Here are some suggested ideas for projects. Treat them for what they
are: suggestions. You can be fairly liberal in your choice of the
topic, as long as there is a connection to the "multiresolution"
theme of the course (you do not even have to adhere strictly
to wavelets). If any of these suggested projects looks interesting
enough to you, or if you have doubts and would like to get further
details on any of them, please do not hesitate to contact us with your
Generally speaking, your project can be in one of three categories:
A literature survey of a significant topic, editorialized to reflect
your own interpretation of the methods used and the results.
Feel free to work in groups of up to 2 or 3, but the scope of the work must
be proportional and individual contributions should be made clear.
A comparison of two or more different techniques (one of which could
be your own suggested alternative) for a novel or significant
application or functionality, where you should provide a critique of
An original "researchy" project where you tackle unsolved interesting
problems that are any combination of theoretical, algorithmic or
experimental in nature.
A software implementation type project, for multimedia applications
of wavelets. It may include programming in JAVA, building software
packages and libraries, etc.
A word on how different type of projects will be evaluated is due.
You should not assume that literature review type projects are considered
to be of less value than projects in which original research is performed.
The literature reviews offered here as possible projects are very closely
related to the research work the instructor and the TA are doing, and
so you are encouraged to consider these. A well done survey is a lot of
work, will give you the opportunity of studying in-depth some topic related
to the course, and will give us valuable information for our research work.
However, since less creativity is required in this case, not very good
jobs will receive less credit than not very good research jobs, provided
that in the latter some progress is made towards solving the problem that
was being tackled.
As you will see, there is a bias in the suggested projects towards (image)
compression applications stemming from the field in which the instructor
works, but feel free (in fact you are encouraged) to choose other
application areas as well. An incomplete list of application areas includes:
inverse/scattering applications in electromagnetics
relationships between wavelets and fractals
modulated lapped transforms and fast architectures
signal/image processing applications like
denoising/deblurring/restoration/interpolation/inverse halftoning, etc.
detection of transient signals/edges/features using wavelets
multimedia applications like web browsing/retrieval/caching/networking
communications applications like xDSL/digital broadcast/multicast/
interference suppression/wireless transmission
joint source-channel coding of speech/images/video for transmission
over noisy channels
multigrid/rendering applications in computer graphics
applications in partial differential equations
discrete wavelet basis for multicarrier communications
non-uniform and multi-band sampling
time-frequency diversity in wideband wireless communications
applications of wavelets in super-resolution
image and video modeling in wavelet domain
application of multiresolution techniques for low power VLSI
Note also that a few ideas for projects for the course appear in your
textbook by Vetterli and Kovacevic (pp. 459-460).
(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
(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
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
(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
(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
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
(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
(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
(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
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
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.