European Conference on Computer Vision (ECCV), 2026

Learning Riemannian Geometry on 3D Objects

¹ Imperial College London ² Peking University

Paper Supplementary

Method overview from raw 3D point samples to a geometric embedding field and exponential and logarithmic map predictions — **Figure 1.** The method takes raw point samples without requiring manifold connectivity, encodes them into a continuous Geometric Embedding Field, and conditions lightweight decoders for exponential and logarithmic map queries.

Abstract

Computing exponential and logarithmic maps on raw geometries is a fundamental yet challenging task in geometry processing, often hindered by the lack of explicit connectivity and the presence of noise. We present a learning-based framework that computes these maps efficiently by learning a continuous geometric embedding field directly from discrete point sets. The field encapsulates the intrinsic local geometry governing exponential and logarithmic maps. A sparse octree-based CNN encodes the input into a high-dimensional feature volume; for an arbitrary query point, its embedding modulates two specialized triplane-based neural networks that predict the maps in a single forward pass. This formulation avoids a requirement for mesh connectivity and is evaluated under irregular sampling, noise, non-manifold input geometry, non-rigid shapes, and real scans.

Keywords: Geometry Processing · Geometric Learning · Neural Fields

Contributions

1
A continuous geometric embedding field learned from discrete point sets for efficient evaluation of exponential and logarithmic maps.
2
A geometry-aware conditioning mechanism and geometric regularization terms that adapt a shared model to diverse local geometries without explicit connectivity.
3
Evaluation of accuracy, runtime, and robustness on remeshed and raw ShapeNet data, non-uniform and noisy samples, DFAUST, and Kinect-v2 scans.

Method

Geometric embedding and hyper-modulation

The pipeline separates a one-time shape encoding stage from repeated local map queries. A sparse octree U-Net produces a continuous volumetric feature field. At a source point p, the interpolated embedding z_p generates source-specific triplanes for ExpNet and LogNet.

Input: Coarse boundary point samples
Backbone: Sparse octree-based 3D U-Net
Local representation: Embedding-conditioned triplanes
Query coordinates: Relative ambient 3D coordinates
Outputs: Exp_p(v) and Log_p(q)
Connectivity: Not required at inference

Hyper-modulation architecture with triplane generators and small MLP decoders for exponential and logarithmic maps — **Figure 2.** The local geometric embedding generates separate triplane feature fields for exponential and logarithmic maps. Relative coordinates query these planes, and small MLPs produce the final outputs.

Local coordinate representation

The networks operate on relative coordinates in the ambient 3D coordinate system rather than on an arbitrarily oriented 2D tangent basis. This avoids basis-rotation ambiguity while retaining tangent-aligned inputs for ExpNet.

Training data filtering

Samples associated with non-injective regions are filtered when multiple distinct tangent vectors reach the same surface location. This excludes ambiguous supervision near or beyond the cut locus.

Query complexity

After the embedding field has been computed, each map evaluation consists of triplane interpolation and a small number of matrix multiplications. Queries can be batched and parallelized.

Results

Quantitative and qualitative evaluation

For logarithmic maps, the paper reports mean distance distortion E_MDD and mean angular error E_θ. For exponential maps, it reports average Euclidean endpoint error E_Exp. Lower is better for all three metrics.

Logarithmic map comparisons on airplane, bunny, bust, and bear point-cloud inputs — **Figure 5.** Logarithmic-map comparisons on point-cloud inputs. The rows include thin and highly curved structures; meshes are used only for visualization.

Table 1

Main benchmark

Runtime is average CPU time per query, measured on an Apple M4 Max without GPU parallelism. Best values within each input group and operator are bold.

Input	N	Vertices	Logarithmic map				Exponential map
Input	N	Vertices	Method	E_MDD ↓	E_θ ↓	Time (s) ↓	Method	E_Exp ↓	Time (s) ↓
ShapeNet	1000	16.6k	Heat	0.250	16.7	0.178	Tracing	0.036	0.396
			Proj.	0.115	14.6	0.001	Proj.	0.049	0.003
			Ours	0.110	11.1	0.012	Ours	0.026	0.019
RawShapeNet	200	10.0k	Heat	0.314	21.3	0.101	Tracing	0.050	0.374
			Proj.	0.166	17.1	0.001	Proj.	0.048	0.003
			Ours	0.169	15.4	0.013	Ours	0.036	0.020
Non-uniform	200	15.0k	Heat	0.339	22.4	0.051	Tracing	0.043	0.415
			Proj.	0.155	16.3	0.0005	Proj.	0.047	0.003
			Ours	0.152	14.0	0.011	Ours	0.030	0.019
Noisy	200	16.6k	Heat	0.326	21.1	0.101	Tracing	0.060	0.374
			Proj.	0.203	19.4	0.001	Proj.	0.055	0.003
			Ours	0.202	17.8	0.013	Ours	0.038	0.020

Interpretation. The proposed method has the lowest angular error and exponential-map error in all four input conditions. It has the lowest distance distortion in three conditions; on RawShapeNet, projection is slightly lower (0.166 versus 0.169). Projection is the fastest baseline, whereas the learned method is substantially faster than Heat or tracing while retaining higher accuracy on most reported metrics.

Table 2

DFAUST and Kinect-v2 generalization

The checkpoint trained for the ShapeNet experiments is used without retraining or fine-tuning.

Logarithmic map
Method	DFAUST E_MDD ↓	DFAUST E_θ ↓	Kinect E_MDD ↓	Kinect E_θ ↓
Heat	0.23	25.47	0.12	8.45
Proj.	0.17	19.04	0.11	9.29
Ours	0.17	16.51	0.11	8.37

Exponential map, E_Exp ↓
Method	DFAUST	Kinect-v2
Tracing	0.033	0.063
Proj.	0.071	0.064
Ours	0.043	0.060

Analysis

Robustness, extensions, and scope

Cut locus visualization on the Stanford Bunny — **Cut-locus filtering.** Non-injectivity creates ambiguous Log supervision; the training pipeline filters such samples.

Sampling signed distance and occupancy representations into intermediate point clouds — **Other 3D representations.** The supplementary material describes sampling SDFs and occupancy fields into point clouds using projection or bisection before applying the same pipeline.

Limitations stated in the paper

Training targets are produced by classical numerical methods, so the quality of the supervision bounds the learned model.
Finite octree and triplane resolution can smooth fine-scale geometric detail.
Large-scale parameterization currently requires a separate stitching algorithm rather than a single globally consistent end-to-end model.

Failure case near a thin articulated wrist and hand transition — **Representative failure case.** Thin articulated regions can be difficult when surfaces are close in Euclidean distance but differ in intrinsic geodesic behavior.

Resources

Paper and citation

The PDFs are included locally so the page can be deployed as a self-contained GitHub Pages project.

Paper PDF Supplementary PDF

BibTeX

@inproceedings{pang2026riemannian,
  title     = {Learning Riemannian Geometry on 3D Objects},
  author    = {Pang, Bo and Foti, Simone and Birdal, Tolga},
  booktitle = {European Conference on Computer Vision},
  year      = {2026}
}

Acknowledgements

T. Birdal was supported by a UKRI Future Leaders Fellowship (MR/Y018818/1). S. Foti and T. Birdal were supported by the EPSRC Project GNOMON (EP/X011364/1). S. Foti was also supported by the Turing AI Fellowship MAGAL (EP/Z534699/1).