In applied mathematical analysis, shearlets are a multiscale framework which allows efficient encoding of
anisotropic
Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's phys ...
features in
multivariate
Multivariate may refer to:
In mathematics
* Multivariable calculus
* Multivariate function
* Multivariate polynomial
In computing
* Multivariate cryptography
* Multivariate division algorithm
* Multivariate interpolation
* Multivariate optical c ...
problem classes. Originally, shearlets were introduced in 2006
[ for the analysis and ]sparse approximation Sparse approximation (also known as sparse representation) theory deals with sparse solutions for systems of linear equations. Techniques for finding these solutions and exploiting them in applications have found wide use in image processing, signal ...
of functions . They are a natural extension of wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
s, to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images, since wavelets, as isotropic objects, are not capable of capturing such phenomena.
Shearlets are constructed by parabolic scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
, shearing, and translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
applied to a few generating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...
s. At fine scales, they are essentially supported within skinny and directional ridges following the parabolic scaling law, which reads ''length² ≈ width''. Similar to wavelets, shearlets arise from the affine group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself.
It is a Lie group if is the real or complex field or quaternions.
...
and allow a unified treatment of the continuum and digital situation leading to faithful implementations. Although they do not constitute an orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For ex ...
for , they still form a frame
A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent.
Frame and FRAME may also refer to:
Physical objects
In building construction
* Framing (co ...
allowing stable expansions of arbitrary functions .
One of the most important properties of shearlets is their ability to provide optimally sparse approximations (in the sense of optimality in [) for cartoon-like functions . In imaging sciences, ''cartoon-like functions'' serve as a model for anisotropic features and are compactly supported in while being apart from a closed piecewise singularity curve with bounded curvature. The decay rate of the -error of the -term shearlet approximation obtained by taking the largest coefficients from the shearlet expansion is in fact optimal up to a log-factor:][
:
where the constant depends only on the maximum curvature of the singularity curve and the maximum magnitudes of , and . This approximation rate significantly improves the best -term approximation rate of wavelets providing only for such class of functions.
Shearlets are to date the only directional representation system that provides sparse approximation of anisotropic features while providing a unified treatment of the continuum and digital realm that allows faithful implementation. Extensions of shearlet systems to are also available. A comprehensive presentation of the theory and applications of shearlets can be found in.][
]
Definition
Continuous shearlet systems
The construction of continuous shearlet systems is based on ''parabolic scaling matrices''
:
as a mean to change the resolution, on ''shear matrices''
:
as a means to change the orientation, and finally on translations to change the positioning.
In comparison to curvelet
Curvelets are a non- adaptive technique for multi-scale object representation. Being an extension of the wavelet concept, they are becoming popular in similar fields, namely in image processing and scientific computing.
Wavelets generalize the Fo ...
s, shearlets use shearings instead of rotations, the advantage being that the shear operator leaves the integer lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or gri ...
invariant in case , i.e., This indeed allows a unified treatment of the continuum and digital realm, thereby guaranteeing a faithful digital implementation.
For the ''continuous shearlet system'' generated by is then defined as
:
and the corresponding ''continuous shearlet transform'' is given by the map
:
Discrete shearlet systems
A discrete version of shearlet systems can be directly obtained from by discretizing the parameter set There are numerous approaches for this but the most popular one is given by
:
From this, the ''discrete shearlet system'' associated with the shearlet generator is defined by
:
and the associated ''discrete shearlet transform'' is defined by
:
Examples
Let be a function satisfying the ''discrete Calderón condition'', i.e.,
:
with and
where denotes the Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of For instance, one can choose to be a Meyer wavelet. Furthermore, let be such that 1, 1
Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "O ...
/math> and
:
One typically chooses to be a smooth bump function
In mathematics, a bump function (also called a test function) is a function f: \R^n \to \R on a Euclidean space \R^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported. The set of all ...
. Then given by
:
is called a ''classical shearlet''. It can be shown that the corresponding discrete shearlet system constitutes a Parseval frame for consisting of bandlimited
Bandlimiting is the limiting of a signal's frequency domain representation or spectral density to zero above a certain finite frequency.
A band-limited signal is one whose Fourier transform or spectral density has bounded support.
A bandlimi ...
functions.[
Another example are compactly supported shearlet systems, where a compactly supported function can be chosen so that forms a ]frame
A frame is often a structural system that supports other components of a physical construction and/or steel frame that limits the construction's extent.
Frame and FRAME may also refer to:
Physical objects
In building construction
* Framing (co ...
for .[ In this case, all shearlet elements in are compactly supported providing superior spatial localization compared to the classical shearlets, which are bandlimited. Although a compactly supported shearlet system does not generally form a Parseval frame, any function can be represented by the shearlet expansion due to its frame property.
]
Cone-adapted shearlets
One drawback of shearlets defined as above is the directional bias of shearlet elements associated with large shearing parameters.
This effect is already recognizable in the frequency tiling of classical shearlets (see Figure in Section #Examples), where the frequency support of a shearlet increasingly aligns along the -axis as the shearing parameter goes to infinity.
This causes serious problems when analyzing a function whose Fourier transform is concentrated around the -axis.
To deal with this problem, the frequency domain is divided into a low-frequency part and two conic regions (see Figure):
:
The associated ''cone-adapted discrete shearlet system'' consists of three parts, each one corresponding to one of these frequency domains.
It is generated by three functions and a ''lattice sampling'' factor
:
where
:
with
:
The systems and basically differ in the reversed roles of and .
Thus, they correspond to the conic regions and , respectively.
Finally, the ''scaling function'' is associated with the low-frequency part .
Applications
* Image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...
and computer sciences
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
[
** ]Denoising
Noise reduction is the process of removing noise from a signal. Noise reduction techniques exist for audio and images. Noise reduction algorithms may distort the signal to some degree. Noise rejection is the ability of a circuit to isolate an un ...
** Inverse problems
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
** Image enhancement
Image editing encompasses the processes of altering images, whether they are digital photographs, traditional photo-chemical photographs, or illustrations. Traditional analog image editing is known as photo retouching, using tools such as ...
** Edge detection
Edge detection includes a variety of mathematical methods that aim at identifying edges, curves in a digital image at which the image brightness changes sharply or, more formally, has discontinuities. The same problem of finding discontinuiti ...
** Inpainting
Inpainting is a conservation process where damaged, deteriorated, or missing parts of an artwork are filled in to present a complete image. This process is commonly used in image restoration. It can be applied to both physical and digital art ...
** Image separation
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
* PDEs[
** Resolution of the wavefront set
** ]Transport equations
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
* Coorbit theory
In mathematics, coorbit theory was developed by Hans Georg Feichtinger and Karlheinz Gröchenig around 1990.H. G. Feichtinger and K. Gröchenig. "Banach spaces related to integrable group representations and their atomic decompositions, II" Monatsh ...
, characterization of smoothness spaces[
* Differential geometry: ]manifold learning
Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low- ...
Generalizations and extensions
* 3D-Shearlets [
* -Shearlets ][
* Parabolic molecules ][
* Cylindrical Shearlets ][
]
See also
* Wavelet transform
In mathematics, a wavelet series is a representation of a square-integrable ( real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal ...
* Curvelet transform
* Contourlet transform
* Bandelet transform
* Chirplet transform
* Noiselet transform
References
{{reflist, refs=
[ Kutyniok, Gitta, and Demetrio Labate, eds. ''Shearlets: Multiscale analysis for multivariate data''. Springer, 2012, {{isbn, 0-8176-8315-1]
[Guo, Kanghui, and Demetrio Labate. "The construction of smooth Parseval frames of shearlets." Mathematical Modelling of Natural Phenomena 8.01 (2013): 82–105.
{{cite web, url= http://www3.math.tu-berlin.de/numerik/mt/www.shearlet.org/papers/shear_construction.pdf , title=PDF ]
[Kittipoom, Pisamai, ]Gitta Kutyniok
Gitta Kutyniok (born 1972) is a German applied mathematician known for her research in harmonic analysis, deep learning, compressed sensing, and image processing. She has a Bavarian AI Chair for "Mathematical Foundations of Artificial Intelligen ...
, and Wang-Q Lim. "Construction of compactly supported shearlet frames." Constructive Approximation 35.1 (2012): 21–72.
{{cite arXiv, title=PDF , eprint=1003.5481 , last1=Kittipoom , first1=P. , last2=Kutyniok , first2=G. , last3=Lim , first3=W. , class=math.FA , year=2010
[Guo, Kanghui, and Demetrio Labate. "Optimally sparse multidimensional representation using shearlets." SIAM Journal on Mathematical Analysis 39.1 (2007): 298–318.
{{cite web, url= http://www3.math.tu-berlin.de/numerik/mt/mt/www.shearlet.org/papers/OSMRuS.pdf , title=PDF ]
[ Kutyniok, Gitta, and Wang-Q Lim. "Compactly supported shearlets are optimally sparse." Journal of Approximation Theory 163.11 (2011): 1564–1589.
{{cite web, url= http://www.math.tu-berlin.de/fileadmin/i26_fg-kutyniok/wangQ/paper/JAT-D-10-00168_final.pdf , title=PDF ]
[ Kutyniok, Gitta, Jakob Lemvig, and Wang-Q Lim. "Optimally sparse approximations of 3D functions by compactly supported shearlet frames." SIAM Journal on Mathematical Analysis 44.4 (2012): 2962–3017.
{{cite arXiv, title=PDF , eprint=1109.5993 , last1=Kutyniok , first1=Gitta , last2=Lemvig , first2=Jakob , last3=Lim , first3=Wang-Q , class=math.FA , year=2011 ]
[Guo, Kanghui, ]Gitta Kutyniok
Gitta Kutyniok (born 1972) is a German applied mathematician known for her research in harmonic analysis, deep learning, compressed sensing, and image processing. She has a Bavarian AI Chair for "Mathematical Foundations of Artificial Intelligen ...
, and Demetrio Labate. "Sparse multidimensional representations using anisotropic dilation and shear operators." Wavelets and Splines (Athens, GA, 2005), G. Chen and MJ Lai, eds., Nashboro Press, Nashville, TN (2006): 189–201.
{{cite web, url= http://www3.math.tu-berlin.de/numerik/mt/mt/www.shearlet.org/papers/SMRuADaSO.pdf , title=PDF
[Purnendu Banerjee and B. B. Chaudhuri, “Video Text Localization using Wavelet and Shearlet Transforms”, In Proc. SPIE 9021, Document Recognition and Retrieval XXI, 2014 (doi:10.1117/12.2036077).{{cite book, title=Document Recognition and Retrieval XXI , arxiv=1307.4990, last1=Banerjee , first1=Purnendu , last2=Chaudhuri , first2=B. B. , editor1-first=Bertrand, editor1-last=Coüasnon, editor2-first=Eric K, editor2-last=Ringger, chapter=Video text localization using wavelet and shearlet transforms, year=2013 , volume=9021, pages=90210B, doi=10.1117/12.2036077, s2cid=10659099]
[Donoho, David Leigh. "Sparse components of images and optimal atomic decompositions." Constructive Approximation 17.3 (2001): 353–382.
{{cite news, title=PDF , citeseerx = 10.1.1.379.8993]
[Grohs, Philipp and Kutyniok, Gitta. "Parabolic molecules." Foundations of Computational Mathematics (to appear)
{{cite arXiv, title=PDF , eprint=1206.1958 , last1=Grohs , first1=Philipp , last2=Kutyniok , first2=Gitta , class=math.FA , year=2012 ]
[{{cite journal , title=Optimally Sparse Representations of Cartoon-Like Cylindrical Data , journal=The Journal of Geometric Analysis , date=2020-08-10 , last1=Easley , first1=Glenn R. , last2=Guo , first2= Kanghui , last3=Labate , first3=Demetrio , last4=Pahari , first4=Basanta R. , volume=39 , issue=9 , pages=8926–8946 , doi=10.1007/s12220-020-00493-0 , s2cid=221675372 , url=https://link.springer.com/article/10.1007/s12220-020-00493-0 , accessdate=2022-01-22 ]
[{{cite journal , title= Smooth projections and the construction of smooth Parseval frames of shearlets , journal=Advances in Computational Mathematics , date=2019-10-29 , last1=Bernhard , first1=Bernhard G. , last2=Labate , first2=Demetrio , last3=Pahari , first3=Basanta R. , volume=45 , issue=5–6 , pages=3241–3264 , doi=10.1007/s10444-019-09736-3 , s2cid=210118010 , url=https://link.springer.com/article/10.1007/s10444-019-09736-3 , accessdate=2022-01-22 ]
External links
Homepage of Gitta Kutyniok
Homepage of Demetrio Labate
Image processing
Time–frequency analysis
Signal processing
Wavelets