Jacket's Wavelets


LOCAL
Nondecimated Complex Wavelet Spectral Tools with Applications
Taewoon Kong and Brani Vidakovic
In this paper we propose spectral tools based on nondecimated complex wavelet transforms
implemented by their matrix formulation. This nondecimated complex wavelet spectra utilizes
both real and imaginary parts of complexvalued wavelet coefficients via their modulus and phases.
A structural redundancy in nondecimated wavelets and a componential redundancy
in complex wavelets act in a synergy when extracting waveletbased informative descriptors.
In particular, we suggest an improved way of separating signals and images based on their
scaling indices in terms of spectral slopes and information contained in the phase in order
to improve performance of classification. We show that performance of the proposed method is
significantly improved when compared with procedures based on standard versions of wavelet
transforms or on realvalued wavelets. It is also worth mentioning that the matrixbased
nondecimated wavelet transform can handle signals of an arbitrary size and in 2D case,
rectangular images of possibly different and nondyadic dimensions. This is in contrast to
the standard wavelet transforms where algorithms for handling objects of nondyadic dimensions
requires either data preprocessing or customized algorithm adjustments. To demonstrate the use
of defined spectral methodology we provide two examples of application on realdata problems:
classification of visual acuity using scaling in pupil diameter dynamic in time and diagnostic
and classification of digital mammogram images using the fractality of digitized images of
the background tissue. The proposed tools are contrasted with the traditional wavelet based
counterparts.
Full version of the paper is available at arXiv:
Full Version.
In the spirit of reproducible research, Taewoon compiled
the package with MATLAB codes used for calculations WavmatCND.zip.
Empirical Waveletbased Estimation for Nonlinear Additive Regression
Models.
German A. Schnaidt Grez and
Brani Vidakovic
Additive regression models are actively researched in the statistical field because of their usefulness in the
analysis of responses determined by nonlinear relationships with multivariate predictors. In this kind of
statistical models, the response depends linearly on unknown functions of predictor variables and typically,
the goal of the analysis is to make inference about these functions.
In this paper, we consider the problem of Additive Regression with random designs from a novel viewpoint:
we propose an estimator based on an orthogonal projection onto a multiresolution space using empirical
wavelet coefficients that are fully data driven. In this setting, we derive a meansquare consistent estimator
based on periodic wavelets on the interval [0, 1]. For construction of the estimator, we assume that the joint
distribution of predictors is nonzero and bounded on its support; We also assume that the functions belong
to a Sobolev space and integrate to zero over the [0,1] interval, which guarantees model identifiability and
convergence of the proposed method. Moreover, we provide the L2 risk analysis of the estimator and derive
its convergence rate.
Theoretically, we show that this approach achieves good convergence rates when the dimensionality of
the problem is relatively low and the set of unknown functions is sufficiently smooth. In this approach, the
results are obtained without the assumption of an equispaced design, a condition that is typically assumed
in most waveletbased procedures.
Finally, we show practical results obtained from simulated data, demonstrating the potential applicability
of our method in the problem of additive regression models with random designs.
Keywords: Wavelets, nonparametric regression, functional data analysis, robust statistical modeling
Full version of the paper with Appendices and Proofs is available at arXiv:
Full Version.
Waveletbased scaling indices for breast cancer diagnostics
Tonya Roberts, Mimi Newell, William Auffermann, and Brani Vidakovic
Mammography is routinely used to screen for breast cancer (BC). However, the radiological
interpretation of mammogram images is complicated by the heterogeneous nature of normal breast
tissue and the fact that cancers are often of the same radiographic density as normal tissue.
In this work, we use wavelets to quantify spectral slopes of BC cases and controls and
demonstrate their value in classifying images. In addition, we propose asymmetry statistics
to be used in forming features which improve the classification result. For the best
classification procedure, we achieve approximately 77% accuracy (sensitivity=73%, specificity=84%)
in classifying mammograms with and without cancer.
Manuscript can be found here,
and
the refrerence is:
Roberts, T., Newell, M., Auffermann, W., and Vidakovic, B. (2017).
Waveletbased scaling indices for breast cancer
diagnostics.
Statistics in Medicine, 36, 12, 19892000, DOI: 10.1002/sim.7264
ESTIMATION OF THE HURST EXPONENT USING TRIMEAN ESTIMATORS ON NONDECIMATED WAVELET COEFFICIENTS
Chen Feng and Brani Vidakovic
Hurst exponent
is an important feature summarizing the noisy highfrequency data
when the inherent scaling pattern cannot be described by standard
statistical models. In this paper, we study the robust estimation
of Hurst exponent by applying a general trimean estimator on nondecimated
wavelet coefficients of the transformed data. Our waveletbased methods
provide a robust way to estimate Hurst exponent and increase the prediction
precision especially when there exists outlier coefficients, outlier
multiresolution levels, and within level dependencies. The properties
of the proposed Hurst exponent estimators are studied both theoretically
and numerically. Compared with other standard waveletbased methods
(Veitch and Abry (VA) method, Soltani, Simard, and Boichu (SSB)
method, medianbased estimators MEDL and MEDLA, and Theiltype (TT)
weighted regression method), our methods reduce the variance of the
estimators by not sacrificing the prediction precision in most cases.
Supplementary material can be found here.
A preliminary version of this paper was posted on
arXiv https://arxiv.org/abs/1709.08775.
ROBUST WAVELETBASED ASSESSMENT OF SCALING WITH APPLICATIONS
Erin K. Hamilton, Minkyoung Kang, Seonghye Jeon, Pepa Ramírez Cobo, Kichun Sky Lee, and Brani Vidakovic
A number of approaches have dealt with statistical assessment of selfsimilary,
and many of those are based on multiscale concepts. Most rely on certain distributional
assumptions which are usually violated by real data traces, often characterized by
large temporal or spatial mean level shifts, missing values or extreme observations.
A novel, robust approach based on Theiltype weighted regression is proposed for
estimating selfsimilarity in twodimensional data (images).
The method is compared to two traditional estimation techniques
that use wavelet decompositions; {ordinary least squares} (OLS)
and AbryVeitch bias correcting estimator (AV). As an application,
the suitability of the robust approach is illustrated in the classification
of digitized mammogram images as cancerous or noncancerous. The diagnostic
employed here is based on the properties of image backgrounds, which is
typically an unused modality in breast cancer screening. Classification
results show nearly 68% of accuracy, varying slightly with the choice
of wavelet basis, and the range of multirseolution levels used.
This paper is under revision for Communicatons in Statistics.
Because of the size of the paper the part of simulations results is deferred to this electronic
Appendix B .
MATLAB codes for generating 1D and 2D fractional Brownian motions are:
MakeFBM.m and
MakeFBM2D.m , respectively.
The 1D fBm is generated by scaling the modulus and randomizing the phase
of gaussians in FFT, while
the 2D fBm is authored by Olivier Barriere.
WavmatND: A MATLAB Package for NonDecimated Wavelet Transform and its Applications
Minkyoung Kang and Brani Vidakovic
A nondecimated wavelet transform (NDWT) is a popular version of wavelet transforms
because of its many advantages in applications. The inherent redundancy of this
transform proved beneficial in tasks of signal denoising and scaling assessment.
To facilitate the use of NDWT, we built a MATLAB package, WavmatND, in which transforms are
done by matrix multilication, and which has three
novel features: First, for signals of moderate size the proposed method reduces computation
time of the NDWT by replacing repetitive convolutions with matrix multiplications.
Second, submatrices of an NDWT matrix can be rescaled, which enables a straightforward
inverse transform. Finally, the method has no constraints on a size of the input signal
in one or in two dimensions, so signals of nondyadic length and rectangular twodimensional
signals with nondyadic sides can be readily transformed. We provide illustrative
examples and a tutorial to assist users in application of this standalone package.
The manuscript can be found HERE and
the package with all MATLAB codes are zipped in WavmatND.zip.
CHARACTERIZING EXONS AND INTRONS BY REGULARITY OF NUCLEOTIDE STRINGS
Tonya Woods, Thanawadee Preeprem, Kichun Lee, Woojin Chang, and Brani Vidakovic
Translation of nucleotides into a numeric form has been approached in many ways and has allowed researchers to investigate the properties of proteincoding sequences and noncoding sequences. Typically, more pronounced longrange correlations and increased regularity were found in introncontaining genes and in nontranscribed regulatory DNA sequences, compared to cDNA sequences or intronless genes. The regularity is assessed by
spectral tools defined on numerical translates.
In most popular approaches of numerical translation the resulting spectra depend on the assignment of numerical values to nucleotides.
Our contribution is to propose and illustrate a spectra which remains invariant to the translation rules used in traditional approaches.
We outline a methodology for representing sequences of DNA nucleotides as numeric matrices in order to analytically investigate important structural characteristics of DNA. This representation allows us to compute the 2dimensional wavelet transformation and assess regularity characteristics of the sequence via the slope of the wavelet spectra. In addition to computing a global slope measure for a sequence, we can apply our methodology for overlapping sections of nucleotides to obtain an ``evolutionary slope."
To illustrate our methodology, we analyzed 376 gene sequences from the first chromosome of the honeybee.
For the genes analyzed, we find that introns are significantly more regular
(lead to more negative spectral slopes) than exons, which agrees with the results
from the literature where regularity is measured on ``DNA walks."
However, unlike DNA walks where the nucleotides are assigned numerical
values depending on nucleotide characteristics (purinepyrimidine, weakstrong
hydrogen bonds, ketoamino, etc.) or other spatial assignments, the proposed spectral
tool is invariant to the assignment of nucleotides. Thus, ambiguity in numerical
translation of nucleotides is eliminated.
This paper is open access
and can be found HERE .
MATLAB files supporting the paper are:
evcdsplot.m and
evdnaslopec.m.
These two files are used for computing the cumulative evolutionary slope of a sequence
and for creating the plots with exons, introns, and combination regions.
The "evdnaslopec.m" program requires the use of WavMat.m function.
Reference:
Woods, T., Preeprem, T., Lee, K., Chang, W., and Vidakovic, B. (2016).
Characterizing Exons and Introns by Regularity of
Nucleotide Strings. Biology Direct, 11, 6, 117;
DOI: 10.1186/s1306201601087
A Constrained Wavelet Smoother for Pathway Identification Tasks in Systems Biology
Sepideh Dolatshahi, Brani Vidakovic, and Eberhard O. Voit
Metabolic time series data are being generated with increasing frequency, because they
contain enormous information about the pathway from which the metabolites derive. This
information is not directly evident, though, and must be extracted with advanced computational
means. One typical step of this extraction is the estimation of slopes of the time courses
from the data. Since the data are almost always noisy, and the noise is typically amplified
in the slopes, this step can become a critical bottleneck. Several smoothers have been proposed
in the literature for this purpose, but they all face the potential problem that smoothed time series
data no longer correspond to a system that conserves mass throughout the measurement time period.
To counteract this issue, we are proposing here a smoother that is based on wavelets and,
through an iterative process, converges to a massconserving, smooth representation of the
metabolic data. The degree of smoothness is user defined. We demonstrate the method with some
didactic examples and with the analysis of actual measurements characterizing the glycolytic
pathway in the dairy bacterium Lactococcus lactis. MATLAB code for the constrained smoother
is available as a supplement.
Paper is here,
and the reference is:
Dolatshahi, S., Vidakovic, B., and Voit, E. O. (2014).
A constrained wavelet smoother for pathway identification tasks in systems biology.
Computers and Chemical Engineering, 71, 728733.
doi: 10.1016/j.compchemeng.2014.07.019.
DENOISING BY BAYESIAN MODELING IN THE DOMAIN OF DISCRETE SCALE MIXING 2D
COMPLEX WAVELET TRANSFORMS
Norbert Remenyi, Orietta Nicolis, Guy Nason, and Brani Vidakovic
Wavelet shrinkage methods that use complexvalued wavelets provide additional insights
to shrinkage process compared to standardly used realvalued wavelets.
Typically, a locationtype statistical model with an additive noise is posed on the observed wavelet coefficients
and the true signal/image part is estimated as the location parameter. Under such
approach the wavelet shrinkage becomes equivalent to a location estimation in the wavelet domain.
The most popular type of models imposed on the wavelet coefficients are Bayesian. This popularity is
well justified: Bayes rules are typically well behaved shrinkage rules, prior information about the signal can be
incorporated in the shrinkage procedure, and adaptivity of Bayes rules can be achieved by
datadriven selection of model hyperparameters. Several papers considering Bayesian wavelet
shrinkage with complex wavelets are available. For example, Jean Marc Lina and coauthors
focus on image denoising, in which the phase of the observed wavelet coefficients is preserved,
but the modulus of the coefficients is shrunk by a Bayes rule.
The procedure introduced in Barber and Nason in 2004 modifies both the phase and modulus of
wavelet coefficients by a bivariate shrinkage rule.
We propose a Bayesian model in the domain of a complex scalemixing discrete unitary, compactly
supported wavelets that generalizes the method in Barber and Nason to 2D signals. In estimating
the signal part the model to allowed to modify both phase and modulus. The choice of wavelet transform
is motivated by the symmetry / antisymmetry of decomposing wavelets, which is possible only in the
complex domain under condition of orthogonality (unitarity) and compact support.
Symmetry is considered a desirable property of wavelets, especially when dealing with images.
The 2D discrete scale mixing wavelet transform is computed by left and rightmultiplying the
image by a wavelet matrix W and its Hermitian transpose, respectively.
Mallat's algorithm to perform this task is not used, but it is implicit in the construction of matrix $W.$
The resulting shrinkage procedures cSMEB and cMOSMEB are based on empirical Bayes approach and
utilize nonzero covariances between real and imaginary parts of the wavelet coefficients.
We discuss the possibility of phasepreserving shrinkage in this framework.
Overall, the methods we propose are calculationally efficient and provide excellent denoising
capabilities when contrasted with comparable and standardly used waveletbased techniques.
A MATLAB toolbox developed by Norbert Remenyi
cSMEB2.zip
illustrates cSMEB and cMOSMEB shrinkage.
Reference:
Remenyi, N., Nicolis, O., Nason, G., and Vidakovic, B. (2014).
Image Denoising With 2D ScaleMixing Complex Wavelet Transforms.
IEEE Transactions on Image Processing, 23, 12, 51655174.
LAMBDA NEIGHBORHOOD WAVELET SHRINKAGE
A waveletbased denoising methodology based on total energy of a neighboring
pair of coefficients plus their ‘‘parental’’ coefficient is proposed. The model is based on a Bayesian
hierarchical model using a contaminated exponential prior on the total mean energy in a
neighborhood of wavelet coefficients. The hyperparameters in the model are estimated by
the empirical Bayes method, and the posterior mean, median and Bayes factor are obtained
and used in the estimation of the total mean energy. Shrinkage of the neighboring coefficients
are based on the ratio of the estimated and the observed energy. It is shown that the
methodology is comparable and often superior to several existing and established wavelet
denoising methods that utilize neighboring information, which is demonstrated by extensive
simulations on a standard battery of test functions. An application to realword data
set from inductance plethysmography is also considered.
A MATLAB toolbox developed by Norbert Remenyi
LNWS.zip
illustrates the methodology.
The toolbox supports the manuscript:
Lambda Neighborhood Wavelet Shrinkage, by Norbert Remenyi and Brani
Vidakovic
Reference:
Reményi, N. and Vidakovic, B. (2013).
Λneighborhood wavelet shrinkage.
Computational Statistics \& Data Analysis, 57, 1, 404416,
doi:10.1016/j.csda.2012.07.008
WAVELET SHRINKAGE WITH DOUBLE WEIBULL PRIORS
Bayesian wavelet shrinkage standardly employs pointmass at zero contamination priors
for the signal part in nonparametric regression problems. In this paper a competitive
methodology is achieved with a simple prior based on Double Weibull distribution
without point mass at zero, but with a singularity at 0.
A MATLAB toolbox developed by Norbert Remenyi
DWWS.zip
illustrates the methodology.
The toolbox supports the manuscript:
Wavelet Shrinkage with Double Weibull Prior, by Norbert Remenyi and Brani
Vidakovic
Reference:
Reményi, N. and Vidakovic, B. (2015).
Wavelet Shrinkage with Double Weibull Prior.
Communications in Statistics  Simulation and Computation,
44, 1, 88104.
DENSITY ESTIMATION WHEN DATA ARE SIZEBIASED: WAVELETBASED MATLAB TOOLBOX
Often researchers need to estimate the density in the presence of
sizebiased data. The waveletbased MATLAB toolbox
biased.zip
performs debiasing and estimattes density by smoothed linear projection wavelet esimator.
The toolbox supports the manuscript:
Wavelet Density Estimation for Stratified SizeBiased Sample, by Pepa Ramirez and Brani
Vidakovic.
Reference:
Ramírez, P. and Vidakovic, B. (2010).
Wavelet density estimation for stratified sizebiased sample.
Journal of Statistical Planning and Inference, 140, 2, 419  432.
LPM: Bayesian Wavelet Thresholding based on Larger Posterior Mode
This project explores the thresholding rules induced by a variation of
the Bayesian MAP principle. The MAP rules are Bayes actions that
maximize the posterior. Under the proposed model the posterior is
neither unimodal or bimodal. The proposed rule is thresholding and always
picks the mode of the posterior larger in absolute value, thus the name
LPM. We demonstrate that the introduced shrinkage performs
comparably to several popular shrinkage techniques. Exact risk
properties of the thresholding rule are explored. We
provide extensive simulational analysis and apply the proposed
methodology to reallife experimental data coming from the field
of Atomic Force Microscopy (AFM).
You could try the LPM thresholding if your MATLAB has access to
WaveLab Module.
MATLAB mfiles, MATHEMATICA nbfiles, data, and figures are zipped in the following archive file:
The manuscript (draft version in PDF) describing the introduced thresholding is here:

Larger Posterior Mode Wavelet Thresholding and Applications.
The authors are Luisa Cutillo, Yoon Young Jung, Fabrizio Ruggeri, and Brani Vidakovic.
The provided files in LPM.zip are sufficient to fully
REPRODUCE this manuscript.
Reference:
Cutillo, L., Jung, Y.Y., Ruggeri, F., and Vidakovic, B. (2008).
Larger Posterior Mode Wavelet Thresholding and Applications,
Journal of Statistical Planning and Inference, 138, 37583773.
Hunting for Dominant StraightLine Features in Nanoscale Images Using Wavelets
Ilya Lavrik, PhD Graduate in Statistics at ISyE,
has developed Matlab Toolbox which searches for significant straight line alignments of
molecular structures in nanoscale images.
The methodology is based on Hough and wavelet transforms.
The software, excellent front end, and manual are available for download:
 nsia.zip NANOSCALE IMAGE ANALYSIS 
zipped archive of NanoLab, a matlab suite with an excellent front.
 images.zip Several Genuine Nanoscale Images
needed for the NSIA. The file is about 13.5MB in size.
 manual.pdf Manual for NSIA
Comments welcome!
Partial support of this project by the
Georgia Institute of Technology Molecular Design Institute, under
prime contract N000149511116 from the Office of Naval
Research. Partial support for this work was also provided by
National Security Agency Grant NSA E2460R at ISyE.
The manuscript (draft version in PDF) describing the methodology can be found here:
 LINEAR FEATURE IDENTIFICATION AND INFERENCE IN NANOSCALE
IMAGES
BLFDR and BaFDR: Bayesian Wavelet Thresholding based on False Discovery Rate
Here are matlab programs authored by Ilya Lavrik, PhD Graduate in Statistics at ISyE,
that implement two versions of wavelet thresholding based on Bayesian
False Discovery Rate: BLFDR  method that uses Bayesian model and matches generalized Efron and Tibshirani's
LFDR in the wavelet domain and BaFDR  method that is based on ordering of posterior probabilities of
hypotheses that coefficients are ``not interesting.''
The programs below require WAVELAB Module.
 bafdr.m BaFDR function.
 blfdr.m BLFDR function.
 blfdr_fixed.m BLFDR fixed function.
 b_factor.m Bayes Factor from BAMS model.
 example1.m Example that plots smoothed versions of
test signals using BLFDR, BLFDRfixed, and BaFDR.
 example2.m Example that calculates MSE, Var and Bias
for noisy test signal estimation using BLFDR, BLFDRfixed, and BaFDR.
The manuscript (draft version in PDF) describing the introduced thresholding is here:
 Bayesian False Discovery Rate Shrinkage.
The authors are Ilya Lavrik, Yoon Young Jung, Fabrizio Ruggeri, and Brani Vidakovic.
THE WAVELETS PUZZLE
Fun with Wavelets!
A rare R. J. Journet dexterity glass top puzzle*.
*Journet Dexterity Puzzles are glass top dexterity puzzles,
made between 1890 and 1960 by an English manufacturer called R. J. Journet
(affectionately known as RJ's).
The diversity of subjects in these puzzles is humungous, ranging
from an R.A.F trip to bomb Hitler in Berlin to Alice in Wonderland's Tea Party.
To complete a puzzle you have to shake,
rattle or roll it. They are called dexterity puzzles
because you have to be dexterous (skillful in using hands) to complete them.
Some are easy, some difficult, and some next to impossible.
WAVELET MATRIX IN MATLAB

Here a matlab routine to form a matrix performing discrete
orthogional wavelet transformation. Once the matrix W
is generated, the transformation d is obtained
by multiplying the data vector y, d=W * y.
Of course, y = W' * d.
The mfile
WavMat.m
should be put in yourmatlabpath/toolbox/wavelab/Orthogonal/,
although if you know the filter, no wavelab is needed  the mfile is
a stand alone.
> dat = [1 0 3 2 1 0 1 2];
> filter = [sqrt(2)/2 sqrt(2)/2];
> W = WavMat(filter,2^3,3);
> wt = W * dat' %should be [sqrt(2)  sqrt(2)  1 1  ...
% 1/sqrt(2) 5/sqrt(2) 1/sqrt(2)  1/sqrt(2) ]
> data = W' * wt % should return you to the 'dat'
If the matrix size exceeds 1024 x 1024, an average PC is getting slow.
WavMat.m could be optimized.
[In Ver. 1.2 built 12/1/04, functions 'modulo' and 'reverse' replaced by builtin
functions. Thanks to Mr Deniz Sodiri for pointing out the original posting was not
stand alone.]
Some readers asked for the algorithm: Here are
three pages
describing it.
AN OPEN PROBLEM OR EASY EXERCISE?

Recently, working on Convex Rearrangements of wavelet filtered
selfsimilar processes I looked at compactly
supported orthogonal wavelet filters and for some
"empirical evidence"
could not find a proof. Let me know if you have an answer.
A NICE 2D DATA TO NOISE/DENOISE
Here is a 2D data set free to use for tasks of image wavelet processing.
This is a 3072 x 2048 (3.2MB, jpg) digital photo of
von Klaus. Von Klaus is a two year old purebreed
[AKC WR021286/04] Doberman Pinscher var. Warlock born in Marietta, Georgia.
Although he looks quite intimidating, von Klaus is a gentle, playful, and devoted dog.
The big (> 6 megapixel) JPG photo is imported to MATLAB using
klaus.m mfile
and 6 gray scale images of various dimensions are made.
Here are all the 6 as an EPS file. (>11MB)
BOOTSTRAPPING WAVELETS
 That is to say:
WAVESTRAPPING.
Also, a noname article in Popular Mechanics [June 2004],
but the two guys look very much
like my graduate student Bin Shi and myself!
The same from
BRASIL!
LOCAL WAVELET RESEARCH PAPERS
WAVELETS AND STATISTICS: A REPOSITORY OF MANUSCRIPTS
This page was doemant from 1999. Since some good folks wanted their paper linked  I decided to
keep this page updated!
December, 1417, 2004
at Consiglio Nazionale
delle Ricerche
Istituto di Matematica Applicata
e Tecnologie Informatiche
(Milano Department  formerly CNRIAMI)
BAMSLP (Bayesian Adaptive Multiresolution Shrinker of Log Periodogram)
The matlab files that implement the BAMSLP shrinker and a few
examples of its use are zipped into archive
BAMSP.zip .
The software is tested with MATLAB6.5.
The the theory behind the software and a paper
describing Bayesianly induced wavelet shrinkage
of LogPeriodogram, can be found
HERE
in the PDF format.
Discrete Complex Orthogonal Wavelet Transformation
Here is some history.
We started these mfiles when
JeanMarc Lina from University of Montreal
was visiting Duke University.
At that time JML, being a complex wavelet guru, guided
Gaby Katul and me
how to do forward complex DISCRETE wavelet transformation.
We originally used complex filters from Lina's papers and then discovered that
Barry G. Sherlock
now at UNCCharlotte
made an mfunction a la Donoho's MakeONFilter.m for producing Daubechies complex
filters.
Getting into the complex wavelet domain was easy compared to returning back to the ``time'' domain.
We made an mfile for inverse transformation but it was less than perfect...(a nice
way to say it did not work as it was supposed to).
Claudia Angelini, a bright graduate student visiting GaTech from
Napoli's CNR, took a
look at our pluses and minuses and fixed the inverse transform in a second.
So here they are:
FWTC_PO.m will mimic FWT_PO with complex filters and
return the complex discrete wavelet transformation, two vectors [re, im] for the real and immaginary parts.
IWTC_PO.m will get you back from the complex wavelet
domain to the space of original discrete data.
Finally, complex Daubechies wavelet filters are made by Sherlock's
MakeCONFilter.m.
To make Complex tools work just add FWTC_PO.m, IWTC_PO.m,
and MakeCONFilter.m to ~/wavelab/Orthogonal/.
2D Continuous Wavelet Transformations
In the Spring of 2001 Xiaoming Huo and myself teamtaught a graduate course on wavelets
at GaTech. We had about 15 graduate students coming from various
Tech's departments.
Heejong Yoo, graduating PhD student from ECE, was an excellent programmer interested in
implementing 2D Continuous Wavelet Transformation in his classproject.
The idea came from commercial software Crittech Psilets 3.0; we decided to make a free
clone!
The theory behind the transformation is trivial: One (listably) multiplies
the 2D object with the sampled fixed level 2D wavelet in the Fourier domain and
then Fourierinverts the product!
Heejong's project is a standalone MATLAB program (no wavelab needed) with an excellent GUI.
Zipped directory with all files needed to run the CWT2D is
Project.zip
and the PPT presentation of the project is:
Cont2DWT.ppt .
Only 2D Mexican hat is available right now.
If you prefer the Wavelab environment, than you can add the function
CWT2.m to ~/wavelab/Continuous/
3D Discrete Wavelet Transformation (Orthogonal, Tensor Product)
This pair of transformations naturally generalizes WaveLab's FWT2_PO.m and IWT2_PO.m. This is a
part of waveletproject of Vicki Yang, gifted graduate student at ISyE who took a course on
wavelets with me. She was interested in wavelet processing of 3D signals with applications.
The forward and inverse transformations are:
FWT3_PO.m, for transforming the data to the wavelet domain, and
IWT3_PO.m, for inversetransforming the data back to the time domain.
The function needed here is
cubelength.m that is a 3D counterpart of Donoho's
quadlength.m utilized by the 2D pair.
You will see that transforms are conceptually and algorithmically easy, and it would
be quite starightforward to construct FWT4_PO, FWT5_PO, ... and their inverses.
Now, both FWT3_PO and IWT3_PO transformations act on 3D data sets and such objects are difficult to visualize.
We made several data
related programs.
(i)
Make3DData.m will make 3D ball with inscribed octahedron.
Both bodies the ball and the octahedron are inscribed in a cube of (dyadic) side N.
The noise can be added to both boundaries and interiors of objects.
(ii) DDD2Movie.m will make a movie from the 3D object
taking frames along the dimension of choice. This is handy for viewing the 3D objects via their
2D cuts.
(iii) A small script
test3d.m will take a 64 x 64 x 64 noisy object, view it,
transfer it to the wavelet domain, view it again, threshold the object, view it, return the thresholded
object to the original domain, and
view it. For some misterious reasons, DDD2Movie.m will show the movie itself while recording,
and builtin matlab function
movie will show the movie twice! Be ready to watch the objects 4 x 3 = 12 times...
To make 3D tools work just add FWT3_PO.m, IWT3_PO.m,
and cubelength.m to ~/wavelab/Orthogonal/
This extension is final project in an undergraduate wavelet research course submitted by
graduating ISyE student, Daphne Lai. Daphne added more Daubechies', Symmlets, and Coiflets,
as well as some new filters. All added filters are numerically stable.
Dyadlike 2D Tools
Standard WaveLab mfunction dyad.m extracts particular level in the discrete
wavelet transformation. If, for example, n=2^J, and the Discrete Wavelet Transfirmation is
WT, the finest level is indexed by dyad(J1), and extracted from WT as WT(dyad(J1)).
I needed dyadlike tools for 2D wavelet transformations. A simple generalization is
dyad2.m and it needs in addition to dyad.m
the `complement' function dyadc.m .
The following matlab script shows use of dyad2:
>> pict = MakeImage('StickFigure',128);
>> wf = MakeONFilter('Haar',1);
>> wpict = FWT2_PO(pict, 5, wf);
>> [diagx, diagy] = dyad2(6,'d');
>> diag_det = wpict(diagx, diagy);
>> imagesc(diag_det)
Some Shortcomings of WaveLab.
There is one problem with FWT_PO.m and its inverse in WaveLab that needs a fix!
The problem propagates to other transformations, notably 2D, etc.
It is well known that any scaling filter H=(h_0, ... ,h_N) can be matched with many
quadrature mirror filters  high pass counterparts G. ``Wavelet polygamy 
one father and many mothers.''
For example $g_n = (1)^{n+x} h_{y  n},$ where $x=0,1$ and
$y$ is arbitrary integer, is a valid QM wavelet filter.
And not all the wavelet bases share the same ``proper''
translation and sign of G defined by $x$ and $y$
Not all H filters start
with $h_0$! For example, proper start for Coiflet 1 (6 tap filter) is
$h_{2}.$
WaveLab does not allow for such flexibility. And although
the reconstructions are perfect, the wavelet domain objects are
circularly shifted.
For example, if a period of a SINE function is sampled
and transformed by wavelet transformation of depth 3 (log(n)L=3), the resulting
transformation should result in scaling coefficients that are degraded SINE function.
This example shows that improper filter alignment causes smoothpart SINE
to shift. To see this, please run the
exercise under WaveLab.
One may ask, why should we care when the reconstruction is perfect?
The proper alignment is critical, because of simulational aspects of wavelets.
Often one
starts with the wavelet domain, feeds the empty levels with (simulated) coefficients
and reconstructs. And if the alignment is not proper various
problems and anomalies can occur.
This could be an interesting project for a devoted grad student!
Please take a look for an excellent solution of this alignment problem by
UviWave software
from University of Vigo.
Unusual procedure is MirrorFilt.m. In it the high pass filter
G is formed as
g = ( (1).^(1:length(h)) ) .* h;
This leads to an orthogonal transformation, but more common
filter g is obtained by
g =  reverse( (1).^(1:length(h)) .* h );
In my version of Wavelab I modified MirrorFilt.m.
DaubechiesLagarias Algorithm in Matlab
Calculate the value of \phi_{jk}(x_0) or \psi_{jk}(x_0) at ANY
point x_0 for ANY orthonormal basis at ANY precision without
going through Mallat's algorithm.
The blurb DL.pdf describes the algorithm.
The matlab programs used are:
MakePollen1.m,
MakePollen2.m,
Phijk.m,
Psijk.m, and mscript
DLtest.m.
Linear Regression Estimator where the kernel is the defined by wavelets...Soon!
FWT2_POE and IWT2_POE for Rectangular Images of Dyadic Sides
In 2D tensor product wavelet transformations, traditionally the
inputs are square images of a dyadic side. Since performing the 2D transformation amounts
to subsequent application of 1D transformations on rows and columns of an image, the
restriction to square dimensions is inessential.
Here are slight extensions of standard wavelab's FWT2_PO.m and IWT2_PO.m, the functions:
FWT2_POE.m and
IWT2_POE.m .
The pair FWT2_POE, IWT2_POE will do the 2D wavelet transformation and its inverse
on rectangular images with dyadic sides.
The mfile quadlength.m needed by FWT2_PO.m and IWT2_PO.m should be replaced by
pow2length.m .
All three files should reside in ...\Wavelab\Orthogonal\ directory.
Now, take a look how the rectangular Lena ( lena21.eps or
lena21.pdf) looks
in the wavelet domain, ( lena21w.eps or
lena21w.pdf).
Data file is lena21.mat.
The choice of size 256 x 512, rather than more interesting 512 x 256 or quite
exciting 1024 x 256, was made by
flipping a coin;).
FWT_PO and inverse for inputs of ANY SIZE
Practical Hints on Running
WaveLab (when the names collide).
Wavelethistory Curiosity
A wavelethistory curiosity I found interesting.
Chapter 4 spanning 70 pages of the book ``Time Series
Analysis and Applications''
by Enders A. Robinson is titled: Wavelet Composition
of Time Series.
The curios thing is that the book is published in 1981!!!
Robinson's wavelets indeed have some of the wavelet spirit.
A quote from page 84:
``...The wavelets arrive in succession, and each wavelet
eventually dies out. The wavelets all have the same basic
form and shape, but the strength or impetus of each wavelet
is random and uncorrelated with the strength of the other
wavelets...
...Despite the foreordained death of any individual wavelet,
the timeseries does not die. The reason is that a new wavelet
is born each day to take the place of the one that does die.
On any given day, the timeseries is composed of many living
wavelets, all of a different age,some young, others old.''
The chapter then formally describes the theory and
practice of Robinson's atomic decompositions.
Reference: Robinson, Enders (1981). Time Series Analysis and Applications.
Houston, Goose Pond Press 628p.
Library of Congress Catalog 8181825
A kind note from Laurent Duval: I am not esp. surprised since i consider Robinson (at least partly) as a geophysicist. One of the early mention i found in geophysics is:
N. Ricker, A note on the determination of the viscocity of shale from the measurement of the wavelet breadth, Geophysics, Society of Exploration Geophysicists, vol. 06, pp. 254258, 1941. See for instance:
http://www.laurentduval.eu/sivawitswhereisthestarlet.html
The word wavelet has gone through a chain: wavelet (geophysics) > ondelettes (geophysics) > wavelet (as we know it). Even earliest spurs are in Huygens.
Les Houches Center of Physics
PHYSICS  SIGNAL  PHYSICS
On the links between nonlinear physics and information sciences
September 813, 2002
INFO
On the links between nonlinear physics and information sciences
Journal Applied Stochastic Models in Business and Industry
[Wiley InterScience ISSN 15241904,
http://www.interscience.wiley.com ]
is considering a special issue on Wavelets and Other Multiscale Methods:
Theory and Applications.
Contributions for the Special Issue that are
good balance of theory and applications of wavelets and other
multiscale methods in industry, finance, and applied sciences
are invited.
Ascii Text
Napoli Wavelet School, Spring 2001
Poster
BAMS Bayesian Adaptive Multiresolution
Smoother; Matlab Demo Program; needs
WaveLab Software
bams.m
uses function
bayesrule.m
Supporting Manuscript
0006
Brani Vidakovic and Fabrizio Ruggeri
BAMS Method: Theory and Simulations
See also the implementation by
Antoniadis, A., Bigot, J. & Sapatinas, T.
BAMS: Matlab Front End (No Wavelab Necessary)
Dr Bin Shi, my former graduate student, made a simple frontend that demonstrates
BAMS shrinkage in MATLAB. As of now, the only signal is doppler and,
as tradditionally done, the standard normal noise is added
to the rescaled signal to achieve desired SNR.
The programs below should be on MATLAB's path and Wavelab is not needed.
Wavelets in Statistics Week at CNRIAMI, Milano
Eight Lectures!
STATISTICS 294, ISDS, Duke University, Fall 1999
Statistics in Time/Scale and Time/Frequency Models
BOOK: STATISTICAL MODELING BY WAVELETS,
by Brani Vidakovic,
Wiley Series in Probability and Statistics; ISBN: 0471293652, pp. 381.

Supplemental WEB page [data sets, program codes, resources, reference updates, and more] is
under preparation.
Please check the site:
wiley.html for the leatest updates.
VOLUME: BAYESIAN INFERENCE IN WAVELET BASED MODELS,
SpringerVerlag, Lecture Notes in Statistics 141.
(ISBN 0387988858)
 Peter Müller
and Brani Vidakovic
are editors a volume on Bayesian inference in the
wavelet (multiscale) domain. The volume is just out of press [June 1999] and some of the contributors include: Abramovich, Aguilar, Albertson, Berliner, Bultheel, Chipman, Clyde, Corradi, Cressie, George, Huang, Jansen, Kalifa, Katul, Kohn, Kolaczyk, Krim, Leporini, Lynch, Mallat, Marron, Milliff, Müller, Nowak, Ogden, Pastor, Pensky, Pesquet, Rios Insua, Rodriguez, Ruggeri, Sapatinas, Simoncelli, Vannucci, Vidakovic, Wang, Wikle, Wolfson, and Yau.
The backcover of the volume reads:
This volume provides a thorough introduction and reference for any researcher who is interested in Bayesian inference for waveletbased models. To achieve this goal, the book starts with an extensive introductory chapter providing a self contained introduction to the use of wavelet decompositions, and the relation to Bayesian inference. The remaining papers in this volume are divided into six parts: independent prior modeling; decision theoretic aspects; dependent prior modeling; spatial models using bivariate wavelet bases; empirical Bayes approaches; and case studies. Chapters are written by experts who published the original research papers establishing the use of wavelet based models in Bayesian inference.
('97, '98 ) Statistics 294 at ISDS, Duke
 The course STA 294 is a ``special topic''
course. In Spring 1997 (1998) the topic (one of the two topics) was
WAVELETS in STATISTICS
Workshop on Wavelets in Statistics at Duke University (October 1213, 1997)
Gabriel Katul on TURBULENCE and Wavelets.
BAYES and WAVELETS (A Review)
Wavelets for Kids (A tutorial written in December 1994)
Nice Data Sets, S+ Wavelet Programs, etc.
Miscellanea
L I N K S
WAVELET APPLETS (Need Javaenabled browser)
LINKS TO SOME NICE AND INFORMATIVE WAVELET PAGES:
(Please send an update if your page has changed the address)
 Jelena at CMU and
Wim
at Bell Labs.

TIMEFREQUENCY TOOLBOX
by Francois Auger, Olivier Lemoine, Paulo Gonçalvès and Patrick Flandrin.
 RICE DSP Folks: A Bank of Papers!

A Friendly Guide to Wavelets.
by Gerald Kaiser
 WAVELET A Cornucopia  Wavelet Page maintained by Andreas Uhl.
 DMOZ An open project
listing of researchers in the wavelet wavelet world.

Wavelets in Statistics  University of Bristol Keep your eye on this site! This team of excellent
researchers is upfront in a range of novelwavelet applications.
 Donoho (and collaborators) papers at Playfair.
 Wavelet Digest Homepage
 Wavelet NetCare at Washington University (X.S.Wang).
 Wavelets at Imager
 CREW * A Standard for Image Compression Ricoh
 Wavelet Image Compression Example. Meteosat Images
 The
Wavelet Project at Intelligent Engineering Systems Laboratory, MIT. Super Lenna Image
 Wavelet resources page Amara Graps' Wavelet Page.
 Numerical Harmonic Analysis Group, Vienna University, Au
 Multiresolution Signal Processing, University of Minnesota
 WavBox Software by C. Taswell
 Multirate Signal Processing Group, University of Wisconsin  Madison
 Steven Baum's Wavelet Page A Brief Guide to Wavelet Sources
 McCody Wavelet Page + Tsunami Plus Wavelet Library.
 YAHOO Wavelet Page.
 Baharav and Leviatan Papers.
 There is a wavelet in your future... Simone Santini says.
 Sandip's Wavelet Home Page
 ARGONNE NL Wavelets and Image Processing Papers.
 JeanMichel Mangen's Wavelet Home Page
 A Fast Discrete Periodic Wavelet Transform Page, by Neil H. Getz
 Wavelet Warriors at Dartmouth.
 Wavelet Group Karlsruhe
 Wavelets in Geophysics (A. Davis Page).
 Demo on 2 D wavelet packets. By John R. Smith and ShihFu Chang.
 Wavelets! Wavelets!! by Zhaobo Meng.
 Magasa's Wavelets
 Wavelets at UVIGO (GST Group)
 Jun's Wavelets
 Geoff Davis' Page
 EPIC  Efficient Pyramid Image Coder Designed by: Eero Simoncelli and Edward Adelson.
 PhysNum  Montreal.
 Wolfgang Dahmen's Veroffentlichungen und Reports.
 Juan Restrepo's papers on Wtransforms.
 Tony Cai and STAT690W: Wavelets And Statistical Function Estimation at Purdue University.
 Brad Lucier's Home Page with wavelet
 Maarten Jansen wavelet papers.
 Peter Shroder's Wavelet Research Page.
 Terry Tao's Wavelet Papers.
 Marseilles Wavelet School France, July 21  26, 1997.
 Marina Vannucci's Wavelet Papers at University of Kent, UK.
 Grenoble Group [Statistics and Stohastic Modeling]
 A Practical Guide to Wavelet Analysis Christopher Torrence and Gilbert P. Compo.
 Plotting and Scheming with Wavelets by Colm Mulcahy.
 JPEGs, Wavelets, Fractals, ZLIBs by Jian Zhang.
 Surfing the Wavelets by Joshua Altmann.
FTP SITES
medialab.MIT (QMF)
XWPL 
WPLIB (Wave Packets)
canada
ftpserver at Washington University
ftpserver at Stanford U
niv. II Taswell's WavBox
ftpserver at Chalmers University Expwavelet
ftpserver at Vienna University
ftpserver at
Missouri University  Physics Dept. WaveLib  a C and Matlab library of wavelet functions
Wavelet Toolbox in Khoros 2.0.1 Khoros
Matlab based Toolkit For Signal Compression Truong Nguyen.
BIBLIOGRAPHIES ON WAVELETS