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. Starting in 2014, my projects are part of the Center for Applied Mathematics Projects (CAMP) at GVSU. Current areas of focus for CAMP are 3D printing, data analytics and visualization, wavelets, and applied mathematics competitions. Watch this page for updates. I enjoy working on interesting applications of mathematics to various areas of science, medicine, and engineering. Many 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: 2013-14: 3D Printing. Student Researchers: Melissa Sherman-Bennett and Sylvanna Krawczyk, along with further work done in 2014 by EFA with Lindsay Czap. 2014: NASA-Nex Challenge. (No collaborators.)
2014: Ordinary Generating Functions of Context-Free Grammars. Student Researcher: Tanner Swett
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 with Wavelets. 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: original article and Prof. Austin's feature column. With my colleague: Prof. David Austin. 2004: Identifying Airplanes in Aerial Photographs Using Wavelets. Student Researchers: Kevin Brink and Drew Colthorp. 2003: Wavelets (Introduction). With my colleague: Prof. Steven Schlicker 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. 1997: Qualitative Analysis of a Singularly-Perturbed System of Differential Equations Related to the van der Pol Equations. (No collaborators, but thank you to my thesis advisor, Jane Smiley Cronin Scanlon.) 1996: A Mathematician Catches A Baseball. (No collaborators.) There is an unfortunate typo in this article. 1996: Research in Differential Games. Student Researcher: David Szurley 1996: Detecting A Leak In An Underground Storage Tank. With Prof. Simon Tavener. 1994: Applications Über Alles: Mathematics for the Liberal Arts. (No collaborators.)Books: Discovering Wavelets, with Prof. Steven Schlicker (1999). Advanced Engineering Mathematics with Profs. Merle Potter and Jack Goldberg (2005).About Wavelets ...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). |

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