Lukas Sawatzki

Transcription

1 Philipps-University Marburg Classical Generalized Shearlet Summer School on Applied Harmonic Analysis Genova

2 Classical Generalized Shearlet Classical Generalized Shearlet

3 Why? - Goal: Analyze functions f L 2 (R d ) - Decompose these functions in suitable blocks - Wavelets: Isotropic blocks with drawbacks for anisotropic structures Classical Generalized Shearlet - Workaround: Develop new systems like ridgelets, curvelets, contourlets, etc. - Or:

4 Why? Advantages: - Anisotropic structure - Promising numerical results - Abstract coorbit theory is applicable Classical Generalized Shearlet

5 Let a R, s R d 1 and t R d, then for a mother-shearlet ψ the shearlets are defined as ψ (a,s,t) (x) := det A 1 2 ψ(a 1 a Ss 1 (x t)), Classical where and A a := ( a 0 T d 1 0 d 1 sign(a) a 1/d I d 1 S s := ( 1 s T 0 d 1 I d 1 ). ) Generalized Shearlet

6 Classical Let G be a group, H a Hilbert space, π : G U(H) a unitary representation and ψ an admissable vector. Then the voice transform of a function f in g G is defined as V π f (g) := f, π(g)ψ H. For a function space Y on G the coorbit spaces are then defined as Co(Y ) := {f S : f, π( )ψ H Y }. Classical Generalized Shearlet

7 Examples - Taking G = R R d with the wavelet transform ( ) x b π(a, b)ψ(x) = a 1/2 ψ a as well as specific weights yields the homogeneous Besov spaces Ḃs 1/2 1/p p,p (R d ). - The reduced Heisenberg group H d r = R d R d T with certain specifications leads to the modulation spaces M s p,p(r d ). Classical Generalized Shearlet - The full shearlet group R R d 1 R d gives us the so-called shearlet coorbit spaces.

8 Generalized Assume we have - a locally compact Hausdorff space X with measure dµ, - a family of functions F = {ψ x } x X forming a tight continuous frame for L 2 (R d ), i.e. for some A > 0. A f L 2 = X f, ψ x 2 dµ(x) Then, the associated voice transform is defined by Classical Generalized Shearlet V F : L 2 (R d ) L 2 (X, µ), V F f (x) := f, ψ x.

9 Reproducing kernel The reproducing kernel is defined via R F : X X C, R F (x, y) := V F (ψ y )(x) = ψ y, ψ x. We then have the reproducing identity R F (V F f ) = V F f Classical Generalized Shearlet for all f L 2 (R d ).

10 Kernel spaces For a kernel K the associated kernel operator is defined via K(f )(x) := K(x, y)f (y) dµ(y). Now define some kernel spaces via { K A q,m = max ess sup K(x, )m(x, ) L q, x X Lemma (Feise, S. (2017)) X ess sup K(, y)m(, y) L q y X Let K A q,m < for every q > 1, then we have the continuous embeddings }. Classical Generalized Shearlet for all 1 < p < r. K(L p,v (X, µ)) L r,v (X, µ)

11 Generalized Assume R F A q,m for all q > 1. Then, consider the spaces H τ,v := {f L 2 (R d ) : V F f L τ,v (X, µ)} for τ > 1 and with the norm f H τ,v := V F f L τ,v these spaces are Banach spaces densely embedded in L 2 (R d ) with F = {ψ x } x X H τ,v. Now we extend the voice transform by Classical Generalized Shearlet V F,τ f (x) = f, ψ x (Hτ,v ) H τ,v to the anti-dual (H τ,v ) L 2 (R d ).

12 Definition coorbit spaces Define the coorbit spaces as Co F,τ (L p,v ) := {f (H τ,v ) : V F,τ f L p,v (X, µ)} and equipped with the norm f Co F,τ (L p,v ) := V F,τ f L p,v these spaces are Banach spaces. Furthermore we have an isometric isomorphism Co F,τ (L p,v ) {F L p,v (X, µ) : R F F = F } Classical Generalized Shearlet induced by V F,τ.

13 Consider X = ( { } R d 1 R d) ( [ 1, 1] R d 1 R d) equipped with the measure F (x) dµ(x) := F (, s, t) dsdt X R d R d 1 1 da + F (a, s, t) dsdt. R d R d 1 1 a d+1 Furthermore define F = {ψ x } x X via ψ (,s,t) := Φ(Ss 1 ( t)), ψ (a,s,t) := det A a 1 2 Ψ(A 1 a Ss 1 ( t)), Classical Generalized Shearlet A a := ( a 0 T d 1 0 d 1 sign(a) a 1/d I d 1 ), S s := ( 1 s T 0 d 1 I d 1 ).

14 Shearlet Under certain assumptions this family imposes a tight frame for L 2 (R d ). Theorem (Feise, S. (2017)) Let Ψ L 1 L 2 be an admissable shearlet and let Φ L 1 L 2 be such that Rd 1 ˆΦ(y, σ) 2 y y d 1 dσ + ˆΨ(ξ 1, ξ) 2 R d 1 y ξ 1 d dξ 1 d ξ = 1 for almost every y R, then F is a continuous Parseval frame for L 2 (R d ), i.e. X f, ψ x 2 dµ(x) = f L 2 (R d ) 2, f L 2 (R d ). Classical Generalized Shearlet

15 Remember the reproducing kernel R F (x, y) = ψ y, ψ x. Classical Generalized Theorem (Feise, S. (2017)) Let ˆΦ and ˆΨ have specific compact supports, then we have R F A q,mv Shearlet for all q > 1. Hence, the generalized coorbit theory is applicable.

16 shearlet coorbit spaces The inhomogeneous shearlet transform is defined as SH F f (x) = f, ψ x. Then, for 1 p < and 1 < τ 2 with p < τ we define the inhomogeneous shearlet coorbit spaces as SC r F,τ,p := Co F,τ (L p,vr ) = {f (H τ,vr ) : SH F f L p,vr (X, µ)} and equipped with the norm Classical Generalized Shearlet f SC r F,τ,p := SH F f L p,vr (X, µ) these spaces are Banach spaces.

17 Under certain assumptions the following properties hold: - SC r F,τ,p SCr F,τ,q for p < q, - SC r F,τ,p SCs F,τ,p for r < s, - SC r F,τ,p = SCr G,τ,p if for the Gramian kernel it holds G(F, G) A 1,mvr, Classical Generalized Shearlet - SC r F,τ,p = SCr F,σ,p, for p < σ, τ <.

18 Thank you for your attention! Classical Generalized Shearlet

19 S. Dahlke, F. De Mari, P. Grohs, D. Labate. Harmonic and Applied Analysis. Applied and Numerical Harmonic Analysis, Birkhäuser, 2015, ISBN M. Fornasier, H. Rauhut. Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl., 11(3): , H. Rauhut, T. Ullrich. Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type. J. Funct. Anal., 260(11): , Classical Generalized Shearlet F. Feise, L. Sawatzki. shearlet coorbit spaces, preprint 2017.