dartmoose

[1] https://home.dartmouth.edu/news/2011/12/rudolph-red-nosed-re...

The nearest sort of application of this kind that I can think of is Walk on Spheres (cool recent paper https://www.cs.cmu.edu/~kmcrane/Projects/MonteCarloGeometryP...), but in that case the authors do a closest point query, not a distance to the nearest surface. There's a subtle difference here because you actually need the closest point (not just the distance) so that you can look up a boundary condition. It's not clear to me how you could make use of an SDF in this setting, although maybe there's an interesting research direction there.

Both differentiable rendering with an SDF (i.e. SDFDiff https://arxiv.org/pdf/1912.07109.pdf) and with a neural SDF(DIST http://b1ueber2y.me/projects/DIST-Renderer/) use automatic differentiation to compute the gradient of the rendering/image-generating process. Section 3.4 in the DIST paper discusses where back prop comes into all of this. Basically your surface is defined by the weights of some neural net (i.e. a neural SDF) and you need to know the gradient of your image with respect to those weights.

For a variety of reasons, these surface representations are easier to handle than triangle meshes which break certain desirable properties for differentiation and require extra care as a result (see edge sampling https://people.csail.mit.edu/tzumao/diffrt/ for an early example of the challenges of doing differentiable rendering with a triangle mesh).

* Instead of using a single latent feature per each implicit surface, the authors propose using a "volume" of latent features per each surface. This allows for the NNs to better capture the geometric detail while remaining relatively shallow. The result is a more accurate and faster to compute neural SDF. Contrary to the claim of another comment, the neural SDF alone is not the interesting part of this paper--the prior works points to at least three other papers that have explored the idea of representing an SDF with a neural net: Park et al.'s DeepSDF https://arxiv.org/pdf/1901.05103.pdf, Mescheder et al's Occupancy Networks http://www.cvlibs.net/publications/Mescheder2019CVPR.pdf, and Chen et al's Learning Implicit Fields https://arxiv.org/pdf/1812.02822.pdf All very interesting papers.

* When I say a "volume" of latent features I specifically mean a voxel-grid where the corners of each voxel are latent features and any position X has a corresponding feature Z which is simply the trilinear interpolation of the features on the corners of the voxel. As the authors mention, they try to keep this sparse by leaving any voxel that does not contain the surface "empty".

The authors use an octree to create L different feature volumes. As L becomes larger, the resolution of the feature volume increases which means that more fine grained details can be encoded as features.

Finally, the authors describe a rendering procedure that makes use of their LOD model (still need to read this part more thoroughly).

Some additional thoughts:

Why are SDFs useful at all?

One comment suggests this is a form of "compression" but meshes have a far smaller memory footprint and are computationally less expensive to render. Ray tracing is extremely fast, largely due to the fact that as a primitive operation in graphics so much time and energy has been invested into understanding how to make it faster with various acceleration structures, like BVHs.

So are SDFs actually useful?

Yes. Triangle or polygon meshes are great when you have them, but are terribly challenging to work with for reconstruction tasks. For instance, you effectively have to pause occasionally during reconstruction to fix your mesh up so that it isn't complete garbage (triangles with small angles, self-intersections, extremely lopsided side lengths, etc). SDFs support arbitrary topology painlessly, which is why they show up so much in reconstruction/computer vision.

So why do we need neural nets to represent them?

I think the primary reason you use a neural network to represent a signed distance function is because it's a more efficient representation than storing the SDF in some sort of grid structure (maybe someone else has more thoughts on this?). As a side benefit, it can simplify any sort of differentiable rendering since the surface itself already is represented in a manner that is naturally differentiable via back-propagation.