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DiGS : Divergence guided shape implicit neural representation for unoriented point clouds

Yizhak Ben-Shabat, Chamin Hewa Koneputugodage, Stephen Gould

arXiv:2106.10811v3cs.CVcs.AIcs.LG

TL;DR

Raw point clouds are often unoriented, making normal-based shape-INR learning difficult, and initialization also matters for reconstruction. DiGS applies divergence guidance and geometric initialization to learn from raw unoriented points; the paper reports state-of-the-art results against other methods without normal supervision and performance comparable to methods using normals.

  • Problem

    Raw scan point clouds typically lack normal vectors, while normal estimation produces noisy, unoriented predictions that challenge normal-based shape-INR learning.

    Source: §1 · p. 1

  • Method

    DiGS learns a shape INR from raw points using an annealed soft divergence constraint and geometric initialization for sinusoidal networks.

    Source: §1 · pp. 1–2, §3 · p. 3

  • Results

    DiGS reports state-of-the-art results against methods without normal supervision and results comparable to methods using normals.

    Source: §1 · p. 2, §7 · p. 8

  • Takeaways & Limitations

    DiGS reconstructs high-fidelity shapes from raw unoriented point clouds without preprocessing, while retaining some smoothing and thin-feature limitations.

    Source: §7 · p. 8

    DiGS can smooth details and miss thin structures, especially where few points are available and normals are absent.

    Source: §6.3 · p. 8

Abstract

from arXiv · show

Shape implicit neural representations (INRs) have recently shown to be effective in shape analysis and reconstruction tasks. Existing INRs require point coordinates to learn the implicit level sets of the shape. When a normal vector is available for each point, a higher fidelity representation can be learned, however normal vectors are often not provided as raw data. Furthermore, the method's initialization has been shown to play a crucial role for surface reconstruction. In this paper, we propose a divergence guided shape representation learning approach that does not require normal vectors as input. We show that incorporating a soft constraint on the divergence of the distance function favours smooth solutions that reliably orients gradients to match the unknown normal at each point, in some cases even better than approaches that use ground truth normal vectors directly. Additionally, we introduce a novel geometric initialization method for sinusoidal INRs that further improves convergence to the desired solution. We evaluate the effectiveness of our approach on the task of surface reconstruction and shape space learning and show SOTA performance compared to other unoriented methods. Code and model parameters available at our project page https://chumbyte.github.io/DiGS-Site/.

1. Introduction

DiGS learns a shape INR directly from raw unoriented points, using a divergence constraint and geometric initialization. The paper reports results on par with, and sometimes better than, normal-supervised methods, and significantly better than other unoriented methods.

1. Introduction

  • Raw scan point clouds are typically unoriented, while normal estimates are noisy and unoriented, challenging methods that rely on normal-based supervision.Some methods can omit normal supervision, but the paper reports that their performance drops significantly without it.Source: §1 · p. 1
  • DiGS uses raw 3D points without preprocessing and softly constrains the divergence of the signed-distance-function gradient field, annealing the constraint during training.The approach is motivated by low divergence nearly everywhere in the signed distance function's gradient vector field.Source: §1 · p. 1
  • Two geometric initialization methods set the initial zero level set approximately to a sphere, with one preserving controllable high frequencies.The methods are designed for sinusoidal implicit networks.Source: §1 · p. 2
  • DiGS performs significantly better than other unoriented methods and on par with, sometimes better than, methods using normals.Evaluations cover surface reconstruction and shape space learning on SRB, DFAUST, and ShapeNet, which include challenging shape and scan properties.Source: §1 · p. 2
  • The paper's contributions are divergence-guided shape-INR learning with soft second-order derivative constraints and two geometric initializations for sinusoidal networks.Source: §1 · p. 2

2. Related-work

Prior work spans classical and neural surface reconstruction, with shape INRs commonly trained as signed-distance or occupancy functions. DiGS builds on normal-supervised and sign-agnostic approaches while addressing initialization for sinusoidal networks.

Surface reconstruction

  • Surface reconstruction methods range from classical triangulation, implicit-function methods, and Poisson approaches to neural networks that map parameters, grids, or INRs to shapes.The related work also notes a differentiable point-to-mesh optimization layer.Source: §2 · p. 2

Shape implicit neural representations (INRs)

  • Shape INRs represent shapes as signed-distance, occupancy, or unified functions, and SIREN's continuous second derivatives support higher-order supervision.SIREN uses periodic sine activations; DeepSDF and SAL use ReLU activations, which do not provide the same higher-order derivative property described here.Source: §2 · p. 2
  • Normal supervision improves INR performance, but raw scans lack ground-truth normals and estimated normals are noisy; the paper reports that DiGS retains performance better without normals.The related-work discussion says IGR and SIREN performance drops significantly when their normal-loss term is removed.Source: §2 · p. 2
  • Concurrent methods include IDF's high/low-frequency decomposition and PHASE's density-based representation, with PHASE+FF omitting ground-truth normal supervision.Source: §2 · p. 2

Initializing shape INRs

  • SIREN's general-purpose initialization lacks geometric grounding and can cause ghost geometries; DiGS proposes two geometric initializations and finds high frequencies important for fine detail.The proposed initializations extend geometric initialization to periodic activations.Source: §2 · p. 3

3. Outline of Divergence Guided Shape INRs

DiGS uses a smooth-to-sharp training schedule to learn an implicit representation without normal information. It starts from a geometrically initialized sphere, then progressively relaxes divergence guidance to recover detail.

3. Outline of Divergence Guided Shape INRs

  • Training progresses through high-divergence, annealing-divergence, and low-divergence phases after geometric initialization.The phases take 50%, 25%, and 25% of iterations, respectively; high divergence quickly and robustly learns smooth representations but can oversmooth reconstructions.Source: §3 · p. 3
  • Geometric initialization starts from a sphere with an SDF sign pattern while retaining controllable high frequencies.Source: §3 · p. 3
  • The high-divergence phase favors a smooth coarse shape and delays fitting fine details; annealing then permits detail while maintaining smoothly changing normals.Source: §3 · p. 3
  • The low-divergence phase permits very fine details, including sharp corners, and encourages interpolation of point samples subject to the Eikonal term.Figure 2 presents results at four training iterations, with DiGS above SIREN without normals.Source: §3 · p. 3, Figure 2 · p. 3

4. Geometric initialization for SIRENs

The section develops SIREN initializations that begin with sphere-like signed-distance level sets while retaining a controllable way to introduce high frequencies. The multi-frequency variant modifies the geometric initialization to address its low-frequency output.

4. Geometric initialization for SIRENs

  • The geometric initialization sets the initial signed distance function to approximately an r-radius sphere by approximating a signed squared norm and subtracting r.The network parameters are chosen so its transformed output approximates the squared norm within the unit ball.Source: §4 · p. 3
  • The multilayer construction approximately preserves activation norms across layers, then applies the single-hidden-layer construction to produce sphere-like level sets.Small Gaussian perturbations are added to constant parameters to facilitate learning.Source: §4 · pp. 3–4
  • MFGI introduces controlled high frequencies in selected first-layer rows and scales subsequent weights to retain the geometric initialization.The basic initialization keeps activations within the first sine period; MFGI uses a frequency factor of 30 for the remaining rows.Source: §4 · p. 4
  • The 2D visualization shows sphere-like level sets for the geometric initialization, while MFGI adds tunable noise; the compared SIREN initialization has near-zero SDF and gradient norm.The MFGI noise is controlled by its frequency and scaling parameters.Source: §4 · p. 4, Figure 3 · p. 4

5. DiGS Loss

The section adds a divergence penalty to the SIREN loss as an unsupervised second-order constraint on the learned signed distance function. In a toy problem, this constraint reduces function variability and improves consistency with the Eikonal constraint.

5.1. Existing loss function components and setup

  • The baseline SIREN loss combines manifold, normal, Eikonal, and non-manifold penalties, while the no-normal comparison removes the normal term.The paper notes that discrete surface constraints admit infinitely many solutions and that normal supervision helps guide the function space, but requires extra supervision.Source: §5.1 · pp. 4–5

5.2. Second order unsupervised constraints

  • The DiGS loss adds an annealed divergence penalty to the SIREN loss without normal supervision, penalizing the magnitude of the SDF gradient field's divergence off the surface.The paper sets λ_div to 100 and uses τ as the annealing factor.Source: §5.2 · p. 5
  • The divergence constraint requires activations with nonzero second derivatives, such as SIRENs, and does not affect ReLU-based networks.The method therefore depends on an architecture that supports the second-order constraint.Source: §5.2 · p. 5

5.3. Minimising divergence as regularisation

  • The authors argue divergence regularization reduces variability across the space more effectively than the Eikonal term alone because it constrains local regions through second-order information.The Eikonal term already determines Dirichlet energy at sampled points, making a separate energy loss redundant there.Source: §5.3 · p. 5
  • Adding the divergence constraint reduces mean Dirichlet energy and makes the learned function more faithful to the Eikonal constraint in the toy problem.The contour visualization shows less variable level sets with more consistent spacing when divergence is included.Source: §5.3 · p. 6, Figure 5 · p. 6

6. Experiments

The experiments evaluate surface reconstruction and shape-space learning across benchmark shapes, ShapeNet, a scene, and DFaust. DiGS performs strongly without normal supervision, while smoothing and ghost geometry remain limitations in some settings.

6. Experiments

  • The experiments use marching cubes to extract zero-level-set meshes and generally follow IGR’s mesh-generation procedure and SIREN’s architecture and hyperparameters.The evaluation includes SRB, ShapeNet, a scene, and DFaust; metrics include Chamfer and Hausdorff distances, squared Chamfer, and IoU.Source: §6 · p. 6

6.1. Surface reconstruction

  • Without normals, DiGS is consistently better than state-of-the-art methods on SRB Chamfer and Hausdorff metrics, and achieves results similar to methods using normal supervision.DiGS removes ghost geometries that affect SIREN and IGR without normal information.Source: §6.1 · p. 6, Table 2 · p. 7, Figure 6 · p. 7
  • On ShapeNet, DiGS without normals outperforms other methods without normal supervision on both squared Chamfer and IoU, but internal structures can produce ghost geometry.Its mean IoU remains better than other methods, while mean squared Chamfer is competitive but worse than methods such as SIREN.Source: §6.1 · p. 7, Table 3 · p. 7
  • For scene reconstruction, DiGS avoids the ghost geometries seen with SIREN without normal supervision, but its smooth output loses fine details such as sofa legs and picture frames.The smoothing is desirable in planar regions such as ceilings, floors, and tables.Source: §6.1 · p. 7, Figure 8 · p. 7

6.2. Shape Space Learning

  • On DFaust shape-space learning, DiGS+n has lower mean registration-and-scan-to-reconstruction distances than IGR and slightly smaller median distances.DiGS+n produces less ghost geometry in reconstruction-to-registration-and-scan comparisons, while DiGS without normals captures human shapes but is oversmoothed and has large ghosts.Source: §6.2 · p. 8, Table 4 · p. 8

6.3. Limitations

  • DiGS is limited in capturing very thin structures and can smooth regions with few points, including sofa legs and wall pictures.The authors relate these challenges to sparse points and the absence of normal information.Source: §6.3 · p. 8

7. Conclusion

The conclusion presents DiGS as a method for raw unoriented point clouds and reports competitive reconstruction performance, alongside limitations involving smoothing and thin features. It also identifies missing data and thin features as challenges for existing methods.

7. Conclusion

  • DiGS represents raw unoriented point clouds without preprocessing and reports state-of-the-art results against methods without normal supervision, with performance comparable to methods using normals.The conclusion also describes a geometrically motivated initialization for sinusoidal representation networks that preserves high frequencies.Source: §7 · p. 8
  • DiGS can smooth reconstructions and struggle with thin features; the authors state that all methods, including DiGS, struggle with missing data and thin features.The paper suggests local self-similarity as a direction for future work on these regions.Source: §7 · p. 8
  • The authors identify point-cloud density and initialization as additional directions for improving shape representation networks.Source: §7 · p. 8

A. Supplementary Material

The supplement outlines supporting theory, initialization analyses, and additional experiments for DiGS.

A. Supplementary Material

A.1.1 SDF properties

The passage compares scalar fields associated with the signed distance function, noting that divergence is small in most regions but spikes near geometric features.

A.1.1 SDF properties

  • The SDF’s gradient norm is one everywhere, while the curl of its gradient field is zero everywhere.Source: §A.1.1 · p. 11
  • The divergence has low magnitude across most areas and spikes at centers, skeletons, and corners.Source: §A.1.1 · p. 11
  • The divergence spikes diffuse quickly away from those locations.Source: §A.1.1 · p. 11

A.1.2 Understanding the divergence constraint

The divergence constraint regularizes the learned distance function: it encourages smoother, more uniform contours, while omitting it can produce sign errors. Its effect depends on how points are sampled, and applying the divergence loss away from surface samples helps preserve fine surface detail.

A.1.2 Understanding the divergence constraint

  • The divergence theorem equates volume-integrated divergence with surface flux, and local divergence reflects how much the gradient field moves toward or away from a point.Low divergence magnitude indicates that gradient directions change little around that point, while the divergence at a point is influenced by its surrounding region.Source: §A.1.2 · p. 11
  • Dirichlet energy measures function smoothness through the integral of squared gradient magnitude.The section presents this energy as a way to quantify SDF complexity.Source: §A.1.2 · p. 11
  • For 2D shapes, divergence can spike at centers, skeletons, or corners and diffuse quickly away from them.These are features of the ground-truth divergence field shown in Figure 10.Source: Figure 10 · p. 12
  • Minimizing the absolute functional derivative, expressed through the divergence loss, seeks low Dirichlet energy among SDFs that satisfy surface interpolation and the eikonal constraint.The constrained solution need not be harmonic, since most SDFs are not harmonic.Source: §A.1.2 · pp. 11–12
  • The divergence loss is evaluated on random spatial samples, not the heavily sampled surface points, because including surface samples would drastically smooth the surface.Fine detail can involve high SDF variability near the surface and low variability farther away.Source: §A.1.2 · p. 12
  • With the divergence term, contours are smoother and more uniformly spaced; without it, sign errors can occur, while clustered constraints can bias the learned function.The toy-problem visualizations report these patterns across repeated runs.Source: §A.1.2 · pp. 12–13, Figure 11 · p. 13

A.1.3 Understanding the divergence constraint in 2D

The 2D experiments show that the useful divergence constraint schedule depends on shape geometry: annealing helps the detailed Koch snowflake, while the smooth circle favors an unannealed constraint. Initialization also affects sign consistency and fine-detail reconstruction.

A.1.3 Understanding the divergence constraint in 2D

  • The comparison uses circle, L-shaped, and Koch snowflake shapes, with heatmaps visualizing learned distance, eikonal, divergence, curl, and unsigned-distance error.The sine-activation network has four layers of 128 units and is trained for 10K epochs.Source: §A.1.3 · p. 13
  • For the circle, an unannealed divergence constraint performs best, whereas starting with the constraint and annealing it works best for the sharp-edged Koch snowflake.For the snowflake, the constraint guides learning toward a smoothed geometry before annealing allows a tighter fit.Source: §A.1.3 · p. 14
  • SIREN without normals exhibits ghost geometries, or spurious zero level sets, while DiGS does not.Source: §A.1.3 · p. 14
  • Initialization significantly affects distance-function sign and the ability to reconstruct fine detail, particularly for the snowflake.Source: §A.1.3 · p. 14

A.1.4 Second order supervision constraint

This section describes DiGS’s curvature supervision, geometric initialization, training and evaluation setup, and reports reconstruction results and ablations across SRB, ShapeNet, and scenes, alongside a DFaust shape-space experiment. It examines performance without normals and how outcomes vary with architecture and task.

A.1.4 Second order supervision constraint

  • The second-order supervised DiGS loss penalizes disagreement between surface mean curvature and the divergence of the vector field.Normal and curvature estimates are noisy and depend strongly on the size of the local neighborhood, so adding these signals does not guarantee better performance.Source: §A.1.4 · p. 14

A.2.1 Further initialization visualizations

  • The geometric initialization produces sphere-like level sets with smoother Eikonal and divergence terms than the compared initialization, and is important for shape space learning.The visualization compares the proposed initialization, including its multi-frequency variant, with the method of Sitzmann et al. [37].Source: §A.2.1 · p. 17, Figure 14 · p. 17

A.2.2 Proofs

  • The initialization proof approximates the squared norm with a SIREN construction and applies a signed square-root transform to obtain a sphere-like distance function.The transform uses ε = 10^-8 for numerical stability and a continuous, sub-differentiable derivative; the propositions extend the construction to deeper networks.Source: §A.2.2 · pp. 17–18

A.2.3 DiGS training procedure comparison

  • DiGS gradually refines a noisy-sphere SDF, reducing incorrect orientations and ghost geometry, whereas SIREN without normals fits quickly but can retain inconsistently oriented surfaces and early ghost geometry.The visualizations compare training stages on 2D and 3D shapes; SIREN without normals has no zero level set at initialization in the 2D examples.Source: §A.2.3 · p. 18, Figure 16 · p. 19, Figure 17 · p. 19

A.3.1 Evaluation metrics

  • Evaluation uses Chamfer and Hausdorff distances for surface accuracy and volumetric IoU for occupancy; ShapeNet uses squared Chamfer, and IoU estimates sample 100,000 points [29].The SRB reports distances to the simulated scans as a measure of overfitting to noisy input.Source: §A.3.1 · p. 20, Table 6 · p. 21

A.3.2 Experimental setup and timing performance

  • DiGS uses the standard SIREN parameter count and has the same inference cost, but divergence computation increases training time; it remains faster and smaller than IGR.The reported reconstruction experiments use SIREN MLPs, with network depth and width varied by task.Source: §A.3.2 · p. 20, Table 5 · p. 20

A.3.3 Surface reconstruction on SRB

  • On the SRB, DiGS achieves the best reconstruction among methods without normal supervision, improves over vanilla SIREN with normal supervision, and is comparable to other normal-supervised methods.Qualitative results also show DiGS improving on methods without ground-truth normals and producing results comparable to normal-based methods.Source: Table 6 · p. 21, Figure 18 · p. 21, §A.3.3 · p. 22
  • SRB ablations show that annealing improves performance, L1 divergence penalties outperform L2, and mean-curvature supervision provides no benefit.Linear and step annealing perform similarly; L1 permits sparse high-divergence locations, which may correspond to sources or sinks in the vector field.Source: Table 7 · p. 22, Table 8 · p. 22, §A.3.3 · p. 22
  • On SRB, multi-frequency geometric initialization with step decay and an L1 divergence penalty performs consistently better than the other ablations, all trained without normals.The divergence loss requires smooth activations and is targeted at high-frequency architectures such as SIRENs; paired with SoftPlus, it reduced performance by 0.99 dC and 6.34 dH on SRB.Source: §A.3.3 · p. 23

A.3.4 Surface reconstruction on ShapeNet

  • DiGS reconstructs ShapeNet shapes on par with NSP, while SIREN without normals produces ghost geometry and SAL oversmooths or misses thin surfaces.For loudspeakers, normal supervision causes more internal ghost geometry, yielding worse results than without normals.Source: §A.3.4 · p. 23, Figure 19 · p. 25, Figure 20 · p. 26

A.3.5 Scene Reconstruction

  • For scene reconstruction, the method uses an eight-layer network with 512 units, trained for 100K iterations on 10M oriented points with 15K random points sampled per iteration.The supplied material describes a multi-angle low-resolution video and notes that a high-resolution video is available externally.Source: §A.3.5 · p. 23

A.3.4 Surface reconstruction on ShapeNet

  • On ShapeNet, squared Chamfer distance and IoU generally have better means and medians without normals, while normal supervision improves medians but not always means.The class-by-class breakdown is in Table 9.Source: §A.3.4 · p. 23, Table 9 · p. 27

A.3.6 Shape Space Learning

  • On DFaust, DiGS without normals learns multiple human shapes whereas IGR without normals fails; with normals, DiGS+n has fewer ghost geometries but oversmooths fine details compared with IGR.IGR with normals captures more detail but exhibits ghost geometries, orientation changes, and missing large surfaces.Source: Figure 21 · p. 28

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