On the Structure of Anisotropic Frames

Transcription

1 On the Structure of Anisotropic Frames P. Grohs ETH Zurich, Seminar for Applied Mathematics ESI Modern Methods of Time-Frequency Analysis

2 Motivation P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 2

3 Geometric Multiscale Analysis Multiscale representation systems which are sensitive to anisotropic features and which provide sparse approximations thereof. Such constructions are typically based on parabolic scaling and a directional transform such as rotation. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 3

4 Curvelets P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 4

5 Curvelets P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 5

6 Curvelets P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 6

7 Curvelets P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 7

8 Curvelets P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 8

9 Curvelets P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 9

10 A Zoo of Transforms Hart Smith (1998): Transform based on parabolic scaling for discretization of linear wave equations Candès and Dohoho (1999): Curvelets for sparse representations of cartoon images. Candès and Demanet (2004): Curvelets for sparse representations of wave propagators. Labate, Lim, Kutyniok and Weiss (2005): Shearlets, where rotation is replaced by shearing, ensuring uniform treatment of continuous and digital transform. Initial construction bandlimited. Kittipoom, Kutyniok and Lim (2012), Dahlke, Steidl and Teschke (2011): Construction of compactly supported shearlet frames. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 10

11 Approximation Properties All these different systems possess optimally sparse approximation properties for functions exhibiting singularities on lower-dimensional manifolds. This has been shown separatly for each system in a list of papers. Question: Is it really necessary to go through these proofs for each single system, or might a higher-level viewpoint be useful? P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 11

12 Goal Meta-Theorem All frame systems based on parabolic scaling (specifically curvelets and shearlets) posses the exact same approximation properties, whenever the generating functions are sufficiently smooth, as well as localized in space and frequency. Concrete results, i.e., questions of type: given C N basis function with M directional vanishing moments, create frame from rotation/shearing + anisotropic dilation, what are the approximation properties? P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 12

13 Main Ideas Introduce class of function systems (m λ ) λ Λ L 2 (R 2 ) which includes all known systems based on parabolic scaling. Show that any two function systems within a class possess equivalent approximation properties. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 13

14 Previous Work Curvelet/Shearlet molecules (Candès-Demanet 2004, Guo-Labate 2008) Unification of curvelet/shearlet-type systems based on rotation/shearing Does not include shearlet/curvelet-based methods Nonquantitative only basis functions with infinitely many vanishing moments and superpolynomial decay in space not valid in practice Coorbit molecules (Dahlke-Steidl-Teschke , Gröchenig-Piotrowski 2008), see Gabi Steidl s talk Wed 10:00-11:00. Powerful framework for systems built from group representation Does not include cone adapted shearlets or curvelet-type constructions Abstract concept P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 14

15 Coorbit Molecules Group G, representation π, window g, well-spread point-set (x λ ) λ Λ G, envelope function H more specific. A system (m λ ) λ Λ L 2 (R d ) is a system of coorbit molecules if m λ, π(z)g H(z 1 x λ ) (1) Can be defined for (non cone-adapted) shearlets, but: What is H?? How to apply to cone-adapted shearlet or curvelet-type systems?? How to verify (1) for concrete system?? Need more concrete approach in terms of analytical properties of the family (m λ ) λ Λ and more general approach in terms of the types of directional transforms allowed. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 15

16 Parabolic Molecules P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 16

17 Basic Structure System (m λ ) λ Λ where every m λ is associated with a scale, a direction and a location. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 17

18 Parametrization Phase Space P := R + T R 2 Definition A parametrization consists of an index set Λ and a mapping Φ Λ : Λ P, Φ Λ (λ) := (s λ, θ λ, x λ ). Canonical parametrization: { Λ 0 := (j, l, k) Z 4 : j 0, l = 2 j/2,..., 2 j/2 }, Φ 0 ((j, l, k)) := (j, πl2 j/2, R θλ D 2 s λ k) ( cos(θ) sin(θ) R θ := sin(θ) cos(θ) ) D a := ( a 0 0 a ). P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 18

19 Canonical Parametrization Figure: Left: frequency tiling associated with canonical parametrization. Right: Translational grids. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 19

20 Definition of Parabolic Molecules Definition Let (Λ, Φ Λ ) be a parametrization. A family (m λ ) λ Λ is called a family of parabolic molecules of order (R, M, N 1, N 2 ) if it can be written as m λ (x) = 2 3s λ/4 a (λ) (D 2 s λ R θλ (x x λ )) such that ( β â (λ) M (ξ) min 1, 2 s λ + ξ sλ/2 N ξ 2 ) ξ 1 ξ 2 N 2 for all β R. The implicit constants are uniform over λ Λ. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 20

21 A Closer Look β â (λ) (ξ) min for all β R means that a (λ)... ( 1, 2 s λ + ξ s λ/2 ξ 2 ) M ξ N 1 ξ 2 N 2... possesses M (almost) vanishing moments in the x 1 -direction... is of smoothness N 1 + N 2 in x 2 and N 1 in x 1... decays like x R in space. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 21

22 Frequency Localization Figure: Left: The weight function ( M min 1, 2 s λ + ξ sλ/2 ξ 2 ) ξ N 1 ξ 2 N 2 for s λ = 3, M = 3, N 1 = N 2 = 2. Right: Approximate Frequency support of a corresponding molecule ˆm λ with θ λ = π/4. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 22

23 Second Generation Curvelets Pick window functions W (r), V (t) which are both real, nonnegative, C s.t. [ ] 1 supp W 2, 2, supp V [ 1, 1] and W (2 j r) 2 = 1, j Z Define in polar coordinates V (t l) 2 = 1. l Z ˆγ (j,0,0) (r, ω) := 2 3j/4 W ( 2 j r ) V ( ) 2 j/2 ω, γ (j,l,k) ( ) := γ (j,0,0) ( Rθ(j,l,k) ( x(j,l,k) )), With appropriate modifications for j = 0 define the second-generation curvelet frame Γ 0 := { γ λ : λ Λ 0}. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 23

24 No big surprise: Lemma Γ 0 constitutes a system of parabolic molecules of arbitrary order. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 24

25 Similar Systems Hart Smith s parabolic frame Borup and Nielsen s construction Curvelet molecules All based on rotation. How about shearing? P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 25

26 Shearlets Index set { Λ σ := (ε, j, l, k) Z 2 Z 4 : ε {0, 1}, j 0, l = 2 j 2,, 2 j } 2, and the shearlet system Σ := {σ λ : λ Λ σ }, with σ (ε,0,0,k) ( ) = ϕ( k), σ (ε,j,l,k) ( ) = 2 3j/4 ψj,l,k ε ( D ε 2 S ε j l,j k ), j 1, where Da 0 = D a, Da 1 := diag( ( ) 1 l2 j/2 a, a), S l,j := and 0 1 (. Sl,j 1 = Sl,j) 0 P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 26

27 Shearlets Definition We call Σ a system of shearlet molecules of order (R, M, N 1, N 2 ) if the functions ϕ, ψ 0 j,l,k, ψ1 j,l,k satisfy ( M β ˆψε j,l,k (ξ 1, ξ 2 ) min 1, 2 s λ + ξ 1+ε + 2 sλ/2 ξ 2 ε ) ξ N 1 ξ 2 ε N 2 for every β N 2 with β R. Essentially same definition as curvelets, only with rotation replaced by shearing. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 27

28 Shearlet Molecules are Parabolic Molecules Define shearlet parametrization ( ) Φ σ (λ) = (s λ, θ λ, x λ ) := j, επ/2 + arctan( l2 j/2 ), (Sl ε ) 1 D ε 2 k, j λ Λ σ. Lemma (G-Kutyniok (2012)) Assume that the shearlet system Σ constitutes a system of shearlet molecules of order (R, M, N 1, N 2 ). Then Σ constitutes a system of parabolic molecules of order (R, M, N 1, N 2 ), associated to the parametrization (Λ σ, Φ σ ). P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 28

29 Shearlets This implies that the following systems constitute systems of parabolic molecules: bandlimited shearlets compactly supported shearlets shearlet molecules as introduced by Guo and Labate (2008). P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 29

30 Summary Theorem (G-Kutyniok (2012)) Known systems to date based on parabolic scaling are parabolic molecules (in particular curvelets and bandlimited and compactly supported shearlets). P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 30

31 Almost Orthogonality P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 31

32 Index Distance Function Definition For two points (s λ, θ λ, x λ ), (s µ, θ µ, x µ ) P define the parabolic distance ω (λ, µ) := 2 s λ s µ (1 + 2 s λ 0 d (λ, µ)), and d (λ, µ) := θ λ θ µ 2 + x λ x µ 2 + e λ, x λ x µ. where λ 0 = argmin(s λ, s µ ) and e λ = (cos(θ λ ), sin(θ λ )). Introduced by H. Smith (1998) and Candès-Demanet (2004) in the context of wave propagation. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 32

33 Almost Orthogonality Theorem (G-Kutyniok (2012)) Let (m λ ) λ Λ, (p µ ) µ M be two systems of parabolic molecules of order (R, M, N 1, N 2 ) with R 2N, M > 4N 5 4, N 1 2N + 3 4, N 2 2N. Then m λ, p µ ω ((s λ, θ λ, x λ ), (s µ, θ µ, x µ )) N. Similar (albeit nonquantitative) results for curvelet molecules in Candès-Demanet (2004) and for shearlets in Guo-Labate (2008). Proof by hard analysis. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 33

34 Applications P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 34

35 1. Sparsity Equivalence P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 35

36 Sparsity Equivalence Definition Let (m λ ) λ Λ and (p µ ) µ M be frame systems, and let 0 < p 1. Then (m λ ) λ Λ and (p µ ) µ M are sparsity equivalent in l p, if ( m λ, p µ ) λ Λ,µ Λ 0 <. lp l p Theorem (G-Kutyniok (2012)) Assume that (m λ ) λ Λ is a system of parabolic molecules associated with Λ of order (R, M, N 1, N 2 ) such that R 2 2 p, M > 4 2 p 5 4, N p + 3 4, N p. Then (m λ ) λ Λ is sparsity equivalent to Γ 0 in l p. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 36

37 Cartoon Approximation Theorem (G-Kutyniok (2012)) Assume that (m λ ) λ Λ is a system of parabolic molecules of order (R, M, N 1, N 2 ) such that (i) (m λ ) λ Λ constitutes a frame for L 2 (R 2 ), (ii) Λ is k-admissible for all k > 2, (iii) it holds that definition R 6, M > , N , N 2 6. Then the frame (m λ ) λ Λ possesses an almost best N-term approximation rate of order N 1+ε, ε > 0 arbitrary for cartoon images. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 37

38 2. Function Spaces P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 38

39 Function Spaces With the curvelet frame Γ 0 := { γ j,l,k : (j, l, k) Λ 0} introduced above, following Borup-Nielsen (2007), define for p, q, α > 1 the function spaces G α p,q given by the norm f G α p,q := ( 2 αj j 0,l k ) 1/p q f, γ j,l,k p 1/q. (2) P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 39

40 Equivalence Result Theorem (G-Kutyniok (2012)) Let Σ = {σ λ : λ Λ} be a frame for L 2 (R 2 ) with dual frame Σ = { σ λ : λ Λ}. Assume further that Σ, Σ are both parabolic molecules of arbitrary order. Then 1/p q 1/q 1/p q 1/q f G α p,q 2αj f, σ λ p 2αj f, σ λ p. j 0 λ Λ j j 0 λ Λ j Can be made quantitative. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 40

41 Localization In general it is hard to establish that dual frame consists of parabolic molecules, but it is possible to show localization results for dual frames: Theorem (G (2012, ACHA)) Let Σ = {σ λ : λ Λ} be a frame for L 2 (R 2 ) with dual frame Σ = { σ λ : λ Λ}. Assume that σ λ, σ µ Cω(λ, µ) N. Then there exists N + depending (in an explicit way) on N, C such that σ λ, σ µ ω(λ, µ) N+. go to proof P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 41

42 Summary Thorough understanding of ingredients of representation systems needed for sparse approximation General framework where results can directly be transferred without giving proofs for every single system Guideline in designing new representation systems Further work (jointly with G. Kutyniok, E. King): Continuous parameters, further properties (seperation, Radon inversion,...), higher dimensions,... How can group-property be relaxed in coorbit theory? P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 42

43 Literature P. Grohs and G. Kutyniok, Parabolic Molecules., Research report No , SAM, ETH Zurich, P. Grohs, Intrinsic Localization of Anisotropic Frames., Appl. Comp. Harmon. Anal., 2012, to appear. E. Candès and L. Demanet, The curvelet representation of wave propagators is optimally sparse., Comm. Pure Appl. Math., Vol. 58, pp , (2006). D. Labate, L. Mantovani and P. Negi, Shearlet Smoothness Spaces., Preprint, S. Dahlke, G. Kutyniok, G. Steidl and G. Teschke, Shearlet Coorbit Spaces and Associated Banach Frames., Appl. Comp. Harmon. Anal., Vol. 27, pp (2009). S. Dahlke, G. Steidl and G. Teschke, Shearlet Coorbit Spaces: compactly supported analyzing shearlets, traces and embeddings., J. Fourier Anal. Appl., Vol. 17, pp (2011). L. Borup and M. Nielsen, Frame decompositions of decomposition spaces., J. Fourier Anal. Appl., Vol. 13, pp (2007). M. Frazier, B. Jawerth and G. Weiss. Littlewood-Paley Theory and the Study of Function Spaces. AMS (1991). P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 43

44 The End Questions? P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 44

45 Definition Λ is k-admissible if sup µ Λ 0 λ Λ ω(λ, µ) k <. return to talk P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 45

46 G locally compact group. π : G U(H), the unitary transforms of a Hilbert space H, irreductible and square-integrable: ψ H s.t. G π(z)ψ, ψ 2 dµ L (z) <. Window g 0 B w := { h H : h, π(z)h W L (L, L 1,w (G)) } definition. Points (x λ ) λ Λ are U-dense ( λ Λ x λu = G) and relatively separated definition. Envelope H W R (L, L 1,w (G)) definition return to talk P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 46

47 w submultiplicative weight: w(xy) w(x)w(y). Then { W L (L, L 1,w (G)) := f L,loc : sup f (y) L 1,w (G) y xq where Q G is a relatively compact neighborhood of the identity. return }, P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 47

48 w submultiplicative weight: w(xy) w(x)w(y). Then { W R (L, L 1,w (G)) := f L,loc : sup f (y) L 1,w (G) y Qx where Q G is a relatively compact neighborhood of the identity. return }, P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 48

49 (x λ ) λ Λ separated if x λ Q x λ Q = for all λ λ and some compact neighborhood Q G of the identity. Relatively separated if finite union of separated sets. return P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 49

50 Banach Algebra Viewpoint Definition Define for N N the Banach Algebra B N := { A : l 2 (Λ) l 2 (Λ) : a λ,λ ω(λ, λ ) N for all λ, λ Λ } with norm A BN := inf { C 0 : a λ,λ C 0 ω(λ, λ ) N for all λ, λ Λ }. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 50

51 Moore-Penrose Pseudoinverse Definition Assume that A is a symmetric matrix with a spectral gap, i.e. σ 2 (A) 0 [A, B] 0 < A < B. Its Moore-Penrose pseudoinverse A + is defined by A + kera = 0 A + rana A rana = I rana. Lemma Consider a frame Σ = (σ λ ) λ Λ for a Hilbert space H. Then the Gram matrix (Σ, Σ) H is symmetric and possesses a spectral gap (A, B being the frame constants). Its Moore-Penrose pseudoinverse satisfies ) A + = ( Σ, Σ. H P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 51

52 Tool I: Landweber Iteration Lemma The matrix A + can be computed via a Landweber-type iteration by the formula A + = β k N ( I βa 2 ) k A, (3) where β = 2 A 2 +B 2. Furthermore, we have that σ 2 ( (I βa 2 ) k A ) [ Br k, Br k ], with r := B2 A 2 A 2 + B 2 < 1. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 52

53 Tool II: Boundedness of Multiplication Lemma Assume that A B N+L with L large enough (but completely explicit). Then we have for all B B N such that AB is symmetric the estimate AB BN C A BN+L B BN. Argument applies to very general classes of ω (also used e.g. for wavelets) Proof is very complicated... P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 53

54 Main Result Theorem Assume A B N+L as before. Then A + B N +, where ) log (1 + β A 2 C 2 N + BN+L = N 1 log(r) 1 P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 54

55 Proof We use the Landweber-type iteration and write A + = β k N A (k), where A (k) := ( I βa 2) k A. Use two different estimates for the entries of A. a (k) λ,λ P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 55

56 First Estimate We claim that A (k) BN (1 + β A 2 B N+L C 2 ) k A BN Proof by induction. Obviously true for k = 0. Assume k 1. Then A (k) = (1 βa 2 )A (k 1). By our multiplication theorem we have AA (k 1) BN C A BN+L A (k 1) BN and, using the same argument βa 2 A (k 1) BN βc 2 A 2 B N+L A (k 1) BN. Using the triangle inequality yields the claim. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 56

57 Second Estimate We claim that To see this, write a (k) λ,λ r k. a (k) λ,λ = (Ae λ, e λ) use Cauchy-Schwartz and spectral properties to obtain A (k) e λ e λ A (k) Br k. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 57

58 Summing Up We have a + λ,λ = k=0 a (k) λ,λ. Use above two estimates to get for any k 0 k 0 a + λ,λ (1 + β A BN+L C 2 ) k ω(λ, λ ) N + r k. k=0 k=k 0 +1 Geometric summation gives a + λ,λ (1 + β A BN+L C 2 ) k 0 ω(λ, λ ) N + r k 0. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 58

59 Final Step Put D := 1 + β A BN+L C 2. Want to find k 0, N + such that both D k 0 ω(λ, λ ) N and r k 0 can be bounded by ω(λ, λ ) N+. Estimate for r k 0 yields Estimate for D k 0 ω(λ, λ ) yields Eliminating k 0 yields This proves the theorem. k 0 = N + log(ω(λ, λ )) log(r) k 0 = (N N + ) log(ω(λ, λ )). log(d) N + = N(1 log(d) log(r) ) 1 P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 59

60 Frame Result Using the fact that the Gramian of the dual frame is the Moore-Penrose pseudoinverse of the Gramian of the primal frame we have shown the following: Theorem (G (2012)) Let Σ = {σ λ : λ Λ} be a frame for L 2 (R 2 ) with dual frame Σ = { σ λ : λ Λ}. Assume that σ λ, σ µ Cω(λ, µ) N. Then there exists N + depending (in an explicit way) on N, C such that σ λ, σ µ ω(λ, µ) N+. P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 60

61 Remarks N + depends on N, A BN+L and the spectrum of A. Dependence on A BN+L is annoying. Similar results exist for wavelet Riesz bases. Our results are a generalization which applies also to anisotropic frames. Localization is essential in the study of function spaces but also in numerical purposes. For instance, localization ensures fast matrix-vector multiplication and is also essential in the study of adaptive methods to solve PDEs. return to talk P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 61