B. G. Bodmann, G. Kutyniok and X. Zhunag
Coarse Qunatization with the Fast Digital Shearlet Transform.
to appear in Wavelets XI (San Diego, CA, 2011), SPIE Proc.
Abstract:
The fast digital shearlet transform (FDST) was recently introduced as
a means to analyze natural images
efficiently, owing to the fact that those are typically governed by
cartoon-like structures. In this paper, we
introduce and discuss a first-order hybrid sigma-delta quantization
algorithm for coarsely quantizing the shearlet
coefficients generated by the FDST. Radial oversampling in the
frequency domain together with our choice for
the quantization helps suppress the reconstruction error in a similar
way as first-order sigma-delta quantization
for finite frames. We provide a theoretical bound for the
reconstruction error and confirm numerically that the
error is in accordance with this theoretical decay.
B. Han, G. Kutyniok, and Z. Shen
Adaptive Multiresolution Analysis Structures and Shearlet Systems.
SIAM J. Numer. Anal. 49 pp. 1921-1946, (2011).
Abstract:
Shearlet tight frames have been extensively studied during the last years due to their optimal approximation properties of cartoon-like images and their unified treatment of the continuum and digital setting. However, these studies only concerned shearlet tight frames generated by a band-limited shearlet, whereas for practical purposes compact support in spatial domain is crucial.
In this paper, we focus on cone-adapted shearlet systems which - accounting for stability questions - are associated with a general irregular set of parameters. We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a frame and provide explicit estimates for their frame bounds. Secondly, exploring these results and using specifically designed wavelet scaling functions and filters, we construct a family of cone-adapted shearlet frames consisting of compactly supported shearlets. For this family, we derive estimates for the ratio of their frame bounds and prove that they provide optimally sparse approximations of cartoon-like images.
G. Kutyniok, W.-Q Lim, and X. Zhuang
Digital Shearlet Transforms
Shearlets: Multiscale Analysis for Multivariate Data, Springer, to appear. (2011).
Abstract:
Over the past years, various representation systems which sparsely approximate functions governed by
anisotropic features such as edges in images have been proposed. We exemplarily mention the systems of contourlets,
curvelets, and shearlets. Alongside the theoretical development of these systems, algorithmic realizations of the
associated transforms were provided. However, one of the most common shortcomings of these frameworks is the lack
of providing a unified treatment of the continuum and digital world, i.e., allowing a digital theory to be a
natural digitization of the continuum theory. In fact, shearlet systems are the only systems so far which satisfy
this property, yet still deliver optimally sparse approximations of cartoon-like images. In this chapter, we
provide an introduction to digital shearlet theory with a particular focus on a unified treatment of the
continuum and digital realm. In our survey we will present the implementations of two shearlet transforms, one
based on band-limited shearlets and the other based on compactly supported shearlets. We will moreover
discuss various quantitative measures, which allow an objective comparison with other directional transforms and
an objective tuning of parameters. The codes for both presented transforms as well as the framework for quantifying
performance are provided in the Matlab toolbox ShearLab.
G. Kutyniok, J. Lemvig, and W.-Q Lim
Shearlets and Optimally Sparse Approximations
Shearlets: Multiscale Analysis for Multivariate Data, Springer, to appear.
Abstract:
Multivariate functions are typically governed by anisotropic features such as edges
in images or shock fronts in solutions of transport-dominated equations. One major goal both
for the purpose of compression as well as for an efficient analysis is the provision of optimally
sparse approximations of such functions. Recently, cartoon-like images were introduced in 2D and
3D as a suitable model class, and approximation properties were measured by considering the
decay rate of the $L^2$ error of the best $N$-term approximation. Shearlet systems are to date
the only representation system, which provide optimally sparse approximations of this model class
in 2D as well as 3D. Even more, in contrast to all other directional representation systems, a
theory for compactly supported shearlet frames was derived which moreover also satisfy this
optimality benchmark. This chapter shall serve as an introduction to and a survey about sparse
approximations of cartoon-like images by band-limited and also compactly supported shearlet frames
as well as a reference for the state-of-the-art of this research field.
G. Kutyniok, M. Shahram, and X. Zhuang
ShearLab: A Rational design of a digital parabolic scaling algorithm.
Submitted (2011).
Abstract:
Multivariate problems are typically governed by anisotropic features such as edges in images. A common bracket of most of the various directional representation systems which have been proposed to deliver sparse approximations of such features is the utilization of parabolic scaling.
One prominent example is the shearlet system. Our objective in this paper is three fold: We firstly develop a digital theory which rationally designed in the sense that it is the digitization of the existing shearlet theory for continuous data. This implicates that shearlet theory
provides a unified treatment of both the continuum and digital realm. Secondly, we analyze the utilization of pseudo-polar grids and the pseudo-polar Fourier transform for digital implementations of parabolic scaling algorithms. We derive an isometric pseudo-polar Fourier transform by careful weighting of the pseudo-polar grid, allowing exploitation of its adjoint for the inverse transform. This leads to a digital implementation of the shearlet transform; an accompanying Matlab toolbox called ShearLab is provided. And, thirdly, we introduce various quantitative measures for digital parabolic scaling algorithms in general, allowing one to tune parameters and objectively improve the implementation as well as compare different directional transform implementations. The usefulness of such measures is
exemplarily demonstrated for the digital shearlet transform.
G. Kanghui and D. Labate.
Optimally Sparse Representations of 3D Data with C2 Surface Singularities using Parseval Frames of Shearlets.
Submitted (2010).
Abstract:
This paper introduces a new Parseval frame of shearlets for the representation of 3D data, which is especially
designed to handle geometric features such as discontinuous boundaries with very high efficiency. This new system of shearlets
forms a multiscale pyramid of well-localized waveforms at various locations and orientations, which become increasingly waferlike
at fine scales. We prove that the new 3-D shearlet representation exhibits essentially optimal approximation properties for
tri-variate functions f which are smooth away from discontinuities along C2 surfaces. Specifically, the N-term approximation
f^S_N obtained by selecting the N largest coefficients of the shearlet expansion of f satisfies the asymptotic estimate
||f - f^S_N||^2 \approx N^{-1} (logN)2; as N \to \infty
Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outperforms wavelet
approximations, which only yields a N^{-1/2} rate. This result extends to the 3D setting the (essentially) optimally sparse
approximation results obtained by the authors using 2D shearlets and by Cand`es and Donoho using curvelets. The result
presented in this paper is the first nonadaptive construction to provide provably optimal approximation properties (up to a
loglike factor) for a large class of 3-dimensional data.
D. L. Donoho, G. Kutyniok, Morteza Shahram and Xiaosheng Zhuang.
A Rational Design of Discrete Shearlet Transform.
SampTA'11 (Singapore, 2011), Proc., to appear.
Abstract:
In this paper, we first develop a digital shearlet theory which is rationally designed in the
sense that it is the digitalization of the existing shearlet theory for continuum data.
This shows that shearlet theory indeed provides a unified treatment for the continuum
and digital realm. Secondly, we discuss our implementation of the associated digital
shearlet transform. This software package called ShearLab is also rationally designed by providing various
performance measures quantifying precision of the reconstruction, tightness of the
frame, robustness of the shearlet transform, and other properties. Such quantitative
performance metrics allow us to tune parameters and objectively improve our
implementation as well as compare different directional transform implementations.
G. Kutyniok, J. Lemvig and W.-Q Lim.
Optimally Sparse Approximations of Multivariate Functions Using Compactly Supported Shearlet Frames.
SampTA'11 (Singapore, 2011), Proc., to appear.
Abstract:
In this paper, we introduce pyramid-adapted shearlet systems for the
three-dimensional setting, and show how one can construct frames for
L^2(\R^3) with this particular shearlet structure. We then
introduce a generalized three-dimensional cartoon-like image model
class of piecewise C^2 smooth functions with discontinuities on a
C^\alpha smooth surface with 1<\alpha <= 2 and show that
pyramid-adapted shearlet systems provide a nearly optimally sparse
approximation error rate within this model class measured by means
of non-linear, best n-term approximations.
G. Kanghui and D. Labate.
Analysis and Detection of Surface Discontinuities using the 3D Continuous
Shearlet Transform.
Appl. Comput. Harmon. Anal., to appear (2010).
Abstract:
Directional multiscale transforms such as the shearlet and curvelet transforms have
emerged in recent years for their ability to capture the geometrical information associated
with the singularity sets of bivariate functions and distributions. In particular,
it was shown that the continuous shearlet transform provides a precise geometrical
characterization for the boundary curves of general planar regions. No specifc results,
however, were known so far in higher dimensions. In this paper, we extend
this framework for the analysis of singularities to the 3-dimensional setting, and
show that the 3-dimensional continuous shearlet transform precisely characterizes
the boundary set of solid regions in R3 by identifying both its location and local
orientation.
G. Kanghui and D. Labate.
Optimally Sparse 3D Approximations using Shearlet Representations.
Electronic Research Announcements in Mathematical Sciences, 17 pp 126-138, (2010).
Abstract:
This paper introduces a new Parseval frame, based on the 3{D shearlet representation, which is
especially designed to capture geometric features such as discontinuous boundaries with very high efficiency.
We show that this approach exhibits essentially optimal approximation properties for 3D functions f which
are smooth away from discontinuities along C2 surfaces. In fact, the N term approximation f^S_N obtained
by selecting the N largest coefficients from the shearlet expansion of f satisfies the asymptotic estimate
||f - f^S_N||^2 \approx N^{-1} (logN)2; as N \to \infty
Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outper-
forms wavelet approximations, which only yields a N^{-1/2} rate. Indeed, the wavelet approximation rate was
the best published nonadaptive result so far and the result presented in this paper is the first nonadaptive
construction which is provably optimal (up to a loglike factor) for this class of 3D data.
Our estimate is consistent with the corresponding 2D (essentially) optimally sparse approximation
results obtained by the authors using 2D shearlets and by Candes and Donoho using curvelets.
G. Kutyniok and D. Labate.
Shearlets. The First Five Year.
Oberwolfach Report No. 44/2010 (2010).
Abstract:
Over the last 20 years, multiscale methods and wavelets have revolutionized
the field of applied mathematics by providing an efficient means
for encoding isotropic phenomena. Directional multiscale systems, particularly
shearlets, are now having the same dramatic impact on the encoding of
multivariate signals. Since its introduction about five years ago, the theory of
shearlets has rapidly developed and gained wide recognition as the superior
way of achieving a truly unified treatment in both the continuum and digital
setting. By now, shearlet analysis has reached maturity as a research field,
with deep mathematical results, efficient numerical methods, and a variety of
high-impact applications. The main goal of the Mini-Workshop Shearlets was
to gather the world's experts in this field in order to foster closer interaction,
attack challenging open problems, and identify future research directions.
G. Kutyniok and W. Lim.
Image Separation Using Wavelets and Shearlets.
Curves and Surfaces (Avignon, France, 2010), Lecture Notes in Computer Science, Springer, to appear (2011).
Abstract:
In this paper, we present an image separation method for separating images into point- and curvelike
parts by employing a combined dictionary consisting of wavelets and shearlets utilizing the fact that
they sparsely represent point and curvilinear singularities, respectively. Our methodology is based
on the very recently introduced mathematical theory of geometric separation, which shows that highly
precise separation of the morphologically distinct features of points and curves can be achieved
by $\ell^1$ minimization. We further provide and discuss an efficient numerical scheme to solve the
associated optimization problem. Finally, we present some experimental results showing the effectiveness
of our algorithm.
G. Kutyniok, J. Lemvig and W. Lim.
Compactly Supported Shearlets.
Approximation Theory XIII (San Antonio, TX, 2010), Springer, to appear.
Abstract:
Shearlet theory has become a central tool in analyzing and
representing 2D data with anisotropic features. Shearlet systems are
systems of functions generated by one single generator with
parabolic scaling, shearing, and translation operators applied to
it, in much the same way wavelet systems are dyadic scalings and
translations of a single function, but including a precise
control of directionality. Of the many directional representation
systems proposed in the last decade, shearlets are among the most
versatile and successful systems. The reason for this being an
extensive list of desirable properties: shearlet systems can be
generated by one function, they provide precise resolution of
wavefront sets, they allow compactly supported analyzing elements,
they are associated with fast decomposition algorithms, and they
provide a unified treatment of the continuum and the digital realm.
The aim of this paper is to introduce some key concepts in
directional representation systems and to shed some
light on the success of shearlet systems as directional
representation systems. In particular, we will give an overview of the
different paths taken in shearlet theory with focus on separable and
compactly supported shearlets in 2D and 3D. We will present constructions
of compactly supported shearlet frames in those dimensions as well as discuss
recent results on the ability of compactly supported shearlet frames
satisfying weak decay, smoothness, and directional moment conditions to
provide optimally sparse approximations of cartoon-like images in 2D as well as in 3D.
Finally, we will show that these compactly supported shearlet systems provide
optimally sparse approximations of an even generalized model of cartoon-like
images comprising of $C^2$ functions that are smooth apart from
piecewise $C^2$ discontinuity edges.
G. Kutyniok, and W. Lim.
Shearlets on Bounded Domains.
Approximation Theory XIII (San Antonio, TX, 2010), Springer, to appear.
Abstract:
Shearlet systems have so far been only considered as a means to analyze $L^2$-functions defined on $\RR^2$, which
exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial
differential equations the function to be
analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain.
Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that
compactly supported shearlet frames satisfying some weak decay and smoothness conditions, when
orthogonally projected onto the bounded domain, do provide (almost)
optimally sparse approximations of elements belonging to this model class.
P. Kittipoom, G. Kutyniok, and W. Lim.
Construction of Compactly Supported Shearlet Frames.
Constr. Approx.
35 pp. 21-72 (2012).
Abstract:
Shearlet tight frames have been extensively studied during the last years due to their optimal
approximation properties of cartoon-like images and their unified treatment of the continuum
and digital setting. However, these studies only concerned shearlet tight frames generated by
a band-limited shearlet, whereas for practical purposes compact support in spatial domain is
crucial.
In this paper, we focus on cone-adapted shearlet systems which
-- accounting for stability questions -- are associated with a general irregular set of parameters.
We first derive sufficient conditions for such cone-adapted irregular shearlet systems to form a
frame and provide explicit estimates for their frame bounds. Secondly, exploring these results
and using specifically designed wavelet scaling functions and filters,
we construct a family of cone-adapted shearlet frames consisting of compactly supported
shearlets. For this family, we derive
estimates for the ratio of their frame bounds and prove that they provide optimally
sparse approximations of cartoon-like images.
G. Kutyniok and W. Lim.
Compactly Supported Shearlets are Optimally Sparse
J. Approx. Theory 163 pp. 1564-1589, (2011).
Abstract:
Cartoon-like images, i.e., C^2 functions which are smooth apart from
a C^2 discontinuity curve, have by now become a standard model for measuring sparse (non-linear)
approximation properties of directional representation systems. It was already shown that
curvelets, contourlets, as well as shearlets do exhibit (almost) optimally sparse approximation
within this model. However, all those results are only applicable to band-limited
generators, whereas, in particular, spatially compactly supported generators are of
uttermost importance for applications.
In this paper, we now present the first complete proof of (almost) optimally sparse approximations
of cartoon-like images by using a particular class of directional representation systems, which
indeed consists of compactly supported elements. This class will be chosen as a subset of
shearlet frames -- not necessarily required to be tight -- with shearlet generators having compact
support and satisfying some weak moment conditions.
P. Kittipoom, G. Kutyniok, and W. Lim.
Irregular Shearlet Frames: Geometry and Approximation Properties.
J. Fourier Anal. Appl. 17 pp. 604-639, (2011).
Abstract:
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(\R^2), 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 a oversampled shearlet systems and co-shearlet systems, to illustrate
the usefulness of our geometric approach to the construction of shearlet frames.
D. L. Donoho and G. Kutyniok.
Geometric Separation using a Wavelet-Shearlet Dictionary.
SAMPTA'09, Marseille : France (2009).
Abstract:
Astronomical images of galaxies can be modeled as a superposition of pointlike and curvelike structures. Astronomers typically face the problem of extracting those components as accurate as possible. Although this problem seems unsolvable -- as there are two unknowns for every datum -- suggestive empirical results have been achieved by employing a dictionary consisting of wavelets and curvelets combined with l_1 minimization techniques. In this paper we present a theoretical analysis in a model problem showing that accurate geometric separation can be achieved by l_1 minimization. We introduce the notions of cluster coherence and clustered sparse objects as a machinery to show that the underdetermined system of equations can be stably solved by l_1 minimization. We prove that not only a radial wavelet-curvelet dictionary achieves nearly-perfect separation at all sufficiently fine scales, but, in particular, also an orthonormal wavelet-shearlet dictionary, thereby proposing this dictionary as an interesting alternative for geometric separation of pointlike and curvelike structures. To derive this final result we show that curvelets and shearlets are sparsity equivalent in the sense of a finite p-norm (0 < p <= 1) of the cross-Grammian matrix.
S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke.
Shearlet Coorbit Spaces and associated Banach Frames.
Appl. Comput. Harmon. Anal. 27 pp. 195-214, (2009).
Abstract:
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.
G. Kutyniok and T. Sauer.
Adaptive Directional
Subdivision Schemes and Shearlet Multiresolution Analysis.
SIAM J. Math. Anal. 41 pp. 1436-1471, (2009).
Abstract:
In this paper, we propose a solution for a fundamental problem in
computational harmonic analysis, namely, the construction of a
multiresolution analysis with directional components. We will do
so by constructing subdivision schemes which provide a means to
incorporate directionality into the data and thus the limit
function. We develop a new type of non-stationary bivariate
subdivision schemes, which allow to adapt the subdivision process
depending on directionality constraints during its performance, and
we derive a complete characterization of those masks for which these
adaptive directional subdivision schemes converge. In addition, we
present several numerical examples to illustrate how this scheme
works. Secondly, we describe a fast decomposition associated with a
sparse directional representation system for two dimensional data,
where we focus on the recently introduced sparse directional
representation system of shearlets. In fact, we show that the
introduced adaptive directional subdivision schemes can be used as a
framework for deriving a shearlet multiresolution analysis with
finitely supported filters, thereby leading to a fast shearlet
decomposition.
F. Colonna, G. R. Easley, K. Guo, and D. Labate.
Radon Transform Inversion using the Shearlet Representation.
Appl. Comput. Harmon. Anal., 29(2), pp. 232-250, (2010)
Abstract:
The inversion of the Radon transform is a classical ill-posed problem where some method of regularization must be applied in order to accurately recover the objects of interest from the observable data. A well-known consequence of the traditional regularization methods is that some important features to be recovered are lost, as evident in imaging applications where the regularized reconstructions are blurred versions of the original. In this paper, we show that the affine-like system of functions known as the shearlet system can be applied to obtain a highly effective reconstruction algorithm which provides near-optimal rate of convergence in estimating a large class of images from noisy Radon date. This is achieved by introducing a shearlet-based decomposition of the Radon operator and applying a thresholding scheme on the noisy shearlet transform coefficients. For a given noise level, the proposed shearlet shrinkage method can be tuned so that the estimator will attain the essentialy optimal mean square error. Several numerical demonstarations show that its performance improves upon similar competitive strategies based on wavelets and curvelets.
K. Guo, and D. Labate.
Characterization and analysis of edges using the Continuous Shearlet Transform.
SIAM on Imaging Sciences 2, pp. 959-986, (2009).
Abstract:
This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors
and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar
regions. This study is motivated by imaging applications, where
such boundary curves represent the edges of points, including corner points and junctions, where the edge curves exhibit abrupt
changes in tangent or curvature. Our results encompass and greatly extend previous results based on the shearlet and curvelet transform which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature.
S. Yi, D. Labate, G. R. Easley, and H. Krim.
A Shearlet Approach to Edge Analysis and Detection.
IEEE Trans. Image Process. 18 (5) pp. 929-941, (2009).
Abstract:
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 coners and junctions.
K. Guo, D. Labate and W. Lim.
Edge Analysis and identification using the Continuous Shearlet Transform.
Appl. Comput. Harmon. Anal., pp. 31, (2009).
Abstract:
It is well known that the continuous wavelet transform has the ability to identify the set of singularities of a function or distribution. It was shown that certain multidimensional generalization of the wavelet transform are useful to capture additional information about the geometry of the singularities of an underlying function. In this paper, we show that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, allows one to exactly identify the location and orientation of the edges of planar objects. In particular, we show that one can use the asymptotic decay of the shearlet coefficients to exactly characterize the location and orientation of the smooth singularities of a piecewise smooth function. 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.
G. R. Easley, D. Labate, and F. Colonna.
Shearlet Based Total Variation for Denoising.
IEEE Trans. Image Process. 18 (2) pp. 260-268, (2009).
Abstract:
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 shearlets representation by constraining the residual coefficinets using a projected adaptive total variation scheme in the shearlet domain. We also analyze the performance of a shearlets-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.
G. Easley, W. Lim, and D. Labate.
Sparse Directional Image Representations using the Discrete Shearlet Transform.
Appl. Comput. Harmon. Anal. 25 pp. 25-46, (2008).
Abstract: It is now widely acknowledged that traditional wavelets are not very effective in dealing with multidimensional signals containing distributed discontinuities. To achieve
a more efficient representation one has to use basis elements with much higher
directional sensitivity.
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.
G. Kutyniok and D. Labate.
Resolution of the Wavefront Set using Continuous Shearlets.
Trans. Amer. Math. Soc. 361 pp. 2719-2754, (2009).
Abstract: It is known that the Continuous Wavelet Transform of a function f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to precisely identify the wavefront set of a distribution. In this paper, we employ the
framework of affine systems to construct the so-called Continuous
Shearlet Transform, which is defined by SH_f(a,s,t) = <f,ψ_{ast}>. The analyzing elements ψ_{ast} are dilated and translated copies of a single generating function \psi, where
the dilation matrices form a two-parameter matrix group consisting
of products of parabolic scaling and shear matrices.
We show that the elements ψ_{ast} form a system of smooth functions at continuous scales a > 0, locations t \in R^2, and oriented along lines of slope s \in R in the frequency domain. We then prove
that the Continuous Shearlet Transform does exactly resolve the
wavefront set of a distribution f. Finally, we point out several
variations of this approach.
G. Kutyniok and T. Sauer.
From Wavelets to Shearlets and back again.
Approximation Theory XII (San Antonio, TX, 2007), Nashboro Press, Nashville, TN (2007), to appear.
Abstract:
In this paper we will study the Continuous Shearlet Transform from a wavelet
point of view, and show how this perspective can be used to derive a new
geometric interpretation of this transform providing the possibility
for FFT-based fast methods to compute the Continuous Shearlet Transform.
K. Guo and D. Labate.
Representation of Fourier Integral Operators using Shearlets.
J. Fourier Anal. Appl. 14 pp. 327-371, (2008)
Abstract:
The traditional methods of time-frequency and multiscale analysis have been successfully employed
for representing most classes of pseudodifferential operators. However these methods are not equally
effective in dealing with Fourier Integral Operators in general. In this paper, we show that the shearlets,
recently introduced by the authors and their collaborators, provide very efficient representations for a
large class of Fourier Integral Operators. The shearlets are an a±ne-like system of well-localized waveform
at various scales, locations and orientations, which are particularly efficient in representing anisotropic
functions. Using this approach, we prove that the matrix representation of a Fourier Integral Operator
with respect to a Parseval frame of shearlets is sparse and well-organized. This fact recovers a similar
result recently obtained by Candes and Demanet using curvelets. The results presented in this paper
show that directional multiscale representations (such as curvelets and shearlets) are a most powerful
tool in the study of those functions and operators where more traditional multiscale methods are unable
to provide the appropriate geometric analysis in the phase space.
S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H.-G. Stark, and G. Teschke.
The Uncertainty Principle associated with the Continuous Shearlet Transform.
Int. J. Wavelets Multiresolut. Inf. Process., to appear.
Abstract:
In this paper we study the continuous Shearlet transform aiming 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 {\em 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 L^2-spaces. Moreover, we study whether the minimizers satisfy
the admissibility condition, thereby proposing a method to balance between
the minimizing and the admissibility property.
K. Guo and D. Labate.
Optimally Sparse Multidimensional Representation using Shearlets.
SIAM J. Math Anal. 39 pp. 298-318, (2007).
Abstract: In this paper we show that the shearlets, an a±ne-like system of functions recently introduced by the authors and their collaborators, are essentially optimal in representing 2-dimensional functions f that are C2 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
||f-f_N^S||_2^2 \approx N^{-2} (\log N)^3as N \to \infty,
which is essentially optimal, and greatly outperforms the corresponding asymptotic approximation rate N^{-1}
associated with wavelet approximations.
Unlike the curvelets, that have similar sparsity properties, the 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.
G. Kutyniok and D. Labate.
Construction of Regular and Irregular Shearlets.
J. Wavelet Theory and Appl. 1 pp. 1-10, (2007).
Abstract: In this paper, we
study the construction of irregular shearlet
systems, i.e., systems of the form SH(&psi,&Lambda) = {
a^{-3/4} &psi(A_a^{-1}S_s^{-1}(x-t)) : (a,s,t) ∈ &Lambda},$
where &psi ∈ L^2(R^2), &Lambda is an arbitrary sequence in
R^+ x R x R^2, A_a is a parabolic scaling
matrix and S_s a shear matrix. These systems are obtained by
appropriately sampling the Continuous Shearlet Transform. We derive
sufficient conditions for such a discrete system to form a frame
for L^2(R^2), and provide explicit estimates for the frame
bounds. Among the examples of such discrete systems, one is the
Parseval frame of shearlets previously introduced by the authors,
which is optimal in approximating 2-D smooth functions with
discontinuities along C^2-curves. This study provides the
framework for the construction of a variety of discrete directional
multiscale systems with the ability to detect orientations inherited
from the Continuous Shearlet Transform.
K. Guo, G. Kutyniok, and D. Labate.
Sparse Multidimensional Representations using Anisotropic Dilation and Shear Operators.
Wavelets and Splines (Athens, GA, 2005), Nashboro Press, Nashville, TN pp. 189-201, (2006).
Abstract: Recent advances in applied mathematics and signal processing have shown that, in order to obtain sparse representations of multi-dimensional functions and signals, one has to use representation elements distributed not only at various scales and locations -- as in classical wavelet theory -- but also at various directions. In this paper, we show that we obtain a construction having exactly these properties by using the framework of affine systems. The representation elements that we obtain are generated by translations, dilations, and shear transformations of a single mother function, and are called shearlets. The shearlets provide optimally sparse representations for 2-D functions that are smooth away from discontinuities along curves. Another benefit of this approach is that, thanks to their mathematical structure, these systems provide a Multiresolution analysis similar to the one associated with classical wavelets, which is very useful for the development of fast algorithmic implementations.
D. Labate, W. Lim, G. Kutyniok, and G. Weiss.
Sparse multidimensional representation using shearlets.
Wavelets XI (San Diego, CA, 2005), 254-262, SPIE Proc. 5914, SPIE, Bellingham, WA, (2005).
Abstract: 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.