Given a shape and a small amount of information about its boundary (informally, how much of the inside of the shape can we see nearby if we are standing on the edge?), when is it possible to uniquely identify and reconstruct the shape just from these boundary measurements?
In particular, we consider the area of the intersection of a disk and the shape for disks centered on the boundary. If the radius is allowed to shrink to zero (as in the usual definition for densities), the problem becomes easy. We focus on nonasymptotic densities where we have a minimum disk size and show that many shapes are reconstructible under this regime. This allows us to provide useful results for image analysis and similar applications. This is joint work with K. Vixie, T. Asaki and K. Sonnanburg.
Sharif graduated from Walla Walla in 2005 and Ken & Tom put him in touch with Kevin Vixie and his summer internship program at Los Alamos. Sharif followed Kevin to Washington State University where he is now pursuing a Ph.D. in mathematics. None of this would have been possible without Ken & Tom!