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Implicit Geometric Regularization for Learning Shapes

Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, Yaron Lipman

arXiv:2002.10099v2cs.LGcs.CVcs.GRstat.ML

TL;DR

The paper addresses how to learn high-fidelity implicit neural shape representations directly from raw point clouds, with or without normals, rather than relying on pre-computed representations or explicit zero-level-set losses. It introduces a simple loss combining point fitting and unit-gradient encouragement, whose optimization implicitly favors smooth, plausible surfaces. In practice the method achieves state-of-the-art representations with greater detail and fidelity, while its theory is limited to the linear case and the method is sensitive to noisy normals.

  • Problem

    The paper addresses the need to compute implicit neural representations directly from raw point clouds, with or without normals, instead of relying on pre-computed implicit representations or explicit zero-level-set losses.

  • Method

    The method fits an MLP to vanish on input points and optionally match normals while encouraging unit-norm spatial gradients, using implicit geometric regularization to select plausible zero level sets.

  • Results

    The method produces state-of-the-art implicit neural representations with significantly more detail and higher fidelity than alternative techniques, and linear analysis proves plane reproduction under the stated conditions.

  • Takeaways & Limitations

    A simple loss can yield favorable smooth, high-fidelity implicit surfaces directly from raw data despite infinitely many signed-distance-function minima.

  • Takeaways & Limitations

    The paper does not provide a full theory for the nonlinear phenomenon, and identifies sensitivity to noisy normals as the method’s main practical limitation.

Abstract

from arXiv · show

Representing shapes as level sets of neural networks has been recently proved to be useful for different shape analysis and reconstruction tasks. So far, such representations were computed using either: (i) pre-computed implicit shape representations; or (ii) loss functions explicitly defined over the neural level sets. In this paper we offer a new paradigm for computing high fidelity implicit neural representations directly from raw data (i.e., point clouds, with or without normal information). We observe that a rather simple loss function, encouraging the neural network to vanish on the input point cloud and to have a unit norm gradient, possesses an implicit geometric regularization property that favors smooth and natural zero level set surfaces, avoiding bad zero-loss solutions. We provide a theoretical analysis of this property for the linear case, and show that, in practice, our method leads to state of the art implicit neural representations with higher level-of-details and fidelity compared to previous methods.

1. Introduction

The paper introduces a way to learn high-fidelity implicit neural shape representations directly from raw point clouds, avoiding pre-computed supervision or explicit zero-level-set losses. Its simple optimization favors smooth, plausible surfaces and is supported theoretically in the linear case.

  • Background: Implicit neural representations encode shapes through neural-network level sets, with network parameters tied directly to shape rather than a fixed volumetric discretization.Previous approaches commonly relied on 3D supervision from known or pre-computed implicit representations.
  • Problem: The paper targets signed-distance-like functions learned directly from point clouds, with or without normal information, for plausible surfaces defined by the raw data.The input consists of points X and optionally normals N.
  • Problem: Earlier raw-data methods either sampled the zero level set or explicitly regularized it, while pre-training reconstruction can hinder collective learning across related shapes.Pre-computing an implicit surface from raw data is also described as difficult and can prevent information sharing between shapes.
  • Contribution: The proposed paradigm uses implicit geometric regularization to obtain smooth, high-fidelity zero level sets without 3D supervision or a direct zero-level-set loss.The method fits an MLP to point-cloud data while encouraging unit-norm gradients.
  • Theory: For approximately planar point clouds, linear-model analysis proves that randomized gradient descent converges with probability one to an approximate signed distance function while avoiding bad critical solutions.The paper calls this theoretical property plane reproduction.
  • Results: In practice, the method produces state-of-the-art implicit neural representations with more detail and higher fidelity than alternative techniques.Experiments cover 3D point clouds and collections of shapes learned directly from raw data.

2. Method

The method trains an MLP with a loss that fits point values and optional normals while enforcing unit gradient norms. Despite infinitely many signed-distance-function minima, stochastic gradient optimization tends to select smooth, plausible zero level sets.

  • Method: The MLP is trained to approximate a signed distance function to a plausible surface defined by the input point cloud and optional normals.The network represents the surface as its zero level set.
  • Loss: The loss encourages the network to vanish on input points and, when normals are available, aligns its spatial gradients with the supplied normals.The normal-fitting term is active when normal data exists.
  • Loss: The Eikonal term encourages the spatial gradients of the network to have unit Euclidean norm over samples from a distribution in R3.This term is motivated by the Eikonal equation and signed-distance-function structure.
  • Geometric regularization: Because discrete point-boundary data admit infinitely many solutions, the optimization problem is ill-posed and requires a mechanism to select a favorable zero level set.Any well-behaved domain containing the points on its boundary can provide a valid Eikonal solution.
  • Optimization behavior: Stochastic gradient optimization tends to produce functions close to signed distance functions with smooth, plausible zero level sets, as reflected by equidistant level sets in planar examples.The zero level sets approximate the input data while avoiding alternative zero-loss solutions.
  • Implementation: The spatial gradient can be computed alongside the MLP output through chain-rule propagation across layers, or more simply with automatic differentiation.Figure 3 illustrates a network computing both f and ∇_xf.

3. Previous work and discussion

Prior 3D shape-learning methods use grids, parametric charts, hybrid structures, or neural PDE solvers, each with distinct representation or boundary-condition constraints. The paper frames discrete-point Eikonal learning as ill-posed, with implicit geometric regularization selecting a favorable solution.

  • Representations: Regular grid volumetric functions define shapes only at grid points, requiring interpolation and cubic-size, geometry-independent grids.Normal approximation additionally requires divided differences.
  • Representations: Neural parametric surfaces model shapes with local or global charts, but consistent atlas coverage is challenging and global parametrizations can have high distortion.Their main advantage is easy sampling, whereas implicit representations ensure perfectly overlapping coverage by construction.
  • Representations: Hybrid representations decompose solids into unions of convex sets, enabling relatively easy transformation between explicit and implicit forms.
  • PDE perspective: Neural PDE methods differ because the paper's discrete point boundary conditions do not satisfy the Eikonal equation's required boundary conditions.Any well-behaved domain containing the points on its boundary can therefore provide a valid initial condition.
  • PDE perspective: From the PDE perspective, the paper's problem is ill-posed with infinitely many solutions, while implicit geometric regularization selects a favorable one.

4. Analysis of the linear model and plane reproduction

The linear analysis studies how the training loss behaves around noisy hyperplanes and establishes plane reproduction under suitable regularization and optimization conditions. Gradient descent avoids bad critical points and converges to one of the signed-distance global minima.

  • Linear model: The linear model uses f(x; w) = w^T x to analyze the paper's loss and its critical points.The analysis is conducted in R^d, with squared error and the normal-data term omitted for simplicity.
  • Plane reproduction: For point clouds near a hyperplane, the analysis targets convergence to an approximate signed distance function of that hyperplane.The data are modeled as hyperplane samples plus deviations bounded by ϵ.
  • Critical points: For λ > cϵ^2, the loss has two global minima corresponding to approximate signed distance functions, while other critical points are saddles or local maxima.The two minima differ by sign.
  • Convergence: Random initialization and sufficiently small step size yield convergence with probability one to an approximate signed distance function of the hyperplane.
  • Geometric interpretation: The preferred minima approximate the hyperplane normal when deviations from planarity are small.The analysis gives λ1 ≤ cϵ and ||u1 − n|| ≤ c′ϵ, and relates sufficient λ to local curvature in the MLP case.
  • Convergence: Theorem 2 states that gradient descent avoids bad critical points with probability one.Because the loss is nonnegative and diverges as the parameter norm grows, the iteration converges, up to the constant step size, to one of two global minima.

5. Implementation and evaluation details

The implementation represents shapes with skip-connected multilayer perceptrons, samples the training distribution around point clouds, and extracts level sets with Marching Cubes. Evaluation uses Chamfer and Hausdorff distances between point sets.

  • Architecture: The shape network has 8 layers, 512 hidden units per layer, and one input-to-middle skip connection.Weights use geometric initialization, with λ = 0.1 and τ = 1.
  • Sampling: The training distribution averages a uniform distribution with Gaussians centered at the input points.Each Gaussian has standard deviation equal to the distance to the 50th nearest neighbor.
  • Level-set extraction: The trained MLP's zero or other level set is extracted using Marching Cubes on uniform grids with ℓ ∈ {256, 512}.
  • Evaluation: Quantitative evaluation uses Chamfer and Hausdorff distances between two point sets.The reported metric collection includes one-sided Chamfer and Hausdorff distances.

6. Model evaluation

The evaluation tests signed-distance-function reproduction and compares geometric fidelity against regression-based implicit representations. The method produces visually accurate level sets and more detail than regression.

  • SDF approximation: The model is evaluated on reproducing signed distance functions for a plane, sphere, and Bimba surface.The experiments use no normals and sample fresh points at every iteration.
  • SDF approximation: Table 1 reports mean ± standard deviation of relative error at 100k random points against known signed distance functions.Relative error compares the predicted function f(x;θ) with the ground-truth signed distance s(x).
  • SDF approximation: Figure 4 visualizes equispaced positive and negative level sets, with the zero level set shown in white.These level sets provide a visual validation of the predictions.
  • Fidelity and level of details: The fidelity comparison trains the proposed method and regression baselines on 250k sampled points from two raw human scans.Regression uses an approximated signed distance function and an L1 loss.
  • Fidelity and level of details: The proposed method produces considerably more level-of-detail than regression on the tested human scans.The paper attributes this potentially to fitting surface points directly and to implicit regularization improving level-set fidelity.

7. Experiments

Experiments evaluate surface reconstruction and learned shape spaces from raw scans. The method achieves strong reconstruction quality, preserves detail, generalizes plausibly to unseen humans, and supports meaningful shape interpolation.

  • 7.1. Surface reconstruction: Surface reconstruction is evaluated from scanned point clouds with or without normals against ground-truth meshes on a benchmark dataset.The scans may contain defects and artifacts introduced by 3D acquisition.
  • 7.1. Surface reconstruction: The benchmark compares the method with DGP using Chamfer, Hausdorff, and one-sided point-cloud-to-reconstruction distances.These metrics assess reconstruction-to-ground-truth accuracy and fit to the input points.
  • 7.1. Surface reconstruction: The method improves upon DGP on 4 out of 5 dataset models when evaluated against ground-truth meshes.DGP fits the input data better on average, while the proposed method performs better in Hausdorff distance; DGP also leaves uncovered surface areas.
  • 7.2. Learning shape space: Shape-space learning uses high-resolution D-Faust scans of 10 humans in multiple poses, with random and unseen-human train-test splits.The multi-shape model uses an auto-decoder with a separate 256-dimensional latent vector for each training example.
  • 7.2. Learning shape space: Reconstructions show more detail than SAL, and unseen-human tests produce plausible approximations despite training on only 8 human shapes.The method predicts held-out shapes by optimizing each test shape’s latent vector for 800 iterations.
  • 7.2. Learning shape space: Coverage analysis shows improvement over SAL except at larger errors in one reconstruction-to-registration direction.The larger-error behavior is attributed to a common reconstructed base, while unseen-human splits have higher reconstruction error rates than random splits.
  • 7.2. Learning shape space: Interpolated latent vectors yield averaged human shapes that blend body shape and pose.Figure 12 visualizes these averaged shapes in blue against training examples in gray.

8. Conclusions

The method learns high-fidelity implicit neural shape representations directly from raw data using a simple loss whose optimization favors favorable signed distance functions. Theory covers the linear case, while experiments report more detailed level sets than previous methods.

  • The method learns high-fidelity implicit neural representations directly from raw data.
  • Its simple loss has infinitely many signed distance functions as minima, yet gradient descent tends to select a favorable one.
  • The linear analysis proves convergence to an approximate signed distance function while avoiding bad critical points.
  • The method produces neural level sets with significantly more details than previous work.
  • The method remains sensitive to noisy normals, motivating future work on robustness.

A.1. Network Architecture.

The experiments use an Auto-Decoder architecture with fully connected layers and softplus activations. The latent dimension differs by application, and reconstruction and shape-space learning use distinct training configurations.

  • The experiments use the Auto-Decoder architecture proposed by Park et al. (2019).
  • FC+ denotes a fully connected layer followed by softplus activation, with β = 100.
  • The architecture uses a skip connection from the input to the fourth layer, and L denotes the latent vector size.
  • For shape reconstruction, the latent size is L = 0; for the shape-space experiment, it is L = 256.
  • Shape Reconstruction: Shape reconstruction uses ADAM for 100k iterations at learning rate 0.0001, sampling 128^2 input points per iteration.
  • Shape Space Learning: Shape-space learning uses four GPUs, 1k epochs, scheduled learning-rate decay, and batches of 32 shapes with 128^2 sampled points each.

B. Additional Results

Additional shape-space experiments evaluate reconstruction on both training and unseen test point clouds using a random train-test split. Figure 14 presents these results alongside raw scans and SAL outputs.

  • The additional experiment reports reconstruction results for both training and unseen test point clouds.
  • The train-test partition uses a random split.
  • Figure 14 compares registration, raw scans, the method’s results, and SAL results in each row.

C.1. Plane Reproduction using Liapunov Function

For the linear plane-reproduction analysis, random-initialized gradient flow converges almost surely to one of two global minima associated with the signed distance function to the data hyperplane. A Liapunov function establishes the convergence argument.

  • Plane Reproduction using Liapunov Function: The analysis considers noiseless point data spanning a hyperplane H of dimension d − 1 that contains the origin.
  • Plane Reproduction using Liapunov Function: With random initialization, gradient flow converges with probability one to one of two global minima corresponding to the signed distance function to H.
  • Plane Reproduction using Liapunov Function: The gradient-flow analysis is performed in transformed coordinates using ℓ(q) from equation 7.
  • Plane Reproduction using Liapunov Function: The limiting model approximates the signed distance function to the hyperplane in transformed coordinates.
  • Plane Reproduction using Liapunov Function: The proof assumes random initialization from a continuous distribution, so the initial point is almost surely not orthogonal to e1.
  • Plane Reproduction using Liapunov Function: The Liapunov function is nonnegative, vanishes only at the fixed point, decreases along the flow, and has bounded level sets.
  • Plane Reproduction using Liapunov Function: The boundedness argument uses h(q) ∈ [0, 1) and finite-radius circular level sets, while the decreasing property follows from a positive semidefinite matrix argument.
  • Plane Reproduction using Liapunov Function: The proof differs from the theorem’s discrete-gradient-descent argument because it analyzes continuous gradient flow directly.
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