Generalized Shearlets and Representation Theory Emily J. King - - PowerPoint PPT Presentation

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Generalized Shearlets and Representation Theory

Emily J. King

Laboratory of Integrative and Medical Biophysics National Institute of Child Health and Human Development National Institutes of Health Norbert Wiener Center University of Maryland

February Fourier Talks February 18, 2011

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Outline

1

Overview

2

Shearlets and reproducing groups

3

Shearlets in L2(Rk)

4

Reducibility and coorbit spaces

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Wavelet definition

Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : Rk → C, y ∈ Rk, and A ∈ GL(R, d) define the following (unitary)

• perators

Tyf(x) = f(x − y) and DAf(x) = | det A|−1/2f(A−1x). Let H be a locally compact Hausdorff topological group, and let π : H → GL(R, k) be a continuous homomorphism. Define G = Rk ⋊π H which has product (y, a) · (z, b) = (y + π(a)z, ab). One unitary representation ν of G (the wavelet representation) is ν(y, a) = TyDπ(a). We consider systems of the form {TyDAψ(x) : A ∈ M ≤ GL(R, d), y ∈ Rk}.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Wavelet definition

Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : Rk → C, y ∈ Rk, and A ∈ GL(R, d) define the following (unitary)

• perators

Tyf(x) = f(x − y) and DAf(x) = | det A|−1/2f(A−1x). Let H be a locally compact Hausdorff topological group, and let π : H → GL(R, k) be a continuous homomorphism. Define G = Rk ⋊π H which has product (y, a) · (z, b) = (y + π(a)z, ab). One unitary representation ν of G (the wavelet representation) is ν(y, a) = TyDπ(a). We consider systems of the form {TyDAψ(x) : A ∈ M ≤ GL(R, d), y ∈ Rk}.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Wavelet definition

Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : Rk → C, y ∈ Rk, and A ∈ GL(R, d) define the following (unitary)

• perators

Tyf(x) = f(x − y) and DAf(x) = | det A|−1/2f(A−1x). Let H be a locally compact Hausdorff topological group, and let π : H → GL(R, k) be a continuous homomorphism. Define G = Rk ⋊π H which has product (y, a) · (z, b) = (y + π(a)z, ab). One unitary representation ν of G (the wavelet representation) is ν(y, a) = TyDπ(a). We consider systems of the form {TyDAψ(x) : A ∈ M ≤ GL(R, d), y ∈ Rk}.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Wavelet definition

Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : Rk → C, y ∈ Rk, and A ∈ GL(R, d) define the following (unitary)

• perators

Tyf(x) = f(x − y) and DAf(x) = | det A|−1/2f(A−1x). Let H be a locally compact Hausdorff topological group, and let π : H → GL(R, k) be a continuous homomorphism. Define G = Rk ⋊π H which has product (y, a) · (z, b) = (y + π(a)z, ab). One unitary representation ν of G (the wavelet representation) is ν(y, a) = TyDπ(a). We consider systems of the form {TyDAψ(x) : A ∈ M ≤ GL(R, d), y ∈ Rk}.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Motivation

Commonly multidimensional data was analyzed using tensor products of 1-dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R2. Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation

• f the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang

2011] and much work has been done to integrate shearlet theory into wavelet theory.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Motivation

Commonly multidimensional data was analyzed using tensor products of 1-dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R2. Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation

• f the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang

2011] and much work has been done to integrate shearlet theory into wavelet theory.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Motivation

Commonly multidimensional data was analyzed using tensor products of 1-dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R2. Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation

• f the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang

2011] and much work has been done to integrate shearlet theory into wavelet theory.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Motivation

Commonly multidimensional data was analyzed using tensor products of 1-dimensional wavelet systems. Information about directional characteristics is desirable. Contourlets [Do/Vetterli 2003], curvelets and ridgelets [Cand` es/Guo 2002], bandlets [Mallat/Pennec 2005], wedgelets [Donoho 1999], and shearlets [Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005] have been suggested to solve the problem over R2. Shearlets have an associated group structure, coupled with a multi-resolution analysis [Kutyniok/Sauer 2009]. Thus, various algebraic tools may be exploited. Shearlets also resolve the wavefront set [Kutyniok/Labate 2009]; that is, shearlets can pick out non-smooth parts of a signal. There exists a digital implementation

• f the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang

2011] and much work has been done to integrate shearlet theory into wavelet theory.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Shearlets

Definition (Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005) Given ψ ∈ L2(R2), the continuous shearlet system is {TyD(SℓAa)−1ψ = a−3/4ψ(A−1

a S−1 ℓ (· − y)) : a > 0, ℓ ∈ R, y ∈ R2},

where Aa = a √a

• and Sℓ =

1 ℓ 1

• .
• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Sheared . . .

Figure: Linear Algebra and its Applications, 3rd ed., David C. Lay

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Sheared . . . sheep

Figure: Linear Algebra and its Applications, 3rd ed., David C. Lay

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Shearlet tiling

Figure: http://www.shearlet.org/theory.html

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Reproducing groups

Definition Assume that for all f ∈ L2(Rd), f =

• G

f, µ(g)φµ(g)φdg, (1) where G is a locally compact group, µ is a unitary representation, dx is a Haar measure of G, φ is a suitable window in L2(Rd), and the integral is interpreted weakly. We shall call a function φ which satisfies Eqn (1) a reproducing function for G. If such a φ exists, we call G a reproducing group.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Reproducing groups

Definition Assume that for all f ∈ L2(Rd), f =

• G

f, µ(g)φµ(g)φdg, (1) where G is a locally compact group, µ is a unitary representation, dx is a Haar measure of G, φ is a suitable window in L2(Rd), and the integral is interpreted weakly. We shall call a function φ which satisfies Eqn (1) a reproducing function for G. If such a φ exists, we call G a reproducing group.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Examples of reproducing functions

Example Let f ∈ L2(R). It is well known that (Inversion of the Short-Time Fourier Transform) f = f, TxMξφTxMξφdxdξ whenever

• |φ(x)|2dx = 1.

(Calder´

• n admissibility / inversion of the Continuous Wavelet

Transform) f = ∞

• f, TxDsφTxDsφdx ds

s2 whenever

∞ |ˆ φ(x)|2 dx

x = 1. Such a φ is called a continuous wavelet.

In [DeMari/Nowak 2001], all reproducing subgroups of R2 ⋊ Sp(1, R) are characterized. Further work in a series of papers [Cordero/DeMari/Nowak/Tobacco 2000s].

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Examples of reproducing functions

Example Let f ∈ L2(R). It is well known that (Inversion of the Short-Time Fourier Transform) f = f, TxMξφTxMξφdxdξ whenever

• |φ(x)|2dx = 1.

(Calder´

• n admissibility / inversion of the Continuous Wavelet

Transform) f = ∞

• f, TxDsφTxDsφdx ds

s2 whenever

∞ |ˆ φ(x)|2 dx

x = 1. Such a φ is called a continuous wavelet.

In [DeMari/Nowak 2001], all reproducing subgroups of R2 ⋊ Sp(1, R) are characterized. Further work in a series of papers [Cordero/DeMari/Nowak/Tobacco 2000s].

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Examples of reproducing functions

Example Let f ∈ L2(R). It is well known that (Inversion of the Short-Time Fourier Transform) f = f, TxMξφTxMξφdxdξ whenever

• |φ(x)|2dx = 1.

(Calder´

• n admissibility / inversion of the Continuous Wavelet

Transform) f = ∞

• f, TxDsφTxDsφdx ds

s2 whenever

∞ |ˆ φ(x)|2 dx

x = 1. Such a φ is called a continuous wavelet.

In [DeMari/Nowak 2001], all reproducing subgroups of R2 ⋊ Sp(1, R) are characterized. Further work in a series of papers [Cordero/DeMari/Nowak/Tobacco 2000s].

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Nota bene

The wavelet representation of R ⋊ R+, the affine group ax + b (the group underlying the wavelet transform), over L2(R) admits two irreducible subrepresentations. Over L2(Rk), it is infinitely many. Let L be a non-trivial proper subspace of L2(R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ ∈ L2(R) be defined as ˆ ψ(x) = x−3/8✶[0,1]. Then ∞ | ˆ ψ(x)|2 dx

x = ∞, so span f does not contain a continuous

wavelet.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Nota bene

The wavelet representation of R ⋊ R+, the affine group ax + b (the group underlying the wavelet transform), over L2(R) admits two irreducible subrepresentations. Over L2(Rk), it is infinitely many. Let L be a non-trivial proper subspace of L2(R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ ∈ L2(R) be defined as ˆ ψ(x) = x−3/8✶[0,1]. Then ∞ | ˆ ψ(x)|2 dx

x = ∞, so span f does not contain a continuous

wavelet.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Nota bene

The wavelet representation of R ⋊ R+, the affine group ax + b (the group underlying the wavelet transform), over L2(R) admits two irreducible subrepresentations. Over L2(Rk), it is infinitely many. Let L be a non-trivial proper subspace of L2(R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ ∈ L2(R) be defined as ˆ ψ(x) = x−3/8✶[0,1]. Then ∞ | ˆ ψ(x)|2 dx

x = ∞, so span f does not contain a continuous

wavelet.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Nota bene

The wavelet representation of R ⋊ R+, the affine group ax + b (the group underlying the wavelet transform), over L2(R) admits two irreducible subrepresentations. Over L2(Rk), it is infinitely many. Let L be a non-trivial proper subspace of L2(R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ ∈ L2(R) be defined as ˆ ψ(x) = x−3/8✶[0,1]. Then ∞ | ˆ ψ(x)|2 dx

x = ∞, so span f does not contain a continuous

wavelet.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Continuous shearlet group definition

Definition Consider the matrix group M = {SℓAa : ℓ ∈ Rk−1, a > 0} where Sℓ is the shearing matrix Sℓ =      1 ; i = j ℓj−1 ; 2 ≤ j ≤ k, i = 1 ; else   , and and Aa is the dilation matrix with diagonal a(2k−i−1)/[2(k−1)], Aa =               a . . . a(2k−3)/[2(k−1)] . . . ... . . . a1/2      , Theorem (K./Czaja 2010) Rk ⋊ M is a reproducing group under the wavelet representation (equivalently, metaplectic representation).

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Continuous shearlet group definition

Definition Consider the matrix group M = {SℓAa : ℓ ∈ Rk−1, a > 0} where Sℓ is the shearing matrix Sℓ =      1 ; i = j ℓj−1 ; 2 ≤ j ≤ k, i = 1 ; else   , and and Aa is the dilation matrix with diagonal a(2k−i−1)/[2(k−1)], Aa =               a . . . a(2k−3)/[2(k−1)] . . . ... . . . a1/2      , Theorem (K./Czaja 2010) Rk ⋊ M is a reproducing group under the wavelet representation (equivalently, metaplectic representation).

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Other generalizations

For k = 3 (Dahlke/Steidl/Teschke 2009) Aa =   a a1/3 a1/3   (resolution of hyperplane singularities, coorbit space theory) (Guo/Labate 2010) Aa =   a a1/2 a1/2   (optimally sparse reps) (K/Czaja 2010) Aa =   a a3/4 a1/2   (investigation of Wick calculus of Toeplitz operators)

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Other generalizations

For k = 3 (Dahlke/Steidl/Teschke 2009) Aa =   a a1/3 a1/3   (resolution of hyperplane singularities, coorbit space theory) (Guo/Labate 2010) Aa =   a a1/2 a1/2   (optimally sparse reps) (K/Czaja 2010) Aa =   a a3/4 a1/2   (investigation of Wick calculus of Toeplitz operators)

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Other generalizations

For k = 3 (Dahlke/Steidl/Teschke 2009) Aa =   a a1/3 a1/3   (resolution of hyperplane singularities, coorbit space theory) (Guo/Labate 2010) Aa =   a a1/2 a1/2   (optimally sparse reps) (K/Czaja 2010) Aa =   a a3/4 a1/2   (investigation of Wick calculus of Toeplitz operators)

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Definitions

Definition A unitary representation π mapping a locally compact group G with Haar measure to a Hilbert space H is called square-integrable if it is irreducible and there exists an f ∈ H such that

• G

|f, π(g)f|2 dg < ∞ Definition We define the Hardy spaces H±(Rk) = {f ∈ L2(Rk) : supp ˆ f ⊆ ˙ Rk

±}.

Clearly, L2(Rk) = H+(Rk) ⊕ H−(Rk).

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Definitions

Definition A unitary representation π mapping a locally compact group G with Haar measure to a Hilbert space H is called square-integrable if it is irreducible and there exists an f ∈ H such that

• G

|f, π(g)f|2 dg < ∞ Definition We define the Hardy spaces H±(Rk) = {f ∈ L2(Rk) : supp ˆ f ⊆ ˙ Rk

±}.

Clearly, L2(Rk) = H+(Rk) ⊕ H−(Rk).

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Reducibility

Theorem (Fabec/´ Olafsson 2003) Let Rk ⋊ M have wavelet representation ν. A non-zero closed subspace L of L2(Rk) is invariant under ν iff L = {f ∈ L2(Rk) : supp ˆ f ⊆ S} for some measurable, M t invariant S ⊆ Rk having positive measure. Moreover, this subpsace is irreducible if and only if S is egodic. Corollary (K. / Wojtek 2010) µ(TDS)k and µ(CSGk) over H±(Rk) are each square-integrable. N.B.: If negative dilations are allowed, these groups become irreducible.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Reducibility

Theorem (Fabec/´ Olafsson 2003) Let Rk ⋊ M have wavelet representation ν. A non-zero closed subspace L of L2(Rk) is invariant under ν iff L = {f ∈ L2(Rk) : supp ˆ f ⊆ S} for some measurable, M t invariant S ⊆ Rk having positive measure. Moreover, this subpsace is irreducible if and only if S is egodic. Corollary (K. / Wojtek 2010) µ(TDS)k and µ(CSGk) over H±(Rk) are each square-integrable. N.B.: If negative dilations are allowed, these groups become irreducible.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Co-orbit space theory

Theorem (K. 2010) For 0 < a0 < a1 and 0 < bi, 2 ≤ i ≤ k, let ψ be a Schwartz function such that supp ˆ ψ ⊆ ([−a1, −a0] ∪ [a0, a1]) × (×k

i=2[−bi, bi]). Then

• (CSG)k

|ψ, ν(g)ψ|dg < ∞. Corollary (K. 2010) Denote H1 = f ∈ L2(Rk) : f, ν(·)ψ ∈ L1((CSG)k), with anti-dual H∼

1 . H1 is non-empty, and moreover the coorbit spaces

(shearlet Besov spaces) SkCp = {f ∈ H∼

1 : ψ, ν(g)ψ ∈ Lp((CSG)k))}

are well-defined.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Co-orbit space theory

Theorem (K. 2010) For 0 < a0 < a1 and 0 < bi, 2 ≤ i ≤ k, let ψ be a Schwartz function such that supp ˆ ψ ⊆ ([−a1, −a0] ∪ [a0, a1]) × (×k

i=2[−bi, bi]). Then

• (CSG)k

|ψ, ν(g)ψ|dg < ∞. Corollary (K. 2010) Denote H1 = f ∈ L2(Rk) : f, ν(·)ψ ∈ L1((CSG)k), with anti-dual H∼

1 . H1 is non-empty, and moreover the coorbit spaces

(shearlet Besov spaces) SkCp = {f ∈ H∼

1 : ψ, ν(g)ψ ∈ Lp((CSG)k))}

are well-defined.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Continuous Shearlets

Theorem (K. 2010) Let C = {(ξ1ξ2, . . . ξk) ∈ Rk : |ξ1| ≥ 2 and

• ξi

ξ1

• ≤ 1 for 2 ≤ i ≤ k}.

Choose ψ1 ∈ L2(Rk) with ∞ | ˆ ψ1(aξ)|2 da

a = 1 for a.a. ξ ∈ R and

for 2 ≤ i ≤ k, ψi ∈ L2(Rk) with ψi = 1. Define ψ ∈ L2(Rk) by ˆ ψ = ˆ ψ1(ξ1) ˆ ψ2(ξ2/ξ1) . . . ˆ ψk(ξk/ξ1). Then ψ is a continuous generalized shearlet, and if further, supp ˆ ψ1 ⊆ [−2, −1/2] ∪ [1/2, 2] and for 2 ≤ i ≤ k, supp ˆ ψi ⊆ [−1, 1], then ψ is a generalized shearlet over L2(C)∨.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Continuous Shearlets

Theorem (K. 2010) Let C = {(ξ1ξ2, . . . ξk) ∈ Rk : |ξ1| ≥ 2 and

• ξi

ξ1

• ≤ 1 for 2 ≤ i ≤ k}.

Choose ψ1 ∈ L2(Rk) with ∞ | ˆ ψ1(aξ)|2 da

a = 1 for a.a. ξ ∈ R and

for 2 ≤ i ≤ k, ψi ∈ L2(Rk) with ψi = 1. Define ψ ∈ L2(Rk) by ˆ ψ = ˆ ψ1(ξ1) ˆ ψ2(ξ2/ξ1) . . . ˆ ψk(ξk/ξ1). Then ψ is a continuous generalized shearlet, and if further, supp ˆ ψ1 ⊆ [−2, −1/2] ∪ [1/2, 2] and for 2 ≤ i ≤ k, supp ˆ ψi ⊆ [−1, 1], then ψ is a generalized shearlet over L2(C)∨.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Continuous Shearlets

Theorem (K. 2010) Let C = {(ξ1ξ2, . . . ξk) ∈ Rk : |ξ1| ≥ 2 and

• ξi

ξ1

• ≤ 1 for 2 ≤ i ≤ k}.

Choose ψ1 ∈ L2(Rk) with ∞ | ˆ ψ1(aξ)|2 da

a = 1 for a.a. ξ ∈ R and

for 2 ≤ i ≤ k, ψi ∈ L2(Rk) with ψi = 1. Define ψ ∈ L2(Rk) by ˆ ψ = ˆ ψ1(ξ1) ˆ ψ2(ξ2/ξ1) . . . ˆ ψk(ξk/ξ1). Then ψ is a continuous generalized shearlet, and if further, supp ˆ ψ1 ⊆ [−2, −1/2] ∪ [1/2, 2] and for 2 ≤ i ≤ k, supp ˆ ψi ⊆ [−1, 1], then ψ is a generalized shearlet over L2(C)∨.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Translation-dilation-shearing definition

Definition For k ≥ 1, we define (TDS)k =

• At,ℓ,y =
• t−1/2Sℓ/2

t−1/2BySℓ/2 t1/2(tS−ℓ/2)

• : t > 0, ℓ ∈ Rk−1, y ∈ Rk
• where for y = t(y1, y2, · · · , yk) ∈ Rk.

By =            yk ; i = j = k yj ; i = k, j < k yi ; i < k, j = k ; else    

i,j

=      . . . y1 . . . y2 . . . . . . ... . . . y1 y2 . . . yk      , For k ≥ 2 and ℓ = t(ℓ1, ℓ2, . . . ℓk−1) ∈ Rk−1, Sℓ is the shearing matrix Sℓ =      1 ; i = j ℓj ; 1 ≤ i ≤ k − 1, j = k ; else  

i,j

=        1 . . . ℓ1 1 . . . ℓ2 . . . . . . ... . . . . . . 1 ℓk−1 . . . 1        For k = 1, we formally define Sℓ for ℓ ∈ R0 to be simply 1.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Representations

Proposition (K. and Czaja 2010) The mapping ν defined on each (TDS)k by ν(At,ℓ,y) = TyDt−1(tSℓ) is a unitary representation, which we shall call the wavelet representation. The mapping µ defined on each (TDS)k for f ∈ L2(Rk) by µ(At,ℓ,y)f(x) = tk/4e−iπByx,xf(t1/2S−ℓ/2x). is a unitary representation, which we shall call the metaplectic representation. Proposition (K. and Czaja 2010) ν and µ are equivalent representations.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Representations

Proposition (K. and Czaja 2010) The mapping ν defined on each (TDS)k by ν(At,ℓ,y) = TyDt−1(tSℓ) is a unitary representation, which we shall call the wavelet representation. The mapping µ defined on each (TDS)k for f ∈ L2(Rk) by µ(At,ℓ,y)f(x) = tk/4e−iπByx,xf(t1/2S−ℓ/2x). is a unitary representation, which we shall call the metaplectic representation. Proposition (K. and Czaja 2010) ν and µ are equivalent representations.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Representations

Proposition (K. and Czaja 2010) The mapping ν defined on each (TDS)k by ν(At,ℓ,y) = TyDt−1(tSℓ) is a unitary representation, which we shall call the wavelet representation. The mapping µ defined on each (TDS)k for f ∈ L2(Rk) by µ(At,ℓ,y)f(x) = tk/4e−iπByx,xf(t1/2S−ℓ/2x). is a unitary representation, which we shall call the metaplectic representation. Proposition (K. and Czaja 2010) ν and µ are equivalent representations.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Reproducing subgroup

Theorem (K. and Czaja 2010) f2

L2(Rk) =

• (TDS)k

|f, µ(At,ℓ,y)φ|2 dt tk+1 dydℓ for all f ∈ L2(Rk) if and only if 2−k =

• Rk

+

|φ(y)|2 dy y2k

k

=

• Rk

+

|φ(−y)|2 dy y2k

k

=

• R2

+

φ(y)φ(−y) dy y2k

k

.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Continuous shearlet group definition

Definition (CSG)k = {Sa,ℓ,y =

• Σ−ℓAa

ByΣ−ℓAa

tΣℓA −1 a

• : a > 0, ℓ ∈ Rk−1, y ∈ Rk},

where Σℓ is the shearing matrix      1 ; i = j

ℓk−i 2

; 1 ≤ i ≤ k − 1, j = k ; else  

i,j

=        1 . . .

ℓk−1 2

1 . . .

ℓk−2 2

. . . ... . . . . . . 1

ℓ1 2

. . . 1        , Aa is the dilation matrix

• a(1−i)/2(k−1)

; i = j ; else

• =

              1 . . . a−1/2(k−1) . . . ... . . . a−1/2      , and By is            y1 ; i = j = k yk+1−j ; i = k, j < k yk+1−i ; i < k, j = k ; else    

i,j

=      . . . yk . . . yk−1 . . . . . . ... . . . yk yk−1 . . . y1      .

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Reproducing subgroup

Theorem (K. and Czaja 2010) If φ ∈ L2(Rk), then f2

L2(Rk) =

• (CSG)k

|f, µ(Sa,ℓ,y)φ|2 da ak+1 dydℓ for all f ∈ L2(Rk) if and only if 1 2k =

• ˙

Rk

+

|φ(y)|2 dy y2k

k

=

• ˙

Rk

+

|φ(−y)|2 dy y2k

k

=

• ˙

Rk

+

φ(y)φ(−y) dy y2k

k

.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Admissibility condition

Theorem (Kutyniok/Labate 2007) Let G be a subset of GLk(R) and define Λ = {(M, y) : M ∈ G, y ∈ Rk}. Then for all f ∈ L2(Rk, f =

• Rn
• G

f, TyDMψψdλ(M)dy if and only if ∆(ψ)(ξ) =

• G

| ˆ ψ(tMξ)|2| det M|dλ(M) = 1 a.a. ξ. Proposition (K 2010) The Calder`

• n admissibility condition for (CSG)k is

∆(ψ)(ξ) =

• Rk−1

∞ | ˆ ψ(tSℓAaξ)|2a−k/4−1dadℓ = 1 a.a. ξ. We call such a ψ a continuous generalized shearlet.

• E. J. King

Generalized Shearlets

Overview Shearlets and reproducing groups Shearlets in L2(Rk) Reducibility and coorbit spaces

Admissibility condition

Theorem (Kutyniok/Labate 2007) Let G be a subset of GLk(R) and define Λ = {(M, y) : M ∈ G, y ∈ Rk}. Then for all f ∈ L2(Rk, f =

• Rn
• G

f, TyDMψψdλ(M)dy if and only if ∆(ψ)(ξ) =

• G

| ˆ ψ(tMξ)|2| det M|dλ(M) = 1 a.a. ξ. Proposition (K 2010) The Calder`

• n admissibility condition for (CSG)k is

∆(ψ)(ξ) =

• Rk−1

∞ | ˆ ψ(tSℓAaξ)|2a−k/4−1dadℓ = 1 a.a. ξ. We call such a ψ a continuous generalized shearlet.

• E. J. King

Generalized Shearlets