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Radon transform intertwines shearlets and wavelets

We prove that the unitary affine Radon transform intertwines the quasi-regular representation of a class of semidirect products, built by shearlet dilation groups and translations, and the tensor product of a standard wavelet representation with a wavelet-like representation. This yields a formula for shearlet coefficients that involves only integral transforms applied to the affine Radon transform of the signal, thereby opening new perspectives in the inversion of the Radon transform.


Introduction
The use of wavelets in signal analysis and computer vision has proved almost optimal for one-dimensional signals in many ways, and the mathematics behind classical wavelets has reached a high degree of elaboration. In higher dimensions, however, the picture is less clear and this partially explains the huge class of representations that has been introduced over 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 anisotropic features, to provide optimal sparse representations, to detect singularities and to be stable against noise, see [9] for an overview and a complete list of references. From the purely mathematical perspective, their construction is based on the well-established theory of square-integrable representations [10], just as wavelets are, and because of this many powerful mathematical tools are available. As far as applications are concerned, their effectiveness has been tested primarily in image processing, where many efficient algorithms have been designed using them (see [9,11] and the website http://www.shearlab.org/ for further details and references).
Thus, in some sense, shearlets behave for high-dimensional signals as wavelets do for 1Dsignal, and it is therefore natural to try to understand if the many strong connections are a consequence of some general mathematical principle.
The purpose of this paper is to address this issue, and to give a partial answer, showing that the link between the shearlet transform and wavelets is the unitary Radon transform in affine coordinates, because it actually intertwines the shearlet representation with a tensor product of two wavelet representations. This fact can be exploited to show that by carefully choosing the mother shearlet it is possible to obtain the classical shearlet coefficients 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 that parametrizes all the hyperplanes parallel to a given one (for the two dimensional case see Fig. 1), followed by a convolution operator with a scale-dependent filter in the variables of the hyperplane. As the shearlet transform admits an inversion formula, it is in principle possible to invert the Radon transform of a given signal by means of it and the aforementioned operations.
For two-dimensional signals, our results, which have been announced in [12], can be described as we now explain. In order to formulate them precisely, we recall the definition of the three main ingredients, namely wavelets, shearlets and the Radon transform. The wavelet group is R ⋊ R × with law (b, a)(b ′ , a ′ ) = (b + ab ′ , aa ′ ). The square integrable wavelet representation W acts on L 2 (R) by W b,a ψ(x) = |a| −1/2 ψ x − b a and the wavelet transform, defined by W ψ f (b, a) = f, ψ b,a , is a multiple of an isometry from L 2 (R) to L 2 (R ⋊ R × ) provided that ψ satisfies the Calderón condition, see (22) below. Next, denote by S the (parabolic) shearlet group, namely and where the vectors are understood as column vectors. The group S acts on L 2 (R 2 ) via the shearlet representation, namely ). The shearlet transform is then S ψ f (b, s, a) = f, S b,s,a ψ , and is a multiple of an isometry provided 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 1 (R 2 ) is the function R aff f : An important fact is that it is possible to define a version of R aff as a unitary map on L 2 (R 2 ). First, it is necessary to compose it with the Riesz-type operator I that we now describe. Its natural domain is the dense subspace of L 2 (R 2 ) where g denotes the Fourier transform of g. The densely defined, self-adjoint unbounded operator I : i.e. a Fourier multiplier in the second variable. It is not hard to show that for all f in the dense subspace of L 2 (R 2 ) the Radon transform R aff f belongs to D and that the map f −→ IR aff f from A to L 2 (R 2 ) extends to a unitary map, denoted by Q, from L 2 (R 2 ) onto itself.
In the two dimensional case our main formula reads now where the meaning of the dummy variable v is Our second most important result is the formula The 1D-wavelet χ 1 and the 1D-filter φ 2 are related to the shearlet admissible vector ψ by the following relations The first equality shows that 2πχ 1 = Hψ ′ 1 is the Hilbert transform H of the weak derivative of ψ 1 .
Equation (2) shows that for any signal f in L 1 (R d )∩L 2 (R d ) the shearlet coefficients can be computed 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 t where χ 1 is given by (3); c) convolve the result with the scale-dependent filter where φ 2 is given by (4) and the convolution is computed with respect to the variable v, that is Finally, since S is a square-integrable representation, there is a reconstruction formula, namely where the integral converges in the weak sense. Note that S ψ f depends on f only through its Radon transform R aff f , see (2). The above equation allows to reconstruct an unknown signal 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 not new. Indeed, it is known that ridgelets are constructed via wavelet analysis in the Radon domain [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 Radon transform is at the root of the proof that shearlets are able to detect the wavefront set of a 2D signal [17].
Our contribution is to clarify this relation from the point of view of non-commutative harmonic 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 Führ [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 various ingredients, namely the groups, the representations, the Radon transform and the unitary extensions that need to be defined. In Section 3 we state and prove the main results.

Preliminaries
2.1. Notation. We briefly introduce the notation. We set R × = R \ {0}. The Euclidean norm of a vector v ∈ R d is denoted by |v| and its scalar product with w ∈ R d by v · w. For any p ∈ [1, +∞] we denote by L p (R d ) the Banach space of functions f : R d → C, which are p-integrable with respect to the Lebesgue measure dx and, if p = 2, the corresponding scalar product and norm are ·, · and · , respectively. The Fourier trasform is denoted by F both on L 2 (R d ) and on L 1 (R d ), where it is defined by If G is a locally compact group, we denote by L 2 (G) the Hilbert space of square-integrable functions with respect to a left Haar measure on G. We denote the (real) general linear group of size d × d by GL(d, R) and by T(d, R) the closed subgroup of unipotent upper triangular matrices.
If H is a closed subgroup of GL(d, R), the semidirect product G = R d ⋊ H is the product is the natural linear action of the matrix h on the column vector b.

Shearlet dilation groups.
In this section we introduce the groups in which we are interested. This family includes the groups introduced by Führ in [18,19], and called generalized shearlet dilation groups for the purpose of generalizing the standard shearlet group introduced in [6,20].
The group S is called the shearing sugroup of H and D is called the diagonal complement or scaling subgroup of H.
Several observations are in order. First of all, if one requires the shearing subgroup S to be Abelian, then one obtains the class introduced by Führ, with a slightly more general definition. This has inspired Definition 1.
Since the map B is continuous, S is automatically connected, and hence by Theorem 3.6.2 in [21], it is closed and simply connected. By construction the elements of H are of the form Furthermore, since the diagonal matrices of GL(d, R) normalize T(d, R), then H is the semidirect product of S and D.
Finally, the assumption that S is a subgroup normalized by D forces the maps B and Λ to satisfy some equalities. Indeed, since then S is a group if and only if for every u, v ∈ R d−1 . Since the compatibility of D with S is equivalent to asking for the following condition to hold for all a = 0 and all s ∈ R d−1 : It follows that H is diffeomorphic as a manifold to R d−1 × R × , so that we can identify the element h s,a with the pair (s, a). With this identification the product law amounts to (s, a)(s ′ , a ′ ) = Λ(a) −1 s ′ + t B(Λ(a) −1 s ′ )s, aa ′ .
We stress that, in general, S is not isomorphic as a Lie group to the additive Abelian group R d−1 , unless S is the standard shearlet group introduced in [20], see the examples below.
Remark. It should be clear that a slightly larger class would be obtained by allowing for diagonal matrices of the form The case µ 0 = 0, however, is uninteresting because any shearlet dilation group corresponding to this choice never admits admissible vectors [19,22]. But then a simple change of variables permits to assume µ 0 = 1, as we did, and to set λ j = µ j − 1.
Remark. In [19] the authors introduce the notion of shearlet dilation group by means of structural properties and then prove that in the case when S is Abelian they can be parametrized 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−1 , which satisfies all the above properties. In this case, s → h s,1 defines a group isomorphism between R d−1 and the Abelian group S.
Clearly, any choice of the weights λ 1 , . . . , λ d−1 is compatible with (12). In particular, if we choose as D the group of matrices 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 and that consequently all the equalities in (10) hold. This case corresponds to Toeplitz shearlet groups (see [24]).
Not all dilation matrices as in (7) are compatible with (12). In [19] it is shown that for any fixed λ 1 .
Example (A non-Abelian shearlet dilation group). The matrices give rise to a non-Abelian shearlet group S. Indeed, it is easily checked that , a product which is not Abelian in the third coordinate. Evidently, 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 by the matrices Further, [X(q, p, t), X(q ′ , p ′ , t ′ )] = X(0, 0, qp ′ − pq ′ ) exhibits the Lie algebra of S as the three dimensional Heisenberg Lie algebra. A straightforward calculation shows that for any choice of λ ∈ R the diagonal matrices Conversely, these are easily seen to be the only rank-one dilations that normalize the matrices B(u). In conclusion, the group D consisting of the matrices together with S give rise to the non-Abelian shearlet dilation group H = SD. It is worth observing that the dilations in D are not the standard dilations of the Heisenberg group. Indeed, the Lie algebra of D consists of the diagonal matrices A λ (τ ) = diag(τ, (λ + 1)τ, (2λ + 1)τ, (3λ + 1)τ ) and shows that these homogeneous dilations are not the standard dilations of the Heisenberg Lie algebra (see [25], p. 620).
2.3. The shearlet representation and admissible vectors. From now on we fix a group G = R d ⋊ H where H is a shearlet group as in Definition 1 and we parametrize its elements as (b, s, a). By (13) we get that a left Haar measure of H is where db is the Lebesgue measure on R d and the last equality holds true since 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 are the elements of L 2 (R d ) satisfying We recall that a unitary representation π of G acting on a Hilbert space H is square integrable 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 → f, π(b, h)ψ , takes values in L 2 (G) and in such case it is a multiple of an isometry, denoted by Proof of Theorem 2. The proof is an immediate consequence of the following result due to Führ, 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 0 ∈ R d such that where saying that O ξ 0 has full measure means that its complement has Lebsegue measure zero. In such case, a vector ψ is admissible if and only if In our setting, with the choice ξ 0 = (1, 0, . . . , 0) we have that To compute the admissible vectors, notice that by (18) Theorem 2 states the surprising fact that the admissibility condition is the same for all generalized shearlet dilation groups. A canonical choice is to assume that where and ψ 2 ∈ L 2 (R d−1 ). However, other choices are available and, in particular, it is possible to build shearlets with compact support in space [27]. We finally recall that, since the representation S is square-integrable, we have the weakly-convergent reproducing formula 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| −2 dbda. It acts on L 2 (R) by means of the square-integrable representation The corresponding wavelet transform is W ψ : L 2 (R) → L 2 (W), given by which is a multiple of an isometry provided that ψ ∈ L 2 (R) satisfies the admissibility condition, namely the Calderón equation, and, in such a case, ψ is called an admissible wavelet.
2.5. 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). As mentioned above, we identify H with R d−1 × 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 Observe that H acts naturally on R d and its (right) dual action is This implies that H acts naturally on Hence we can define the quasi-regular representation of H acting on L 2 (R d−1 ) by means of where we recall that λ D = λ 1 + · · · + λ d−1 . In general, V is not irreducible, but we can always define the voice transform associated to a fixed vector ψ ∈ L 2 (R d−1 ), namely the mapping V ψ : 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 2 (R d ) becomes whereas the group H is the affine group R d−1 ⋊ R × in dimension d − 1 and V is the corresponding wavelet representation which is not irreducible unless d = 2. Furthermore, the voice transform can be written as convolution operator 2.6. The affine Radon transform. In this section we recall the definition and the main properties of the Radon transform. Then we introduce the particular restriction of the 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 1 (R d ) by following the approach in [29], see also [30] as a classical reference. Given f ∈ L 1 (R d ) its Radon transform is the function where m is the Euclidean measure on the hyperplane and the equality (25) holds for almost all (n, t) ∈ (R d \ {0}) × R. We add some comments. Definition (25) makes sense since, given n ∈ R d \ {0} Fubini theorem gives that so that for almost all t ∈ R the integral n·x=t |f (x)|dm(x) is finite and R(n, t) is well defined.
This means that Rf is completely defined by choosing a representative (n, t) for each hyperplane (n : t), i.e. by choosing a suitable system of coordinates on the affine Grassmannian The canonical choice [30] is given by parametrizing P d−1 with its two-fold covering S d−1 , where S d−1 is the unit sphere in R d .

We are interested in another restriction of the Radon transform. For all
Definition 3. Given f ∈ L 1 (R d ), the affine Radon transform of f is the function R aff f : Remark. The transform R aff is obtained from R by parametrizing the projective space The complement of U 0 is the set of horizontal hyperplanes, those for which the normal vector has the first component equal to zero (see Figure 1 for the 2-dimensional case). The set of pairs (v, t) such that (n(v) : t) ∈ U 0 is negligible, so that R aff f completely defines Rf . In A we recall the relation between the affine Radon transform and the usual Radon transform in polar coordinates.
The next proposition, whose proof can be found in [29], summarizes the behaviour of the Radon transform under affine linear actions. The translation and dilation operators act on a function f : R d → C as respectively, for b ∈ R d and A ∈ GL(d, R). Both operators map each L p (R d ) onto itself and D A is normalized to be an isometry on L 1 (R d ).
Proposition 4. Given f ∈ L 1 (R d ), the following properties hold true: We now state a crucial result in Radon transform theory in its standard version. Below we prove two variations that are taylored to our setting but are also of some independent interest.
for all n ∈ R d \ {0} and all τ ∈ R.
Here the Fourier transform on the right hand side is in R d , whereas the operator F on the left hand side is 1-dimensional and acts on the variabile t. We repeat this slight abuse of 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 1 (R d ) ∩ L 2 (R d ).
Proposition 6 (Fourier slice theorem, 2). Define ψ : Proof. By Proposition 5 we know that for all v ∈ R d−1 the affine Radon transform R aff f (v, ·) is in L 1 (R) and satisfies We start by proving that the function τ By hypothesis we know that so that there exists a negligible set E ⊆ R d−1 such that (29) follows by the Fourier inversion formula in L 2 (R).
It is possible to extend the affine Radon transform R aff to L 2 (R d ) as a unitary map. However, this raises some technical issues, that are addressed in the next section.

The unitary extension. Consider the subspace
a Fourier multiplier with respect to the last variable. Since τ → |τ | d−1 2 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-adjoint injective operator.
Remark. The operator I is related to the inverse of the Riesz potential with exponent where I 0 is the inverse of the standard Riesz potential defined by

Furthermore, D is invariant under translations and dilations by matrices of the form
Lemma 7. For all b ∈ R d and A as in (32) it holds Proof. The first of relations (33) is a consequence of the fact that Precisely, for all f ∈ D we have that This proves (33).
The space D becomes a pre-Hilbert space with respect to the scalar product , for all f ∈ D. Hence I is an isometric operator from D, with the new scalar product (34), to L 2 (R d−1 × R). Since I is self-adjoint and injective, Ran(I) is dense in L 2 (R d−1 × R). Hence, by standard arguments, it extends uniquely to a unitary operator, denoted I , from the completion H of D onto L 2 (R d−1 × R).
To extend R aff to L 2 (R d ) as a unitary operator, note that, by Proposition 5 with n = n(v), the affine Radon transform of where ξ 1 is the first component of the vector ξ ∈ R d . Therefore requiring that R aff f belongs to which is dense in L 2 (R d ) since it contains the functions whose Fourier transform is smooth and has compact support disjoint from the hyperplane ξ 1 = 0. By definition of A, We shall need a suitable formulation of the main result in Radon transform theory, namely the following version of Theorem 4.1 in [30]. For the sake of completeness we include the proof in B.
Theorem 8. The affine Radon transform extends to a unique unitary operator from L 2 (R d ) onto H, denoted with R and, hence, Q = I R is a unitary operator from L 2 (R d ) onto L 2 (R d−1 × 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 the literature. The proof is given in B.

The Intertwining Theorem and its consequences
3.1. 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 a triple (b, s, a) ∈ R d × R d−1 × R × and S b,s,a is as in (16).
Theorem 10. The unitary operator Q intertwines the shearlet representation with the tensor product of two unitary representations, precisely for every f ∈ L 2 (R d ).
Proof. By density, it is enough to prove the equality on A. We shall use throughout the fact that R aff f ∈ D for every f ∈ A and that D is invariant under all translations and under the dilations described in (32). Since for all (b, s, a) ∈ G it holds (b, s, a) = (b, 0, 1)(0, s, 1)(0, 0, a), it is sufficient to prove the equality for each of the three factors. For f ∈ A and b ∈ R d we have Since I commutes with translations, I ⊗W n(v)·b,1 = T (0,n(v)·b) implies For f ∈ A and a ∈ R × we have Rf ( t h 0,a n(v), t).
A direct calculation gives t h 0,a n(v) = a 1 0 0 Λ(a)

The behavior of the Radon transform under linear operations implies that
Since where the matrix A is of the form and because of the behavior of the operator I under dilations, we obtain by Proposition 4 we obtain the following string of equalities: Finally, so that the behavior of I under dilations implies Therefore, by IR aff S b,s,a f = IR aff S b,0,1 S 0,s,1 S 0,0,a f, equation (36) follows applying the relations obtained above.
3.2. The admissibility conditions. In this subsection we discuss the admissibility conditions and some of their consequences.
Our objective is to obtain an expression for the shearlet transform that makes use of formula (36). To this end, we start by looking for natural conditions that guarantee that ψ ∈ L 2 (R d ) is an admissible vector for the shearlet representation S, namely that it satisfies (17).
Equation (36) suggests that a good choice for the admissible vector ψ is of the form . If this is the case, then by (35) it follows that so that F ψ factorizes as where we assume that ψ 2 ∈ L 2 (R d−1 ) and ψ 1 ∈ L 2 (R). Equation (37) is the canonical choice of admissible vectors given by (19). Furthermore, so that the assumption that This, together with the fact that ψ 1 ∈ L 2 (R), implies that ψ 1 belongs to the domain of the differential operator I 0 (see (31)). Therefore With the choice (37) the admissibility condition (17) reduces to From now on we fix ψ ∈ L 2 (R d ) of the form (37) with ψ 1 ∈ L 2 (R) satisfying and ψ 2 ∈ L 2 (R d−1 ).
Corollary 11. Under the assumptions (39), for every L 2 (R d ) Proof. For all f ∈ L 2 (R d ) and (b, s, a) ∈ G where in the last equality we have used the fact that φ 1 is an admissible wavelet. This is true because by (38) which are both finite.
Equation (41) shows that the shearlet coefficients S ψ f (b, s, a) can be computed in terms of the unitary Radon transform Qf , which involves the pseudo-differential operator I and it is difficult to compute numerically. However, if f ∈ L 1 (R d ) ∩ L 2 (R d ), there is yet a different way to express the shearlet transform. To this end we need to choose ψ in such a way that Qψ is in the domain of the operator I, that is, in such a way that Assuming this and recalling that Qψ = F ψ 2 ⊗ I 0 ψ 1 we obtain This shows that Qψ is in the domain of I if and only if ψ 1 satisfies the additional condition In this case, by (38)
Since f ∈ L 1 (R d ) ∩ L 2 (R d ), Proposition 6 and Proposition 9 imply that for almost all Since F (Qf (v, ·)) ∈ L 2 (R) for almost all v ∈ R d−1 , the above equality implies that R aff f (v, ·) is in the domain of I 0 and, by definition of I 0 , By assumption φ 1 is in the domain of I 0 and the same property holds true for W n(v)·b,a φ 1 .
Since I 0 is self-adjoint, we get by taking into account that I 0 W n(·)·b,a = |a| − d−1 2 W n(·)·b,a I 0 . Setting from (43) we finally get Observe that we have used the fact that χ 1 is an admissible wavelet, too, the proof is analogous to the proof that ψ 1 is such. As for (41) we can rewrite the above formula by using the voice transform of H, i.e.
Observe that formulas (41) and (45) can be also written in terms of the polar Radon transform using relation (51).
Equation (45) shows that for any signal f ∈ L 1 (R d ) ∩ L 2 (R d ) the shearlet coefficients can be computed by means of three classical transforms: first compute the affine Radon transform R aff f , then apply the wavelet transform to the last variable where χ 1 is given by (44), and, finally, "mock-convolve" with respect to the variable v Note that the "mok-convolution" reduces to the standard convolution in R d−1 when Notice that the shearlet coefficients S γ ψ f (b, s, a) depend on f only through its affine Radon transform R aff f . Therefore Equation (21) allows to reconstruct any unknown signal f ∈ L 1 (R d ) ∩ L 2 (R d ) from its Radon transform by computing the shearlet coefficients by means of (45). Finally, it is worth observing that this reconstruction does not involve the differential operator I as applied to the signal. Hence, another interesting aspect of our result is that it could open the way to new methods for inverting the Radon transform, a very important issue in applications. Indeed, this result leads to an inversion formula for the 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 where S γ b,s,a is given by (23) and V s,a is the wavelet representation in dimension d − 1 as in (24). Therefore in the case of the standard shearlet group Theorem 10 shows that the unitary operator Q intertwines the shearlet representation S γ with the tensor product of two wavelet representations.
For the sake of clarity we write the above equation for d = 2 and in terms of the Radon transform in polar coordinates S γ ψ f (x, y, s, a) = |a| where x, y, s ∈ R, a ∈ R × and f ∈ L 1 (R d ) ∩ L 2 (R 2 ). As mentioned in the introduction, χ 1 = I 2 0 ψ 1 , see (38), so that the admissible 1D-wavelet χ 1 is proportional to the Hilbert transform H of the weak derivative of ψ 1 , which is the first factor of the shearlet admissible vector ψ(ξ 1 , ξ 2 ) = ψ 1 (ξ 1 ) ψ 2 (ξ 2 /ξ 1 ).
unless v = 0, in which case η(θ) can be any vector in S d−2 . Then, item (i) of Proposition 4 gives that R aff f (v, t) = 1 1 + |v| 2 R pol f θ, t 1 + |v| 2 (51) and this is the relation that we need. Thus, the Plancherel theorem and the Fourier slice theorem give that for any f ∈ A 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 )) ⊥ = {0} in D. Take then ϕ ∈ D such that ϕ, R aff f D = 0 for all f ∈ A. By the definition of the scalar product on D and the Fourier slice theorem we have that where t ξ = (ξ 1 , tξ ). Therefore, if ϕ, R aff f D = 0 for all f ∈ A, then F ϕ ξ ξ 1 , · (ξ 1 ) = 0 almost everywhere. However, and hence ϕ = 0 in D. Therefore R aff : A → H has dense image in H and we can extend it to a unique unitary operator R from L 2 (R d ) onto H. Hence, Q = I R is a unitary operator from L 2 (R d ) onto L 2 (R d−1 × R).
Proof of Proposition 9. We start by observing that (35) is true if f ∈ A, by the Fourier slice theorem and by the definition of Q. Take now f ∈ L 2 (R d ). By density there exists a sequence (f n ) n ∈ A such that f n → f in L 2 (R d ). Since Q is unitary from L 2 (R d ) onto L 2 (R d−1 × R) and I ⊗F is unitary from L 2 (R d−1 × R) into itself, where I is the identity operator, (I ⊗F )Qf n → (I ⊗F )Qf in L 2 (R d−1 × R). Since f n ∈ A, for almost every τ n(v)). So that, passing to a subsequence if necessary, |τ | d−1 2 F f n (τ n(v)) → (I ⊗F )Qf (v, τ ) for almost every (v, τ ) ∈ R d−1 × R. Therefore for almost every (v, τ ) ∈ R d−1 × R, where the last equality holds true using a subsequence if necessary.