Radon transform intertwines shearlets and wavelets
aa r X i v : . [ m a t h . F A ] M a r RADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS
F. BARTOLUCCI, F. DE MARI, E. DE VITO, AND F. ODONE
Abstract.
We prove that the unitary affine Radon transform intertwines the quasi-regular representation of a class of semidirect products, built by shearlet dilation groupsand translations, and the tensor product of a standard wavelet representation with awavelet-like representation. This yields a formula for shearlet coefficients that involvesonly integral transforms applied to the affine Radon transform of the signal, therebyopening new perspectives in the inversion of the Radon transform. Introduction
The use of wavelets in signal analysis and computer vision has proved almost optimal forone-dimensional signals in many ways, and the mathematics behind classical wavelets hasreached a high degree of elaboration. In higher dimensions, however, the picture is lessclear and this partially explains the huge class of representations that has been introducedover the years to handle high dimensional problems, such as directional wavelets [1],ridgelets [2], curvelets [3], wavelets with composite dilations [4], contourlets [5], shearlets[6], reproducing groups of the symplectic group [7], Gabor ridge functions [7] and mocklets[8] – to name a few.Among them, shearlets stand out because of their ability to efficiently capture anisotropicfeatures, to provide optimal sparse representations, to detect singularities and to be stableagainst noise, see [9] for an overview and a complete list of references. From the purelymathematical perspective, their construction is based on the well-established theory ofsquare-integrable representations [10], just as wavelets are, and because of this manypowerful mathematical tools are available. As far as applications are concerned, their ef-fectiveness has been tested primarily in image processing, where many efficient algorithmshave been designed using them (see [9, 11] and the website for further details and references).Thus, in some sense, shearlets behave for high-dimensional signals as wavelets do for 1D-signal, and it is therefore natural to try to understand if the many strong connections area consequence of some general mathematical principle.The purpose of this paper is to address this issue, and to give a partial answer, showingthat the link between the shearlet transform and wavelets is the unitary Radon transformin affine coordinates, because it actually intertwines the shearlet representation with atensor product of two wavelet representations. This fact can be exploited to show thatby carefully choosing the mother shearlet it is possible to obtain the classical shearletcoefficients as a sequence of operations performed on the Radon transform of the signal,namely a one-dimensional wavelet transform, with respect to the “sliding” coordinate
Key words and phrases. shearlets; wavelets; Radon transform. that parametrizes all the hyperplanes parallel to a given one (for the two dimensionalcase see Fig. 1), followed by a convolution operator with a scale-dependent filter in thevariables of the hyperplane. As the shearlet transform admits an inversion formula, it isin principle possible to invert the Radon transform of a given signal by means of it andthe aforementioned operations.For two-dimensional signals, our results, which have been announced in [12], can bedescribed as we now explain. In order to formulate them precisely, we recall the definitionof the three main ingredients, namely wavelets, shearlets and the Radon transform. Thewavelet group is
R ⋊ R × with law ( b, a )( b ′ , a ′ ) = ( b + ab ′ , aa ′ ). The square integrablewavelet representation W acts on L ( R ) by W b,a ψ ( x ) = | a | − / ψ (cid:0) x − ba (cid:1) and the wavelet transform, defined by W ψ f ( b, a ) = h f, ψ b,a i , is a multiple of an isometryfrom L ( R ) to L ( R ⋊ R × ) provided that ψ satisfies the Calder´on condition, see (22)below. Next, denote by S the (parabolic) shearlet group, namely R ⋊ ( R ⋊ R × ) withmultiplication ( b, s, a )( b ′ , s ′ , a ′ ) = ( b + N s A a b ′ , s + | a | / s ′ , aa ′ )where A a = a (cid:20) | a | − / (cid:21) , N s = (cid:20) − s (cid:21) and where the vectors are understood as column vectors. The group S acts on L ( R ) viathe shearlet representation, namely S b,s,a f ( x ) = | a | − / f ( A − a N − s ( x − b )) . The shearlet transform is then S ψ f ( b, s, a ) = h f, S b,s,a ψ i , and is a multiple of an isometryprovided that an admissibility condition on ψ is satisfied [13, 14], see (17) below. Finally,the Radon transform in affine coordinates of a signal f ∈ L ( R ) is the function R aff f : R → C defined by R aff f ( v, t ) = Z R f ( t − vy, y ) d y, ( v, t ) ∈ R . An important fact is that it is possible to define a version of R aff as a unitary map on L ( R ). First, it is necessary to compose it with the Riesz-type operator I that we nowdescribe. Its natural domain is the dense subspace of L ( R ) D = n g ∈ L ( R ) : Z R | ξ | | b g ( ξ , ξ ) | d ξ d ξ < + ∞ o , where b g denotes the Fourier transform of g . The densely defined, self-adjoint unboundedoperator I : D → L ( R ) is defined by d ( I g )( ξ , ξ ) = | ξ | b g ( ξ , ξ ) , ( ξ , ξ ) ∈ R , i.e. a Fourier multiplier in the second variable. It is not hard to show that for all f inthe dense subspace of L ( R ) A = n f ∈ L ( R d ) ∩ L ( R ) : Z R | b f ( ξ , ξ ) | | ξ | d ξ d ξ < + ∞ o , ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 3 the Radon transform R aff f belongs to D and that the map f aff f from A to L ( R ) extends to a unitary map, denoted by Q , from L ( R ) onto itself.In the two dimensional case our main formula reads now Q S b,s,a f = (cid:0) W s, | a | / ⊗ I (cid:1) W (1 , v ) · b,a Q f (1)where the meaning of the dummy variable v is W (1 , v ) · b,a g ( v, t ) = | a | − g (cid:18) v, t − (1 , v ) · ba (cid:19) . Our second most important result is the formula S ψ f ( x, y, s, a ) = | a | − Z R W χ (cid:0) R aff f ( v, • ) (cid:1) ( x + vy, a ) φ (cid:16) v − s | a | / (cid:17) d v, (2)provided that f ∈ L ( R d ) ∩ L ( R d ) and ψ is of the form b ψ ( ξ , ξ ) = c ψ ( ξ ) c ψ (cid:16) ξ ξ (cid:17) . The 1 D -wavelet χ and the 1 D -filter φ are related to the shearlet admissible vector ψ bythe following relations c χ ( ξ ) = | ξ | c ψ ( ξ ) (3) φ ( ξ /ξ ) = c ψ ( ξ /ξ ) . (4)The first equality shows that 2 πχ = Hψ ′ is the Hilbert transform H of the weak deriv-ative of ψ .Equation (2) shows that for any signal f in L ( R d ) ∩ L ( R d ) the shearlet coefficients can becomputed by means of three classical trasforms. Indeed, in order to obtain S ψ f ( x, y, s, a )one can:a) compute the Radon transform R aff f ( v, t ) of the original signal f ;b) apply the wavelet transform with respect to the variable tG ( v, b, a ) = W χ (cid:0) R aff f ( v, · ) (cid:1) ( b, a ) , (5)where χ is given by (3);c) convolve the result with the scale-dependent filterΦ a ( v ) = φ (cid:18) − v | a | / (cid:19) , where φ is given by (4) and the convolution is computed with respect to the variable v , that is S ψ f ( x, y, s, a ) = ( G ( • , x + • y, a ) ∗ Φ a ) ( s ) . Finally, since S is a square-integrable representation, there is a reconstruction formula,namely f = Z S S ψ f ( x, y, s, a ) S x,y,s,a ψ d x d y d s d a | a | , (6) F. BARTOLUCCI, F. DE MARI, E. DE VITO, AND F. ODONE where the integral converges in the weak sense. Note that S ψ f depends on f only throughits Radon transform R aff f , see (2). The above equation allows to reconstruct an unknownsignal f from its Radon transform by computing the shearlet coefficients by means of (6).The fact that the Radon transform does play a prominent role in this circle of ideas is notnew. Indeed, it is known that ridgelets are constructed via wavelet analysis in the Radondomain [15], Gabor frames are defined as the directionally-sensitive Radon transforms[7], discrete shearlet frames are used to invert the Radon transform [16] and the Radontransform is at the root of the proof that shearlets are able to detect the wavefront set ofa 2D signal [17].Our contribution is to clarify this relation from the point of view of non-commutativeharmonic analysis. We are actually able to prove a rather general result, Theorem 3,which generalizes (1) and holds for the class of groups that were introduced by F¨uhr[18, 19] and that are known as shearlet dilation groups .The paper is organized as follows. In Section 2 we present in full detail all the variousingredients, namely the groups, the representations, the Radon transform and the unitaryextensions that need to be defined. In Section 3 we state and prove the main results.2. Preliminaries
Notation.
We briefly introduce the notation. We set R × = R \ { } . The Euclideannorm of a vector v ∈ R d is denoted by | v | and its scalar product with w ∈ R d by v · w . Forany p ∈ [1 , + ∞ ] we denote by L p ( R d ) the Banach space of functions f : R d → C , whichare p -integrable with respect to the Lebesgue measure d x and, if p = 2, the correspondingscalar product and norm are h· , ·i and k · k , respectively. The Fourier trasform is denotedby F both on L ( R d ) and on L ( R d ), where it is defined by b f ( ξ ) = F f ( ξ ) = Z R d f ( x )e − πi ξ · x d x, f ∈ L ( R d ) . If G is a locally compact group, we denote by L ( G ) the Hilbert space of square-integrablefunctions with respect to a left Haar measure on G . We denote the (real) general lineargroup of size d × d by GL( d, R ) and by T( d, R ) the closed subgroup of unipotent uppertriangular matrices.If H is a closed subgroup of GL( d, R ), the semidirect product G = R d ⋊ H is the product R d × H with group operation( b , h )( b , h ) = ( b + h [ b ] , h h ) , where b , b ∈ R d , h , h ∈ H and where h [ b ] is the natural linear action of the matrix h on the column vector b .2.2. Shearlet dilation groups.
In this section we introduce the groups in which we areinterested. This family includes the groups introduced by F¨uhr in [18, 19], and calledgeneralized shearlet dilation groups for the purpose of generalizing the standard shearletgroup introduced in [6, 20].
Definition 1. A shearlet dilation group H <
GL( d, R ) is a subgroup of the form H = SD ,where ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 5 (i) S is a Lie subgroup of T( d, R ) consisting of matrices of the form (cid:20) − t s B ( s ) (cid:21) with s ∈ R d − and B : R d − → T( d − , R ) a smooth map;(ii) D is the one-parameter subgroup of GL( d, R ) consisting of the diagonal matrices a diag(1 , | a | λ , . . . , | a | λ d − ) = a (cid:20) a ) (cid:21) (7)as a ranges in R × . Here ( λ , . . . , λ d − ) is a fixed vector in R d − .The group S is called the shearing sugroup of H and D is called the diagonal complementor scaling subgroup of H .Several observations are in order. First of all, if one requires the shearing subgroup S tobe Abelian, then one obtains the class introduced by F¨uhr, with a slightly more generaldefinition. This has inspired Definition 1.Since the map B is continuous, S is automatically connected, and hence by Theorem 3.6.2in [21], it is closed and simply connected. By construction the elements of H are of theform h s,a = h s, h ,a = a (cid:20) − t s Λ( a )0 B ( s )Λ( a ) (cid:21) . (8)Furthermore, since the diagonal matrices of GL( d, R ) normalize T( d, R ), then H is thesemidirect product of S and D .Finally, the assumption that S is a subgroup normalized by D forces the maps B and Λto satisfy some equalities. Indeed, since (cid:20) − t u B ( u ) (cid:21) (cid:20) − t v B ( v ) (cid:21) = (cid:20) − t ( v + t B ( v ) u )0 B ( u ) B ( v ) (cid:21) , then S is a group if and only if B (0) = I d − (9) B ( u ) B ( v ) = B ( v + t B ( v ) u ) (10) B ( u ) − = B ( − t B ( u ) − u ) (11)for every u, v ∈ R d − . Since (cid:20) a ) (cid:21) (cid:20) − t s B ( s ) (cid:21) (cid:20) a ) − (cid:21) = (cid:20) t (Λ( a ) − s )0 Λ( a ) B ( s )Λ( a ) − (cid:21) the compatibility of D with S is equivalent to asking for the following condition to holdfor all a = 0 and all s ∈ R d − :Λ( a ) B ( s )Λ( a ) − = B (Λ( a ) − s ) . (12)It follows that H is diffeomorphic as a manifold to R d − × R × , so that we can identify theelement h s,a with the pair ( s, a ). With this identification the product law amounts to( s, a )( s ′ , a ′ ) = (cid:0) Λ( a ) − s ′ + t B (Λ( a ) − s ′ ) s, aa ′ (cid:1) . (13) F. BARTOLUCCI, F. DE MARI, E. DE VITO, AND F. ODONE
We stress that, in general, S is not isomorphic as a Lie group to the additive Abeliangroup R d − , unless S is the standard shearlet group introduced in [20], see the examplesbelow. Remark.
It should be clear that a slightly larger class would be obtained by allowing fordiagonal matrices of the formsign( a ) diag( | a | µ , | a | µ , . . . , | a | µ d − ) , a ∈ R × . The case µ = 0, however, is uninteresting because any shearlet dilation group correspond-ing to this choice never admits admissible vectors [19, 22]. But then a simple change ofvariables permits to assume µ = 1, as we did, and to set λ j = µ j − Remark.
In [19] the authors introduce the notion of shearlet dilation group by meansof structural properties and then prove that in the case when S is Abelian they can beparametrized as in Definition 1.We now give three examples. If S is Abelian a full characterization is provided in [19],see also [22] for a connection with a suitable class of subgroups of the symplectic group. Example (The standard shearlet group) . A possible choice for B is the map B ( s ) = I d − ,which satisfies all the above properties. In this case, s h s, defines a group isomorphismbetween R d − and the Abelian group S .Clearly, any choice of the weights λ , . . . , λ d − is compatible with (12). In particular, ifwe choose as D the group of matrices A a = a (cid:20) | a | γ − I d − (cid:21) ⇐⇒ Λ( a ) = | a | γ − I d − a ∈ R × , where γ ∈ R is a fixed parameter, then we obtain the d -dimensional shearlet group,usually denoted S γ , and, often, the parameter γ is chosen to be 1 /d [20, 23]. Example (The Toeplitz shearlet group) . Another important example arises when B ( s )is the Toeplitz matrix B ( s ) = T (ˆ s ) = − s − s . . . − s d − − s − s ...... . . . . . . . . . ...... . . . 1 − s . . . . . . , (14)where ˆ s = t ( s , . . . , s d − ). It is easy to see that T (ˆ u ) T (ˆ v ) = T (ˆ u♯ ˆ v ) where(ˆ u♯ ˆ v ) i := u i + v i + X j + k = i v j u k , i = 1 , . . . d − λ k = kλ , k = 2 , . . . , d − λ . ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 7
Example (A non-Abelian shearlet dilation group) . The matrices g ( u , u , u ) = − u − u − u − u − u − u as u = ( u , u , u ) ranges in R give rise to a non-Abelian shearlet group S . Indeed, it iseasily checked that g ( u , u , u ) g ( v , v , v ) = g ( u + v , u + v − u v , u + v − u ( v + 12 v )) , a product which is not Abelian in the third coordinate. Evidently, B ( u ) = − u − u − u is a smooth function of u . The group S is isomorphic to the standard Heisenberg group,as is most clearly seen at the level of Lie algebra. Indeed, the Lie algebra of S is given bythe matrices X ( q, p, t ) = q p t q p , because X ( q, p, t ) = 0 and henceexp( X ( q, p, t )) = I + X ( q, p, t ) + 12 X ( q, p, t )= q p + q t + qp q p = g ( − q, − ( p + 12 q ) , − ( t + 12 qp )) . Further, [ X ( q, p, t ) , X ( q ′ , p ′ , t ′ )] = X (0 , , qp ′ − pq ′ )exhibits the Lie algebra of S as the three dimensional Heisenberg Lie algebra. A straight-forward calculation shows that for any choice of λ ∈ R the diagonal matricesΛ( a ) = | a | λ | a | λ | a | λ normalize B ( u ) because Λ( a ) B ( u )Λ( a ) − = B (Λ( a ) − u ). Conversely, these are easily seento be the only rank-one dilations that normalize the matrices B ( u ). In conclusion, thegroup D consisting of the matrices a (cid:20) a ) (cid:21) together with S give rise to the non-Abelian shearlet dilation group H = SD . It isworth observing that the dilations in D are not the standard dilations of the Heisenberg F. BARTOLUCCI, F. DE MARI, E. DE VITO, AND F. ODONE group. Indeed, the Lie algebra of D consists of the diagonal matrices A λ ( τ ) = diag( τ, ( λ +1) τ, (2 λ + 1) τ, (3 λ + 1) τ ) and[ A λ ( τ ) , X ( q, p, t )] = X ( − λτ q, − λτ p, − λτ t )shows that these homogeneous dilations are not the standard dilations of the HeisenbergLie algebra (see [25], p. 620).2.3. The shearlet representation and admissible vectors.
From now on we fix agroup G = R d ⋊ H where H is a shearlet group as in Definition 1 and we parametrize itselements as ( b, s, a ). By (13) we get that a left Haar measure of H isd h = | a | λ D − d s d a where λ D = λ + . . . + λ d − and d s , d a are the Lebesgue measures of R d − and R × . As aconsequence, a left Haar measure on G isd g = d b d h | det h s,a | = | a | − ( d +1) d b d s d a where d b is the Lebesgue measure on R d and the last equality holds true since | det h s,a | = | a | d + λ D . The quasi-regular representation of G on L ( R d ) is S b,s,a f ( x ) = | a | − d + λD f ( h − s,a ( x − b )) . (16)The next result generalizes Theorem 4.12 in [19] to the case when S is not Abelian. Theorem 2.
The representation S is square-integrable and its admissible vectors ψ arethe elements of L ( R d ) satisfying < C ψ = Z R d |F ψ ( ξ ) | | ξ | d d ξ < + ∞ , (17) where ξ = ( ξ , ξ ′ ) ∈ R × R d − . We recall that a unitary representation π of G acting on a Hilbert space H is square inte-grable if it is irreducible and if there exists a (non-zero) element ψ ∈ H , called admissible vector, such that the associated voice transform , i.e. the linear map f
7→ h f, π ( b, h ) ψ i ,takes values in L ( G ) and in such case it is a multiple of an isometry, denoted by W ψ : H → L ( G ). Proof of Theorem 2.
The proof is an immediate consequence of the following result dueto F¨uhr, see [26] and the references therein. The quasi-regular representation of R d ⋊ H is square integrable if and only if there exists a vector ξ ∈ R d such that(i) the dual orbit O ξ = { t hξ ∈ R d : h ∈ H } is open and it is of full measure,(ii) the stabilizer H ξ = { h ∈ H : t hξ = ξ } is compact,where saying that O ξ has full measure means that its complement has Lebsegue measurezero. In such case, a vector ψ is admissible if and only if Z H |F ψ ( t hξ ) | d h < + ∞ . (18) ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 9
In our setting, with the choice ξ = (1 , , . . . ,
0) we have that t h s,a ξ = t h ,at h s, . . . = a (cid:20) a ) s (cid:21) so that O ξ = R × × R d − , which is of full measure, and H ξ is trivial. Hence S is square-integrable.To compute the admissible vectors, notice that by (18) Z H |F ψ ( t hξ ) | d h = Z R d − × R × |F ψ ( a Λ( a ) s, a ) | | a | λ D − d s d a = Z R d − × R × |F ψ ( ξ , ξ ′ ) | | ξ | d d ξ d ξ ′ with the change of variables a = ξ and s = Λ( ξ ) − ξ ′ /ξ . (cid:3) Theorem 2 states the surprising fact that the admissibility condition is the same for allgeneralized shearlet dilation groups. A canonical choice is to assume that F ψ ( ξ , ξ ′ ) = F ψ ( ξ ) F ψ ( ξ ′ /ξ ) (19)where ψ ∈ L ( R ) satisfies Z R × |F ψ ( ξ ) | | ξ | d ξ < + ∞ . (20)and ψ ∈ L ( R d − ). However, other choices are available and, in particular, it is possibleto build shearlets with compact support in space [27]. We finally recall that, since therepresentation S is square-integrable, we have the weakly-convergent reproducing formula[28] f = 1 C ψ Z G S ψ f ( b, s, a ) S b,s,a ψ d b d s d a | a | d +1 . (21)2.4. Wavelet Transform.
We recall that the one-dimensional affine group W is R ⋊ R × endowed with the product ( b, a )( b ′ , a ′ ) = ( b + ab ′ , aa ′ )and left Haar measure | a | − d b d a . It acts on L ( R ) by means of the square-integrablerepresentation W b,a f ( x ) = | a | − f ( x − ba ) . The corresponding wavelet transform is W ψ : L ( R ) → L ( W ), given by W ψ f ( b, a ) = h f, W b,a ψ i , which is a multiple of an isometry provided that ψ ∈ L ( R ) satisfies the admissibilitycondition, namely the Calder´on equation,0 < Z R |F ψ ( ξ ) | | ξ | d ξ < + ∞ (22)and, in such a case, ψ is called an admissible wavelet. The quasi-regular representation of H . Consider now the shearlet dilation group H , with S and D its shearing and dilation subgroups, respectively (see Definition 1). Asmentioned above, we identify H with R d − × R × as manifolds and sometimes denote by( s, a ) the element h s,a of H . Recall that by (13) the product law is then( s, a )( s ′ , a ′ ) = (Λ( a ) − s ′ + t B (Λ( a ) − s ′ ) s, aa ′ ) . Observe that H acts naturally on R d and its (right) dual action is t h s,a (cid:20) v v (cid:21) = a (cid:20) v Λ( a )( t B ( s ) v − s ) (cid:21) . This implies that H acts naturally on P d − = ( R d \ { } ) / ∼ as well. By identifying R d − with { (1 , v ) : v ∈ R d − } / ∼ we get that H acts on R d − as t h s,a .v = Λ( a )( t B ( s ) v − s ) . Hence we can define the quasi-regular representation of H acting on L ( R d − ) by meansof V s,a f ( v ) = | a | λD f (Λ( a )( t B ( s ) v − s )) , where we recall that λ D = λ + · · · + λ d − . In general, V is not irreducible, but we canalways define the voice transform associated to a fixed vector ψ ∈ L ( R d − ), namely themapping V ψ : L ( R d − ) −→ C ( H ) defined by V ψ f ( s, a ) = h f, V s,a ψ i , where C ( H ) is the space of continuous functions on H . Example (The standard shearlet group, continued) . For the classical shearlet group S γ the shearlet representation on L ( R d ) becomes S γb,s,a f ( x ) = | a | − γ ( d − f ( A − a S − s ( x − b )) , (23)whereas the group H is the affine group R d − ⋊ R × in dimension d − V is thecorresponding wavelet representation V s,a f ( v ) = | a | ( d − γ − f (cid:18) v − s | a | − γ (cid:19) , (24)which is not irreducible unless d = 2. Furthermore, the voice transform can be written asconvolution operator V ψ f ( s, a ) = | a | ( d − γ − Z R d − f ( v ) ψ (cid:18) v − s | a | − γ (cid:19) d v = f ∗ Ψ a ( s )where Ψ a ( v ) = | a | ( d − γ − ψ (cid:18) − v | a | − γ (cid:19) . ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 11
The affine Radon transform.
In this section we recall the definition and themain properties of the Radon transform. Then we introduce the particular restriction ofthe Radon transform in which we are interested, the so-called affine Radon transform ,obtained by parametrizing the space of hyperplanes by affine coordinates.We first define the Radon transform on L ( R d ) by following the approach in [29], seealso [30] as a classical reference. Given f ∈ L ( R d ) its Radon transform is the function R f : ( R d \ { } ) × R → C defined by R f ( n, t ) = 1 | n | Z n · x = t f ( x ) d m ( x ) , (25)where m is the Euclidean measure on the hyperplane( n : t ) := { x ∈ R d : n · x = t } (26)and the equality (25) holds for almost all ( n, t ) ∈ ( R d \ { } ) × R . We add some comments.Definition (25) makes sense since, given n ∈ R d \ { } Fubini theorem gives that Z R d | f ( x ) | d x = Z R (cid:18)Z n · x = t | f ( x ) | d m ( x ) (cid:19) d t < + ∞ , so that for almost all t ∈ R the integral R n · x = t | f ( x ) | d m ( x ) is finite and R ( n, t ) is welldefined.Furthermore, each pair ( n, t ) ∈ ( R d \ { } ) × R defines the hyperplane ( n : t ) by meansof (26). Clearly, the correspondence between parameters ( n, t ) and hyperplanes is notbijective. Indeed ( n ′ , t ′ ) and ( n, t ) determine the same hyperplane if and only if thereexists λ ∈ R × such that n ′ = λn and t ′ = λt and this equivalence relation motivates thenotation ( n : t ) for the hyperplane in (26). Because of the factor 1 / | n | in (25), R f is apositively homogenous function of degree − i.e. for all λ ∈ R × R f ( λn, λt ) = | λ | − R f ( n, t ) . (27)This means that R f is completely defined by choosing a representative ( n, t ) for eachhyperplane ( n : t ), i.e. by choosing a suitable system of coordinates on the affine Grass-mannian { hyperplanes of R d } ≃ P d − × R . The canonical choice [30] is given by parametrizing P d − with its two-fold covering S d − ,where S d − is the unit sphere in R d .We are interested in another restriction of the Radon transform. For all v ∈ R d − set t n ( v ) = (1 , t v ). Definition 3.
Given f ∈ L ( R d ), the affine Radon transform of f is the function R aff f : R d − × R → C given by R aff f ( v, t ) = R f ( n ( v ) , t )= 1 p | v | Z n ( v ) · x = t f ( x )d m ( x ) = Z R d − f ( t − v · y, y )d y. (28) Remark.
The transform R aff is obtained from R by parametrizing the projective space P d − with affine coordinates. Indeed, the map ( v, t ) ( n ( v ) : t ) is a diffeomorphism of π xy n ( v ) n ( v ) · ( x, y ) = t ( t, Figure 1. space of hyperplanes parametrized by affine coordinates (2-dimensional case) R d − × R onto the open subset U = { ( n : t ) : ∃ λ ∈ R × s.t. λn ∈ π } , where π = { n ( v ) : v ∈ R d − } . The complement of U is the set of horizontal hyperplanes ,those for which the normal vector has the first component equal to zero (see Figure 1 forthe 2-dimensional case). The set of pairs ( v, t ) such that ( n ( v ) : t ) U is negligible, sothat R aff f completely defines R f . In A we recall the relation between the affine Radontransform and the usual Radon transform in polar coordinates.The next proposition, whose proof can be found in [29], summarizes the behaviour of theRadon transform under affine linear actions. The translation and dilation operators acton a function f : R d → C as T b f ( x ) = f ( x − b ) , D A f ( x ) = | det A | − f (cid:0) A − x (cid:1) , respectively, for b ∈ R d and A ∈ GL( d, R ). Both operators map each L p ( R d ) onto itselfand D A is normalized to be an isometry on L ( R d ). Proposition 4.
Given f ∈ L ( R d ) , the following properties hold true:(i) R T b f ( n, t ) = R f ( n, t − n · b ) , for all b ∈ R d ;(ii) R D A f ( n, t ) = R f ( t An, t ) , for all A ∈ GL ( d, R ) . We now state a crucial result in Radon transform theory in its standard version. Belowwe prove two variations that are taylored to our setting but are also of some independentinterest.
Proposition 5 (Fourier slice theorem, 1) . For any f ∈ L ( R d ) F ( R f ( n, · ))( τ ) = F f ( τ n ) . for all n ∈ R d \ { } and all τ ∈ R . ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 13
Here the Fourier transform on the right hand side is in R d , whereas the operator F onthe left hand side is 1-dimensional and acts on the variabile t . We repeat this slight abuseof notation in other formulas below.In the next formulation, written for the affine Radon transform, the function f to which R aff is applied is taken in L ( R d ) ∩ L ( R d ). Proposition 6 (Fourier slice theorem, 2) . Define ψ : R d − × ( R \ { } ) → R d by ψ ( v, τ ) = τ n ( v ) . For every f ∈ L ( R d ) ∩ L ( R d ) there exists a negligible set E ⊆ R d − such that forall v E the function R aff f ( v, · ) is in L ( R ) and satisfies R aff f ( v, · ) = F − [ F f ◦ ψ ( v, · )] . (29) Proof.
By Proposition 5 we know that for all v ∈ R d − the affine Radon transform R aff f ( v, · ) is in L ( R ) and satisfies F ( R aff f ( v, · ))( τ ) = F f ◦ ψ ( v, τ ) , τ ∈ R . We start by proving that the function τ
7→ F f ◦ ψ ( v, τ ) is in L ( R ), that is Z R |F f ◦ ψ ( v, τ ) | d τ < + ∞ . The map ψ : R d − × ( R \ { } ) → R d , defined by ψ ( v, τ ) = τ n ( v ), is a diffeomorphismonto the open set V = { ξ ∈ R d : ξ = 0 } with Jacobian J ψ ( v, τ ) = τ d − . By hypothesiswe know that k f k = Z R d |F f ( ξ ) | d ξ = Z R d − Z R |F f ◦ ψ ( v, τ ) | | τ | d − d τ d v < + ∞ , so that there exists a negligible set E ⊆ R d − such that C f := Z R |F f ◦ ψ ( v, τ ) | | τ | d − d τ < + ∞ for all v E . Therefore, for all v E it holds Z R |F f ◦ ψ ( v, τ ) | d τ = Z | τ |≤ |F f ◦ ψ ( v, τ ) | d τ + Z | τ | > | τ | d − | τ | d − |F f ◦ ψ ( v, τ ) | d τ ≤ kF f k ∞ + Z R |F f ◦ ψ ( v, τ ) | | τ | d − d τ ≤ k f k + C f < + ∞ . Hence the function t
7→ R aff f ( v, t ) is in L ( R d ) ∩ L ( R d ) and (29) follows by the Fourierinversion formula in L ( R ). (cid:3) It is possible to extend the affine Radon transform R aff to L ( R d ) as a unitary map.However, this raises some technical issues, that are addressed in the next section.2.7. The unitary extension.
Consider the subspace D = (cid:8) f ∈ L ( R d − × R ) : Z R d − × R | τ | d − |F f ( ξ, τ ) | d ξ d τ < + ∞ (cid:9) of L ( R d − × R ) and define the operator I : D → L ( R d − × R ) by F I f ( ξ, τ ) = | τ | d − F f ( ξ, τ ) , (30) a Fourier multiplier with respect to the last variable. Since τ
7→ | τ | d − is a strictly positive(almost everywhere) Borel function on R , the spectral theorem for unbounded operators,see Theorem VIII.6 of [31], shows that D is dense and that I is a positive self-adjointinjective operator. Remark.
The operator I is related to the inverse of the Riesz potential with exponent( d − / L ( R ). Indeed, if ψ ∈ L ( R d − ) and if ψ ∈ L ( R ) is such that Z R | τ | d − |F ψ ( τ ) | d τ < + ∞ , then ψ ⊗ ψ ∈ D , because Z R d − × R | τ | d − |F ( ψ ⊗ ψ )( ξ, τ ) | d ξ d τ = Z R d − |F ψ ( ξ ) | d ξ Z R | τ | d − |F ψ ( τ ) | d τ < + ∞ , so that I ( ψ ⊗ ψ ) = ψ ⊗ I ψ , where I is the inverse of the standard Riesz potential defined by F I ψ ( τ ) = | τ | d − F ψ ( τ ) . (31)Furthermore, D is invariant under translations and dilations by matrices of the form A = (cid:20) A v a (cid:21) , (32)where A ∈ GL( d − , R ) , v ∈ R d − , a ∈ R × . Lemma 7.
For all b ∈ R d and A as in (32) it holds I T b = T b I , I D A = | a | − d − D A I . (33) Proof.
The first of relations (33) is a consequence of the fact that F T b f ( ξ, τ ) = e − πib · ξ F f ( ξ, τ )for all f ∈ L ( R d − × R ). Precisely, for all f ∈ D we have that F I T b f ( ξ, τ ) = | τ | d − F T b f ( ξ, τ )= | τ | d − e − πib · ξ F f ( ξ, τ )= e − πib · ξ F I f ( ξ, τ )= F T b I f ( ξ, τ ) , whence I T b = T b I . The second follows from F D A f ( ξ, τ ) = F f ( t A ( ξ, τ )). Indeed, for all f ∈ D F I D A f ( ξ, τ ) = | τ | d − F D A f ( ξ, τ )= | τ | d − F f ( t A ( ξ, τ ))= | τ | d − F f ( t A ξ + τ v, aτ )= | τ | d − | aτ | − d − F I f ( t A ξ + τ v, aτ )= | a | − d − F D A I f ( ξ, τ ) . ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 15
This proves (33). (cid:3)
The space D becomes a pre-Hilbert space with respect to the scalar product h f, g i D = hI f, I g i = Z R d − × R | τ | d − F ( f ( v, · ))( τ ) F ( g ( v, · ))( τ ) d v d τ. (34)Furthermore, || f || D = h f, f i D = hI f, I f i = ||I f || , for all f ∈ D . Hence I is an isometric operator from D , with the new scalar product (34),to L ( R d − × R ). Since I is self-adjoint and injective, Ran( I ) is dense in L ( R d − × R ).Hence, by standard arguments, it extends uniquely to a unitary operator, denoted I ,from the completion H of D onto L ( R d − × R ).To extend R aff to L ( R d ) as a unitary operator, note that, by Proposition 5 with n = n ( v ),the affine Radon transform of f ∈ L ( R d ) ∩ L ( R d ) belongs to L ( R d − × R ) if and onlyif is finite the integral Z R d − × R |R aff f ( v, t ) | d v d t = Z R d − Z R |F ( R aff f ( v, · ))( τ ) | d τ d v = Z R d − × R |F f ( τ, τ v ) | d τ d v = Z R d |F f ( ξ ) | | ξ | d − d ξ, where ξ is the first component of the vector ξ ∈ R d . Therefore requiring that R aff f belongs to L ( R d − × R ) is equivalent to Z R d |F f ( ξ ) | | ξ | d − d ξ < + ∞ . We denote by A = { f ∈ L ( R d ) ∩ L ( R d ) : Z R d |F f ( ξ ) | | ξ | d − d ξ < + ∞} , which is dense in L ( R d ) since it contains the functions whose Fourier transform is smoothand has compact support disjoint from the hyperplane ξ = 0. By definition of A , R aff f ∈ L ( R d − × R ) for all f ∈ A .We shall need a suitable formulation of the main result in Radon transform theory, namelythe following version of Theorem 4.1 in [30]. For the sake of completeness we include theproof in B. Theorem 8.
The affine Radon transform extends to a unique unitary operator from L ( R d ) onto H , denoted with R and, hence, Q = I R is a unitary operator from L ( R d ) onto L ( R d − × R ) . As mentioned above, we need yet another generalization of the Fourier slice theorem(Proposition 5). We think that it is perhaps known, but we could not locate it in theliterature. The proof is given in B.
Proposition 9 (Fourier slice theorem, 3) . For all f ∈ L ( R d ) F ( Q f ( v, · ))( τ ) = | τ | d − F f ( τ n ( v )) (35) for almost every ( v, τ ) ∈ R d − × R . The Intertwining Theorem and its consequences
The main Theorem.
We recall that the group G is the semidirect product G = R d ⋊ H where H = SD is the shearlet dilation group, S is the shearing subgroup and D the scaling subgroup of H , as in Definition 1. Each element in G is parametrized by atriple ( b, s, a ) ∈ R d × R d − × R × and S b,s,a is as in (16). Theorem 10.
The unitary operator Q intertwines the shearlet representation with thetensor product of two unitary representations, precisely Q S b,s,a f ( v, t ) = ( V s, a ⊗ W n ( v ) · b,a ) Q f ( v, t ) (36) for every f ∈ L ( R d ) .Proof. By density, it is enough to prove the equality on A . We shall use throughoutthe fact that R aff f ∈ D for every f ∈ A and that D is invariant under all translationsand under the dilations described in (32). Since for all ( b, s, a ) ∈ G it holds ( b, s, a ) =( b, , , s, , , a ), it is sufficient to prove the equality for each of the three factors.For f ∈ A and b ∈ R d we have R aff S b, , f ( v, t ) = R aff T b f ( v, t )= R T b f ( n ( v ) , t )= R f ( n ( v ) , t − n ( v ) · b )= R aff f ( v, t − n ( v ) · b )= (I ⊗ W n ( v ) · b, ) R aff f ( v, t ) . Since I commutes with translations, I ⊗ W n ( v ) · b, = T (0 ,n ( v ) · b ) implies IR aff S b, , f ( v, t ) = I (I ⊗ W n ( v ) · b, ) R aff f ( v, t ) = (I ⊗ W n ( v ) · b, ) IR aff f ( v, t ) . For f ∈ A and a ∈ R × we have R aff S , ,a f ( v, t ) = | a | d + λD R aff D h ,a f ( v, t )= | a | d + λD R D h ,a f ( n ( v ) , t )= | a | d + λD R f ( t h ,a n ( v ) , t ) . A direct calculation gives t h ,a n ( v ) = a (cid:20) a ) (cid:21) (cid:20) v (cid:21) = a (cid:20) a ) v (cid:21) = an (Λ( a ) v ) . ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 17
The behavior of the Radon transform under linear operations implies that R aff S , ,a f ( v, t ) = | a | d + λD R f (cid:18) an (Λ( a ) v ) , a ta (cid:19) = | a | d + λD − R aff f (cid:18) Λ( a ) v, ta (cid:19) = | a | d − ( V , a ⊗ W ,a ) R aff f ( v, t ) . Since ( V , a ⊗ W ,a ) = | a | − λD )2 D A , where the matrix A is of the form A = (cid:20) Λ( a ) − a (cid:21) , and because of the behavior of the operator I under dilations, we obtain IR aff S , ,a f ( v, t ) = I| a | d − ( V , a ⊗ W ,a ) R aff f ( v, t )= ( V , a ⊗ W ,a ) IR aff f ( v, t ) . Finally, let s = t ( s , . . . , s d − ) ∈ R d − . Then R aff S ,s, f ( v, t ) = R aff D h s, f ( v, t )= R D h s, f ( n ( v ) , t ) = R f ( t h s, n ( v ) , t ) . Since t h s, n ( v ) = (cid:20) − s t B ( s ) (cid:21) (cid:20) v (cid:21) = (cid:20) t B ( s ) v − s (cid:21) = n (cid:0) t B ( s ) v − s (cid:1) , by Proposition 4 we obtain the following string of equalities: R aff S ,s, f ( v, t ) = R f (cid:0) n (cid:0) t B ( s ) v − s (cid:1) , t (cid:1) = R aff f (cid:0) t B ( s ) v − s, t (cid:1) = ( V s, ⊗ I) R aff f ( v, t ) . Finally, ( V s, ⊗ I) = | a | − λD )2 T ( − ( t B ( s )) − s, D A , where A = (cid:20) t B ( s ) −
00 1 (cid:21) , so that the behavior of I under dilations implies IR aff S ,s, f ( v, t ) = I ( V s, ⊗ I) R aff f ( v, t )= ( V s, ⊗ I) IR aff f ( v, t ) . Therefore, by IR aff S b,s,a f = IR aff S b, , S ,s, S , ,a f, equation (36) follows applying the relations obtained above. (cid:3) The admissibility conditions.
In this subsection we discuss the admissibility con-ditions and some of their consequences.Our objective is to obtain an expression for the shearlet transform that makes use offormula (36). To this end, we start by looking for natural conditions that guaranteethat ψ ∈ L ( R d ) is an admissible vector for the shearlet representation S , namely that itsatisfies (17).Equation (36) suggests that a good choice for the admissible vector ψ is of the form Q ψ = φ ⊗ φ where φ ∈ L ( R ), φ ∈ L ( R d − ). If this is the case, then by (35) it follows that φ ( v ) F φ ( τ ) = | τ | d − F ψ ( τ n ( v ))so that F ψ factorizes as F ψ ( τ, τ v ) = F ψ ( τ ) F ψ ( v ) , (37)where we assume that ψ ∈ L ( R d − ) and ψ ∈ L ( R ). Equation (37) is the canonicalchoice of admissible vectors given by (19). Furthermore, F φ ( τ ) = | τ | d − F ψ ( τ ) φ ( v ) = F ψ ( v ) , so that the assumption that ψ ∈ L ( R d − ) is automatically satisfied. Since φ ∈ L ( R ),then Z R | τ | d − |F ψ ( τ ) | d τ < + ∞ . This, together with the fact that ψ ∈ L ( R ), implies that ψ belongs to the domain ofthe differential operator I (see (31)). Therefore φ = I ψ . (38)With the choice (37) the admissibility condition (17) reduces to0 < Z R |F ψ ( τ ) | | τ | d τ < + ∞ . From now on we fix ψ ∈ L ( R d ) of the form (37) with ψ ∈ L ( R ) satisfying Z R | τ | d − |F ψ ( τ ) | d τ < + ∞ , < Z R |F ψ ( τ ) | | τ | d τ < + ∞ , (39)and ψ ∈ L ( R d − ). Corollary 11.
Under the assumptions (39) , for every L ( R d ) S ψ f ( b, s, a ) = V φ ( W φ ( Q f ( v, t ))( n ( v ) · b, a )) ( s, a ) (40) ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 19
Proof.
For all f ∈ L ( R d ) and ( b, s, a ) ∈ G S ψ f ( b, s, a ) = h f, S b,s,a ψ i = hQ f, Q S b,s,a ψ i = hQ f, ( V s,a ⊗ W n ( · ) · b,a ) Q ψ i = hQ f, ( V s,a ⊗ W n ( · ) · b,a )( φ ⊗ φ ) i = hQ f, V s,a φ ⊗ W n ( · ) · b,a φ i = Z R d − × R Q f ( v, τ ) V s,a φ ( v ) W n ( v ) · b,a φ ( τ ) d v d τ = Z R d − (cid:18)Z R Q f ( v, τ ) W n ( v ) · b,a φ ( τ ) d τ (cid:19) V s,a φ ( v ) d v = Z R d − W φ ( Q f ( v, • ))( n ( v ) · b, a ) V s,a φ ( v ) d v, (41)where in the last equality we have used the fact that φ is an admissible wavelet. This istrue because by (38) Z R |F φ ( τ ) | | τ | d τ = Z R |F I ψ ( τ ) | | τ | d τ ≤ Z < | τ | < |F ψ ( τ ) | | τ | d τ + Z | τ |≥ | τ | d − |F ψ ( τ ) | d τ ≤ Z R |F ψ ( τ ) | | τ | d τ + Z R | τ | d − |F ψ ( τ ) | d τ, which are both finite. (cid:3) Equation (41) shows that the shearlet coefficients S ψ f ( b, s, a ) can be computed in termsof the unitary Radon transform Q f , which involves the pseudo-differential operator I andit is difficult to compute numerically. However, if f ∈ L ( R d ) ∩ L ( R d ), there is yet adifferent way to express the shearlet transform. To this end we need to choose ψ in sucha way that Q ψ is in the domain of the operator I , that is, in such a way that Z R d − × R | τ | d − |F Q ψ ( v, τ ) | d v d τ < + ∞ . Assuming this and recalling that Q ψ = F ψ ⊗ I ψ we obtain Z R d − × R | τ | d − |F Q ψ ( v, τ ) | d v d τ = Z R d − × R | τ | d − |F ( F ψ ⊗ I ψ )( v, τ ) | d v d τ = Z R d − |F ψ ( v ) | d v Z R | τ | d − |F I ψ ( τ ) | d τ = || ψ || Z R | τ | d − |F ψ ( τ ) | d τ. This shows that Q ψ is in the domain of I if and only if ψ satisfies the additional condition Z R | τ | d − |F ψ ( τ ) | d τ < + ∞ . (42) In this case, by (38) I Qψ = I ( φ ⊗ φ ) = φ ⊗ I φ . Corollary 12.
Under the assumptions (39) and (42) , S ψ f ( b, s, a ) = | a | − d − V φ (cid:0) W χ ( R aff f ( v, t ))( n ( v ) · b, a ) (cid:1) ( s, a ) . for all f ∈ L ( R d ) ∩ L ( R d ) .Proof. For all f ∈ L ( R d ) ∩ L ( R d ) and ( b, s, a ) ∈ G S ψ f ( b, s, a ) = Z R d − (cid:18)Z R Q f ( v, τ ) W n ( v ) · b,a φ ( τ ) d τ (cid:19) V s,a φ ( v ) d v = Z R d − hQ f ( v, · ) , W n ( v ) · b,a φ i V s,a φ ( v ) d v. (43)Since f ∈ L ( R d ) ∩ L ( R d ), Proposition 6 and Proposition 9 imply that for almost all v ∈ R d − , R aff f ( v, · ) is in L ( R ) and F ( Q f ( v, · ))( τ ) = | τ | d − F f ( τ, τ v ) = | τ | d − F R aff f ( v, · )( τ ) . Since F ( Q f ( v, · )) ∈ L ( R ) for almost all v ∈ R d − , the above equality implies that R aff f ( v, · ) is in the domain of I and, by definition of I , Q f ( v, · )( τ ) = I R aff f ( v, · ) . By assumption φ is in the domain of I and the same property holds true for W n ( v ) · b,a φ .Since I is self-adjoint, we get hQ f ( v, · ) , W n ( v ) · b,a φ i = hR aff f ( v, · ) , I W n ( v ) · b,a φ i = | a | − d − hR aff f ( v, · ) , W n ( v ) · b,a I φ i , by taking into account that I W n ( · ) · b,a = | a | − d − W n ( · ) · b,a I . Setting χ = I φ = I ψ , i.e. F χ ( τ ) = | τ |F ψ ( τ ) , (44)from (43) we finally get S ψ f ( b, s, a ) = (45) | a | λD +1 − d Z R d − W χ ( R aff f ( v, · ))( n ( v ) · b, a ) φ (Λ( a )( t B ( s ) v − s )) d v Observe that we have used the fact that χ is an admissible wavelet, too, the proof isanalogous to the proof that ψ is such. As for (41) we can rewrite the above formula byusing the voice transform of H , i.e. S ψ f ( b, s, a ) = | a | − d − V φ (cid:0) W χ ( R aff f ( v, t ))( n ( v ) · b, a ) (cid:1) ( s, a ) . (cid:3) ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 21
Observe that formulas (41) and (45) can be also written in terms of the polar Radontransform using relation (51).Equation (45) shows that for any signal f ∈ L ( R d ) ∩ L ( R d ) the shearlet coefficientscan be computed by means of three classical transforms: first compute the affine Radontransform R aff f , then apply the wavelet transform to the last variable G ( v, b, a ) = W χ ( R aff f ( v, · ))( n ( v ) · b, a ) , where χ is given by (44), and, finally, “mock-convolve” with respect to the variable v S γψ f ( b, s, a ) = Z R d − G ( v, b, a )Φ a ( s − t B ( s ) v ) d v with the scale-dependent filter Φ a ( v ) = φ ( − Λ( a ) v ) . Note that the “mok-convolution” reduces to the standard convolution in R d − when B ( t ) = I d − .Notice that the shearlet coefficients S γψ f ( b, s, a ) depend on f only through its affine Radontransform R aff f . Therefore Equation (21) allows to reconstruct any unknown signal f ∈ L ( R d ) ∩ L ( R d ) from its Radon transform by computing the shearlet coefficients bymeans of (45). Finally, it is worth observing that this reconstruction does not involve thedifferential operator I as applied to the signal. Hence, another interesting aspect of ourresult is that it could open the way to new methods for inverting the Radon transform, avery important issue in applications. Indeed, this result leads to an inversion formula forthe Radon transform based on the shearlet and the wavelet transforms. Example (The standard shearlet group, continued) . For the classical shearlet group S γ ,(36) becomes Q S γb,s,a f ( v, t ) = ( V s,a ⊗ W n ( v ) · b,a ) Q f ( v, t ) , (46)where S γb,s,a is given by (23) and V s,a is the wavelet representation in dimension d − Q intertwines the shearlet representation S γ with the tensor product oftwo wavelet representations.For a fixed admissible vector ψ ∈ L ( R d ) of the form (37) with ψ ∈ L ( R ) satisfying (39)and ψ ∈ L ( R d − ), equation (41) becomes S γψ f ( b, s, a )= | a | ( d − γ − Z R d − W φ ( Q f ( v, • ))( n ( v ) · b, a ) φ (cid:18) v − s | a | − γ (cid:19) d v, (47)for any f ∈ L ( R d ) and ( b, s, a ) ∈ G . Assuming that ψ satisfies the additional condition(42), for any f ∈ L ( R d ) ∩ L ( R d ) and ( b, s, a ) ∈ G , equality (45) becomes S γψ f ( b, s, a )= | a | ( d − γ − Z R d − W χ ( R aff f ( v, • ))( n ( v ) · b, a ) φ (cid:18) v − s | a | − γ (cid:19) d v, (48)where χ is the admissible vector defined by (44). For the sake of clarity we write the above equation for d = 2 and in terms of the Radontransform in polar coordinates S γψ f ( x, y, s, a )= | a | γ − Z R W χ (cid:18) R pol f (arctan v, •√ v ) (cid:19) ( x + vy, a ) φ (cid:16) v − s | a | − γ (cid:17) d v √ v , where x, y, s ∈ R , a ∈ R × and f ∈ L ( R d ) ∩ L ( R ). As mentioned in the introduction, χ = I ψ , see (38), so that the admissible 1 D -wavelet χ is proportional to the Hilberttransform H of the weak derivative of ψ , which is the first factor of the shearlet admissiblevector b ψ ( ξ , ξ ) = c ψ ( ξ ) c ψ ( ξ /ξ ). Acknowledgement
F. De Mari and E. De Vito are members of the Gruppo Nazionale per l’Analisi Matem-atica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di AltaMatematica (INdAM).
Appendix A. The polar and the affine Radon transform
As mentioned in Section 2.6, the natural restriction of the Radon transform is the polarRadon transform R pol f , which is obtained by restricting R f to the closed subset S d − × R ,where S d − is the unit sphere in R d . Define Θ d − = [0 , π ] d − × [0 , π ). For all θ ∈ Θ d − we write inductively t θ = ( θ , t ˆ θ ) , θ ∈ [0 , π ] , ˆ θ ∈ Θ d − and then we put t η ( θ ) = (cos θ , sin θ t η (ˆ θ )) , where η (ˆ θ ) ∈ S d − corresponds to the previous inductive step. Clearly, the map η :Θ d − → S d − induces a parametrization of the unit sphere in R d . Also, observe that themap Θ d − → P d − given by ( θ, t ) ( η ( θ ) : t ) is a two-fold covering of P d − . Definition 13.
Take f ∈ L ( R d ). The polar Radon transform of f is the function R pol f :Θ d − × R → C defined by R pol f ( θ, t ) = R f ( η ( θ ) , t ) = Z η ( θ ) · x = t f ( x ) d m ( x ) . (49)It is easy to find the relation between R pol and R aff . Using the parametrization η of theunit sphere, we can write any vector n ( v ) as n ( v ) = p | v | η ( θ ). More precisely, thereexists t θ = ( θ , t ˆ θ ) ∈ Θ d − such that(1 , t v ) = p | v | (cos θ , sin θ t η (ˆ θ )) . (50)Equality (50) holds if and only ifcos θ = 1 p | v | , sin θ η (ˆ θ ) = v p | v | . It follows that θ = arccos (cid:16) p | v | (cid:17) ∈ [0 , π , η (ˆ θ ) = v | v | , ADON TRANSFORM INTERTWINES SHEARLETS AND WAVELETS 23 unless v = 0, in which case η (ˆ θ ) can be any vector in S d − . Then, item (i) of Proposition 4gives that R aff f ( v, t ) = 1 p | v | R pol f (cid:16) θ, t p | v | (cid:17) (51)and this is the relation that we need. Appendix B. Other proofs
Proof of Theorem 8.
For the reader’s convenience we adapt the proof of [30] to our con-text. Recall that the map ψ : R d − × ( R \ { } ) → R d , defined by ψ ( v, τ ) = τ n ( v ), is adiffeomorphism onto the open set V = { ξ ∈ R d : ξ = 0 } with Jacobian J Φ( v, τ ) = τ d − .Thus, the Plancherel theorem and the Fourier slice theorem give that for any f ∈ A|| f || = Z V |F f ( ξ ) | d ξ = Z R d − × ( R \{ } ) |F f ( τ n ( v )) | | τ | d − d v d τ = Z R d − × R |F ( R aff f ( v, · ))( τ ) | | τ | d − d v d τ = ||R aff f || D . Thus, R aff f belongs to D for all f ∈ A and R aff is an isometric operator from A into D .We want to prove that R aff : A → H has dense image in H . Since H is the completion of D , it is enough to prove that R aff has dense image in D , that is (Ran( R aff )) ⊥ = { } in D .Take then ϕ ∈ D such that h ϕ, R aff f i D = 0 for all f ∈ A . By the definition of the scalarproduct on D and the Fourier slice theorem we have that h ϕ, R aff f i D = Z R d − × R | τ | d − F ( ϕ ( v, · ))( τ ) F ( R aff f ( v, · ))( τ ) d v d τ = Z R d − × R | τ | d − F ( ϕ ( v, · ))( τ ) F f ( τ n ( v )) d v d τ = Z R d F (cid:16) ϕ (cid:0) ˜ ξξ , · (cid:1)(cid:17) ( ξ ) F f ( ξ )d ξ, where t ξ = ( ξ , t ˜ ξ ). Therefore, if h ϕ, R aff f i D = 0 for all f ∈ A , then F (cid:16) ϕ (cid:0) ˜ ξξ , · (cid:1)(cid:17) ( ξ ) = 0almost everywhere. However, k ϕ k D = Z R d − × R | τ | d − |F ( ϕ ( v, · ))( τ ) | d v d τ = Z R d |F (cid:16) ϕ (cid:0) ˜ ξξ , · (cid:1)(cid:17) ( ξ ) | d ξ and hence ϕ = 0 in D . Therefore R aff : A → H has dense image in H and we can extendit to a unique unitary operator R from L ( R d ) onto H . Hence, Q = I R is a unitaryoperator from L ( R d ) onto L ( R d − × R ). (cid:3) Proof of Proposition 9.
We start by observing that (35) is true if f ∈ A , by the Fourierslice theorem and by the definition of Q . Take now f ∈ L ( R d ). By density there existsa sequence ( f n ) n ∈ A such that f n → f in L ( R d ). Since Q is unitary from L ( R d ) onto L ( R d − × R ) and I ⊗F is unitary from L ( R d − × R ) into itself, where I is the identityoperator, (I ⊗F ) Q f n → (I ⊗F ) Q f in L ( R d − × R ). Since f n ∈ A , for almost every( v, τ ) ∈ R d − × R (I ⊗F ) Q f n ( v, τ ) = (I ⊗F ) IR aff f n ( v, τ )= | τ | d − (I ⊗F ) R aff f n ( v, τ )= | τ | d − F f n ( τ n ( v )) . So that, passing to a subsequence if necessary, | τ | d − F f n ( τ n ( v )) → (I ⊗F ) Q f ( v, τ )for almost every ( v, τ ) ∈ R d − × R . Therefore for almost every ( v, τ ) ∈ R d − × R ,(I ⊗F ) Q f ( v, τ ) = lim n → + ∞ | τ | d − F f n ( τ n ( v )) = | τ | d − F f ( τ n ( v )) , where the last equality holds true using a subsequence if necessary. (cid:3) References [1] J.-P. Antoine, R. 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F. Bartolucci, Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35,Genova, Italy
E-mail address : [email protected] F. De Mari, Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35, Gen-ova, Italy
E-mail address : [email protected] E. De Vito ,Dipartimento di Matematica, Universit`a di Genova, Via Dodecaneso 35, Gen-ova, Italy
E-mail address : [email protected] F. Odone, Dibris, Universit`a di Genova, Via Dodecaneso 35, Genova, Italy
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