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.

 
3D Print From A Photograph of a Hand
2013:  3D Printing.  Student Researchers:  Melissa Sherman-Bennett and Sylvanna Krawczyk.  

2013:  Fighting Fires in Siberia.  Student Researcher:  Beth Bjorkman.  This entry won a 2nd place prize in the 2012-13 "Competitive Mathematical Game" sponsored by the la Fédération Française des Jeux Mathématiques. 

2012:  (1) Matching CGM Days With Wavelets [paper under review], and (2) A Scoring System for Continuous Glucose Monitor Data. (No students involved.)

2011:  Identifying Potholes Using Smartphone Data.  Student Researchers:  Nathan Marculis and SaraJane Parsons (with assistance from Clark Bowman).  This entry was one of the winners of the InnoCentive Boston Pothole Challenge.  For more information, click here.  (This is a wavelets project.)

2009:  Classification of Diabetic Patients Using Wavelets.  Student Researchers:  Robert Castellano and Derek Olsen.  We also have their award-winning MathFest talk available.

Imitating Chuck Close

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
(This is a wavelets project.)

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.

Shape Recognition:  Airplanes


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.



Here is some more information about wavelets:  Wavelet analysis is a modification of Fourier analysis, where functions other than sine and cosine are used as the basis functions. In Fourier analysis, the goal is to decompose a function by thinking of it as a combination of trigonometric functions with different 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 1: An Example from Fourier analysis.


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.

Figure 2: The D4 Scaling Function.



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).

Who is Prof. Edward Aboufadel?

Take a look at my web site.


Subpages (1): 3D Printing