Analysis of Geometric Subdivision Schemes

Transcription

1 Analysis of Geometric Subdivision Schemes Ulrich Reif Technische Universität Darmstadt Erice, September 27, 2013 Joint work with Malcolm Sabin and Tobias Ewald

2 Standard schemes binary linear local uniform real-valued periodic grid p l+1 2i+σ = aσp j i+j, l σ {0, 1}, i Z, pi l R. j n Ulrich Reif / 29

3 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren p l+1 2i+σ = j n a j σ(l)p l i+j, σ {0, 1} d, i Z d, p l i R. Ulrich Reif / 29

4 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren non-uniform: Floater, Cashman, Hormann, Levin, Levin p l+1 2i+σ = j n a j σ(i)p l i+j, σ {0, 1} d, i Z d, p l i R. Ulrich Reif / 29

5 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren non-uniform: Floater, Cashman, Hormann, Levin, Levin arbitrary topology: Peters, Prautzsch, R., Sabin, Zorin p l+1 2i+σ = j <n a j σ(i)p l i+j, σ {0, 1} 2, i N 2 Z n, p l i R 3. Ulrich Reif / 29

6 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren non-uniform: Floater, Cashman, Hormann, Levin, Levin arbitrary topology: Peters, Prautzsch, R., Sabin, Zorin non-linear: Donoho, Floater, Kuijt, Oswald, Schaefer, Yu p l+1 2i+σ = a σ(p l i,..., p l i+n), σ {0, 1} d, i Z d, p l i R. Ulrich Reif / 29

7 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren non-uniform: Floater, Cashman, Hormann, Levin, Levin arbitrary topology: Peters, Prautzsch, R., Sabin, Zorin non-linear: Donoho, Floater, Kuijt, Oswald, Schaefer, Yu non-local: Kobbelt, Unser, Warren, Weimer p l+1 2i+σ = j Z d a j σp l i+j, σ {0, 1} d, i Z d, p l i R. Ulrich Reif / 29

8 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren non-uniform: Floater, Cashman, Hormann, Levin, Levin arbitrary topology: Peters, Prautzsch, R., Sabin, Zorin non-linear: Donoho, Floater, Kuijt, Oswald, Schaefer, Yu non-local: Kobbelt, Unser, Warren, Weimer vector-valued: Conti, Han, Jia, Merrien, Micchelli, Sauer, Zimmermann p l+1 2i+σ = j <n a j σp l i+j, σ {0, 1} d, i Z d, p l i R d. Ulrich Reif / 29

9 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren non-uniform: Floater, Cashman, Hormann, Levin, Levin arbitrary topology: Peters, Prautzsch, R., Sabin, Zorin non-linear: Donoho, Floater, Kuijt, Oswald, Schaefer, Yu non-local: Kobbelt, Unser, Warren, Weimer vector-valued: Conti, Han, Jia, Merrien, Micchelli, Sauer, Zimmermann manifold-valued: Dyn, Grohs, Wallner, Weinmann p l+1 2i+σ = a σ(p l i,..., p l i+n), σ {0, 1} d, i Z d, p l i M. Ulrich Reif / 29

10 Non-standard schemes non-stationary: Beccari, de Boor, Casciola, Dyn, Levin, Romani, Warren non-uniform: Floater, Cashman, Hormann, Levin, Levin arbitrary topology: Peters, Prautzsch, R., Sabin, Zorin non-linear: Donoho, Floater, Kuijt, Oswald, Schaefer, Yu non-local: Kobbelt, Unser, Warren, Weimer vector-valued: Conti, Han, Jia, Merrien, Micchelli, Sauer, Zimmermann manifold-valued: Dyn, Grohs, Wallner, Weinmann geometric: Albrecht, Cashman, Dyn, Hormann, Levin, Romani, Sabin p l+1 2i+σ = g σ(p l i,..., p l i+n), σ {0, 1} d, i Z d, p l i R d. Ulrich Reif / 29

11 Sabin s Circle-preserving subdivision Ulrich Reif / 29

12 Sabin s Circle-preserving subdivision Ulrich Reif / 29

13 Sabin s Circle-preserving subdivision Ulrich Reif / 29

14 Sabin s Circle-preserving subdivision Ulrich Reif / 29

15 Sabin s Circle-preserving subdivision Ulrich Reif / 29

16 Sabin s Circle-preserving subdivision Ulrich Reif / 29

17 Sabin s Circle-preserving subdivision Ulrich Reif / 29

18 Sabin s Circle-preserving subdivision Ulrich Reif / 29

19 GLUE-schemes Definition Let P = (p i ) i Z, p i E := R d. A geometric, local, uniform, equilinear subdivision scheme G : P l P l+1 is characterized by: Ulrich Reif / 29

20 GLUE-schemes Definition Let P = (p i ) i Z, p i E := R d. A geometric, local, uniform, equilinear subdivision scheme G : P l P l+1 is characterized by: G: The scheme G commutates with similarities, G S = S G, S S(E). Ulrich Reif / 29

21 GLUE-schemes Definition Let P = (p i ) i Z, p i E := R d. A geometric, local, uniform, equilinear subdivision scheme G : P l P l+1 is characterized by: G: The scheme G commutates with similarities, G S = S G, S S(E). L: New points depend on finitely many old points. Ulrich Reif / 29

22 GLUE-schemes Definition Let P = (p i ) i Z, p i E := R d. A geometric, local, uniform, equilinear subdivision scheme G : P l P l+1 is characterized by: G: The scheme G commutates with similarities, G S = S G, S S(E). L: New points depend on finitely many old points. U: The same two rules (even/odd) apply everywhere, p l+1 2i+σ = g σ(p l i,..., p l i+m), σ {0, 1}. The functions g σ are C 1,1 in a neighborhood of linear data. Ulrich Reif / 29

23 GLUE-schemes Definition Let P = (p i ) i Z, p i E := R d. A geometric, local, uniform, equilinear subdivision scheme G : P l P l+1 is characterized by: G: The scheme G commutates with similarities, G S = S G, S S(E). L: New points depend on finitely many old points. U: The same two rules (even/odd) apply everywhere, p l+1 2i+σ = g σ(p l i,..., p l i+m), σ {0, 1}. The functions g σ are C 1,1 in a neighborhood of linear data. E: The standard linear chain E = (ie) i Z is dilated by the factor 1/2, G(E) = E/2 Ulrich Reif / 29

24 GLUE-schemes Definition Let P = (p i ) i Z, p i E := R d. A geometric, local, uniform, equilinear subdivision scheme G : P l P l+1 is characterized by: G: The scheme G commutates with similarities, G S = S G, S S(E). L: New points depend on finitely many old points. U: The same two rules (even/odd) apply everywhere, p l+1 2i+σ = g σ(p l i,..., p l i+m), σ {0, 1}. The functions g σ are C 1,1 in a neighborhood of linear data. E: The standard linear chain E = (ie) i Z is dilated by the factor 1/2, G(E) = E/2 Ulrich Reif / 29

25 GLUE-schemes Definition Let P = (p i ) i Z, p i E := R d. A geometric, local, uniform, equilinear subdivision scheme G : P l P l+1 is characterized by: G: The scheme G commutates with similarities, G S = S G, S S(E). L: New points depend on finitely many old points. U: The same two rules (even/odd) apply everywhere, p l+1 2i+σ = g σ(p l i,..., p l i+m), σ {0, 1}. The functions g σ are C 1,1 in a neighborhood of linear data. E: The standard linear chain E = (ie) i Z is dilated by the factor 1/2, G(E) = E/2 + τe. Ulrich Reif / 29

26 Basics: matrix-like formalism Analogous to the representation of linear schemes in terms of pairs of matrices, there exist functions g σ such that p l+1 2i+σ = g σ(p l i ), where p l i = [p l i ;... ; pl i+n 1 ] are subchains of Pl of length n. Constant chains are fixed points, g σ (p) = p if p = 0. Composition of functions g σ is denoted by g Σ = g σl g σ1, Σ = [σ 1,..., σ l ], Σ = l. Let e := [e;... ; ne]. Then g Σ (e) = 2 Σ e + τ Σ e. Ulrich Reif / 29

27 Basics: spaces of chains E Z := R d Z is the space of infinite chains in R d. E n := R d n is the space of chains with n vertices in R d. L n := {p E n : 2 p = 0} is the space of linear chains. Π : E n L n is the orthogonal projector onto L n. For P E Z, let P := sup p i 2, P 1 := P, P 2 := 2 P. i Ulrich Reif / 29

28 Basics: spaces of chains E Z := R d Z is the space of infinite chains in R d. E n := R d n is the space of chains with n vertices in R d. L n := {p E n : 2 p = 0} is the space of linear chains. Π : E n L n is the orthogonal projector onto L n. For p E n, let p := sup p i 2, p 1 := p, p 2 := 2 p. i Ulrich Reif / 29

29 Basics: relative distortion The relative distortion of some chain p E n is defined by p 2 if Πp 1 0 κ(p) := Πp 1 if Πp 1 = 0. Invariance under similarities, κ(p) = κ(s(p)), S S(E). Distortion of infinite chain, κ(p) := sup κ(p i ), p i = [p i ;... ; p i+n 1 ]. i Z Distortion sequence generated by subdivision, κ l := κ(p l ), P l := G l (P). Ulrich Reif / 29

30 Straightening Definition The chain P is straightened by G if κ l is a null sequence; strongly straightened by G if κ l is summable; straightened by G at rate α if 2 lα κ l is bounded. Ulrich Reif / 29

31 Straightening Definition The chain P is straightened by G if κ l is a null sequence; strongly straightened by G if κ l is summable; straightened by G at rate α if 2 lα κ l is bounded. Lemma Let G be a GLUE-scheme. If the chain P is straightened by G, then P 1 Cq l for any q > 1/2; strongly straightened by G, then P 1 Cq l for any q = 1/2; straightened by G at rate α, then P 2 C2 l(1+α). Ulrich Reif / 29

32 Straightening Definition The chain P is straightened by G if κ l is a null sequence; strongly straightened by G if κ l is summable; straightened by G at rate α if 2 lα κ l is bounded. Lemma Let G be a GLUE-scheme. If the chain P is straightened by G, then P 1 Cq l for any q > 1/2; strongly straightened by G, then P 1 Cq l for any q = 1/2; straightened by G at rate α, then P 2 C2 l(1+α). Proof: induction on Σ q-pochhammer symbol Ulrich Reif / 29

33 Straightening Definition The chain P is straightened by G if κ l is a null sequence; strongly straightened by G if κ l is summable; straightened by G at rate α if 2 lα κ l is bounded. Theorem (R. 2013) Let G be a GLUE-scheme. If the chain P is straightened by G, then P l converges to a continuous limit curve; strongly straightened by G, then the limit curve is C 1 and regular; straightened by G at rate α, then the limit curve is C 1,α and regular. Ulrich Reif / 29

34 Convergence Let ϕ be a C k -function which has compact support; constitues a partition of unity, j ϕ( j) = 1. Associate a curve Φ l to the chain P l at stage l by Φ l [P l ] := pj l ϕ(2 l j) l Z Ulrich Reif / 29

35 Convergence Let ϕ be a C k -function which has compact support; constitues a partition of unity, j ϕ( j) = 1. Associate a curve Φ l to the chain P l at stage l by Φ l [P l ] := pj l ϕ(2 l j) l Z If Φ l [P l ] is Cauchy in C 0, then the limit curve Φ[P] := lim l Φ l [P l ] is well defined, continuous, and independent of ϕ. Ulrich Reif / 29

36 Convergence Let ϕ be a C k -function which has compact support; constitues a partition of unity, j ϕ( j) = 1. Associate a curve Φ l to the chain P l at stage l by Φ l [P l ] := pj l ϕ(2 l j) l Z If Φ l [P l ] is Cauchy in C 0, then the limit curve Φ[P] := lim l Φ l [P l ] is well defined, continuous, and independent of ϕ. If Φ l [P l ] is Cauchy in C k, then the limit curve Φ[P] is C k. Ulrich Reif / 29

37 Convergence Let ϕ be a C k -function which has compact support; constitues a partition of unity, j ϕ( j) = 1. Associate a curve Φ l to the chain P l at stage l by Φ l [P l ] := pj l ϕ(2 l j) l Z If Φ l [P l ] is Cauchy in C 0, then the limit curve Φ[P] := lim l Φ l [P l ] is well defined, continuous, and independent of ϕ. If Φ l [P l ] is Cauchy in C k, then the limit curve Φ[P] is C k. Use modulus of continuity to establish Hölder exponent. Ulrich Reif / 29

38 Proximity Given a GLUE-schem G, choose a linear subdivision scheme A with equal shift, i.e., G(E) = AE = (E + τe)/2. Schemes G and A differ by remainder R, R(P) := G(P) AP. Ulrich Reif / 29

39 Proximity Given a GLUE-schem G, choose a linear subdivision scheme A with equal shift, i.e., G(E) = AE = (E + τe)/2. Schemes G and A differ by remainder R, R(P) := G(P) AP. Choose ϕ as limit function of A correspondig to Ulrich Reif / 29

40 Proximity Ulrich Reif / 29

41 Proximity Given a GLUE-schem G, choose a linear subdivision scheme A with equal shift, i.e., G(E) = AE = (E + τe)/2. Schemes G and A differ by remainder R, R(P) := G(P) AP. Choose ϕ as limit function of A correspondig to Dirac data δ j,0 to define curves Φ l [P l ] at level l. Curves at levels l and l + r differ by j ( Φ l+r [P l+r ] Φ l [P l ] ) c 2 ij R(P i ) 0. i=l Ulrich Reif / 29

42 Proximity Given a GLUE-schem G, choose a linear subdivision scheme A with equal shift, i.e., G(E) = AE = (E + τe)/2. Schemes G and A differ by remainder R, R(P) := G(P) AP. Choose ϕ as limit function of A correspondig to Dirac data δ j,0 to define curves Φ l [P l ] at level l. Curves at levels l and l + r differ by j ( Φ l+r [P l+r ] Φ l [P l ] ) c Use bound R(P i ) 0 cκ i q i 2 ij R(P i ) 0. with q = 1/2 in case of strong straightening, and q = 2/3 otherwise. i=l Ulrich Reif / 29

43 Checks for straightening For applications, we need explicit values α, δ such that P is straightened by G at rate α whenever κ(p) δ. Ulrich Reif / 29

44 Checks for straightening Lemma Let Γ l [δ] := κ l (e + d) sup. 0< d 2 δ d 2 If Γ l [δ] < 1 for some l N, then P is straightened by G at rate α = log 2 Γ l [δ] l whenever κ(p) δ. Ulrich Reif / 29

45 Checks for straightening Lemma Let Γ l [δ] := κ l (e + d) sup. 0< d 2 δ d 2 If Γ l [δ] < 1 for some l N, then P is straightened by G at rate α = log 2 Γ l [δ] l whenever κ(p) δ. + A rigorous upper bound on Γ l [δ] can be established using mean value theorem and interval arithmetics. The larger δ, the poorer α. Ulrich Reif / 29

46 Checks for straightening Theorem (R. 2012) Let Γ l [δ] := κ l (e + d) κ k (e + d) sup and Γ k [δ, γ] := max. 0< d 2 δ d 2 δ d 2 γ d 2 If Γ l [δ] < 1 for some l N, and Γ k [δ, γ] < 1 for some k N, then P is straightened by G at rate α = log 2 Γ l [δ] l whenever κ(p) γ. Ulrich Reif / 29

47 Checks for straightening Theorem (R. 2012) Let Γ l [δ] := κ l (e + d) κ k (e + d) sup and Γ k [δ, γ] := max. 0< d 2 δ d 2 δ d 2 γ d 2 If Γ l [δ] < 1 for some l N, and Γ k [δ, γ] < 1 for some k N, then P is straightened by G at rate α = log 2 Γ l [δ] l whenever κ(p) γ. + Rigorous upper bounds on Γ l [δ] and Γ k [δ, γ] via interval arithmetics. + Choose δ as small as possible to get good α. + Choose γ as large as possible to get good range of applicability. Ulrich Reif / 29

48 Differentiation In general, the derivative of a function g : E n E n is represented by n n matrices of dimension d d, each. Ulrich Reif / 29

49 Differentiation In general, the derivative of a function g : E n E n is represented by n n matrices of dimension d d, each. By property G, the derivative of g σ at e has the special form Dg σ (e) q = A σ q Π n + B σ q Π t, σ {0, 1}, where A σ, B σ are (n n)-matrices, and Π t := diag[1, 0,..., 0], Π n := diag[0, 1,..., 1] are (d d)-matrices representing orthogonal projection onto the x-axis and its orthogonal complement. Let A = (A 0, A 1 ) and B = (B 0, B 1 ) denote the linear subdivision schemes corresponding to normal and tangential direction. Ulrich Reif / 29

50 Inheritance of C 1,α -regularity Theorem (R. 2013) Let the linear schemes A and B be C 1,α and C 1,β, resp. If P is straightened by G, then the limit curve Φ[P] is C 1,min(α,β). Proof: Show that lim δ 0 ( lim inf Γl [δ] ) 1/l ( max jsr(a 2 0, A 2 1), jsr(b0 2, B1 2 ) ). l Ulrich Reif / 29

51 Locally linear schemes Definition A GLUE-scheme G is called locally linear if there exist (n n)-matrices A 0, A 1 such that Dg σ (e) q = A σ q Π n + B σ q Π t. Ulrich Reif / 29

52 Locally linear schemes Definition A GLUE-scheme G is called locally linear if there exist (n n)-matrices A 0, A 1 such that Dg σ (e) q = A σ q. In this case the linear scheme A = (A 0, A 1 ) is called the linear companion of G. Ulrich Reif / 29

53 Locally linear schemes Definition A GLUE-scheme G is called locally linear if there exist (n n)-matrices A 0, A 1 such that Dg σ (e) q = A σ q. In this case the linear scheme A = (A 0, A 1 ) is called the linear companion of G. For d = 1, any GLUE-scheme G is locally linear. For d 2, the scheme G is locally linear if A = B. Circle-preserving subvdivion is locally linear, and the standard four-point scheme is its linear companion. Ulrich Reif / 29

54 Inheritance of C 2,α -regularity Theorem (R. 2013) Let G be locally linear, and let the linear companion A be C 2,α. If P is straightened by G, then the limit curve Φ[P] is C 2,α. Proof: Use basic limit function ϕ of A to define curves Φ l [P l ]. Use bound R(P) 0 cκ(p) P 2 on the remainder R(P) := G(P) AP. Ulrich Reif / 29

55 Inheritance of C 3,α -regularity Ulrich Reif / 29

56 Inheritance of C 3,α -regularity... cannot be expected! Ulrich Reif / 29

57 A counter-example Consider g 0 (p l i,..., p l i+3) = 6 32 pl i pl i pl i p l i p l i 2 p l i g 1 (p l i,..., p l i+3) = 1 32 pl i pl i pl i pl i+3 The scheme is locally linear with A 0, A 1 representing quintic B-spline subdivision. However, limit curves Φ [P] are not C 4, and not even C 3. Ulrich Reif / 29

58 Conclusion Geometric subdivision schemes deserve attention. Results apply to a wide range of algorithms. Hölder continuity of first order can be established rigorously by means of a universal computer program (at least in principle, runtime may be a problem). For locally linear schemes, Hölder-regularity of second order can be derived from a linear scheme, defined by the Jacobians at linear data. Regularity of higher order requires new concepts. Ulrich Reif / 29