On the Structure of Anisotropic Frames - univie.ac.at...features and which provide sparse...

On the Structure of Anisotropic FramesP. GrohsETH Zurich, Seminar for Applied Mathematics

ESI Modern Methods of Time-Frequency Analysis

Motivation Parabolic Molecules Almost Orthogonality Applications

Motivation

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


Geometric Multiscale Analysis

Multiscale representation systems which are sensitive to anisotropicfeatures and which provide sparse approximations thereof.

Such constructions are typically based on parabolic scaling and adirectional transform such as rotation.



Curvelets



Curvelets



Curvelets



Curvelets



Curvelets



Curvelets



A Zoo of Transforms

◦ Hart Smith (1998): Transform based on parabolic scaling fordiscretization of linear wave equations

◦ Candès and Dohoho (1999): Curvelets for sparserepresentations of cartoon images.Candès and Demanet (2004): Curvelets for sparserepresentations of wave propagators.

◦ Labate, Lim, Kutyniok and Weiss (2005): Shearlets, whererotation is replaced by shearing, ensuring uniform treatment ofcontinuous and digital transform. Initial construction bandlimited.

◦ Kittipoom, Kutyniok and Lim (2012), Dahlke, Steidl and Teschke(2011): Construction of compactly supported shearlet frames.



Approximation Properties

All these different systems possess optimally sparse approximationproperties for functions exhibiting singularities on lower-dimensionalmanifolds. This has been shown separatly for each system in a list ofpapers.

Question: Is it really necessary to go through these proofs for eachsingle system, or might a higher-level viewpoint be useful?



Goal

Meta-TheoremAll frame systems based on parabolic scaling (specifically curveletsand shearlets) posses the exact same approximation properties,whenever the generating functions are sufficiently smooth, as well aslocalized in space and frequency.

Concrete results, i.e., questions of type: given CN basis function withM directional vanishing moments, create frame from rotation/shearing+ anisotropic dilation, what are the approximation properties?



Main Ideas

Introduce class of function systems (mλ)λ∈Λ ⊂ L2(R2) whichincludes all known systems based on parabolic scaling.

Show that any two function systems within a class possessequivalent approximation properties.



Previous Work

◦ Curvelet/Shearlet molecules (Candès-Demanet 2004,Guo-Labate 2008)

, Unification of curvelet/shearlet-type systems based onrotation/shearing

/ Does not include shearlet/curvelet-based methods/ Nonquantitative – only basis functions with infinitely many

vanishing moments and superpolynomial decay in space ; notvalid in practice

◦ Coorbit molecules (Dahlke-Steidl-Teschke 2008-2012,Gröchenig-Piotrowski 2008), see Gabi Steidl’s talk Wed10:00-11:00.

, Powerful framework for systems built from group representation/ Does not include cone adapted shearlets or curvelet-type

constructions/ Abstract concept



Coorbit Molecules

Group G, representation π, window g, “well-spread” point-set(xλ)λ∈Λ ⊂ G, “envelope function” H more specific . A system(mλ)λ∈Λ ⊂ L2(Rd ) is a system of coorbit molecules if

|〈mλ, π(z)g〉| . H(z−1xλ) (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 thefamily (mλ)λ∈Λ and more general approach in terms of the types ofdirectional transforms allowed.



Parabolic Molecules



Basic Structure

System (mλ)λ∈Λ where every mλ is associated with a scale, adirection and a location.



Parametrization“Phase Space”

P := R+ × T× R2

DefinitionA parametrization consists of an index set Λ and a mappingΦΛ : Λ→ P, ΦΛ(λ) := (sλ, θλ, xλ).Canonical parametrization:

Λ0 :={

(j , l , k) ∈ Z4 : j ≥ 0, l = −2bj/2c, . . . ,2bj/2c},

Φ0((j , l , k)) := (j , πl2−bj/2c,R−θλD2−sλk)

Rθ :=

(cos(θ) sin(θ)− sin(θ) cos(θ)

)Da :=

(a 00√

a

).



Canonical Parametrization

Figure: Left: frequency tiling associated with canonical parametrization.Right: Translational grids.



Definition of Parabolic Molecules

DefinitionLet (Λ,ΦΛ) be a parametrization. A family (mλ)λ∈Λ is called a familyof parabolic molecules of order (R,M,N1,N2) if it can be written as

mλ(x) = 23sλ/4a(λ) (D2sλRθλ (x − xλ))

such that∣∣∣∂β a(λ)(ξ)∣∣∣ . min

(1,2−sλ + |ξ1|+ 2−sλ/2|ξ2|

)M〈|ξ|〉−N1 〈ξ2〉−N2

for all |β| ≤ R. The implicit constants are uniform over λ ∈ Λ.



A Closer Look

∣∣∣∂β a(λ)(ξ)∣∣∣ . min

(1,2−sλ + |ξ1|+ 2−sλ/2|ξ2|

)M〈|ξ|〉−N1 〈ξ2〉−N2

for all |β| ≤ R means that a(λ) ...... possesses M (almost) vanishing moments in the x1-direction... is of smoothness N1 + N2 in x2 and N1 in x1

... decays like 〈x〉−R in space.



Frequency Localization

Figure: Left: The weight function

min(

1, 2−sλ + |ξ1|+ 2−sλ/2|ξ2|)M

〈|ξ|〉−N1 〈ξ2〉−N2 for sλ = 3, M = 3,N1 = N2 = 2. Right: Approximate Frequency support of a correspondingmolecule mλ with θλ = π/4.



Second Generation CurveletsPick window functions W (r), V (t) which are both real, nonnegative,C∞ s.t.

supp W ⊆[

12,2], supp V ⊆ [−1,1]

and ∑j∈Z

W (2−j r)2 = 1,∑l∈Z

V (t − l)2 = 1.

Define in polar coordinates

γ(j,0,0)(r , ω) := 2−3j/4W(2−j r

)V(

2bj/2cω),

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-generationcurvelet frame

Γ0 :={γλ : λ ∈ Λ0} .



No big surprise:

LemmaΓ0 constitutes a system of parabolic molecules of arbitrary order.



Similar Systems

◦ Hart Smith’s parabolic frame◦ Borup and Nielsen’s construction◦ Curvelet molecules

All based on rotation. How about shearing?



ShearletsIndex set

Λσ :={

(ε, j , l , k) ∈ Z2 × Z4 : ε ∈ {0,1}, j ≥ 0, l = −2bj2 c, · · · ,2b

j2 c},

and the shearlet system

Σ := {σλ : λ ∈ Λσ} ,

with

σ(ε,0,0,k)(·) = ϕ(· − k), σ(ε,j,l,k)(·) = 23j/4ψεj,l,k(Dε

2j Sεl,j · −k), j ≥ 1,

where D0a = Da, D1

a := diag(√

a,a), Sl,j :=

(1 l2−bj/2c

0 1

)and

S1l,j =

(S0

l,j

)>.



Shearlets

DefinitionWe call Σ a system of shearlet molecules of order (R,M,N1,N2) if thefunctions ϕ, ψ0

j,l,k , ψ1j,l,k satisfy

|∂βψεj,l,k (ξ1, ξ2)| . min(

1,2−sλ + |ξ1+ε|+ 2sλ/2|ξ2−ε|)M〈|ξ|〉−N1〈ξ2−ε〉−N2

for every β ∈ N2 with |β| ≤ R.

Essentially same definition as curvelets, only with rotation replacedby shearing.



Shearlet Molecules are Parabolic Molecules

Define shearlet parametrization

Φσ(λ) = (sλ, θλ, xλ) :=(

j , επ/2 + arctan(−l2−bj/2c), (Sεl )−1 Dε2−j k

),

λ ∈ Λσ.

Lemma (G-Kutyniok (2012))

Assume that the shearlet system Σ constitutes a system of shearletmolecules of order (R,M,N1,N2). Then Σ constitutes a system ofparabolic molecules of order (R,M,N1,N2), associated to theparametrization (Λσ,Φσ).



Shearlets

This implies that the following systems constitute systems ofparabolic molecules:◦ bandlimited shearlets◦ compactly supported shearlets◦ shearlet molecules as introduced by Guo and Labate (2008).



Summary

Theorem (G-Kutyniok (2012))

Known systems to date based on parabolic scaling are parabolicmolecules (in particular curvelets and bandlimited and compactlysupported shearlets).



Almost Orthogonality



Index Distance Function

DefinitionFor two points (sλ, θλ, xλ), (sµ, θµ, xµ) ∈ P define the parabolicdistance

ω (λ, µ) := 2|sλ−sµ| (1 + 2sλ0 d (λ, µ)) ,

andd (λ, µ) := |θλ − θµ|2 + |xλ − xµ|2 + |〈eλ, xλ − xµ〉|.

where λ0 = argmin(sλ, sµ) and eλ = (cos(θλ), sin(θλ))>.

Introduced by H. Smith (1998) and Candès-Demanet (2004) in thecontext of wave propagation.



Almost Orthogonality


Let (mλ)λ∈Λ, (pµ)µ∈M be two systems of parabolic molecules of order(R,M,N1,N2) with

R ≥ 2N, M > 4N − 54, N1 ≥ 2N +

34, N2 ≥ 2N.

Then|〈mλ,pµ〉| . ω ((sλ, θλ, xλ), (sµ, θµ, xµ))−N

.

Similar (albeit nonquantitative) results for curvelet molecules inCandès-Demanet (2004) and for shearlets in Guo-Labate (2008).

Proof by “hard analysis”.



Applications



1. Sparsity Equivalence



Sparsity EquivalenceDefinitionLet (mλ)λ∈Λ and (pµ)µ∈M be frame systems, and let 0 < p ≤ 1. Then(mλ)λ∈Λ and (pµ)µ∈M are sparsity equivalent in `p, if∥∥∥(〈mλ,pµ〉)λ∈Λ,µ∈Λ0

∥∥∥`p→`p

<∞.


Assume that (mλ)λ∈Λ is a system of parabolic molecules associatedwith Λ of order (R,M,N1,N2) such that

R ≥ 22p, M > 4

2p− 5

4, N1 ≥ 2

2p

+34, N2 ≥ 2

2p.

Then (mλ)λ∈Λ is sparsity equivalent to Γ0 in `p.P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 36


Cartoon Approximation


Assume that (mλ)λ∈Λ is a system of parabolic molecules of order(R,M,N1,N2) such that

(i) (mλ)λ∈Λ constitutes a frame for L2(R2),(ii) Λ is k-admissible for all k > 2, definition

(iii) it holds that

R ≥ 6, M > 12− 54, N1 ≥ 6 +

34, N2 ≥ 6.

Then the frame (mλ)λ∈Λ possesses an almost best N-termapproximation rate of order N−1+ε, ε > 0 arbitrary for cartoon images.



2. Function Spaces



Function Spaces

With the curvelet frame

Γ0 :={γj,l,k : (j , l , k) ∈ Λ0}

introduced above, following Borup-Nielsen (2007), define forp,q, α > 1 the function spaces Gα

p,q given by the norm

‖f‖Gαp,q :=

∑j≥0,l

2αj

(∑k

|〈f , γj,l,k 〉|p)1/p

q1/q

. (2)



Equivalence Result

Theorem (G-Kutyniok (2012))Let Σ = {σλ : λ ∈ Λ} be a frame for L2(R2) with dual frameΣ = {σλ : λ ∈ Λ}. Assume further that Σ, Σ are both parabolicmolecules of arbitrary order. Then

‖f‖Gαp,q∼

∑j≥0

2αj

∑λ∈Λj

|〈f , σλ〉|p

1/p

q1/q

∼

∑j≥0

2αj

∑λ∈Λj

|〈f , σλ〉|p

1/p

q1/q

.

Can be made quantitative.



LocalizationIn general it is hard to establish that dual frame consists of parabolicmolecules, but it is possible to show localization results for dualframes:

Theorem (G (2012, ACHA))

Let Σ = {σλ : λ ∈ Λ} be a frame for L2(R2) with dual frameΣ = {σλ : λ ∈ Λ}. Assume that

|〈σλ, σµ〉| ≤ Cω(λ, µ)−N .

Then there exists N+ depending (in an explicit way) on N, C such that

|〈σλ, σµ〉| . ω(λ, µ)−N+

.

go to proof



Summary

◦ Thorough understanding of ingredients of representationsystems needed for sparse approximation

◦ General framework where results can directly be transferredwithout 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?



Literature

P. Grohs and G. Kutyniok, “Parabolic Molecules.”, Research report No. 2012-14, SAM, ETH Zurich , 2012.http://arxiv.org/abs/1206.1958

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.1472–1528, (2006).

D. Labate, L. Mantovani and P. Negi, “Shearlet Smoothness Spaces.”, Preprint, 2012.

S. Dahlke, G. Kutyniok, G. Steidl and G. Teschke, “Shearlet Coorbit Spaces and Associated Banach Frames.”, Appl. Comp. Harmon. Anal.,Vol. 27, pp. 195–214 (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. 1232–1255 (2011).

L. Borup and M. Nielsen, “Frame decompositions of decomposition spaces.”, J. Fourier Anal. Appl., Vol. 13, pp. 39–70 (2007).

M. Frazier, B. Jawerth and G. Weiss. “Littlewood-Paley Theory and the Study of Function Spaces.” AMS (1991).


http://arxiv.org/abs/1206.1958


The End

Questions?



DefinitionΛ is k -admissible if

supµ∈Λ0

∑λ∈Λ

ω(λ, µ)−k <∞.

return to talk



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 dµL(z) <∞.

Windowg 6= 0 ∈ Bw :=

{h ∈ H : 〈h, π(z)h〉 ∈ WL(L∞,L1,w (G))

}definition .

Points (xλ)λ∈Λ are U-dense (⋃λ∈Λ xλU = G) and relatively

separated definition .Envelope H ∈ WR(L∞,L1,w (G)) definition

return to talk



w submultiplicative weight: w(xy) ≤ w(x)w(y). Then

WL(L∞,L1,w (G)) :=

{f ∈ L∞,loc : sup

y∈xQ|f (y)| ∈ L1,w (G)

},

where Q ⊂ G is a relatively compact neighborhood of the identity.return



w submultiplicative weight: w(xy) ≤ w(x)w(y). Then

WR(L∞,L1,w (G)) :=

{f ∈ L∞,loc : sup

y∈Qx|f (y)| ∈ L1,w (G)

},

where Q ⊂ G is a relatively compact neighborhood of the identity.return



(xλ)λ∈Λ separated if xλQ∩ xλ′Q = ∅ for all λ 6= λ′ and some compactneighborhood Q ⊂ G of the identity. Relatively separated if finiteunion of separated sets. return



Banach Algebra Viewpoint

DefinitionDefine for N ∈ N the Banach Algebra

BN :={

A : l2(Λ)→ l2(Λ) : |aλ,λ′ | . ω(λ, λ′)−N for all λ, λ′ ∈ Λ}

with norm

‖A‖BN:= inf

{C0 : |aλ,λ′ | ≤ C0ω(λ, λ′)−N for all λ, λ′ ∈ Λ

}.



Moore-Penrose PseudoinverseDefinitionAssume that A is a symmetric matrix with a spectral gap, i.e.

σ2(A) ⊂ 0 ∪ [A,B] 0 < A < B.

Its Moore-Penrose pseudoinverse A+ is defined by

A+∣∣kerA = 0 A+

∣∣ranAA

∣∣ranA = I

∣∣ranA.

LemmaConsider a frame Σ = (σλ)λ∈Λ for a Hilbert space H. Then the Grammatrix (Σ,Σ)H is symmetric and possesses a spectral gap (A,B beingthe frame constants). Its Moore-Penrose pseudoinverse satisfies

A+ =(

Σ, Σ)H.



Tool I: Landweber Iteration

LemmaThe matrix A+ can be computed via a Landweber-type iteration bythe formula

A+ = β∑k∈N

(I − βA2)k

A, (3)

where β = 2A2+B2 . Furthermore, we have that

σ2

((I− βA2)k

A)⊆ [−Br k ,Br k ],

with

r :=B2 − A2

A2 + B2 < 1.



Tool II: Boundedness of Multiplication

LemmaAssume that A ∈ BN+L with L large enough (but completely explicit).Then we have for all B ∈ BN 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. forwavelets)

Proof is very complicated...



Main Result

TheoremAssume A ∈ BN+L as before. Then A+ ∈ BN+ , where

N+ = N

1−log(

1 + β‖A‖2BN+L

C2)

log(r)

−1



Proof

We use the Landweber-type iteration and write

A+ = β∑k∈N

A(k),

whereA(k) :=

(I − βA2)k

A.

Use two different estimates for the entries

a(k)λ,λ′

of A.



First EstimateWe claim that

‖A(k)‖BN ≤ (1 + β‖A‖2BN+L

C2)k‖A‖BN

Proof by induction.Obviously true for k = 0. Assume k − 1. Then

A(k) = (1− βA2)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

‖βA2A(k−1)‖BN ≤ βC2‖A‖2BN+L‖A(k−1)‖BN .

Using the triangle inequality yields the claim.P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 56


Second Estimate

We claim that|a(k)λ,λ′ | . r k .

To see this, write|a(k)λ,λ′ | = |(Aeλ,e′λ)|

use Cauchy-Schwartz and spectral properties to obtain

· · · ≤ ‖A(k)eλ‖‖e′λ‖ ≤ ‖A(k)‖ ≤ Br k .



Summing Up

We have

a+λ,λ′ =

∞∑k=0

a(k)λ,λ′ .

Use above two estimates to get for any k0

|a+λ,λ′ | .

k0∑k=0

(1 + β‖A‖BN+LC2)kω(λ, λ′)−N +

∞∑k=k0+1

r k .

Geometric summation gives

|a+λ,λ′ | . (1 + β‖A‖BN+LC

2)k0ω(λ, λ′)−N + r k0 .



Final StepPut D := 1 + β‖A‖BN+LC

2.Want to find k0,N+ such that both Dk0ω(λ, λ′)−N and r k0 can bebounded by ω(λ, λ′)−N+

.Estimate for r k0 yields

k0 = −N+ log(ω(λ, λ′))

log(r)

Estimate for Dk0ω(λ, λ′) yields

k0 = (N − N+)log(ω(λ, λ′))

log(D).

Eliminating k0 yields

N+ = N(1− log(D)

log(r))−1

This proves the theorem.P. Grohs ESI Modern Methods of Time-Frequency Analysis p. 59


Frame Result

Using the fact that the Gramian of the dual frame is theMoore-Penrose pseudoinverse of the Gramian of the primal frame wehave shown the following:

Theorem (G (2012))

Let Σ = {σλ : λ ∈ Λ} be a frame for L2(R2) with dual frameΣ = {σλ : λ ∈ Λ}. Assume that

|〈σλ, σµ〉| ≤ Cω(λ, µ)−N .

Then there exists N+ depending (in an explicit way) on N, C such that

|〈σλ, σµ〉| . ω(λ, µ)−N+

.



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 fastmatrix-vector multiplication and is also essential in the study ofadaptive methods to solve PDEs.

return to talk


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