Mathematics Project Abstract
ANALYSIS OF FAIR SIGNAL SMOOTHING ALGORITHMS
Presenters:
Belinda Chang, Illinois Mathematics and Science Academy, 1500 W. Sullivan Road, Aurora, IL 60506; dreamybq@imsa.edu
David Xia, Illinois Mathematics and Science Academy, 1500 W. Sullivan Road, Aurora, IL 60506; dxia@imsa.edu
Mentor:
Paul Fischer, Ph.D., Chemistry Department, Argonne National Laboratories, 9700 South Cass Avenue, Argonne, IL 60439-4844
Abstract:
Ever since Gabriel Taubin published his landmark abstract on Fair Signal Smoothing, the field of graphical and data manipulation has seen great progress in revolutionizing physics. His relatively simple algorithm allows researchers to take series of graphical data and filter out the noise and extraneous information, allowing researchers to analyze life-like depictions of objects without distortions caused by misrepresentations in the data. While the algorithms only require a few lines of code, the hardest part of this mentorship was the study of Fourier analysis and our attempts at learning both the math and physics behind the algorithms. Our first project was to graphically represent a 3-dimensional carotid artery. After dividing the artery into slices, we took one of the slices, transformed the data into Fourier space by using Fast Fourier Transform (FFT), and removed the high frequency waves that caused the noise. However, this technique could only be used in 2-dimensional data smoothing, which means we had to create a separate algorithm for 3-dimensions. Instead of using cubic splines, which appeared at the time to be the only way to smooth the 3-dimensional data, we used the help of our mentor to transmute the data points into a polygonal mesh and used a simple Laplacian smoothing technique.