Spline Density Functions
Pol del Aguila Pla

• 2021-09-27

Abstract
SDFs are spline functions defined on a regular grid that would really like - in an L2 sense - to look and act like their cousins, probability density functions (PDF). For boring degrees, SDFs are easy to build and commonly found (0: histogram). From degree 2 onwards, things get tricky, their L2 tendencies take them towards bad (negative) places, constraints are needed, and research gets interesting. In this talk, I’ll first motivate our interest on SDFs for PET and other applications. Then, I will give a simplistic overview of the theory of generalised sampling using B-splines, including both orthogonal and oblique projections. I will proceed to show how this applies to building SDFs, and state and empirically showcase their properties. Then, I will briefly summarise the problems of non-negative approximation and the solutions we envision. I will finish showing some empirical results and future directions of research. This presentation includes joint work with Aleix Boquet-Pujadas and Michael Unser