Prof. Edward Aboufadel's Applied Mathematics Projects
This page provides an overview of applied mathematics projects, many that I have done with students, during my career. A significant portion of this work has been done at the Mathematics REU at Grand Valley State University. I enjoy working on interesting applications of mathematics to various areas of science, medicine, and engineering. Most of the projects I have pursued utilize wavelets. As I learned about wavelets in the 1990's, I came across how the FBI uses wavelets, and an application of wavelets to denoising musical recordings. Since 2000, I have been working with students on other applications of wavelets. Wavelet-based filters are applied for basically two reasons: information rearrangement and information reduction. In each of these projects, the use of wavelets is a key step in the method. This is a list of the projects and the people involved, along with links to publications or videos, when available: 2013: Baseball Motion Charts. Student Researcher: Spencer Trautmann.
2009: Classification of Diabetic Patients Using Wavelets. Student Researchers: Robert Castellano and Derek Olsen. We also have their award-winning MathFest talk available. 2008: Creating Chuck Close-like Portraits Using Wavelets. Student Researchers: Clara Madsen and Sarah Boyenger. 2006: Position Coding. Student Researchers: Timothy Armstrong and Liz Smietana. 2005: Analysis of Handwriting Samples Using Wavelets. Student Researchers: Beverly Lytle and Caroline Yang. 2004: A New Method of Computing the Center of Population of the U.S. With my colleague: Prof. David Austin. 2004: Identifying Airplanes in Aerial Photographs Using Wavelets. Student Researchers: Kevin Brink and Drew Colthorp. 2003: Breaking CAPTCHAs Using Wavelets. Student Researchers: Julie Olsen and Jesse Windle. 2002: Wavelet-based Steganography. Student Researcher: Lisa Driskell. 2000: Developing Bivariate Daubechies Wavelets. Student Researchers: Amanda Cox and Amy VanderZee. 1996: A Mathematician Catches A Baseball. (No students involved.) 1996: Detecting A Leak In An Underground Storage Tank. With Prof. Simon Tavener.frequencies. This approach is very useful when dealing with
periodic functions. However, if we wish to analyze non-periodic
functions, then the focus on frequencies can lead to poor results. An
example can be found in Figure 1, where the characteristic function on
the time interval is approximated by a sum of sines and
cosines. Wavelet analysis is designed to better handle non-periodic functions,
because of the focus on the time intervals where functions are
defined.
Figure 2 is a graph of the scaling function known as D4. Scaling functions play a role similar to the sine and cosine functions of Fourier analysis in that other wavelets in a wavelet family are generated from the scaling function. The scaling function is also called the father wavelet.
Some of this material is based upon work supported by the National Science Foundation under Grants No. DMS-0451254, DMS-0137264, DMS-9820221, DMS-1003993, and DMS-1262342. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF). |