Shearlet Group Research Articles

Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces

AbstractWe develop a distributional framework for the shearlet transform and the shearlet synthesis operator , where is the Lizorkin test function space and is the space of highly localized test functions on the standard shearlet group . These spaces and their duals are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector ψ belongs to . Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from to and from to . Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.

Embeddings of shearlet coorbit spaces into Sobolev spaces

We investigate the existence of embeddings of shearlet coorbit spaces associated to weighted mixed Lp-spaces into classical Sobolev spaces in dimension three by using the description of coorbit spaces as decomposition spaces. This different perspective on these spaces enables the application of embedding results that allow the complete characterization of embeddings for certain integrability exponents, and thus provides access to a deeper understanding of the smoothness properties of coorbit spaces, and of the influence of the choice of shearlet groups on these properties. We give a detailed analysis, identifying which features of the dilation groups have an influence on the embedding behavior, and which do not. Our results also allow to comment on the validity of the interpretation of shearlet coorbit spaces as smoothness spaces.

Clifford Valued Shearlet Transform

This paper deals with the construction of $$n=3 \text{ mod } 4$$ Clifford algebra $$Cl_{n,0}$$ -valued admissible shearlet transform using the shearlet group $$(\mathbb {R}^* < imes \mathbb {R}^{n-1}) < imes \mathbb {R}^n$$ , a subgroup of affine group of $${\mathbb {R}}^n$$ . The admissibility conditions for a nonzero Clifford valued square integrable function have been obtained. Various properties such as reconstruction formula, orthogonality relation, isometry and reproducing kernel have been dealt. As an application, Heisenberg type uncertainty principle for Clifford shearlet transform has been derived.

Standard shearlet group in arbitrary space dimensions and projections in its L1-algebra

This paper is devoted to definition standard shearlet group $$\mathbb{S} = {\mathbb{R}^ + } \times {\mathbb{R}^{n - 1}} \times {\mathbb{R}^n}$$, in arbitrary space dimensions and concerned with the projections which are, self adjoint idempotents in the group algebra $${L^1}(\mathbb{S})$$. Actually we determine minimal projections, associated with an open free orbit, in details.

A continuous frame representation for an abstract shearlet group

Let \({\mathbb {S}}\) be a locally compact abstract shearlet group and \(\sigma :{\mathbb {S}}\rightarrow {\mathcal {U}}({\mathcal {H}})\) be a representation of \({\mathbb {S}}\) on a separable Hilbert space \({\mathcal {H}}\). We give a necessary and sufficient condition for the family of representation coefficients to be a continuous shearlet frame. Furthermore relations between continuous shearlet frames associated to a given representation and its irreducible sub-representations are investigated.

Linear dependency of translations and square-integrable representations

Let G be a locally compact group. We examine the problem of determining when nonzero functions in L2(G) have linearly independent left translations. In particular, we establish some results for the case when G has an irreducible, square-integrable, unitary representation. We apply these results to the special cases of the affine group, the shearlet group, and the Weyl–Heisenberg group. We also investigate the case when G has an abelian, closed subgroup of finite index.

Frames arising from irreducible solvable actions I

In this work, we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable Lie groups. In contrast to other well-known techniques such as the coorbit theory, the generalized coorbit theory and other discretization schemes, we make no assumption on the integrability or square-integrability of the representations of interest. Moreover, our scheme produces explicit constructions of frames with precise frame bounds. As an illustration of the scope of our results, we highlight that a large class of representations which naturally occur in wavelet theory and time–frequency analysis is handled by our scheme. For example, the affine group, the generalized Heisenberg groups, the shearlet groups, solvable extensions of vector groups and various solvable extensions of non-commutative nilpotent Lie groups are a few examples of groups whose irreducible representations are handled by our method. The class of representations studied in this work is described as follows. Let G be a simply connected, connected, completely solvable Lie group with Lie algebra g=p+m. Next, let π be an infinite-dimensional unitary irreducible representation of G obtained by inducing a character from a closed normal subgroup P=exp⁡p of G. Additionally, we assume that G=P⋊M, M=exp⁡m is a closed subgroup of G, dμM is a fixed Haar measure on the solvable Lie group M and there exists a linear functional λ∈p⁎ such that the representation π=πλ=indPG(χλ) is realized as acting in L2(M,dμM). Making no assumption on the integrability of πλ, we describe explicitly a discrete subset Γ of G and a vector f∈L2(M,dμM) such that πλ(Γ)f is a tight frame for L2(M,dμM). We also construct compactly supported smooth functions s and discrete subsets Γ⊂G such that πλ(Γ)s is a frame for L2(M,dμM).

Study of shearlet transform using block matrix dilation

In this paper, the shearlet theory is extended from ordinary matrices to block matrices and its properties are investigated. More precisely, the block shearlet group is defined as a semidirect product group and the block shearlet transform is constructed by means of a quasiregular representation of this semidirect product group. In addition, we show that the block shearlet transform can overcome two major issues of the classical shearlet transform: time complexity and redundancy. We also report the result of careful experiments on video denoising which compares our proposed method with state of the art multiscale techniques, including curvelets and 3D shearlet transform.

Resolution of the Wavefront Set Using General Continuous Wavelet Transforms

We consider the problem of characterizing the wavefront set of a tempered distribution $$u\in \mathcal {S}'(\mathbb {R}^{d})$$ in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group $$H\subset \mathrm{GL}(\mathbb {R}^{d})$$ . In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group H, called microlocal admissibility and (weak) cone approximation property. Essentially, microlocal admissibility sets up a systematic relationship between the scales in a wavelet dilated by $$h\in H$$ on one side, and the matrix norm of h on the other side. The (weak) cone approximation property describes the ability of the wavelet system to adapt its frequency-side localization to arbitrary frequency cones. Together, microlocal admissibility and the weak cone approximation property allow the characterization of points in the wavefront set using multiple wavelets. Replacing the weak cone approximation by its stronger counterpart gives rise to single wavelet characterizations. We illustrate the scope of our results by discussing—in any dimension $$d\ge 2$$ —the similitude, diagonal and shearlet dilation groups, for which we verify the pertinent conditions. As a result, similitude and diagonal groups can be employed for multiple wavelet characterizations, whereas for the shearlet groups a single wavelet suffices. In particular, the shearlet characterization (previously only established for $$d=2$$ ) holds in arbitrary dimensions.

Abstract Shearlet Transform

In this paper, the shearlet theory is extended from Euclidean spaces to locally compact groups. More precisely, the abstract shearlet group is defined as a 3-fold semidirect product and the abstract shearlet transform is constructed by means of a quasiregular representation of the semidirect product group. Its properties are investigated and results are illustrated by some examples.

Different Faces of the Shearlet Group

Recently, shearlet groups have received much attention in connection with shearlet transforms applied for orientation sensitive image analysis and restoration. The square integrable representations of the shearlet groups provide not only the basis for the shearlet transforms but also for a very natural definition of scales of smoothness spaces, called shearlet coorbit spaces. The aim of this paper is twofold: first we discover isomorphisms between shearlet groups and other well-known groups, namely extended Heisenberg groups and subgroups of the symplectic group. Interestingly, the connected shearlet group with positive dilations has an isomorphic copy in the symplectic group, while this is not true for the full shearlet group with all nonzero dilations. Indeed we prove the general result that there exist, up to adjoint action of the symplectic group, only one embedding of the extended Heisenberg algebra into the Lie algebra of the symplectic group. Having understood the various group isomorphisms it is natural to ask for the relations between coorbit spaces of isomorphic groups with equivalent representations. These connections are examined in the second part of the paper. We describe how isomorphic groups with equivalent representations lead to isomorphic coorbit spaces. In particular we apply this result to square integrable representations of the connected shearlet groups and metaplectic representations of subgroups of the symplectic group. This implies the definition of metaplectic coorbit spaces. Besides the usual full and connected shearlet groups we also deal with Toeplitz shearlet groups.

Coorbit spaces and wavelet coefficient decay over general dilation groups

We study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square-integrable quasi-regular representation on L 2 ( R d ) \textrm {L}^2(\mathbb {R}^d) . It turns out that these representations are integrable as well, with respect to a wide variety of weights, thus allowing to consistently quantify wavelet coefficient decay via coorbit space norms. We then show that these spaces always admit an atomic decomposition in terms of bandlimited Schwartz wavelets. We exhibit spaces of Schwartz functions contained in all coorbit spaces, and dense in most of them. We also present an example showing that for a consistent definition of coorbit spaces, the irreducibility requirement cannot be easily dispensed with. We then address the question of how to predict wavelet coefficient decay from vanishing moment assumptions. To this end, we introduce a new condition on the open dual orbit associated to a dilation group: If the orbit is temperately embedded, it is possible to derive rather general weighted mixed L p \textrm {L}^{p} -estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function. These estimates have various applications: They provide very explicit admissibility conditions for wavelets and integrable vectors, as well as sufficient criteria for membership in coorbit spaces. As a further consequence, one obtains a transparent way of identifying elements of coorbit spaces with certain (cosets of) tempered distributions. We then show that, for every dilation group in dimension two, the associated dual orbit is temperately embedded. In particular, the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces, where they complement and generalize the known results.

On sufficient conditions and stability of shearlet frame

In 2011, Kittipoom et al. introduced a new class of shearlet generators, and gave many good properties. For example, the new class of shearlet generators is dense in the space of square integrable functions and the shearlet frame generated by a member of the new class of shearlet generators possesses the strong HAP etc. In this paper, our main goal is to study sufficient conditions and the stability of shearlet frame. Specifically, we first adjust the definition of shearlet group introduced by Kittipoom et al. through referring to the definition of shearlet group introduced by Dahlke et al., and make each element of the new class of shearlet generators to be admissible. Secondly, we show that a function from the new class of shearlet generators generates shearlet frames for every sufficiently dense and relatively separated sequence. Finally, we prove that a frame generated by a function from the space remains a frame when the time, scale and shear parameters and the generating function undergo small perturbations. These conclusions make shearlet framework to possess great flexibility and practicality in engineering applications.

Irregular Shearlet Frames: Geometry and Approximation Properties

Recently, shearlet systems were introduced as a means to derive efficient encoding methodologies for anisotropic features in 2-dimensional data with a unified treatment of the continuum and digital setting. However, only very few construction strategies for discrete shearlet systems are known so far.In this paper, we take a geometric approach to this problem. Utilizing the close connection with group representations, we first introduce and analyze an upper and lower weighted shearlet density based on the shearlet group. We then apply this geometric measure to provide necessary conditions on the geometry of the sets of parameters for the associated shearlet systems to form a frame for L 2(ℝ2), either when using all possible generators or a large class exhibiting some decay conditions. While introducing such a feasible class of shearlet generators, we analyze approximation properties of the associated shearlet systems, which themselves lead to interesting insights into homogeneous approximation abilities of shearlet frames. We also present examples, such as oversampled shearlet systems and co-shearlet systems, to illustrate the usefulness of our geometric approach to the construction of shearlet frames.

The Continuous Shearlet Transform in Arbitrary Space Dimensions

This paper is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.

Shearlet coorbit spaces and associated Banach frames

In this paper, we study the relationships of the newly developed continuous shearlet transform with the coorbit space theory. It turns out that all the conditions that are needed to apply the coorbit space theory can indeed be satisfied for the shearlet group. Consequently, we establish new families of smoothness spaces, the shearlet coorbit spaces. Moreover, our approach yields Banach frames for these spaces in a quite natural way. We also study the approximation power of best n-term approximation schemes and present some first numerical experiments.

THE UNCERTAINTY PRINCIPLE ASSOCIATED WITH THE CONTINUOUS SHEARLET TRANSFORM

Finding optimal representations of signals in higher dimensions, in particular directional representations, is currently the subject of intensive research. Since the classical wavelet transform does not provide precise directional information in the sense of resolving the wavefront set, several new representation systems were proposed in the past, including ridgelets, curvelets and, more recently, Shearlets. In this paper we study and visualize the continuous Shearlet transform. Moreover, we aim at deriving mother Shearlet functions which ensure optimal accuracy of the parameters of the associated transform. For this, we first show that this transform is associated with a unitary group representation coming from the so-called Shearlet group and compute the associated admissibility condition. This enables us to employ the general uncertainty principle in order to derive mother Shearlet functions that minimize the uncertainty relations derived for the infinitesimal generators of the Shearlet group: scaling, shear and translations. We further discuss methods to ensure square-integrability of the derived minimizers by considering weighted L2-spaces. Moreover, we study whether the minimizers satisfy the admissibility condition, thereby proposing a method to balance between the minimizing and the admissibility property.