Next, we apply a double reflection, h 1 (x, y) =(1 x, y), to Q, which is a reflection with respect to the ertical line x = 0.5 followed by a reflection with respect to the x axis. Q u h1 Q l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 42

43 Finally, we map the interior of Q to Ω u using the corresponding inerse functions. The interior of Q u is mapped to the interior of the upper cured quadrilateral of Ω u by r (j+1) π n u z 4 nu r π 4. Similarly, r (j 1) π n u z 4 nu r π 4 is the map that takes the interior of Q l to the interior of the lower cured quadrilateral of Ω u. j = f u() Q u Ω u Q l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 43

44 So, ϕ u (p) = r (j 1) π n u z 4 nu r π 4 h 1 r π 4 z n 4 r (i+1) π n (p) if p belongs to the upper cured quadrilateral of Ω u ; otherwise, the transition function is ϕ u (p) = r (j+1) π n u z 4 nu r π 4 h 1 r π 4 z n 4 r (i 1) π n (p). Ω p Q u Ω u Q l Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 44

45 Ω w Ω t t w u Ω Ω u Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 45

46 Now, consider the transition function ϕ w : Ω w Ω w and let p be a point in Ω w. p Ω Ω w Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 46

47 Since [, w] is not an edge of K, we rotate Ω by an angle, θ, that makes [ f (), f (w)] coincide with the x axis. If f (w) = i, for some i = 0, 1,..., 2 n 1, with i odd, then θ = i π n. i Ω p r i π n Ω p Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 47

48 We then take the set r i π (Ω n w ) onto the canonical quadrilateral, L, where L = (0, 0), ( , 0), 2,, 2 2,. 2 p L Ω Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 48

49 We use the map z n 4 to take r i π n (Ω w ) onto L: z n 4 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 49

50 Next, we apply a double reflection, h 2 (x, y) =( 2 x, y), to L, which is a double reflection: a reflection w.r.t the line x = 2/2 followed by a reflection w.r.t the y axis: z n 4 h 2 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 50

51 Next, we apply the map z nw 4 to h 2 z n 4 r i π (Ω n w ): z n 4 h 2 z 4 nw Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 51

52 Finally, we apply a rotation, r j π, to the set z nw 4 n h 2 z n 4 w r i π n (Ω w ): Ω w r j π n w Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 52

53 So, ϕ w (p) = r j π n w z 4 nw h 2 z n 4 r i π n (p). ϕ w (p) p Ω Ω w Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 53

54 What about geometry? The idea is to uniformly sample the canonical quadrilateral corresponding to each quadrilateral face of the input mesh. This is done for each face of the input mesh at a time. i4, j 4 = 14, 3 4 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 54

55 For each parameter point in the canonical quadrilateral, we compute the corresponding 3D point on the Catmull-Clark subdiision surface defined from the input mesh. Use Jos Stam s work! i4, j 4 = 14, 3 4 Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 55

56 Finally, we map the points in the canonical quads to the p-domains. If n is the degree of, then we map 12 n + 1 points to Ω. These are the parameter points which can be iewed as being inside the underlying space of the star, st(, K), of in K. Ω Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 56

57 To map the points from the canonical quadrilateral to Ω, we use the composite map r i π n z n 4 r π, where i identifies the cured quadrilateral in Ω 4 that will contain the point. p (r 3 π n z 4 n r π 4 )(p) i = 3 i = 1 i = 5 i = 7 Ω Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 57

58 For eery point p =(r i π n z n 4 r π )(p) in Ω 4, we define a pair of points, (p, q), where q is the point on the Catmull-Clark subdiision surface associated with the point p. p q p =(r 3 π n z 4 n r π 4 )(p) Ω Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 58

59 We are now in a position to compute the parametrization, θ : Ω E 3, associated with. In order to do so, Ying and Zorin defined a basis of monomials, (x r y s ) r,s, of total degree, r + s, at most d = min{14, n + 1}. Using this basis, they define a polynomial, ψ (x, y), of degree at most d whose coefficients are computed by minimizing the differences ψ (p ) q in the least-squares sense, where (p, q) are the pairs of points we just computed before. The polynomial ψ is the shape function associated with. Note that these functions do not necessarily satisfy the conex hull property, as the Bézier patches we used before. Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 59

60 Finally, we define as θ : Ω E 3 θ (p) = z J (p)(ψ z ϕ z )(p) (γ z ϕ z )(p) z J (p)(γ z ϕ z )(p) for eery p Ω, where J (p) ={u I p Ω u } and γ : Ω R is a bump function. The set J (p) has at least one ertex and at most 4. Note that θ is defined as a conex combination of the alues of (ψ z ϕ z )(p) weighted by (γ z ϕ z )(p), which can be thought as the influence of (ψ z ϕ z )(p) on θ (p). Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 60

61 The bump function, γ z, is defined as a composition of a few maps. First, the point ϕ z (p) is taken to the canonical domain using the map r π z n z 4 r 4 i π, where i identifies the cured quadrilateral of Ω z that contains the point ϕ z (p) in the n z p-domain Ω z. Let (x, y) be the coordinates of the point (r π 4 quad. z n z 4 r i π n z ϕ z )(p) in the canonical ϕ z (p) i = 3 i = 5 i = 1 r π 4 z n z 4 r 3 π n z (x, y) Ω z Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 61

62 We compute the product ξ(x) ξ(y), where ξ : [0, 1] R is the function defined in the preious lecture. So, putting eerything together, we define γ z : Ω z R as follows: γ z (x, y) = where η : [0, 1] 2 R is gien by η r π z n z 4 4 r i π (x, n y), z η(x, y) =ξ(x) ξ(y). Computational Manifolds and Applications (CMA) , IMPA, Rio de Janeiro, RJ, Brazil 62