There’s a new doctor in the house. He goes by the name of Matt Demers, and today I was privileged to see him transition from Almost Dr. Matt, Ph.D. candidate to full-fledged Dr. Matt Demers, Ph.D. Mathematics.
I have known Matt for several years now – first as an undergrad student, then as a graduate student, but most importantly as a friend. As an undergraduate student, I remember grading his work and thinking this kid is going places. Matt’s work was always well thought out, thorough, and stood out from his peers. I was not surprised when Matt decided to do his Masters, and definitely not surprised when he decided to take on the Ph.D.
Beyond his intellectual capabilities (which are vast), Matt is one of those people who somehow can make Mathematics fun, approachable, and attainable to anyone who is fortunate enough to listen to him speak. He presents complicated theory with humour and an almost mischievous anticipation; sort of like a kid opening a gift at Christmas. His eyes twinkle as he carefully unwraps what is before him, anxiously waiting to see what is inside. Except in this case, the present is not for him, it’s for those of us in the audience who are about to be let in on some rather amazing mathematical secret.
Truly, Matt is passionate about Mathematics and that passion is contagious. His students love him; in fact, they rave about him. And because of this, I have had no problem leaving my classes in Matt’s very capable hands whenever I was unable to teach. He was born to be a teacher. Watching him today – as he weaved a tale around his research – reaffirmed this fact for me.
Anyway, I’m thrilled that I can finally say: Congrats Dr. Matt. I’m so happy for you, and I’m incredibly proud of you.
For those interested, I’ve attached Matt’s abstract below.
The use of fractal-based methods in imaging was first popularized with fractal image compression in the early 1990s. In this application, one seeks to approximate a given target image by the fixed point of a contractive operator called the fractal transform. Typically, one uses Local Iterated Function Systems with Grey-Level Maps (LIFSM), where the involved functions map a parent (domain) block in an image to a smaller child (range) block and the grey-level maps adjust the shading of the shrunken block. The fractal transform is defined by the collection of optimal parent-child pairings and parameters defining the grey-level maps. Iteration of the fractal transform on any initial image produces an approximation of the fixed point and, hence, an approximation of the target image. Since the parameters defining the LIFSM take less space to store than the target image does, image compression is achieved.
This thesis extends the theoretical and practical frameworks of fractal imaging to one involving a particular type of multifunction that captures the idea that there are typically many near-optimal parent-child pairings. Using this extended machinery, we treat three application areas. After discussing established edge detection methods, we present a fractal-based approach to edge detection with results that compare favorably to the Sobel edge detector. Next, we discuss two methods of information hiding: first, we explore compositions of fractal transforms and cycles of images and apply these concepts to image-hiding; second, we propose and demonstrate an algorithm that allows us to securely embed with redundancy a binary string within an image. Finally, we discuss some theory of certain random fractal transforms with potential applications to texturing.