## LOCAL

### Non-decimated Quaternion Wavelet Spectral Tools with Applications

Taewoon Kong and Brani Vidakovic

Quaternion wavelets are redundant wavelet transforms generalizing complex-valued non-decimated wavelet transforms. In this paper we propose a matrix-formulation for non-decimated quaternion wavelet transforms and define spectral tools for use in machine learning tasks. Since quaternionic algebra is an extension of complex algebra, quaternion wavelets bring redundancy in the components that proves beneficial in wavelet based tasks. Specifically, the wavelet coefficients in the decomposition are quaternion-valued numbers that define the modulus and three phases. The novelty of this paper is definition of non-decimated quaternion wavelet spectra based on the modulus and phase-dependent statistics as low-dimensional summaries for 1-D signals or 2-D images. A structural redundancy in non-decimated wavelets and a componential redundancy in quaternion wavelets are linked to extract more informative features. In particular, we suggest an improved way of classifying signals and images based on their scaling indices in terms of spectral slopes and information contained in the three quaternionic phases. We show that performance of the proposed method significantly improves when compared to the standard versions of wavelets including the complex-valued wavelets. To illustrate performance of the proposed spectral tools we provide two examples of application on real-data problems: classification of sounds using scaling in high-frequency recordings over time and monitoring of steel rolling process using the fractality of captured digitized images. The proposed tools are compared with the counterparts based on standard wavelet transforms.

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 PackageWavmatQND.zip.

### Non-decimated Complex Wavelet Spectral Tools with Applications

Taewoon Kong and Brani Vidakovic

In this paper we propose spectral tools based on non-decimated complex wavelet transforms implemented by their matrix formulation. This non-decimated complex wavelet spectra utilizes both real and imaginary parts of complex-valued wavelet coefficients via their modulus and phases. A structural redundancy in non-decimated wavelets and a componential redundancy in complex wavelets act in a synergy when extracting wavelet-based 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 real-valued wavelets. It is also worth mentioning that the matrix-based non-decimated wavelet transform can handle signals of an arbitrary size and in 2-D case, rectangular images of possibly different and non-dyadic dimensions. This is in contrast to the standard wavelet transforms where algorithms for handling objects of non-dyadic dimensions requires either data preprocessing or customized algorithm adjustments. To demonstrate the use of defined spectral methodology we provide two examples of application on real-data 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 Wavelet-based Estimation for Non-linear 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 non-linear 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 mean-square 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 non-zero 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 wavelet-based 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, non-parametric regression, functional data analysis, robust statistical modeling

Full version of the paper with Appendices and Proofs is available at arXiv: Full Version.

### Wavelet-based 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). Wavelet-based scaling indices for breast cancer diagnostics. Statistics in Medicine, 36, 12, 1989--2000, 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 high-frequency 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 non-decimated wavelet coefficients of the transformed data. Our wavelet-based methods provide a robust way to estimate Hurst exponent and increase the prediction precision especially when there exists outlier coefficients, outlier multi-resolution levels, and within level dependencies. The properties of the proposed Hurst exponent estimators are studied both theoretically and numerically. Compared with other standard wavelet-based methods (Veitch and Abry (VA) method, Soltani, Simard, and Boichu (SSB) method, median-based estimators MEDL and MEDLA, and Theil-type (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 WAVELET-BASED 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 self-similary, 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 Theil-type weighted regression is proposed for estimating self-similarity in two-dimensional data (images). The method is compared to two traditional estimation techniques that use wavelet decompositions; {ordinary least squares} (OLS) and Abry-Veitch 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 non-cancerous. 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 1-D and 2-D fractional Brownian motions are: MakeFBM.m and MakeFBM2D.m , respectively. The 1-D fBm is generated by scaling the modulus and randomizing the phase of gaussians in FFT, while the 2-D fBm is authored by Olivier Barriere.

### WavmatND: A MATLAB Package for Non-Decimated Wavelet Transform and its Applications

Minkyoung Kang and Brani Vidakovic

A non-decimated 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 non-dyadic length and rectangular two-dimensional signals with non-dyadic sides can be readily transformed. We provide illustrative examples and a tutorial to assist users in application of this stand-alone 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 protein-coding sequences and noncoding sequences. Typically, more pronounced long-range correlations and increased regularity were found in intron-containing genes and in non-transcribed regulatory DNA sequences, compared to cDNA sequences or intron-less 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 2-dimensional 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 (purine-pyrimidine, weak-strong hydrogen bonds, keto-amino, 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, 1--17; DOI: 10.1186/s13062-016-0108-7

### 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 mass-conserving, 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, 728--733. 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 complex-valued wavelets provide additional insights to shrinkage process compared to standardly used real-valued wavelets. Typically, a location-type 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 data-driven 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 scale-mixing discrete unitary, compactly supported wavelets that generalizes the method in Barber and Nason to 2-D 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 2-D discrete scale mixing wavelet transform is computed by left- and right-multiplying 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 cSM-EB and cMOSM-EB are based on empirical Bayes approach and utilize non-zero covariances between real and imaginary parts of the wavelet coefficients. We discuss the possibility of phase-preserving shrinkage in this framework. Overall, the methods we propose are calculationally efficient and provide excellent denoising capabilities when contrasted with comparable and standardly used wavelet-based techniques.

A MATLAB toolbox developed by Norbert Remenyi cSM-EB2.zip illustrates cSM-EB and cMOSM-EB shrinkage.

Reference:
Remenyi, N., Nicolis, O., Nason, G., and Vidakovic, B. (2014). Image Denoising With 2D Scale-Mixing Complex Wavelet Transforms. IEEE Transactions on Image Processing, 23, 12, 5165--5174.

### LAMBDA NEIGHBORHOOD WAVELET SHRINKAGE

A wavelet-based 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 real-word 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). Λ-neighborhood wavelet shrinkage. Computational Statistics \& Data Analysis, 57, 1, 404--416, doi:10.1016/j.csda.2012.07.008

### WAVELET SHRINKAGE WITH DOUBLE WEIBULL PRIORS

Bayesian wavelet shrinkage standardly employs point-mass 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, 88--104.

### DENSITY ESTIMATION WHEN DATA ARE SIZE-BIASED: WAVELET-BASED MATLAB TOOLBOX

Often researchers need to estimate the density in the presence of size-biased data. The wavelet-based 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 Size-Biased Sample, by Pepa Ramirez and Brani Vidakovic.

Reference:
Ramírez, P. and Vidakovic, B. (2010). Wavelet density estimation for stratified size-biased 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 real-life 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 m-files, MATHEMATICA nb-files, 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, 3758--3773.

### Hunting for Dominant Straight-Line 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 nano-scale 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 N00014-95-1-1116 from the Office of Naval Research. Partial support for this work was also provided by National Security Agency Grant NSA E-24-60R at ISyE.
The manuscript (draft version in PDF) describing the methodology can be found here:
• LINEAR FEATURE IDENTIFICATION AND INFERENCE IN NANO-SCALE IMAGES

### 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 m-file WavMat.m should be put in yourmatlabpath/toolbox/wavelab/Orthogonal/, although if you know the filter, no wavelab is needed -- the m-file 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 built-in 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 self-similar 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 m-file 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 no-name 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!

### MINICOURSE IN MILANO WAVELETS AND SELF-SIMILARITY: THEORY AND APPLICATIONS

December, 14-17, 2004 at Consiglio Nazionale delle Ricerche Istituto di Matematica Applicata e Tecnologie Informatiche (Milano Department - formerly CNR-IAMI)

### BAMS-LP (Bayesian Adaptive Multiresolution Shrinker of Log Periodogram)

The matlab files that implement the BAMS-LP 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 Log-Periodogram, can be found HERE in the PDF format.

### Some ADDONS for WaveLab Module

#### Discrete Complex Orthogonal Wavelet Transformation

Here is some history. We started these m-files when Jean-Marc 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 UNC-Charlotte made an m-function 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 m-file 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/.

#### 2-D Continuous Wavelet Transformations

In the Spring of 2001 Xiaoming Huo and myself team-taught 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 class-project. The idea came from commercial software Crit-tech 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 Fourier-inverts 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/

#### 3-D 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 wavelet-project of Vicki Yang, gifted graduate student at ISyE who took a course on wavelets with me. She was interested in wavelet processing of 3-D 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 inverse-transforming the data back to the time domain. The function needed here is cubelength.m that is a 3-D 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 built-in 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/

#### A New Extended MakeONFilter, MakeONFilterExt.m.

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.

Standard WaveLab m-function 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(J-1), and extracted from WT as WT(dyad(J-1)). I needed dyad-like tools for 2-D 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);
>> 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 smooth-part 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.

#### Daubechies-Lagarias 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 m-script DLtest.m.

#### FWT2_POE and IWT2_POE for Rectangular Images of Dyadic Sides

In 2-D tensor product wavelet transformations, traditionally the inputs are square images of a dyadic side. Since performing the 2-D transformation amounts to subsequent application of 1-D 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 2-D wavelet transformation and its inverse on rectangular images with dyadic sides. The m-file 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;).

Planned...

#### Practical Hints on Running WaveLab (when the names collide).

Many soon to be posted, stay tuned...

### Wavelet-history Curiosity

A wavelet-history 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 time-series 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 time-series 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 81-81825

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. 254-258, 1941. See for instance: http://www.laurent-duval.eu/siva-wits-where-is-the-starlet.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 8-13, 2002

INFO On the links between nonlinear physics and information sciences

### CALL FOR PAPERS

Journal Applied Stochastic Models in Business and Industry [Wiley InterScience ISSN 1524-1904, 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

Poster

### BAMS Bayesian Adaptive Multiresolution Smoother; Matlab Demo Program; needs WaveLab Software

bams.m uses function bayesrule.m

Supporting Manuscript

• 00-06 Brani Vidakovic and Fabrizio Ruggeri
BAMS Method: Theory and Simulations

### BAMS: Matlab Front End (No Wavelab Necessary)

Dr Bin Shi, my former graduate student, made a simple front-end 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 CNR-IAMI, Milano

Eight Lectures!

### 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, Springer-Verlag, Lecture Notes in Statistics 141.

(ISBN 0-387-98885-8)
• 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 back-cover of the volume reads: This volume provides a thorough introduction and reference for any researcher who is interested in Bayesian inference for wavelet-based 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

• Archive