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Beyond Wavelet
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The themes of classical wavelets include terms such as compression and effi?
cient representation. Important features which play a role in analysis of functions
in two variables are dilation, translation, spatial and frequency localization and
singularity orientation. Singularities of functions in more than one variable vary in
dimensionality. Important singularities in one dimension are simply points. In two
dimensions zero and one dimensional singularities are important. A smooth singu?
larity in two dimensions may be a one dimensional smooth manifold. Smooth sin?
gularities in two dimensional images often occur as boundaries of physical objects.
Efficient representation in two dimensions is a hard problem and is addressed in
the first six chapters. The next two chapters return to problems of one dimen?
sion where new important results are given. The final two chapters represent a
transition from harmonic analysis to statistical methods and filtering theory but
the goals remain consistent with those of earlier chapters. We have chosen to
title "Beyond Wavelets". We could have used the title, "Pursuing the Promise of
Wavelets". We briefly describe each chapter.
The lead chapter, "Digital Ridgelet Transform based on True Ridge Functions"
by David Donoho and Georgina Flesia addresses the problem of analyzing the
structure of a function of two real variables. It extends work of Donoho and an
associated group of co-workers. Special credit is due to Emmanuel Candes. Donoho
and Candes have constructed a system called curvelets which gives high-quality
asymptotic approximation of singularities. Passage from their continuum study
to one appropriate for applications requires development of digital algorithms to
implement concepts of the continuum study faithfully. A less obvious proposal
than a standard tensor product basis was made earlier by Donoho emphasizing
"wide-sense" ridgelets with localization properties in radial and angular frequency
domains. Wide-sense ridgelets are no longer of strict ridge form but allow the
possibility of an orthonormal set of elements. The theory is related to that of
the Radon transform and to rotation and scaling of images. At the continuum
level these are natural but for digital data issues are problematic. In this chapter a
definiton of digital ridgelet transform is given. The digital transform has structural
relationships strongly analogous to those of the continuum case. The transform
takes a n-by-n array of data in Cartesian coordinates and expands it by a factor
of 4 in creating a coefficient array. This leaves room for further improvementsChapter 2 is a companion chapter to Chapter 1 and continues the study of digi?
tal implementation of ridgelets with ridgelet packets. The two principal approaches
given are the frequency-domain approach and the Radon approach. In the first
approach a recursive dyadic partition of the polar Fourier domain produces a col?
lection of rectangular tiles followed by a tensor basis of windowed sinusoids in
the angular and radial variables for each tile. In the Radon approach transforma?
tion to the Radon domain is followed by using wavelets in the angular variable and
wavelet packets in the second Radon variable. The Radon isometry is important in
this case. The notion of pseudopolar Fast Fourier Transform and a pseudo Radon
isometry called the normalized Slant Stack are discussed and used. In both cases
analysis of image data relies on directionally oriented waveforms. The wavelet
packet and the local sinusoidal packet bases are generalizations of the original
wavelet systems of elements. Ridgelet packets which follow in the spirit of these
systems are highly orientation selective and bear much the same relationship to
ridgelets as do wavelet packets to wavelets.
In Chapter 3, Frangois Meyer and Raphy Coifman create brushlets to address
the problem of describing an image with a library of steerable wavelet packets.
By careful design of the window of the local Fourier basis, brushlets with very
fast decay are obtained. They note that other direct ionally oriented filter banks
have been constructed which a redundancy factor of 2 or 4. This presents a major
hurdle to computing a sparse image representation. By use of a construction in the
Fourier domain they create wavelet packets which are complex valued functions
with a phase. A key ingredient of the construction is a window used for local
Fourier analysis. The window is required to have very fast decay.
Do and Vetterli study image representation in Chapter 4. An observation that
the curvelet transform is defined in the frequency domain leads to the question: "Is
there as spatial domain scheme for refinement which at each generation, doubles
the spatial resolution as well as the angular resolution?" They propose a filter
bank construction that effectively deals with piecewise smooth images with smooth
contours. The resulting image expansion is a frame composed of contour segments,
which are named contourlets. Their work leads to an effective method to implement
the discrete curvelet transform.
Chan and Zhou open discussion of the ENO-wavelet construction in Chapter 5,
by discussing oscillations which emulate the classical Gibbs’ phenomenon. It has
be discovered that the wavelet Gibbs’ phenomenon is generated by using differ?
ence filters across boundaries of discontinuity. ENO is the acronym for the phrase
essential non-oscillatory which represents an approach for suppression of unwanted
oscillations encountered at discontinuities. Rigorous approximation error bounds
are found to depend on the smoothness of function away from discontinuities when
the ENO approach is used. Several applications of the ENO method are given which
include function approximation, image compression and signal denoising.
An explicit model for Bayesian reconstruction of tomographic data is given by
S. Zhao and H. Cai in Chapter 6. Their approach to image analysis is based on an
interesting analogy to classical mechanics. The intensity of each pixel of an image
is modelled by a transverse motion of a "pixtron". The energy for Bayesian tomographic reconstruction is interpreted as the total kinetic energy of the collection
of pixtrons and log-likelihood is interpreted as potential energy restricting motion
of pixtrons. Finally, the use of the minimization of a log-posterior is analogous to
the principle of least action of classical mechanics. The analogy allows them to
show that a Gaussian Markov random field prior can viewed as the kinetic energy
of free motion of pixtrons. The analogy leads to a novel image prior for Bayesian
tomographic reconstruction based on level-set evolution of an image driven by the
mean curvature motion. Their methods are accompanied by applications to brain
slice images which demonstrate algorithms produced by the model.
Chui and Stockier give extensive description of recent developments of spline
wavelets and frames in Chapter 7. Splines have many of the natural features
required in the original design of I. Daubechies for wavelets which result in beauti?
ful formulas. Vanishing moments reflect smoothness. Design of wavelet frames with
vanishing moments requires a series of new ideas. The authors explain why early
design approaches fail to create wavelets with higher orders of vanishing moments
and then provide steps to recover vanishing moments. The method involves the
notion of vanishing moment recover functions. The theory is extended in the direc?
tion of tight spline-wavelet frames with arbitrary knot sequences that allow stacked
knots. Knot Stacking provides local increase in smoothness and can be applied at
the boundaries of bounded intervals and half line segments. This gives greater
flexibility overcoming standard rigid design features of classical wavelets in which
supports are closely tied to the dilation factor of wavelet families. Multi-wavelets
represent a special case of this more general construction.
Chapter 8, "Afl^ne, Quasi-afl[ine and Co-affine Wavelets", by Washington Uni?
versity the group of researchers, is devoted to fully understanding results of Ron
and Shen. Dilations and translation are two characteristic operators used to define
the wavelet pyramid. The question studied asks whether the order in which dila?
tion and translation are applied is important. A subset of the affine group, used in
the wavelet definition, is the set translations followed by dilation. A second subset
of the aflfine group is the set for which dilation is applied first which is followed
by translation. The effects are dramatically diflferent. Ron and Shen found that by
reversing the order of these operators at a ’half-way’ point in the wavelet pyramid
results in a diflferent set of functions and yet they are sufficient to solve the rep)-
resentation problem. This chapter is devoted to understanding this phenomenon
and it is discovered that the choice of Ron and Shen is essentially optimal.
Benichou and Saito search for relations between the related criteria in Chap?
ter 9. Two studies motivate them. Olshausen and Field pioneered an approach
to imaging which investigates representation of natural images emphasizing sparsity
of representation using a large library of photographs of natural images and
computer experiments to derive a set of basis elements for eflficient representation.
Bell and Sejnowski conducted similar studies in which statistical independence
was the major criterion. The pair of studies suggests both the basis derived for
sparse representation and the basis derived under the independence criterion pro?
duce elements eflficient for capture of edges, orientation and location; all features
prominently studied by image researchers. Their study is based on a modest goalthat begin s with an artificial stochastic process , the spike process , from which
they obtain theorem s which give precis e condition s on the sparsity and statistical
independenc e criteria to select the same basis for the spike process .
S. Akkarakaran and P.P. Vaidyanatha n provid e a new direction from previou s
work in Chapter 10. Standard filter banks fall unde r the theory of design and uni?
form filter banks. A nonunifor m filter bank is one whose channe l decimatio n rates
need not all be equal . Most nonunifor m filter bank design s resul t in approximatio n
or near-perfec t reconstructio n which leaves open theoretica l issues for nonunifor m
filter banks. Their study is restricte d to filter banks with integer decimatio n rates.
A set, S, of integer s satisfies maximal decimatio n if the reciprocal s of the inte?
gers sum to unity. They only study filter banks with integer decimatio n rates.
Their study searche s for necessary and sufficien t condition s on S for existence of
a perfect-reconstructio n filter bank belongin g to some class which uses S as its set
of decimators . They presen t examples with condition s which are either sufficient
or necessary but unfortunatel y different . They focus on rational filter banks and
strengthe n known necessary condition s providin g an importan t step to solving the
problem . However, the basic problem remain s unresolved . Necessary and sufficien t
condition s remain unknown . Thus they open an importan t problem and provid e
insigh t toward solving it.
This volum e is a produc t which was conceive d durin g a conferenc e funded
by the National Science Foundatio n and the Conferenc e Board of Mathematical
Sciences at which David Donoho was the principa l speaker in May of 2000 at the
Universit y of Missouri - St. Louis. The title "Beyond Wavelets" is due to David
Donoho. I thank the NSF and the Universit y of Missouri - St. Louis and the suppor t
staff of the Mathematics Departmen t there . A very special thanks is extended to
David Donoho for his continue d suppor t and understanding .
Many contribute d to the success of that conferenc e and to the origina l idea to
develop "Beyond Wavelets". I give thanks to Charles Chui, Raphy Coiftnan, Ingrid
Daubechies , and Joachim Stockier and Shiying Zhao. I thank the contributor s to
the volum e both for their efforts and understanding . I take responsibilit y for the
delays encountere d and beg your forgiveness . Many more deserv e to be mentione d
to whom I extend my thanks anonymously .
Grant Welland
St. Louis, MO
February , 2003.
https://wap.sciencenet.cn/blog-370458-292816.html
上一篇：话说泛函---Hilbert空间[转]
下一篇：2D-FFT (C code)
cient representation. Important features which play a role in analysis of functions
in two variables are dilation, translation, spatial and frequency localization and
singularity orientation. Singularities of functions in more than one variable vary in
dimensionality. Important singularities in one dimension are simply points. In two
dimensions zero and one dimensional singularities are important. A smooth singu?
larity in two dimensions may be a one dimensional smooth manifold. Smooth sin?
gularities in two dimensional images often occur as boundaries of physical objects.
Efficient representation in two dimensions is a hard problem and is addressed in
the first six chapters. The next two chapters return to problems of one dimen?
sion where new important results are given. The final two chapters represent a
transition from harmonic analysis to statistical methods and filtering theory but
the goals remain consistent with those of earlier chapters. We have chosen to
title "Beyond Wavelets". We could have used the title, "Pursuing the Promise of
Wavelets". We briefly describe each chapter.
The lead chapter, "Digital Ridgelet Transform based on True Ridge Functions"
by David Donoho and Georgina Flesia addresses the problem of analyzing the
structure of a function of two real variables. It extends work of Donoho and an
associated group of co-workers. Special credit is due to Emmanuel Candes. Donoho
and Candes have constructed a system called curvelets which gives high-quality
asymptotic approximation of singularities. Passage from their continuum study
to one appropriate for applications requires development of digital algorithms to
implement concepts of the continuum study faithfully. A less obvious proposal
than a standard tensor product basis was made earlier by Donoho emphasizing
"wide-sense" ridgelets with localization properties in radial and angular frequency
domains. Wide-sense ridgelets are no longer of strict ridge form but allow the
possibility of an orthonormal set of elements. The theory is related to that of
the Radon transform and to rotation and scaling of images. At the continuum
level these are natural but for digital data issues are problematic. In this chapter a
definiton of digital ridgelet transform is given. The digital transform has structural
relationships strongly analogous to those of the continuum case. The transform
takes a n-by-n array of data in Cartesian coordinates and expands it by a factor
of 4 in creating a coefficient array. This leaves room for further improvementsChapter 2 is a companion chapter to Chapter 1 and continues the study of digi?
tal implementation of ridgelets with ridgelet packets. The two principal approaches
given are the frequency-domain approach and the Radon approach. In the first
approach a recursive dyadic partition of the polar Fourier domain produces a col?
lection of rectangular tiles followed by a tensor basis of windowed sinusoids in
the angular and radial variables for each tile. In the Radon approach transforma?
tion to the Radon domain is followed by using wavelets in the angular variable and
wavelet packets in the second Radon variable. The Radon isometry is important in
this case. The notion of pseudopolar Fast Fourier Transform and a pseudo Radon
isometry called the normalized Slant Stack are discussed and used. In both cases
analysis of image data relies on directionally oriented waveforms. The wavelet
packet and the local sinusoidal packet bases are generalizations of the original
wavelet systems of elements. Ridgelet packets which follow in the spirit of these
systems are highly orientation selective and bear much the same relationship to
ridgelets as do wavelet packets to wavelets.
In Chapter 3, Frangois Meyer and Raphy Coifman create brushlets to address
the problem of describing an image with a library of steerable wavelet packets.
By careful design of the window of the local Fourier basis, brushlets with very
fast decay are obtained. They note that other direct ionally oriented filter banks
have been constructed which a redundancy factor of 2 or 4. This presents a major
hurdle to computing a sparse image representation. By use of a construction in the
Fourier domain they create wavelet packets which are complex valued functions
with a phase. A key ingredient of the construction is a window used for local
Fourier analysis. The window is required to have very fast decay.
Do and Vetterli study image representation in Chapter 4. An observation that
the curvelet transform is defined in the frequency domain leads to the question: "Is
there as spatial domain scheme for refinement which at each generation, doubles
the spatial resolution as well as the angular resolution?" They propose a filter
bank construction that effectively deals with piecewise smooth images with smooth
contours. The resulting image expansion is a frame composed of contour segments,
which are named contourlets. Their work leads to an effective method to implement
the discrete curvelet transform.
Chan and Zhou open discussion of the ENO-wavelet construction in Chapter 5,
by discussing oscillations which emulate the classical Gibbs’ phenomenon. It has
be discovered that the wavelet Gibbs’ phenomenon is generated by using differ?
ence filters across boundaries of discontinuity. ENO is the acronym for the phrase
essential non-oscillatory which represents an approach for suppression of unwanted
oscillations encountered at discontinuities. Rigorous approximation error bounds
are found to depend on the smoothness of function away from discontinuities when
the ENO approach is used. Several applications of the ENO method are given which
include function approximation, image compression and signal denoising.
An explicit model for Bayesian reconstruction of tomographic data is given by
S. Zhao and H. Cai in Chapter 6. Their approach to image analysis is based on an
interesting analogy to classical mechanics. The intensity of each pixel of an image
is modelled by a transverse motion of a "pixtron". The energy for Bayesian tomographic reconstruction is interpreted as the total kinetic energy of the collection
of pixtrons and log-likelihood is interpreted as potential energy restricting motion
of pixtrons. Finally, the use of the minimization of a log-posterior is analogous to
the principle of least action of classical mechanics. The analogy allows them to
show that a Gaussian Markov random field prior can viewed as the kinetic energy
of free motion of pixtrons. The analogy leads to a novel image prior for Bayesian
tomographic reconstruction based on level-set evolution of an image driven by the
mean curvature motion. Their methods are accompanied by applications to brain
slice images which demonstrate algorithms produced by the model.
Chui and Stockier give extensive description of recent developments of spline
wavelets and frames in Chapter 7. Splines have many of the natural features
required in the original design of I. Daubechies for wavelets which result in beauti?
ful formulas. Vanishing moments reflect smoothness. Design of wavelet frames with
vanishing moments requires a series of new ideas. The authors explain why early
design approaches fail to create wavelets with higher orders of vanishing moments
and then provide steps to recover vanishing moments. The method involves the
notion of vanishing moment recover functions. The theory is extended in the direc?
tion of tight spline-wavelet frames with arbitrary knot sequences that allow stacked
knots. Knot Stacking provides local increase in smoothness and can be applied at
the boundaries of bounded intervals and half line segments. This gives greater
flexibility overcoming standard rigid design features of classical wavelets in which
supports are closely tied to the dilation factor of wavelet families. Multi-wavelets
represent a special case of this more general construction.
Chapter 8, "Afl^ne, Quasi-afl[ine and Co-affine Wavelets", by Washington Uni?
versity the group of researchers, is devoted to fully understanding results of Ron
and Shen. Dilations and translation are two characteristic operators used to define
the wavelet pyramid. The question studied asks whether the order in which dila?
tion and translation are applied is important. A subset of the affine group, used in
the wavelet definition, is the set translations followed by dilation. A second subset
of the aflfine group is the set for which dilation is applied first which is followed
by translation. The effects are dramatically diflferent. Ron and Shen found that by
reversing the order of these operators at a ’half-way’ point in the wavelet pyramid
results in a diflferent set of functions and yet they are sufficient to solve the rep)-
resentation problem. This chapter is devoted to understanding this phenomenon
and it is discovered that the choice of Ron and Shen is essentially optimal.
Benichou and Saito search for relations between the related criteria in Chap?
ter 9. Two studies motivate them. Olshausen and Field pioneered an approach
to imaging which investigates representation of natural images emphasizing sparsity
of representation using a large library of photographs of natural images and
computer experiments to derive a set of basis elements for eflficient representation.
Bell and Sejnowski conducted similar studies in which statistical independence
was the major criterion. The pair of studies suggests both the basis derived for
sparse representation and the basis derived under the independence criterion pro?
duce elements eflficient for capture of edges, orientation and location; all features
prominently studied by image researchers. Their study is based on a modest goalthat begin s with an artificial stochastic process , the spike process , from which
they obtain theorem s which give precis e condition s on the sparsity and statistical
independenc e criteria to select the same basis for the spike process .
S. Akkarakaran and P.P. Vaidyanatha n provid e a new direction from previou s
work in Chapter 10. Standard filter banks fall unde r the theory of design and uni?
form filter banks. A nonunifor m filter bank is one whose channe l decimatio n rates
need not all be equal . Most nonunifor m filter bank design s resul t in approximatio n
or near-perfec t reconstructio n which leaves open theoretica l issues for nonunifor m
filter banks. Their study is restricte d to filter banks with integer decimatio n rates.
A set, S, of integer s satisfies maximal decimatio n if the reciprocal s of the inte?
gers sum to unity. They only study filter banks with integer decimatio n rates.
Their study searche s for necessary and sufficien t condition s on S for existence of
a perfect-reconstructio n filter bank belongin g to some class which uses S as its set
of decimators . They presen t examples with condition s which are either sufficient
or necessary but unfortunatel y different . They focus on rational filter banks and
strengthe n known necessary condition s providin g an importan t step to solving the
problem . However, the basic problem remain s unresolved . Necessary and sufficien t
condition s remain unknown . Thus they open an importan t problem and provid e
insigh t toward solving it.
This volum e is a produc t which was conceive d durin g a conferenc e funded
by the National Science Foundatio n and the Conferenc e Board of Mathematical
Sciences at which David Donoho was the principa l speaker in May of 2000 at the
Universit y of Missouri - St. Louis. The title "Beyond Wavelets" is due to David
Donoho. I thank the NSF and the Universit y of Missouri - St. Louis and the suppor t
staff of the Mathematics Departmen t there . A very special thanks is extended to
David Donoho for his continue d suppor t and understanding .
Many contribute d to the success of that conferenc e and to the origina l idea to
develop "Beyond Wavelets". I give thanks to Charles Chui, Raphy Coiftnan, Ingrid
Daubechies , and Joachim Stockier and Shiying Zhao. I thank the contributor s to
the volum e both for their efforts and understanding . I take responsibilit y for the
delays encountere d and beg your forgiveness . Many more deserv e to be mentione d
to whom I extend my thanks anonymously .
Grant Welland
St. Louis, MO
February , 2003.
https://wap.sciencenet.cn/blog-370458-292816.html
上一篇：话说泛函---Hilbert空间[转]
下一篇：2D-FFT (C code)
