Maths

Exponential Curves on the Similarity Group

The similarity group on the plane consists of transformations which preserve similarity and orientation: if you apply a transformation to a triangle, you get a similar triangle (equal angles) in the same orientation. The spatial projections of exponential curves on the similarity group make logarithmic spirals, as visualised here.

Generators for Pairs of Position-Orientations

In our group, we frequently work with data on position-orientation space. I made a visualisation of exponential curves connecting pairs of position-orientations, based on "Roto-Translation Invariant Metrics on Position-Orientation Space" by Bellaard & Smets.

Diffusion-Shock Filtering

We extended Diffusion-Shock (DS) filtering to Position-Orientation space for SSVM 2025. DS filtering can be used to denoise images and to inpaint missing structures. I made a WebGPU implementation of DS filtering on Euclidean space.

Eikonal PDE

In "Crossing-Preserving Geodesic Tracking on Spherical Images", we model blood vessels as geodesics of Riemannian distances on position-orientation spaces. It turns out that these distances are the (viscosity) solutions of corresponding eikonal PDEs. Here I go over how we can solve the eikonal PDE practically to recover the distance map. I also wrote some shaders so we can see it in practice.

Conway's Game of Life

A WebGPU implementation of Conway's Game of Life.

Flow Matching on Lie Groups

In "Flow Matching on Lie Groups", we developed a new technique for generative modelling on Lie groups, generalising Flow Matching by Lipman et al. As the name implies, this works by flowing data from an initial to a target distribution. Here are some nice videos of this in action.