Abstract: Nuzhat Binte Arif
Mentor: Professor Leslie Cheng
My topic of interest for research this summer is Fourier Analysis. Fourier Analysis, named after French mathematician and physicist Joseph Fourier, is the mathematics of transforming complex waves into simple sine-cosine curves so that they can be better understood and easily analyzed. Fourier Analysis on Euclidean spaces (3-dimensional spaces) has many different applications including better understanding of the study of heat propagation and many other scientific applications in the fields of engineering, partial differential equations, medical imaging, number theory, signal processing, cryptography, numerical analysis, acoustics, optics, geometry and even oceanography.
Since Fourier Analysis is an area of mathematics that is new to me, I am starting out with the basics. I will be reading several texts and literature in the field and then focusing on one particular application: medical imaging. Medical imaging technologies such as X-rays, CAT scans, and ultrasounds have become an integral part of modern medicine and have given a new dimension to it. But such technology uses mathematics to process the data it collects during scans and then transforms it into readable images. Each machine may be different and may have different functions but they all use a basic format of mathematical algorithms to process the data. Since the same logic of Fourier Analysis is used in medical image reconstruction, I will focus my research on it and I will be using these texts to guide me through the initial process:
• “The World According to Wavelets” by Barbara Burke Hubbard
• “Introduction to the Mathematics of Medical Imaging” by Charles L. Epstein
• “The Mathematics of Medical Imaging”- a Beginner’s Guide
It is amazing how seemingly simple mathematics can be used to do something so complex and useful.