Home

Since its introduction in the 1970s, magnetic resonance imaging (MRI) has become an invaluable (and non-invasive) diagnostic tool in medicine. MR imaging incorporates several fascinating ideas from physics, engineering, statistics and computer science. In this REU, we will explore the mathematical underpinnings of MR image reconstruction.

Specifically, we investigate methods of improving the accuracy of MR image reconstructions using edge information. Over the course of the program, we will explore

• Fourier series - the underlying physics of MRI dictates that the MR scanner collects Fourier coefficients of the specimen being imaged.
• The Gibbs Phenomenon - Fourier approximations work very well for smooth and periodic functions; however, when approximating piecewise-smooth functions (such as those encountered in MR scans), they suffer from non-physical oscillations and poor accuracy.
• Edge-Augmented Fourier Sums - we will investigate the relation between edges and Fourier coefficients and devise methods for incorporating edge information in Fourier approximations.
• Edge Detection from Fourier Data - estimating edge (local) information directly from Fourier (global) data is a challenging task!
• 2D Fourier Approximations - preliminary investigations towards extending the 1D methods above to 2D Fourier sums.