Generalized Shearlets and Representation Theory

Transcription

1 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

2 Outline Overview 1 Overview 2 3 4

3 Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : R k C, y R k, and A GL(R, d) define the following (unitary) operators T y f(x) = f(x y) and D A f(x) = det A 1/2 f(a 1 x). Let H be a locally compact Hausdorff topological group, and let π : H GL(R, k) be a continuous homomorphism. Define G = R k 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) = T y D π(a). We consider systems of the form {T y D A ψ(x) : A M GL(R, d), y R k }.

4 Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : R k C, y R k, and A GL(R, d) define the following (unitary) operators T y f(x) = f(x y) and D A f(x) = det A 1/2 f(a 1 x). Let H be a locally compact Hausdorff topological group, and let π : H GL(R, k) be a continuous homomorphism. Define G = R k 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) = T y D π(a). We consider systems of the form {T y D A ψ(x) : A M GL(R, d), y R k }.

5 Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : R k C, y R k, and A GL(R, d) define the following (unitary) operators T y f(x) = f(x y) and D A f(x) = det A 1/2 f(a 1 x). Let H be a locally compact Hausdorff topological group, and let π : H GL(R, k) be a continuous homomorphism. Define G = R k 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) = T y D π(a). We consider systems of the form {T y D A ψ(x) : A M GL(R, d), y R k }.

6 Wavelet definition Definition (Haar 1909, Grossman / Morlet / Mallat / Daubechies 1980s, et multi al.) For f : R k C, y R k, and A GL(R, d) define the following (unitary) operators T y f(x) = f(x y) and D A f(x) = det A 1/2 f(a 1 x). Let H be a locally compact Hausdorff topological group, and let π : H GL(R, k) be a continuous homomorphism. Define G = R k 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) = T y D π(a). We consider systems of the form {T y D A ψ(x) : A M GL(R, d), y R k }.

7 Motivation Overview 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 [Candès/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 R 2. 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 of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory.

8 Motivation Overview 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 [Candès/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 R 2. 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 of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory.

9 Motivation Overview 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 [Candès/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 R 2. 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 of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory.

10 Motivation Overview 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 [Candès/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 R 2. 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 of the shearlet transform [Donoho/Kutyniok/Shahram/Zhuang 2011] and much work has been done to integrate shearlet theory into wavelet theory.

11 Shearlets Overview Definition (Guo/Kutyniok/Labate 2006 and Labate/Lim/Kutyniok/Weiss 2005) Given ψ L 2 (R 2 ), the continuous shearlet system is {T y D (Sl A 1ψ a) = a 3/4 ψ(a 1 a S 1 l ( y)) : a > 0, l R, y R 2 }, ( ) ( ) a 0 where A a = 1 l and S 0 a l =. 0 1

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

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

14 Shearlet tiling Overview Figure:

15 Reproducing groups Definition Assume that for all f L 2 (R d ), f = f, µ(g)φ µ(g)φdg, (1) G 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 L 2 (R d ), 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.

16 Reproducing groups Definition Assume that for all f L 2 (R d ), f = f, µ(g)φ µ(g)φdg, (1) G 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 L 2 (R d ), 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.

17 Examples of reproducing functions Example Let f L 2 (R). It is well known that (Inversion of the Short-Time Fourier Transform) f = f, T x M ξ φ T x M ξ φdxdξ whenever φ(x) 2 dx = 1. (Calderón admissibility / inversion of the Continuous Wavelet Transform) f = 0 f, Tx D s φ T x D s φdx ds s whenever 2 ˆφ(x) 2 dx 0 x = 1. Such a φ is called a continuous wavelet. In [DeMari/Nowak 2001], all reproducing subgroups of R 2 Sp(1, R) are characterized. Further work in a series of papers [Cordero/DeMari/Nowak/Tobacco 2000s].

18 Examples of reproducing functions Example Let f L 2 (R). It is well known that (Inversion of the Short-Time Fourier Transform) f = f, T x M ξ φ T x M ξ φdxdξ whenever φ(x) 2 dx = 1. (Calderón admissibility / inversion of the Continuous Wavelet Transform) f = 0 f, Tx D s φ T x D s φdx ds s whenever 2 ˆφ(x) 2 dx 0 x = 1. Such a φ is called a continuous wavelet. In [DeMari/Nowak 2001], all reproducing subgroups of R 2 Sp(1, R) are characterized. Further work in a series of papers [Cordero/DeMari/Nowak/Tobacco 2000s].

19 Examples of reproducing functions Example Let f L 2 (R). It is well known that (Inversion of the Short-Time Fourier Transform) f = f, T x M ξ φ T x M ξ φdxdξ whenever φ(x) 2 dx = 1. (Calderón admissibility / inversion of the Continuous Wavelet Transform) f = 0 f, Tx D s φ T x D s φdx ds s whenever 2 ˆφ(x) 2 dx 0 x = 1. Such a φ is called a continuous wavelet. In [DeMari/Nowak 2001], all reproducing subgroups of R 2 Sp(1, R) are characterized. Further work in a series of papers [Cordero/DeMari/Nowak/Tobacco 2000s].

20 Nota bene Overview The wavelet representation of R R +, the affine group ax + b (the group underlying the wavelet transform), over L 2 (R) admits two irreducible subrepresentations. Over L 2 (R k ), it is infinitely many. Let L be a non-trivial proper subspace of L 2 (R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ L 2 (R) be defined as ˆψ(x) = x 3/8 1 [0,1]. Then ˆψ(x) 2 dx 0 x wavelet. =, so span f does not contain a continuous

21 Nota bene Overview The wavelet representation of R R +, the affine group ax + b (the group underlying the wavelet transform), over L 2 (R) admits two irreducible subrepresentations. Over L 2 (R k ), it is infinitely many. Let L be a non-trivial proper subspace of L 2 (R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ L 2 (R) be defined as ˆψ(x) = x 3/8 1 [0,1]. Then ˆψ(x) 2 dx 0 x wavelet. =, so span f does not contain a continuous

22 Nota bene Overview The wavelet representation of R R +, the affine group ax + b (the group underlying the wavelet transform), over L 2 (R) admits two irreducible subrepresentations. Over L 2 (R k ), it is infinitely many. Let L be a non-trivial proper subspace of L 2 (R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ L 2 (R) be defined as ˆψ(x) = x 3/8 1 [0,1]. Then ˆψ(x) 2 dx 0 x wavelet. =, so span f does not contain a continuous

23 Nota bene Overview The wavelet representation of R R +, the affine group ax + b (the group underlying the wavelet transform), over L 2 (R) admits two irreducible subrepresentations. Over L 2 (R k ), it is infinitely many. Let L be a non-trivial proper subspace of L 2 (R). Then L contains a reproducing function for the Heisenberg group (STFT) but not necessarily for the affine group (CWT). For example, let ψ L 2 (R) be defined as ˆψ(x) = x 3/8 1 [0,1]. Then ˆψ(x) 2 dx 0 x wavelet. =, so span f does not contain a continuous

24 Continuous shearlet group definition Definition Consider the matrix group M = {S l A a : l R k 1, a > 0} where S l is the shearing matrix 1 ; i = j S l = l j 1 ; 2 j k, i = 1, and 0 ; else and A a is the dilation matrix with diagonal a (2k i 1)/[2(k 1)], a a (2k 3)/[2(k 1)]... 0 A a =..., a 1/2 Theorem (K./Czaja 2010) R k M is a reproducing group under the wavelet representation (equivalently, metaplectic representation).

25 Continuous shearlet group definition Definition Consider the matrix group M = {S l A a : l R k 1, a > 0} where S l is the shearing matrix 1 ; i = j S l = l j 1 ; 2 j k, i = 1, and 0 ; else and A a is the dilation matrix with diagonal a (2k i 1)/[2(k 1)], a a (2k 3)/[2(k 1)]... 0 A a =..., a 1/2 Theorem (K./Czaja 2010) R k M is a reproducing group under the wavelet representation (equivalently, metaplectic representation).

26 Other generalizations For k = 3 (Dahlke/Steidl/Teschke 2009) A a = a a 1/ a 1/3 (resolution of hyperplane singularities, coorbit space theory) a 0 0 (Guo/Labate 2010) A a = 0 a 1/2 0 (optimally sparse 0 0 a 1/2 reps) (K/Czaja 2010) A a = calculus of Toeplitz operators) a a 3/ a 1/2 (investigation of Wick

27 Other generalizations For k = 3 (Dahlke/Steidl/Teschke 2009) A a = a a 1/ a 1/3 (resolution of hyperplane singularities, coorbit space theory) a 0 0 (Guo/Labate 2010) A a = 0 a 1/2 0 (optimally sparse 0 0 a 1/2 reps) (K/Czaja 2010) A a = calculus of Toeplitz operators) a a 3/ a 1/2 (investigation of Wick

28 Other generalizations For k = 3 (Dahlke/Steidl/Teschke 2009) A a = a a 1/ a 1/3 (resolution of hyperplane singularities, coorbit space theory) a 0 0 (Guo/Labate 2010) A a = 0 a 1/2 0 (optimally sparse 0 0 a 1/2 reps) (K/Czaja 2010) A a = calculus of Toeplitz operators) a a 3/ a 1/2 (investigation of Wick

29 Definitions Overview 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 f, π(g)f 2 dg < Definition G We define the Hardy spaces H ± (R k ) = {f L 2 (R k ) : supp ˆf Ṙk ±}. Clearly, L 2 (R k ) = H + (R k ) H (R k ).

30 Definitions Overview 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 f, π(g)f 2 dg < Definition G We define the Hardy spaces H ± (R k ) = {f L 2 (R k ) : supp ˆf Ṙk ±}. Clearly, L 2 (R k ) = H + (R k ) H (R k ).

31 Reducibility Overview Theorem (Fabec/Ólafsson 2003) Let R k M have wavelet representation ν. A non-zero closed subspace L of L 2 (R k ) is invariant under ν iff L = {f L 2 (R k ) : supp ˆf S} for some measurable, M t invariant S R k 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 ± (R k ) are each square-integrable. N.B.: If negative dilations are allowed, these groups become irreducible.

32 Reducibility Overview Theorem (Fabec/Ólafsson 2003) Let R k M have wavelet representation ν. A non-zero closed subspace L of L 2 (R k ) is invariant under ν iff L = {f L 2 (R k ) : supp ˆf S} for some measurable, M t invariant S R k 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 ± (R k ) are each square-integrable. N.B.: If negative dilations are allowed, these groups become irreducible.

33 Co-orbit space theory Theorem (K. 2010) For 0 < a 0 < a 1 and 0 < b i, 2 i k, let ψ be a Schwartz function such that supp ˆψ ([ a 1, a 0 ] [a 0, a 1 ]) ( k i=2 [ b i, b i ]). Then (CSG) k ψ, ν(g)ψ dg <. Corollary (K. 2010) Denote H 1 = f L 2 (R k ) : f, ν( )ψ L 1 ((CSG) k ), with anti-dual H 1. H 1 is non-empty, and moreover the coorbit spaces (shearlet Besov spaces) are well-defined. S k C p = {f H 1 : ψ, ν(g)ψ L p ((CSG) k ))}

34 Co-orbit space theory Theorem (K. 2010) For 0 < a 0 < a 1 and 0 < b i, 2 i k, let ψ be a Schwartz function such that supp ˆψ ([ a 1, a 0 ] [a 0, a 1 ]) ( k i=2 [ b i, b i ]). Then (CSG) k ψ, ν(g)ψ dg <. Corollary (K. 2010) Denote H 1 = f L 2 (R k ) : f, ν( )ψ L 1 ((CSG) k ), with anti-dual H 1. H 1 is non-empty, and moreover the coorbit spaces (shearlet Besov spaces) are well-defined. S k C p = {f H 1 : ψ, ν(g)ψ L p ((CSG) k ))}

35 Continuous Shearlets Theorem (K. 2010) Let C = {(ξ 1 ξ 2,... ξ k ) R k : ξ 1 2 and ξi ξ 1 1 for 2 i k}. Choose ψ 1 L 2 (R k ) with ˆψ 0 1 (aξ) 2 da a = 1 for a.a. ξ R and for 2 i k, ψ i L 2 (R k ) with ψ i = 1. Define ψ L 2 (R k ) by Then ˆψ = ˆψ 1 (ξ 1 ) ˆψ 2 (ξ 2 /ξ 1 )... ˆψ k (ξ k /ξ 1 ). ψ is a continuous generalized shearlet, and if further, supp ˆψ 1 [ 2, 1/2] [1/2, 2] and for 2 i k, supp ˆψ i [ 1, 1], then ψ is a generalized shearlet over L 2 (C).

36 Continuous Shearlets Theorem (K. 2010) Let C = {(ξ 1 ξ 2,... ξ k ) R k : ξ 1 2 and ξi ξ 1 1 for 2 i k}. Choose ψ 1 L 2 (R k ) with ˆψ 0 1 (aξ) 2 da a = 1 for a.a. ξ R and for 2 i k, ψ i L 2 (R k ) with ψ i = 1. Define ψ L 2 (R k ) by Then ˆψ = ˆψ 1 (ξ 1 ) ˆψ 2 (ξ 2 /ξ 1 )... ˆψ k (ξ k /ξ 1 ). ψ is a continuous generalized shearlet, and if further, supp ˆψ 1 [ 2, 1/2] [1/2, 2] and for 2 i k, supp ˆψ i [ 1, 1], then ψ is a generalized shearlet over L 2 (C).

37 Continuous Shearlets Theorem (K. 2010) Let C = {(ξ 1 ξ 2,... ξ k ) R k : ξ 1 2 and ξi ξ 1 1 for 2 i k}. Choose ψ 1 L 2 (R k ) with ˆψ 0 1 (aξ) 2 da a = 1 for a.a. ξ R and for 2 i k, ψ i L 2 (R k ) with ψ i = 1. Define ψ L 2 (R k ) by Then ˆψ = ˆψ 1 (ξ 1 ) ˆψ 2 (ξ 2 /ξ 1 )... ˆψ k (ξ k /ξ 1 ). ψ is a continuous generalized shearlet, and if further, supp ˆψ 1 [ 2, 1/2] [1/2, 2] and for 2 i k, supp ˆψ i [ 1, 1], then ψ is a generalized shearlet over L 2 (C).

38

39 Translation-dilation-shearing definition Definition For k 1, we define (TDS) k { ( ) } t = A t,l,y = 1/2 S l/2 0 t 1/2 B y S l/2 t 1/2 ( t : t > 0, l R k 1, y R k S l/2 ) where for y = t (y 1, y 2,, y k ) R k. B y = y k ; i = j = k y j ; i = k, j < k y i ; i < k, j = k 0 ; else i,j y y 2 = y 1 y 2... y k, For k 2 and l = t (l 1, l 2,... l k 1 ) R k 1, S l is the shearing matrix l 1 1 ; i = j l 2 S l = l j ; 1 i k 1, j = k = ; else i,j l k For k = 1, we formally define S l for l R 0 to be simply 1.

40 Representations Proposition (K. and Czaja 2010) The mapping ν defined on each (TDS) k by ν(a t,l,y ) = T y D t 1 ( t S l ) is a unitary representation, which we shall call the wavelet representation. The mapping µ defined on each (TDS) k for f L 2 (R k ) by µ(a t,l,y )f(x) = t k/4 e iπ Byx,x f(t 1/2 S l/2 x). is a unitary representation, which we shall call the metaplectic representation. Proposition (K. and Czaja 2010) ν and µ are equivalent representations.

41 Representations Proposition (K. and Czaja 2010) The mapping ν defined on each (TDS) k by ν(a t,l,y ) = T y D t 1 ( t S l ) is a unitary representation, which we shall call the wavelet representation. The mapping µ defined on each (TDS) k for f L 2 (R k ) by µ(a t,l,y )f(x) = t k/4 e iπ Byx,x f(t 1/2 S l/2 x). is a unitary representation, which we shall call the metaplectic representation. Proposition (K. and Czaja 2010) ν and µ are equivalent representations.

42 Representations Proposition (K. and Czaja 2010) The mapping ν defined on each (TDS) k by ν(a t,l,y ) = T y D t 1 ( t S l ) is a unitary representation, which we shall call the wavelet representation. The mapping µ defined on each (TDS) k for f L 2 (R k ) by µ(a t,l,y )f(x) = t k/4 e iπ Byx,x f(t 1/2 S l/2 x). is a unitary representation, which we shall call the metaplectic representation. Proposition (K. and Czaja 2010) ν and µ are equivalent representations.

43 Reproducing subgroup Theorem (K. and Czaja 2010) f 2 L 2 (R k ) = f, µ(a t,l,y )φ 2 dt (TDS) k t k+1 dydl for all f L2 (R k ) if and only if 2 k = 0 = R k + R 2 + φ(y) 2 dy y 2k k = φ(y)φ( y) dy. yk 2k R k + φ( y) 2 dy y 2k k

44 Continuous shearlet group definition Definition ( Σ la (CSG) k = {S a,l,y = a 0 B yσ la t a Σ laa 1 where Σ l is the shearing matrix 1 ; i = j lk i 2 ; 1 i k 1, j = k 0 ; else ) : a > 0, l R k 1, y R k }, =..... l1 i,j lk 1 lk 2, A a is the dilation matrix ({ a (1 i)/2(k 1) ; i = j 0 ; else ) 0 a 1/2(k 1)... 0 = a 1/2, and B y is y 1 ; i = j = k y k+1 j ; i = k, j < k y k+1 i ; i < k, j = k 0 ; else y k y k 1 = i,j y k y k 1... y 1

45 Reproducing subgroup Theorem (K. and Czaja 2010) If φ L 2 (R k ), then f 2 L 2 (R k ) = f, µ(s a,l,y )φ 2 da (CSG) k a k+1 dydl for all f L2 (R k ) if and only if 1 2 k = 0 = Ṙ k + Ṙ k + φ(y) 2 dy y 2k k = φ(y)φ( y) dy. yk 2k Ṙ k + φ( y) 2 dy y 2k k

46 Admissibility condition Theorem (Kutyniok/Labate 2007) Let G be a subset of GL k (R) and define Λ = {(M, y) : M G, y R k }. Then for all f L 2 (R k, f = f, T y D M ψ ψdλ(m)dy R n G if and only if (ψ)(ξ) = ˆψ( t Mξ) 2 det M dλ(m) = 1 a.a. ξ. G Proposition (K 2010) The Calderòn admissibility condition for (CSG) k is (ψ)(ξ) = ˆψ( t S l A a ξ) 2 a k/4 1 dadl = 1 a.a. ξ. R k 1 0

47 Admissibility condition Theorem (Kutyniok/Labate 2007) Let G be a subset of GL k (R) and define Λ = {(M, y) : M G, y R k }. Then for all f L 2 (R k, f = f, T y D M ψ ψdλ(m)dy R n G if and only if (ψ)(ξ) = ˆψ( t Mξ) 2 det M dλ(m) = 1 a.a. ξ. G Proposition (K 2010) The Calderòn admissibility condition for (CSG) k is (ψ)(ξ) = ˆψ( t S l A a ξ) 2 a k/4 1 dadl = 1 a.a. ξ. R k 1 0