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Deep Implicit Moving Least-Squares Functions for 3D Reconstruction

Shi-Lin Liu, Hao-Xiang Guo, Hao Pan, Peng-Shuai Wang, Xin Tong, Yang Liu

arXiv:2103.12266v2cs.CVcs.GR

TL;DR

Point sets offer flexible shape modeling but their discrete nature limits continuous, fine geometry, while implicit methods can require costly per-point network evaluation. IMLSNets combine octree-scaffolded point generation with implicit MLS surfaces and network-independent function evaluation. On 3D object reconstruction, they outperform point-set generation and global or local implicit methods.

  • Problem

    Point sets are flexible for 3D generation, but their discrete nature limits continuous, fine geometry.

    Source: §1 · p. 1

  • Method

    IMLSNet uses an octree scaffold to generate MLS points where needed and represents shapes with implicit MLS surfaces.

    Source: §1 · p. 1, §3.2 · p. 3

  • Results

    IMLSNet outperforms point-set generation and global or local implicit methods on 3D object reconstruction.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The approach combines point-set flexibility with the approximation power of implicit surface representations.

    Source: §5 · p. 8

    The input point-cloud octree cannot serve as the scaffold because it may be incomplete, inaccurate, and differ from the target shape.

    Source: §3.2 · p. 3

Abstract

from arXiv · show

Point set is a flexible and lightweight representation widely used for 3D deep learning. However, their discrete nature prevents them from representing continuous and fine geometry, posing a major issue for learning-based shape generation. In this work, we turn the discrete point sets into smooth surfaces by introducing the well-known implicit moving least-squares (IMLS) surface formulation, which naturally defines locally implicit functions on point sets. We incorporate IMLS surface generation into deep neural networks for inheriting both the flexibility of point sets and the high quality of implicit surfaces. Our IMLSNet predicts an octree structure as a scaffold for generating MLS points where needed and characterizes shape geometry with learned local priors. Furthermore, our implicit function evaluation is independent of the neural network once the MLS points are predicted, thus enabling fast runtime evaluation. Our experiments on 3D object reconstruction demonstrate that IMLSNets outperform state-of-the-art learning-based methods in terms of reconstruction quality and computational efficiency. Extensive ablation tests also validate our network design and loss functions.

1. Introduction

Point sets offer flexible 3D shape modeling but produce discrete, rough geometry, while implicit methods provide continuity at the cost of per-point network evaluation. The paper combines point sets with implicit MLS surfaces, using octree scaffolding and localized evaluation to target smooth reconstruction efficiently.

1. Introduction

  • Point sets flexibly model changing topology and complex surfaces, but their discrete geometry is rough and limits fine continuous shape representation.Implicit methods define smooth functions over 3D domains, but require individual network evaluations before surface extraction.Source: §1 · p. 1
  • IMLS interpolates nearby oriented-point planes into signed distances, whose zero level set forms a smooth, continuous surface.The formulation operates within a narrow band around the points.Source: §1 · p. 1
  • An octree scaffold generates variable numbers of MLS points only where needed, with tailored losses regularizing their distribution.The motivation is that dense, evenly distributed points are effective for MLS, while fixed-count prediction consumes resources and is hard to generalize.Source: §1 · p. 1
  • Because the MLS surface is localized near generated points, supervision and evaluation can sample octree nodes rather than the full 3D domain.Once MLS points are embedded, function evaluation is independent of the network, avoiding per-point network evaluation.Source: §1 · p. 2
  • The approach outperforms point-set generation and global or local implicit methods on 3D object reconstruction.The paper also reports ablation tests validating its design choices.Source: §1 · p. 2

2. Related Work

Related work spans point sets, voxel, mesh, patch, primitive, and implicit representations, each with different constraints on detail, topology, or computational cost. The paper combines explicit point-set generation with implicit MLS surfaces, chosen for local, fast evaluation.

2. Related Work

  • Point sets can be limited in detailed geometry when point counts are specified, prompting point upsampling to improve density and shape quality.The cited approaches include and point upsampling works.Source: §2 · p. 2
  • Dense voxels represent occupancy but incur high memory costs at high resolution; sparse voxels such as octrees improve memory and computational efficiency.The cited dense-voxel methods are, and sparse-voxel methods include.Source: §2 · p. 2
  • Meshes and patches can improve shape quality, but predefined topology or resolution constrain meshes, while disconnected patches constrain patch representations.The cited mesh methods are and patch methods.Source: §2 · p. 2
  • The proposed hybrid represents smooth, continuous implicit MLS surfaces while retaining explicit point-set flexibility and efficiency.The paper identifies it as a deep implicit representation combining advantages of point sets and implicit functions.Source: §2 · p. 2
  • Implicit MLS is selected for its fast, local evaluation and direct implicit function, which supports signed-distance supervision.MLS includes projection surfaces defined through iterative projection and implicit MLS surfaces that define a function directly.Source: §2 · p. 2

3. Method

The method builds smooth implicit surfaces from MLS points, then predicts those points with an octree-based scaffold and learned local geometry. It trains the scaffold and surface using sampled SDF supervision and regularizers for point and surface smoothness.

3.1. IMLS surface

  • The IMLS function averages signed distances to nearby oriented tangent planes, and its zero level set defines a smooth surface.With uniform sampling, the cited result states that the surface approximates the original surface geometrically and topologically, and the function approximates its signed distance function.Source: §3.1 · pp. 2–3

3.2. Deep IMLS surface

  • An octree scaffold assigns MLS points to finest non-empty octants, making their total number and positions adaptive to the target surface.The input point-cloud octree is not used directly as the scaffold because it may be incomplete, inaccurate, or differ from the target shape.Source: §3.2 · p. 3
  • For each finest non-empty octant, an MLP predicts MLS-point offsets, normals, and radii, with bounded offsets and radii supporting learned local priors.The offset components are restricted to [−βh, βh], with β set to 1.5 by default; radii are constrained to [lr/2, 2lr].Source: §3.2 · p. 3

3.3. Network structure

  • A U-Net-like O-CNN autoencoder uses output-guided skip connections to preserve input geometry and predict missing regions.The encoder processes an input octree, while the decoder recursively predicts empty or non-empty cells to the specified output depth.Source: §3.3 · p. 3, §3.2 · p. 3

3.4. Loss function design

  • Training uses supervised scaffold and sampled-SDF objectives, alongside octree classification and SDF losses.The SDF loss compares the predicted IMLS surface with the ground-truth SDF; octree status is classified as empty or non-empty.Source: §3.4 · p. 4
  • Local regularizers encourage MLS-point repulsion and smoothness by penalizing tangent-plane projection errors and neighboring radius changes.A small weight decay is also included in the total loss.Source: §3.4 · p. 4

4. Experiments

IMLSNet is evaluated on ShapeNet reconstruction, including unseen classes, against classical and learning-based methods; the experiments also test its design choices. The reported comparisons favor IMLSNet, while ablations identify how octree depth, MLS point count, loss terms, and radius settings affect reconstruction.

4. Experiments

  • Network forwarding averages 30 milliseconds per point-cloud input, while CPU marching cubes averages 1 second per shape, including file I/O.The generated MLS points also provide a preview of the final surface.Source: §4 · p. 5

4.1. 3D object reconstruction from point clouds

  • IMLSNet achieves the best performance among compared methods; results show more detail and better capture of small parts, and its LFD comparison with ConvOccNet favors IMLSNet's visual quality.The evaluation includes 13 ShapeNet classes and five unseen classes.Source: §4.1 · pp. 6–7, Table 1 · p. 7, Table 2 · p. 7
  • On five unseen shape classes, IMLSNet achieves the best result among approaches using local features.Figure 4 orders each comparison as input point, ConvOccNet, IMLSNet, and ground truth.Source: §4.1 · p. 7, Figure 4 · p. 7

4.2. Ablation study

  • Deeper input octrees provide more point information and improve accuracy, while deeper output octrees yield better-detailed geometry.The comparisons include IMLSNet(7,6,1) versus IMLSNet(6,6,1), and IMLSNet(7,7,1) versus shallower-output variants.Source: §4.2 · p. 7, Figure 5 · p. 7
  • Shallow output octrees with larger MLS point counts achieve results similar to deeper octrees with fewer points when total point counts are comparable, but the deeper setting captures small, thin details better.The compared settings include IMLSNet(7,6,2), IMLSNet(7,5,4), and IMLSNet(7,7,1).Source: §4.2 · p. 7, Figure 5 · p. 7
  • Removing each tested loss term degrades surface quality; the surface-gradient term matters most, while removing radius smoothness causes only a slight performance drop.Without the gradient term, IoU and NC are lower and some surface details are lost.Source: §4.2 · pp. 7–8, Table 3 · p. 8
  • Fixing the MLS radius worsens IMLSNet(6,6,1), whereas IMLSNet(7,7,1) performs well with a fixed radius, suggesting adaptive radii are more effective for coarser scaffolds.The fixed-radius variants are denoted cr.(6,6,1) and cr.(7,7,1).Source: §4.2 · p. 8, Table 3 · p. 8
  • Octree-aided direct local-function prediction can recover most surface regions but may produce holes; explicit MLS point generation and IMLS formulation avoid this flaw.The zero-iso surface is not guaranteed to pass through the finest non-empty octant region in the direct-prediction approach.Source: §4.2 · p. 8, Figure 7 · p. 8

5. Conclusion and future work

The paper presents a deep implicit moving least-squares surface technique for 3D reconstruction, combining point-set flexibility with implicit-surface approximation. It identifies higher-order MLS, differentiable rendering, and adaptive point generation as directions for further work.

  • The proposed deep implicit MLS technique combines point-set flexibility with the approximation power of implicit surfaces for 3D reconstruction.The paper reports that extensive tests demonstrate its efficacy and generalization ability.Source: §5 · p. 8
  • Higher-order IMLS variants such as APSS, which replaces planes with spheres, are a possible way to enhance reconstruction quality.Other variants of MLS surfaces are also proposed as extensions.Source: §5 · p. 8
  • Combining IMLS-Net with implicit surface raytracing could enable differentiable MLS prediction using only 2D image supervision.The paper notes that sphere-marching rendering can exploit the MLS points' band regions.Source: §5 · p. 8
  • Adaptive, non-uniform MLS point generation could reduce unnecessary points and further improve reconstruction quality.Predicting adaptive octrees as the scaffold is proposed as a possible approach.Source: §5 · p. 8

A. Training data preparation

Training meshes are normalized and converted into signed-distance grids and samples, with additional near-surface samples at a finer depth.

  • ShapeNet meshes are normalized into a padded [−1, 1]3 box, converted to 256 × 256 × 256 SDF grids, and sampled progressively, with extra near-surface samples at depth 7.The preprocessing uses TSDF fusion to generate watertight meshes and OpenVDB to compute SDF values and gradients.Source: A. Training data preparation · p. 11

B. Evaluation metrics

The evaluation uses mesh-sampled points and several geometric and perceptual metrics; accompanying tables cover network dimensions and reconstruction performance under different settings.

B. Evaluation metrics

  • Ground-truth and predicted meshes are randomly sampled to form point sets, with nearest-point projections and point normals defined for evaluation.The passage defines projections in both directions between the sampled sets and defines n(·) as the operator returning a point's normal vector.Source: B. Evaluation metrics · p. 11
  • The listed measures are L1 Chamfer distance, volumetric IOU, F-Score, and the light field descriptor (LFD).IOU is intersection volume divided by union volume; F-Score combines precision and recall within distance τ, while LFD compares rendered views encoded with Zernike moments and Fourier descriptors.Source: B. Evaluation metrics · p. 11
  • Table 4 lists feature-channel dimensions across octree depths, while Table 5 evaluates network settings for 3D object reconstruction.Source: Table 4 · p. 11, Table 5 · p. 11

C. Network architecture

The architecture discussion reports IMLSNet parameter size and describes a reduced-channel variant intended to lower model size while retaining comparable performance.

C. Network architecture

D. Octree-aided deep local implicit function

The octree-aided deep local implicit function can miss parts of the zero-iso surface even in covered octants; expanding the octree recovers surfaces there, but with higher reconstruction error than IMLSNet.

D. Octree-aided deep local implicit function

E. Evaluation of object reconstruction

Object-reconstruction evaluations report that IMLSNet outperforms other approaches in accuracy and in recovering details and thin regions.

E. Evaluation of object reconstruction

F. Robustness test on noise levels

The robustness test compares IMLSNet(7,7,1) with ConvOccNet under several test-noise levels after training both on noisy data. IMLSNet performs better at lower noise levels and still outperforms ConvOccNet when both methods degrade gracefully at the highest tested level.

F. Robustness test on noise levels

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