# The Fourier Transforms Project

The Fourier transform is an indispensible tool in many disciplines. The multidimensional Fourier is also very powerful and can be written in terms of polar coordinates (in 2D) and spherical polar coordinates (in 3D) instead of the usual Cartesian coordinates. This is particularly useful for problems that are most easily described in curvilinear coordinates. However, a complete interpretation of the standard Fourier operational tool-set of rules in terms of these curvilinear coordinates was missing from the literature. Some results were known, but the important results on shift, multiplication and in particular convolution, were incomplete. I developed this tool-set of rules in 2D polar coordinates and 3D spherical polar coordinates. The work in 2D was published in the Journal of the Optical Society of America . Following up on that article, I wrote a more extended version of this work which is to appear shortly but can also be found here.

The corresponding version in 3D has recently been completed and published as well, and the paper can be seen here. The Cartesian equivalent of these theoretical tools are known to be powerful and applicable to a large range of problems, so they should prove useful to anyone working in curvilinear coordinates. Work is currently ongoing to develop a symbolic computer algebra toolbox to enable symbolic computer calculations with these transforms. Work is also ongoing to develop a discrete version of these multidimensional curvilinear Fourier transforms.

# The Photoacoustics Project

Photoacoustic tomography is a new technique for the visualization of the internal structures of soft tissue. A short-pulsed laser source is used to irradiate the sample. This produces a small temperature rise which induces a pressure inside the sample through thermal expansion. This pressure acts as an acoustic source and generates acoustic waves which are detected by ultrasound transducers outside the sample. This approach was based entirely on pulsed laser excitation. However, a new frequency-domain approach was recently developed that offers many of the benefits of pulsed photoacoustics but requires a different theory for modeling due to the longer timescales of the optical input signal.

To address this modeling need, I developed a theory of long-pulse photoacoustic tomography. I also worked on unifying and continuing the mathematical analysis of the short-pulse photoacoustic signal analysis problem.

In a related but separate area, there has also been a lot of recent interest in thermal imaging. The basic idea of applying heat or cold to an area and observing the resulting temperature change with an infrared camera has led to the development of rapid and relatively inexpensive inspection systems. However, the main drawback to date has been that such an approach provides mainly qualitative results. To advance the quantitative results that are possible via thermal imaging, techniques and algorithms from conventional tomography were modified and adapted to account for the attenuative nature of thermal waves. As a result of this line of research and of the work done on multidimensional Fourier transforms, I showed that the modeling and mathematics of all these tomographic techniques can be unified under a single theoretical approach. This work has recently been submitted for publication and attempts to unify apparently distinct tomographic approaches into a single powerful mathematical theory that the tools of signal theory and processing can be used to solve.

# Modelling and Vibrations

My bread and butter, really. A bunch of stuff has been done in this general area over the years.

Older work dealt with the vibrations of spinning disks. I’ve continued to on various vibrations problems, more recently on modeling and analysis of ice-induced vibrations. This interest in vibrations is also being extended into novel lines of investigation, for example in collaboration with researchers in Human Kinetics to investigate the bouncing of babies in jolly jumpers.

Due to the interest of one my students in the subject, more recent analytical work has focused on the modeling and analysis of the vibrations of a guitar soundboard and brace system, motivated by the goal to improve the manufactured consistency of wooden musical instruments. The background to this work is that traditional methods for making musical instruments has always been considered the only way of tuning and optimizing the individual wood pieces as well as the whole instrument in order to make each instrument sound its best. However, because this complex tuning process is not well understood, it is difficult or impossible to automate the techniques that skilled instrument makers use to consistently produce instruments with good tone.

Recent work by my student involved analyzing a simplified analytical model in order to understand the vibrations of a system consisting of guitar soundboard (plate) and brace. In particular, the question of how changes in the brace affected the modes and frequencies of vibration of the coupled system were investigated. This analysis led to some previously unknown results on why braces tend to be scalloped during hand-manufacture of guitars. This is an interesting and exciting project and we look forward to working on it for a few more years.

# The Mobility Project

We’ve been working on developing a Wearable Mobility Monitoring System (WMMS) using a BlackBerry. A more detailed account of this project is on its own page.

# The Boat Project

This started out because a guy at my sailing club broke his mast and these boats had a history of masts breaking. So one of my students did an undergraduate thesis on it, which can be read here. The same student went on to develop a Data Acquisition tool for a boat. Fancy stuff, but basically we figured out in the first part of the project that we didn’t have any real data for anything and hey, let’s build something to get some numbers, shall we? His ‘red box’ actually worked quite well. He finished his thesis and wrote a paper that was accepted in the IEEE Transactions on Instrumentation and Measurement. You can read the paper here and the thesis here.

# Biomedical Signal Processing

This project is now done. It dealt with signal processing heart signals to identify non-deterministic events. Basically, can we get any info out of a heart signal inspite of the fact that the beating of the heart itself is really loud and NOT what we’re interested in. Conference papers from this work can be found here and here. The journal paper has been accepted and can be read online. The website with the signals, code and all the various documents related to this project can be found here.

## Leave a Reply