Workshop Report: Mathematical Imaging and Surface Processing at the MFO: Romy Williamson

1 Introduction
The Mathematisches Forschungsinstitut Oberwolfach — or MFO — is a Mathematical Institute located deep in the hills of the Black Forest, far away from the distractions of civilisation. It was built by the Nazis in 1944, in a deliberately secluded location, so as to be an unlikely target for Allied bombing. This is ideal, when you want to peacefully concentrate on maths.
I attended the workshop on Mathematical Imaging and Surface Processing, from 2nd to 7th February 2025. This was organised by Mirela Ben Chen, an inspiring figure in the Geometry Processing community, along with Antonin Chambolle and Benedikt Wirth. Throughout the week, we heard a variety of talks from fields including optimal transport, inverse problems, surface representation, rendering, video generation, fluid simulation and more. We heard from researchers whose work is strongly rooted in classical theory, as well as many researchers who are either using, or investigating the properties of, machine learning techniques such as diffusion models.

2 Contribution
My personal contribution was to present my recently-accepted paper, Neural Geometry Processing via Spherical Neural Surfaces. This paper fitted in very well with the Surface Processing side of the workshop, and I noticed links with several of the other talks, particularly:
• Xavier Dennec (Flag Spaces and Geometric Statistics) — this had relevance to the part of my project where I find eigenfunctions of the Laplace-Beltrami operator.
• Nicholas Sharp (SpaceMesh: A Continuous Representation for Learning Manifold Surface Meshes) — Nick’s discussion of various surface representations led very nicely into my presentation of an alternative neural representation.
Please see the project webpage for more information.

3 Interesting People and Talks
These are the talks that stuck in my mind or inspired me the most. Listening to these talks has helped me to figure out what I like the most in terms of research topics and presentation styles, and to take cues from this to steer my own research direction and presentation style.
• Nicholas Sharp (SpaceMesh: A Continuous Representation for Learning Manifold Surface Meshes): excellent presentation style. He is also very engaging and enthusiastic in conversation. I was excited that he talked about halfedege meshes, and he mentioned properties of orbits of subalgebras of halfedge meshes, which I had figured out myself last summer, not realising it was an established thing. I am motivated and inspired to see someone that is very skilled and knowledgeable at classical geometry processing techniques, combining these thoughtfully (not blindly) with neural networks, to create algorithms that are more robust than previous neural techniques, with improved performance over classical methods.
• Albert Chern (Fluid Dynamics with Sub-Riemannian Geometry): I am amazed at his ability to produce such incredible results with such an aesthetic method (no ugly performance tricks and no magic neural net, etc). I plan to learn some Riemannian Geometry so I can understand more. I played a geometric game, trigon Blockus, with Albert Chern and others.(1)  Albert Chern also likes the Nichomachus Identity (related to the year 2025) and we have both independently tried to find a 4-dimensional
visual proof of that, so far without success.
(1) Why is it triangle, and pentagon, not trigon and pentangle?
• Florine Hartwig (Optimal Motion from Shape Change): I was really excited to see this talk, because Niloy and I had seen the original talk eighteen months ago at the Obergurgl Geometry Processing Workshop in Innsbruck, Austria. The original paper provided an elegant framework to predict the global motion of a deformable body in space, given its shape change. The follow-up paper explored how to optimally deform a shape so that its global motion matches a target motion as closely as possible.
They were able to do the optimisation quite elegantly within the framework. I enjoyed talking to Florine. We had some things in common, such as a background in pure maths, and rowing. Her work also relates to Riemannian geometry and I want to understand more.
• Robert Beinert (A Geometric Optimal Transport Framework for 3D Shape Interpolation): I liked this talk very much because I have spent some time in my research thinking about surface correspondence and shape interpolation, from the perspective of neural surfaces, but I had never considered it from an optimal transport point of view. I really liked the method and I was impressed that it worked at all, but I am not entirely convinced about its framing/applications, because it has no semantic knowledge of
the shapes so it can easily go wrong when the shapes have different-enough proportions. I spent some time talking to Robert and Simon Schwarz during the afternoon excursion, and they explained to me the meaning of Habilitation in Germany.
• Mark Gillespie (harmonic functions rendering). Quite interesting talk. It was a ‘walk on spheres’ type of thing, for rendering implicitly defined surfaces, but the assumption is that the function is harmonic, not necessarily an SDF. This fits into the category of ‘questioning a common assumption in the field’. I’m not totally sure how often this is practically useful but it’s a nice problem set up.
• Nicole Feng (heat method geodesics) This is a cool paper. It constructs geodesic distance robustly, by ‘diffusing’ oriented normals for a short time and solving a Poisson equation. I also noticed she dealt with it quite well when some audience were distracting from the talk a bit by being pedantic about what it even means to have signed distance in some odd cases. She joked about it slightly and moved on, it didn’t put her off that they didn’t like one of the examples.
• Zorah Lahner (nuclear fusion) This was a really good example of a talk that everyone found interesting and engaging even though it didn’t have any actual results. I think that is a difficult kind of talk to give and it leaves the speaker vulnerable to a lot of tricky questions. Maybe people liked it partly because the lack of answers made it a good discussion point.

4 Benefit
This workshop has provided me with great academic benefit. I have been exposed to new topics and I now have a better idea where I need to do further reading to improve my background knowledge in several areas. These areas would include Riemannian geometry, optimal transport, and diffusion. Also importantly, I have worked on soft skills such as presenting and networking.
The MFO was a very calm place to think. I enjoyed looking at the maths books in the library and playing the instruments in the music room. I need to be calm in order to think  clearly and be creative. Therefore I appreciated the effort the MFO has made to make such a conducive environment.