@article{yi_labate_easley_krim_2009, title={A Shearlet Approach to Edge Analysis and Detection}, volume={18}, ISSN={["1941-0042"]}, DOI={10.1109/TIP.2009.2013082}, abstractNote={It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions}, number={5}, journal={IEEE TRANSACTIONS ON IMAGE PROCESSING}, author={Yi, Sheng and Labate, Demetrio and Easley, Glenn R. and Krim, Hamid}, year={2009}, month={May}, pages={929–941} } @article{guo_labate_lim_2009, title={Edge analysis and identification using the continuous shearlet transform}, volume={27}, ISSN={["1096-603X"]}, DOI={10.1016/j.acha.2008.10.004}, abstractNote={It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution f . It was recently shown that certain multidimensional generalizations of the wavelet transform are useful to capture additional information about the geometry of the singularities of f . In this paper, we consider the continuous shearlet transform, which is the mapping f ∈ L 2 ( R 2 ) → SH ψ f ( a , s , t ) = 〈 f , ψ ast 〉 , where the analyzing elements ψ ast form an affine system of well localized functions at continuous scales a > 0 , locations t ∈ R 2 , and oriented along lines of slope s ∈ R in the frequency domain. We show that the continuous shearlet transform allows one to exactly identify the location and orientation of the edges of planar objects. In particular, if f = ∑ n = 1 N f n χ Ω n where the functions f n are smooth and the sets Ω n have smooth boundaries, then one can use the asymptotic decay of SH ψ f ( a , s , t ) , as a → 0 (fine scales), to exactly characterize the location and orientation of the boundaries ∂ Ω n . This improves similar results recently obtained in the literature and provides the theoretical background for the development of improved algorithms for edge detection and analysis.}, number={1}, journal={APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS}, author={Guo, Kanghui and Labate, Demetrio and Lim, Wang-Q}, year={2009}, month={Jul}, pages={24–46} } @article{kutyniok_labate_2009, title={Resolution of the wavefront set using continuous shearlets}, volume={361}, DOI={10.1090/s0002-9947-08-04700-4}, abstractNote={It is known that the Continuous Wavelet Transform of a distribution f f decays rapidly near the points where f f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f f . However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by S H ψ f ( a , s , t ) = ⟨ f ψ a s t ⟩ \mathcal {SH}_\psi f(a,s,t) = \langle {f}{\psi _{ast}}\rangle , where the analyzing elements ψ a s t \psi _{ast} are dilated and translated copies of a single generating function ψ \psi . The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements { ψ a s t } \{\psi _{ast}\} form a system of smooth functions at continuous scales a > 0 a>0 , locations t ∈ R 2 t \in \mathbb {R}^2 , and oriented along lines of slope s ∈ R s \in \mathbb {R} in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f f .}, number={5}, journal={Transactions of the American Mathematical Society}, author={Kutyniok, G. and Labate, D.}, year={2009}, pages={2719–2754} } @article{easley_labate_colonna_2009, title={Shearlet-Based Total Variation Diffusion for Denoising}, volume={18}, ISSN={["1941-0042"]}, DOI={10.1109/TIP.2008.2008070}, abstractNote={We propose a shearlet formulation of the total variation (TV) method for denoising images. Shearlets have been mathematically proven to represent distributed discontinuities such as edges better than traditional wavelets and are a suitable tool for edge characterization. Common approaches in combining wavelet-like representations such as curvelets with TV or diffusion methods aim at reducing Gibbs-type artifacts after obtaining a nearly optimal estimate. We show that it is possible to obtain much better estimates from a shearlet representation by constraining the residual coefficients using a projected adaptive total variation scheme in the shearlet domain. We also analyze the performance of a shearlet-based diffusion method. Numerical examples demonstrate that these schemes are highly effective at denoising complex images and outperform a related method based on the use of the curvelet transform. Furthermore, the shearlet-TV scheme requires far fewer iterations than similar competitors.}, number={2}, journal={IEEE TRANSACTIONS ON IMAGE PROCESSING}, author={Easley, Glenn R. and Labate, Demetrio and Colonna, Flavia}, year={2009}, month={Feb}, pages={260–268} } @article{guo_labate_2008, title={Representation of Fourier integral operators using shearlets}, volume={14}, ISSN={["1531-5851"]}, DOI={10.1007/s00041-008-9018-0}, number={3}, journal={JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS}, author={Guo, Kanghui and Labate, Demetrio}, year={2008}, pages={327–371} } @article{easley_labate_lim_2008, title={Sparse directional image representations using the discrete shearlet transform}, volume={25}, ISSN={["1096-603X"]}, DOI={10.1016/j.acha.2007.09.003}, abstractNote={In spite of their remarkable success in signal processing applications, it is now widely acknowledged that traditional wavelets are not very effective in dealing multidimensional signals containing distributed discontinuities such as edges. To overcome this limitation, one has to use basis elements with much higher directional sensitivity and of various shapes, to be able to capture the intrinsic geometrical features of multidimensional phenomena. This paper introduces a new discrete multiscale directional representation called the discrete shearlet transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the discrete shearlet transform is very competitive in denoising applications both in terms of performance and computational efficiency.}, number={1}, journal={APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS}, author={Easley, Glenn and Labate, Demetrio and Lim, Wang-Q}, year={2008}, month={Jul}, pages={25–46} } @article{kutyniok_labate_2007, title={Construction of regular and irregular shearlet frames}, volume={1}, number={1}, journal={Journal of Wavelet Theory and Applications}, author={Kutyniok, G. and Labate, D.}, year={2007}, pages={1–12} } @article{guo_labate_2007, title={Optimally sparse multidimensional representation using shearlets}, volume={39}, ISSN={["1095-7154"]}, DOI={10.1137/060649781}, abstractNote={In this paper we show that shearlets, an affine-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f which are $C^2$ except for discontinuities along $C^2$ curves. More specifically, if $f_N^S$ is the N-term reconstruction of f obtained by using the N largest coefficients in the shearlet representation, then the asymptotic approximation error decays as $\norm{f-f_N^S}_2^2 \asymp N^{-2} (\log N)^3, N \to \infty,$ which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate $N^{-1}$ associated with wavelet approximations. Unlike curvelets, which have similar sparsity properties, shearlets form an affine-like system and have a simpler mathematical structure. In fact, the elements of this system form a Parseval frame and are generated by applying dilations, shear transformations, and translations to a single well-localized window function.}, number={1}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Guo, Kanghui and Labate, Demetrio}, year={2007}, pages={298–318} } @article{guo_labate_2007, title={Sparse shearlet representation of Fourier integral operators}, volume={14}, journal={Electronic Research Announcements in Mathematical Sciences}, author={Guo, K. H. and Labate, D.}, year={2007}, pages={7–19} } @article{guo_labate_2006, title={Some remarks on the unified characterization of reproducing systems}, volume={57}, number={3}, journal={Collectanea Mathematica (Barcelona, Spain)}, author={Guo, K. H. and Labate, D.}, year={2006}, pages={295–307} } @inbook{guo_labate_lim_weiss_wilson_2006, title={The theory of wavelets with composite dilations}, ISBN={0817637788}, DOI={10.1007/0-8176-4504-7_11}, abstractNote={A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L 2(ℝn) under the action of lattice translations and dilations by products of elements drawn from non-commuting sets of matrices A and B. Typically, the members of B are matrices whose eigenvalues have magnitude one, while the members of A are matrices expanding on a proper subspace of ℝn. The theory of these systems generalizes the classical theory of wavelets and provides a simple and flexible framework for the construction of orthonormal bases and related systems that exhibit a number of geometric features of great potential in applications. For example, composite wavelets have the ability to produce “long and narrow” window functions, with various orientations, well-suited to applications in image processing.}, booktitle={Harmonic analysis and applications}, publisher={Boston: Birkhauser}, author={Guo, K. and Labate, D. and Lim, W. and Weiss, G. and Wilson, E.}, year={2006}, pages={231–249} } @article{kutyniok_labate_2006, title={Theory of reproducing systems on locally compact Abelian groups}, volume={106}, DOI={10.4064/cm106-2-3}, abstractNote={A reproducing system is a countable collection of functions fj : j 2 Jg such that a general function f can be decomposed as f = P j2J cj(f)`j, with some control on the analyzing coe-cients cj(f). Several such systems have been introduced very successfully in mathematics and its applications. We present a unifled viewpoint to the study of reproducing systems on locally compact abelian groups G. This approach gives a novel characterization of the Parseval frame generators for a very general class of reproducing systems on L 2 (G). As an application of this result, we obtain a new characterization of Parseval frame generators for Gabor and a-ne systems on L 2 (G).}, journal={Colloquium Mathematicum}, author={Kutyniok, G. and Labate, D.}, year={2006}, pages={197–220} } @article{guo_labate_lim_weiss_wilson_2006, title={Wavelets with composite dilations and their MRA properties}, volume={20}, ISSN={["1096-603X"]}, DOI={10.1016/j.acha.2005.07.002}, abstractNote={Affine systems are reproducing systems of the form A C = { D c T k ψ ℓ : 1 ⩽ ℓ ⩽ L , k ∈ Z n , c ∈ C } , which arise by applying lattice translation operators T k to one or more generators ψ ℓ in L 2 ( R n ) , followed by the application of dilation operators D c , associated with a countable set C of invertible matrices. In the wavelet literature, C is usually taken to be the group consisting of all integer powers of a fixed expanding matrix. In this paper, we develop the properties of much more general systems, for which C = { c = a b : a ∈ A , b ∈ B } where A and B are not necessarily commuting matrix sets. C need not contain a single expanding matrix. Nonetheless, for many choices of A and B , there are wavelet systems with multiresolution properties very similar to those of classical dyadic wavelets. Typically, A expands or contracts only in certain directions, while B acts by volume-preserving maps in transverse directions. Then the resulting wavelets exhibit the geometric properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for multidimensional signal and image processing applications. Our method is a systematic approach to the theory of affine-like systems yielding these and more general features.}, number={2}, journal={APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS}, author={Guo, KH and Labate, D and Lim, WQ and Weiss, G and Wilson, E}, year={2006}, month={Mar}, pages={202–236} } @article{labate_wilson_2005, title={Connectivity in the set of Gabor frames}, volume={18}, ISSN={["1096-603X"]}, DOI={10.1016/j.acha.2004.09.003}, abstractNote={In this paper we present a constructive proof that the set of Gabor frames is path-connected in the L 2 ( R n ) -norm. In particular, this result holds for the set of Gabor–Parseval frames as well as for the set of Gabor orthonormal bases. In order to prove this result, we introduce a construction which shows exactly how to modify a Gabor frame or Parseval frame to obtain a new one with the same property. Our technique is a modification of a method used in [Glas. Mat. 38 (58) (2003) 75–98] to study the connectivity of affine Parseval frames.}, number={1}, journal={APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS}, author={Labate, D and Wilson, E}, year={2005}, month={Jan}, pages={123–136} } @inproceedings{labate_lim_kutyniok_weiss_2005, title={Sparse multidimensional representation using shearlets}, ISBN={0819459194}, DOI={10.1117/12.613494}, abstractNote={In this paper we describe a new class of multidimensional representation systems, called shearlets. They are obtained by applying the actions of dilation, shear transformation and translation to a fixed function, and exhibit the geometric and mathematical properties, e.g., directionality, elongated shapes, scales, oscillations, recently advocated by many authors for sparse image processing applications. These systems can be studied within the framework of a generalized multiresolution analysis. This approach leads to a recursive algorithm for the implementation of these systems, that generalizes the classical cascade algorithm.}, booktitle={Wavelets XI: 31 July-3 August, 2005, San Diego, California, USA}, publisher={Bellingham, Wash. : SPIE}, author={Labate, D. and Lim, W. and Kutyniok, G. and Weiss, G.}, editor={M. Papadakis, A. F. Laine and Unser, M. A.Editors}, year={2005}, pages={254–262} } @article{hernandez_labate_weiss_al._2004, title={Oversampling, quasi-affine frames, and wave packets}, volume={16}, DOI={10.1016/j.acha.2003.12.002}, abstractNote={In [E. Hernández, D. Labate, G. Weiss, J. Geom. Anal. 12 (4) (2002) 615–662], three of the authors obtained a characterization of certain types of reproducing systems. In this work, we apply these results and methods to various affine-like, wave packets and Gabor systems to determine their frame properties. In particular, we study how oversampled systems inherit properties (like the frame bounds) of the original systems. Moreover, our approach allows us to study the phenomenon of oversampling in much greater generality than is found in the literature.}, number={2}, journal={Applied and Computational Harmonic Analysis}, author={Hernandez, E. and Labate, D. and Weiss, G. and al.}, year={2004}, pages={111–147} } @article{guo_labate_lim_weiss_wilson_2004, title={Wavelets with composite dilations}, volume={10}, ISSN={["1079-6762"]}, DOI={10.1090/S1079-6762-04-00132-5}, abstractNote={A wavelet with composite dilations is a function generating an orthonormal basis or a Parseval frame for L 2 ( R n ) L^2({\mathbb R}^n) under the action of lattice translations and dilations by products of elements drawn from non-commuting matrix sets A A and B B . Typically, the members of B B are shear matrices (all eigenvalues are one), while the members of A A are matrices expanding or contracting on a proper subspace of R n {\mathbb R}^n . These wavelets are of interest in applications because of their tendency to produce “long, narrow” window functions well suited to edge detection. In this paper, we discuss the remarkable extent to which the theory of wavelets with composite dilations parallels the theory of classical wavelets, and present several examples of such systems.}, journal={ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Guo, KH and Labate, D and Lim, WQ and Weiss, G and Wilson, E}, year={2004}, pages={78–87} } @inbook{gressman_labate_weiss_wilson_2003, title={Affine, quasi-affine and co-affine wavelets}, ISBN={0127432736}, DOI={10.1016/s1570-579x(03)80036-8}, abstractNote={Abstract “Classical” wavelets are obtained by the action of a particular countable subset of operators associated with the affine group on a function ψ ∈ L 2 (ℝ). More precisely, this set is the collection { D 2j T k : j,k ∈ ℤ}, where T k is the translation by the integer k and D 2j is the (unitary) dilation by 2 j . We thus obtain the discrete wavelet system. Ron and Shen [ 4 ] have shown that by interchanging and renormalizing “half” of the operators in this set one obtains an important collection of systems that can be considered “equivalent” to this affine system. In this paper we show that, in a precise sense, the choice of Ron and Shen is optimal.}, booktitle={Beyond wavelets}, publisher={San Diego, CA: Academic Press}, author={Gressman, P. and Labate, D. and Weiss, G. and Wilson, E.}, year={2003}, pages={215–224} } @article{labate_2002, title={A unified characterization of reproducing systems generated by a finite family}, volume={12}, DOI={10.1007/bf02922050}, abstractNote={This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame generators for shift-invariant subspaces of L2(ℝn). A number of applications of this general result are then obtained, among which are the characterization of tight frames and dual frames for Gabor and wavelet systems.}, number={3}, journal={Journal of Geometric Analysis}, author={Labate, D.}, year={2002}, pages={469–491} } @article{hernandez_labate_weiss_2002, title={A unified characterization of reproducing systems generated by a finite family. II}, volume={12}, DOI={10.1007/bf02930656}, abstractNote={This article presents a general result from the study of shift-invariant spaces that characterizes tight frame and dual frame generators for shift-invariant subspaces of L2(ℝn). A number of applications of this general result are then obtained, among which are the characterization of tight frames and dual frames for Gabor and wavelet systems.}, number={4}, journal={Journal of Geometric Analysis}, author={Hernandez, E. and Labate, D. and Weiss, G.}, year={2002}, pages={615–622} } @article{labate_2001, title={Pseudodifferential operators on modulation spaces}, volume={262}, ISSN={["1096-0813"]}, DOI={10.1006/jmaa.2001.7566}, abstractNote={We establish a connection between certain classes of pseudodifferential operators and Hille–Tamarkin operators. As an application, we find the conditions that guarantee compactness and summability of the eigenvalues of pseudodifferential operators acting on the modulation spaces M p , p .}, number={1}, journal={JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS}, author={Labate, D}, year={2001}, month={Oct}, pages={242–255} } @article{labate_2001, title={Time-frequency analysis of pseudodifferential operators}, volume={133}, ISSN={["0026-9255"]}, DOI={10.1007/s006050170028}, number={2}, journal={MONATSHEFTE FUR MATHEMATIK}, author={Labate, D}, year={2001}, pages={143–156} } @misc{labate_canavero_demarchi_1994, title={A COMPARISON OF FRACTAL DIMENSION AND SPECTRUM COEFFICIENT CHARACTERIZATION OF 1/F-ALPHA NOISE}, volume={31}, ISSN={["1681-7575"]}, DOI={10.1088/0026-1394/31/1/011}, number={1}, journal={METROLOGIA}, author={LABATE, D and CANAVERO, F and DEMARCHI, A}, year={1994}, month={Apr}, pages={51–53} } @inproceedings{dilece_isnardi_labate_canavero_1994, title={Exact SPICE model of field coupling to multiconductor transmission lines}, ISBN={078032000X}, booktitle={1994 International Symposium on Electromagnetic Compatibility, Hotel Sendai Plaza, Sendai, Japan, May 16-20, 1994}, publisher={Tokyo: Institute of Electronics, Information and Communication Engineers; Institute of Electrical Engineers of Japan}, author={Dilece, B. and Isnardi, L. and Labate, D. and Canavero, F. G.}, year={1994}, pages={12–15} } @article{vecchi_labate_canavero_1994, title={FRACTAL APPROACH TO LIGHTNING RADIATION ON A TORTUOUS CHANNEL}, volume={29}, ISSN={["0048-6604"]}, DOI={10.1029/93RS03030}, abstractNote={In this paper the radiation from a geometrically fractal (i.e., arbitrarily irregular) discharge channel is investigated from the point of view of engineering applications of fractal geometry. Numerical results are presented for lightning return stroke radiation, which demonstrate that the time waveform of radiated field (in the Fraunhofer region) is a fractal, and within the framework of employed approximations, it has the same fractal dimension as the channel path. Some implications of this finding are discussed along with the limitations of the model.}, number={4}, journal={RADIO SCIENCE}, author={VECCHI, G and LABATE, D and CANAVERO, F}, year={1994}, pages={691–704} }