Jacket's Wavelets

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:
• LPM.zip

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

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, BLFDR-fixed, and BaFDR.
• example2.m Example that calculates MSE, Var and Bias for noisy test signal estimation using BLFDR, BLFDR-fixed, 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 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

• 2048 x 2048 ascii file klaus2048.asc, and mat file klaus2048.mat. Beware, sizes are 67MB and 33.5MB.
• 1024 x 1024 ascii file klaus1024.asc, and mat file klaus1024.mat. Beware, sizes are 16.7MB and 8MB.
• 512 x 512 ascii file klaus512.asc, and mat file klaus512.mat
• 256 x 256 ascii file klaus256.asc, and mat file klaus256.mat
• 128 x 128 ascii file klaus128.asc, and mat file klaus128.mat
• 64 x 64 ascii file klaus64.asc, and mat file klaus64.mat

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!

LOCAL WAVELET RESEARCH PAPERS

• Some wavelet related preprints from Tech. Also, a bunch of manuscripts writen while I was at Duke (1992-2000). Almost all of them are published at various outlets.

WAVELETS AND STATISTICS: A REPOSITORY OF MANUSCRIPTS

• Check the List

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

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

Some ADDONS for WaveLab Module

Discrete Complex Orthogonal Wavelet Transformation

2-D Continuous Wavelet Transformations

3-D Discrete Wavelet Transformation (Orthogonal, Tensor Product)

A New Extended MakeONFilter, MakeONFilterExt.m.

Dyad-like 2-D Tools

 >> 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.

	g = -( (-1).^(1:length(h)) ) .* h;

        g = - reverse( (-1).^(1:length(h))  .* h );

Daubechies-Lagarias Algorithm in Matlab

Linear Regression Estimator where the kernel is the defined by wavelets...Soon!

FWT2_POE and IWT2_POE for Rectangular Images of Dyadic Sides

FWT_PO and inverse for inputs of ANY SIZE

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

Wavelet-history Curiosity

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

CALL FOR PAPERS

Napoli Wavelet School, Spring 2001

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