(1) Literature review: stochastic multiresolution models based on wavelets.

Real signals are often modeled not deterministically but as stochastic processes, and in this case one is interested in studying the statistical properties of wavelet coefficients obtained by expanding such processes. This project consists of determining what is and what is not known about the statistics of such wavelet coefficients. Work on this includes recently proposed models like Hidden Markov, mixture models, etc.

(2) Literature review: non maximally flat filters.

It is well known that, to obtain smooth wavelets from FIR filters, all the zeros in the lowpass filter have to be placed at pi: as a consequence, the transition band of such filters is very wide, and this may be undesirable for some applications. This project consists of conducting a literature search to determine what has been already done wrt this problem: study the tradeoff between smooth basis functions and a wide transition band vs. less smooth basis functions and a steeper transition band.

(3) Wavelet-based image compression.

Wavelet based techniques represent the current state-of-the-art in image coding. The efficiency of wavelet-based coding methods is in large part due to the use of a space-frequency based data-structure for representation and quantization. The new class of coders includes that of Shapiro; Said and Pearlman; Taubman and Zakhor; Xiong, Ramchandran and Orchard; Joshi, Fisher, and Bamberger; and LoPresto, Ramchandran and Orchard. The use of better context modeling seems to be the key to better performance. This project explores the role of efficient context modeling integrated into efficient wavelet data models like Hidden Markov models and Gaussian mixtures to push the state-of-the-art in the field.

(4) Wavelet based denoising.

Investigate the performance of Wiener filtering vs. Donoho's soft denoising technique, Burrus et al.'s soft denoising on undecimated wavelet data, and spectral subtraction. The study of different noise processes can be included, e.g. additive white gaussian noise, non-white gaussian noise, etc. Nonadditive noise (e.g. quantization noise when coding at low bitrates) must also be explored.

(5) Multidimensional Filter Banks and Wavelets.

Daubechies showed that by iterating filter banks one can obtain continuous wavelet bases (assuming the lowpass filter is regular). In the 1D case, there already exist a number of techniques to design filters with an appropriate degree of regularity. The field of multidimensional wavelets and associated filter banks is, however, quite young. In more than one dimension sampling is described by a lattice and its corresponding basis (matrix). Thus, when using the method of iterated filter banks, one has to deal with taking powers of matrices instead of scalars. But while different matrices can represent the same sampling lattice, taking their powers can lead to vastly different behavior of iterated filters. Therefore, although there have been some initial results on the design of irreducible wavelet bases, a number of questions still remain open.

In one dimension, Daubechies gave a sufficient condition for a filter to be regular (the existence of a continuous wavelet basis is then guaranteed), namely it must possess a certain number of zeros at the aliasing frequency pi (for the case of sampling by 2). In multiple dimensions one would like to follow the same approach, i.e. to impose a zero of an order m at multidimensional aliasing frequencies. The difficult task is precisely how to achieve the above requirement and at the same time have a perfect reconstruction system together with some other requirements, e.g. linear phase.

The project consists in making an overview of the state-of-the-art of multidimensional filter banks and associated wavelets. Possible problems to look at are finding orthogonal linear phase wavelets (for 4 channels separable subsampled by 2), as well playing with fractal tilings of the plane.

(6) Signal Processing with wavelet maxima.

By taking a non-subsampled wavelet transform, one obtains a time-invariant, redundant representation. It is therefore possible to perform some non-linear processing, and still be able to recover the original signal. One such non-linear processing consists in keeping only local maximas (and minima). Reconstruction is iterative, in order to find an estimated reconstruction that has the same maximas (using alternate projections). An interesting application that can be investigated is the compression of signals using only wavelet maxima representations.

(7) Matching pursuits.

Matching pursuits, as proposed by Mallat for time-frequency representation, is related to matching pursuits in statistics, and multistage vector quantization in signal compression. Using a very large dictionary of functions, it successively approximates the signal by removing the contribution from the function in the set which has the largest cross-correlation with the signal, and then applies the same method on the residual. The goal of the project is to play with real signals, and for example, investigate the suitability of matching pursuits for compression. That is, what are good dictionaries for a given signal, and how can one efficiently represent the dictionary entries (as well as the residual if needed).

(8) Adapted bases for compression

There are currently well-studied techniques for optimizing filters of an orthogonal filter bank to maximize coding gain. There also exist fast tree-pruning algorithms to find the best signal-adaptive tree-structured basis using a single prototype set of filters. Jointly optimizing both has recently been studied but using a coding gain criterion to optimize the filter banks: doing a joint optimization using rate-distortion criteria has not been studied.

(9) Nonbinary 2-D adaptive wavelet packet based segmentation.

There is a fast 1-D algorithm to find non-binary time-adaptive wavelet packets that are optimal for compression (this technique is based on dynamic programming). There is a difficulty in extending this to 2-D (for images) and therefore fast heuristics are needed to find good 2-D wavelet-packet non-binary segmentations. This project consists of exploring such fast 2-D segmentation methods.

(10) Hierarchical optical flow representation, non standard regularization.

Optical flow is a popular method in computer vision for motion understanding and analysis. It has recently been used for compressing image sequences as well. This project deals with the use of multiresolution techniques (e.g. the Laplacian pyramid or the wavelet pyramid) for the representation of the optical flow field, and its use for a compression application. It is well known that the optical flow must satisfy an equation (which, not surprisingly, is known as the optical flow equation) relating its time and spatial partial derivatives, whose solution is not unique. Then, the equation has to be "completed" with more terms in order to obtain unique solutions: computer vision people have studied different different regularization techniques, an example of which is the classical smoothness constraint of Horn and Schunk. One original idea that needs be explored is to regularize the optical flow equation in a compression sensible framework: for this application, one is not interested in finding the "true" physical velocities field, but instead one wants to find the field which minimizes reconstruction distortion, within the class of all fields whose representation requires less than a prespecified number of bits.

(11) Communications applications: multicarrier techniques based on wavelets.

The basic idea is to exploit the recently studied adaptive wavelet packet representations in the source compression field and apply them to multicarrier communications using orthogonal signaling. This can lead to arbitrary tilings of the time-frequency plane, optimized in a communications sense (e.g., to combat a specific pattern of noise energy distribution in the time-frequency plane), instead of optimizing for compression (e.g., to obtain the maximum energy compaction in the t-f plane), which the traditional application of tiling algorithms.

(12) Boundary filters.

When dealing with finite-length signals (most real-life signals fall in this category), one needs to deal with boundary effects due to this, while preserving perfect reconstruction. There are several known methods to deal with this problem, e.g., symmetric extension (for linear phase filter banks), circular extension, and the explicit design of boundary filters. This project involves a study of the existing methods, and experimenting to determine the pros and cons of the various methods.

(13) Applications related to the "lifting" construction

Daubechies and Sweldens have recently introduced a constructive parametrization for filter banks and wavelets based on the idea of "lifting". This project explores the state-of-the-art in this design construction including applications to next-generation wavelets defined on 3-D and arbitrary surfaces. The applications of lifting to the design of nonlinear filter banks in a signal adaptive way can also be explored.

(14) Applications related to Time-Frequency Diversity Techniques in Wirless Communications

Signal fading due to variatikons in channel characteristics is one of the major factors limiting the performance of modern wireless communication system. To overcome fading various diversity techniques have been proposed that effectively transmit multiple copies of the signal or the channel. In code division multiple access systems RAKE-receivers exploit multipath diversity. Doppler diversity can be exploited in the same situation. Developing a new framework that exploits joint multipath Doppler diversity in an optimal fashion can be explored.

(15) M-Channel (Input Adapted) Filter Bank Design The discovery of perfect-reconstruction filter banks was in mid-eighties. The problem attracted a lot of attention since then, especially for subband coding purposes. Quite a lot of nice work has been done in this area, using statistical quantizer models, Lloyd Max quantizers, etc. The main focus was on the design of the FIR or IIR filterbanks imposing several different constraints (paraunitary property, FIR constraints, etc.). The goal in the subband coding problem is to minimize the expected mean squared error between the input and output in the presence of quantizers given statistical models of the quantizers and the input. There exist several different versions of this problem which have not been solved. Very similar ideas can also be applied for signal processing applications other than compression, such as design of optimal filter banks for denoising/restoration/etc.

(16) Super Resolution

Super resolution is the problem of increasing the resolution of a given image/video. Several interpolation-like techniques can be utilized to solve such problems. Applications of wavelet-based techniques can be explored.

(17) Nonuniform Sampling and Multirate Filter Banks Bandlimited signals can be recovered with non-uniform sampling if the sampling instants satisfy certain criteria. It turns out that nonuniform sampling can be a good alternative for uniform sampling for various problems.

(18) Multi-wavelets

These are wavelets which are built on more than one scaling functions. A possible project may be to explore the fundamental properties of multi-wavelets and design them considering various constraints.

(19) JAVA

This deals with developing the software for efficient implementation of algorithms based on wavelet transforms using Java. Teaching courses on the internet, web browser-based signal processing algorithms, etc. are possible applications of the project.