Measuring shapes with random Delaunay mosaics

Abstract: Classically, we are used to working with shapes embedded into the square lattice. It might be anything, a pixelated photo of a maple leaf, or a 3d-scanned voxelated image of a human organ. And despite being simple and intuitive, such a way of representation can totally distort the internal geometry of the object. While it is pretty simple to estimate the area of the leaf from a pixelated photograph, there is no straightforward way to compute its perimeter: the pixel approximation of the boundary varies dramatically, depending on how good the object is “aligned” with the lattice. By changing the square lattice for an isotropic mosaic, we can expect that the misalignment problem will fade away, as the isotropic background equalizes all directions. In the talk, we will be moving objects and mosaics in all possible ways, and utilizing the archaic probability theory approach of counting how many points or lines are there in the space, to lead us to the precise answer of how an approximation of a p-dimensional object in d-dimensional space distorts its p-dimensional volume.

mathematical physicscombinatoricsprobabilitystatistics theory

Audience: researchers in the topic

Abu Dhabi Stochastics Seminar