NeuralSRNF: Neural Square Root Normal Fields for the Statistical Shape Analysis and Generation of Nonrigid 3D and 4D Objects

I. INTRODUCTION

• Discrete surfaces make elastic metrics sensitive to discretization quality, while nonlinear bending-and-stretching metrics are too expensive for large-dataset statistical analysis.SRNF methods make analysis more efficient with an L2 metric, but SRNF inversion lacks an analytical solution and numerical inversion is expensive and approximate.Source: §I · p. 1
• NeuralSRNF represents genus-0 surfaces as continuous MLP functions and learns to recover surfaces from SRNF maps, enabling resolution-agnostic functional shape analysis.Its conditional MLP uses DINOv2-encoded features and FiLM; compared with, it is more than 100 times faster and more accurate.Source: §I · p. 2
• NeuralSRNF outperforms state-of-the-art methods in accuracy and computational efficiency across 3D and 4D human, face, and animal datasets.The framework supports geodesics, deformation transfer, statistical summarization, and 3D shape generation.Source: §I · p. 2

II. RELATED WORK

• SRNFs provide an efficient analysis space because their L2 metric approximates a partial elastic metric, but numerical inversion remains expensive and approximate.The related-work discussion traces this approach to Jermyn et al.,.Source: §II · p. 2
• NeuralSRNF adapts the surface-based neural representation of Nizamani et al. to SRNF inversion and is up to 100 times faster than state-of-the-art methods.Prior related methods include physically motivated elastic metrics and latent-space approaches with geometric constraints,,,,,,,,,.Source: §II · pp. 2–3

III. METHOD

• NeuralSRNF represents surfaces in a resolution-agnostic manner and learns to invert SRNF maps more efficiently and accurately than numerical methods.Source: §III · p. 3

A. The SRNF inversion problem

• An SRNF scales the surface normal field by the square root of its magnitude, while normal direction and magnitude relate to local bending and stretching.Source: A. The SRNF inversion problem · p. 3
• The SRNF-space L2 metric corresponds to a partial elastic metric, and geodesics become straight lines that must be mapped back to surfaces for visualization.Source: A. The SRNF inversion problem · p. 3
• SRNF inversion asks when a normal-and-area pair represents a valid surface and whether its differential equation has a unique global solution.Source: A. The SRNF inversion problem · p. 3
• The SRNF representation does not fully measure stretching, and local reconstruction may be nonunique; shape analysis requires a global surface solution.Source: A. The SRNF inversion problem · p. 3
• The evaluation materials include a five-dataset comparison on 100 test samples and qualitative reconstructions of humans and animals against numerical SRNF and SAE.Source: Table I · p. 4, Figure 3 · p. 4

2) Neural Square Root Normal Fields:

• Numerical SRNF inversion requires multiresolution and multiscale surfaces plus careful initialization, limiting its applicability.Source: 1) Neural Square Root Normal Fields · p. 4
• The learned inverse maps a location, its SRNF value, and a shape-specific latent code to the corresponding surface point, allowing one function to represent many shapes.Source: 1) Neural Square Root Normal Fields · p. 4
• The qualitative results show NeuralSRNF inversion on animals, faces, and humans, with heatmaps depicting pointwise distance from ground-truth surfaces.Source: Figure 4 · p. 4
• The conditional MLP uses residual blocks, FiLM conditioning, SoftPlus activations, and a linear projection to produce a smooth, continuous 3D surface representation.Source: 1) Neural Square Root Normal Fields · p. 4
• Precomputed DINOv2 features provide shape-specific latent codes that do not require fine-tuning during inference.Source: 1) Neural Square Root Normal Fields · p. 4

2) Training and losses:

• The MLP is trained by minimizing mean squared error between predicted surface points and their ground-truth correspondences.Source: 2) Training and losses · p. 5
• Because NeuralSRNF is continuous and resolution-agnostic, automatic differentiation can compute differential properties including surface normals.Source: 2) Training and losses · p. 5

B. Geodesics

• Map two surfaces to SRNF space, take the straight line there as their geodesic, then invert its points with NeuralSRNF.The SRNF-space L2 metric is equivalent to the elastic metric in surface space.Source: B. Geodesics · p. 5

C. Deformation transfer

• Add the source deformation field Q(h1) − Q(f1) to target Q(f2), then invert Q(h2) to recover the deformed target surface.The source deformation is transferred without computing and parallel-transporting a geodesic in surface space.Source: C. Deformation transfer · p. 5
• Figure 5 presents registration states together with geodesic computation in SRNF space.Source: Figure 5 · p. 5, C. Deformation transfer · p. 5

B. Geodesics

• Figure 6 reports more natural intermediate surfaces for the proposed geodesics than for the compared state-of-the-art examples.Source: Figure 6 · p. 5

C. Deformation transfer

• For τ from 0 to 1, the SRNF path transports the full source geodesic to the target; τ > 1 extrapolates beyond the given deformation.Source: C. Deformation transfer · p. 5

D. Summary statistics and shape generation

• Figure 7 compares numerical SRNF with NeuralSRNF geodesics and describes head and tail movement as more natural in NeuralSRNF.Source: Figure 7 · p. 6
• Map the input shapes to SRNFs and perform PCA there to avoid repeatedly solving the expensive surface-space mean-shape optimization.The SRNF mean is the average of the input SRNFs.Source: D. Summary statistics and shape… · p. 6
• The eigenvectors and eigenvalues define principal geodesic paths, which NeuralSRNF maps back to surfaces.Source: D. Summary statistics and shape… · p. 6
• Sample coefficients for the leading eigenvectors to generate new surfaces, restricting them to [−1, 1] to keep samples within one standard deviation of the mean.Source: D. Summary statistics and shape… · p. 6
• The geodesic tables define evaluation using pairwise L1 distance on FAUST and DFAUST, and the LIMP metric on FAUST, DFAUST, and COMA.Source: Table II · p. 6, Table III · p. 6

• Evaluation covers 3D and 4D face, human, and animal datasets, including MPI COMA, FAUST, DFAUST, CAPE, and Animal3D.Source: §IV · p. 7
• Surfaces are spherically parameterized using Kurtek et al.’s implementation of the Hoppe and Praun approach.Source: §IV · p. 7
• Training uses AdamW with cosine annealing for 1,000 epochs, batch size 16, and 16×16 point patches, taking 8–24 hours depending on dataset complexity.Source: §IV · p. 7

A. SRNF inversion

• 0.10–0.65 s per surface versus 4.73–69 s for numerical SRNF: NeuralSRNF is 50–100 times faster and performs better across all five datasets on the three reported metrics.The test uses 100 randomly sampled surfaces per dataset and measures mean squared L2 distance between ground-truth and recovered surfaces.Source: A. SRNF inversion · p. 7
• NeuralSRNF has lower inversion error across surface regions, including fingers and non-convex areas, than the state-of-the-art methods shown in Figure 3.Source: A. SRNF inversion · p. 7
• Figure 4 extends the qualitative inversion examples to human bodies, faces, and animals, with reconstruction error shown as a heatmap.Source: A. SRNF inversion · p. 7

B. Geodesics

• NeuralSRNF achieves lower mean, median, and standard deviation pairwise geodesic errors than state-of-the-art methods on the datasets each method supports.Comparisons are restricted to datasets supported by the authors’ released code.Source: B. Geodesics · p. 7
• NeuralSRNF achieves lower LIMP-metric geodesic error than LIMP on FAUST and COMA, although it is not optimized for that error.The evaluation covers 100 pairs from FAUST, DFAUST, and COMA.Source: B. Geodesics · pp. 7–8
• Animal geodesics from NeuralSRNF show natural head bending and fox-tail movement, whereas numerical SRNF inversion produces unnatural shrinkage.Linear interpolation also produces unnatural shrinkage, particularly around the bending animal head.Source: B. Geodesics · p. 9
• NeuralSRNF produces natural human geodesics, while previous methods fail to recover plausible intermediate shapes in the reported arm-bending case.The comparison includes linear interpolation, numerical SRNF, LIMP, NeuroMorph, 4Deform, and an implicit neural surface approach.Source: B. Geodesics · p. 9
• 2 s versus 118 s: NeuralSRNF computes the Figure 6 geodesic about 50 times faster than numerical SRNF.For Figures 8 and 9, it takes 3 s, compared with 7 min for 4Deform and 20 min for the neural implicit surface method; LIMP and NeuroMorph take 1 sec each.Source: B. Geodesics · p. 9
• The comparison excludes because no official implementation, pretrained model, or generated outputs were publicly available at submission.The authors report contacting the authors and attempting an independent reimplementation.Source: B. Geodesics · p. 9

C. Deformation transfer

• NeuralSRNF transfers deformations across human, animal, and facial shapes using linear extrapolation in SRNF space followed by inversion to surfaces.The examples vary the extrapolation parameter τ from 0 to 1.5 in intervals of 0.25.Source: C. Deformation transfer · p. 9
• At τ = 1, the transferred target pose matches the source deformation’s target pose.The transferred result is denoted h2 = α(1), corresponding to the pose of h1.Source: C. Deformation transfer · p. 9
• NeuralSRNF retains natural transferred poses beyond τ = 1, while numerical SRNF inversion produces limb shrinkage that worsens with extrapolation.Source: C. Deformation transfer · p. 9

E. Application to 4D surfaces

• Less than 15 s versus 45 minutes: the NeuralSRNF DFAUST mean looks more natural than the numerical-SRNF mean, which shows arm shrinkage.The example includes four punching sequences and two one-leg-jump sequences.Source: Figure 16 · p. 11, E. Application to 4D surfaces · p. 12

D. Summary statistics and 3D shape generation

• Animal and human datasets yield statistical means and the first three modes of variation, which are mapped back to surfaces using NeuralSRNF.The method also uses the computed means and modes to sample random surfaces in SRNF space before inversion.Source: D. Summary statistics and 3D shape… · p. 11
• Generated human shapes retain dataset features with NeuralSRNF, whereas numerical SRNF results show missing or shrunk limbs; the framework also generates random 3D animals.The human comparison includes close-up views.Source: D. Summary statistics and 3D shape… · p. 11

E. Application to 4D surfaces

• The 4D-face example presents a mean that captures main facial features and the first two principal modes of variation.Source: Figure 17 · p. 12, E. Application to 4D surfaces · p. 12
• For 4D analysis, each frame is represented by an SRNF, so temporal alignment, geodesics, and summary statistics operate on curves in SRNF space before inversion to surfaces.Source: E. Application to 4D surfaces · p. 12
• 15 s versus around 45 min: NeuralSRNF inverts a 4D geodesic sequence 180 times faster than numerical SRNF and is reported as more accurate.The example applies joint temporal alignment and geodesic computation to two punching-action sequences.Source: E. Application to 4D surfaces · p. 12

V. CONCLUSION

• NeuralSRNF is a continuous surface-based neural representation that provides a practical solution to the ill-posed SRNF inversion problem.Source: §V · p. 12
• The framework supports geodesics, deformation transfer, statistical summaries, and shape generation by analyzing shapes in SRNF space with an L2 metric.The L2 metric facilitates these downstream analysis tasks.Source: §V · p. 12
• Up to 100 times faster than existing methods: NeuralSRNF achieves improved accuracy and state-of-the-art 3D shape-analysis results across human-body, face, and animal datasets.Source: §V · p. 12
• The approach requires registered 3D surfaces, separate training for each shape class, and genus-0 surfaces.Source: §V · p. 12