Geometry of Greyscale Images
Project Description
We will use analytic (calculus, matrix theory) and computational approaches to understand the geometry of two dimensional "black-and-white" images such as those generated by CAT scans, MRIs, and ultrasounds. Because such an image can be regarded as the graph of a real-valued function of two variables, basic concepts from the Calculus can be used to identify geometrically significant features in images. Of particular interest to us will be ridge, valley, and connector curves of a function. We will come up with a method for computing such curves for a greyscale image, and we will investigate the extend to which the geometry of the curves (their intersections, etc.) provide geometrically meaningful landmarks for image comparison and shape description.
Background Needed
A student should have had a course in calculus (multivariate calculus is best). Knowing some matrix theory would be a bonus, but isn't necessary. The project will use Matlab, a computational environment, so the student should feel comfortable using computers and new software. Above all, a willingness to think creatively and experiment with mathematical ideas will be a great asset to the project.