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Deep Level Sets: Implicit Surface Representations for 3D Shape Inference
Mateusz Michalkiewicz, Jhony K. Pontes, Dominic Jack, Mahsa Baktashmotlagh, Anders Eriksson
TL;DR
Learned 3D reconstruction methods can produce coarse, inaccurate object boundaries and often require post-processing to extract meshes. This paper predicts continuous, oriented level sets with an end-to-end trainable model and variational geometric loss. Its reconstructions are reported as more accurate and detailed than voxel-based representations.
Problem
Existing approaches struggle to classify object-boundary pixels accurately, producing coarse representations that require post-processing to extract surface meshes.
Source: §1 · p. 2Method
Results
The approach produces more accurate and geometrically detailed reconstructions than voxel-based representations, with reported differences narrowing at higher resolution.
Source: §1 · p. 2, §6 · p. 7, §5.2 · p. 7Takeaways & Limitations
The level-set formulation supports geometric regularizers and is presented as a flexible complement to existing approaches for shape inference.
Source: §1 · p. 2, §5.1 · p. 7, §6 · p. 7Higher-resolution shape inference and segmentation remain directions for future work.
Source: §6 · p. 7
Abstract
from arXiv · showhide
Existing 3D surface representation approaches are unable to accurately classify pixels and their orientation lying on the boundary of an object. Thus resulting in coarse representations which usually require post-processing steps to extract 3D surface meshes. To overcome this limitation, we propose an end-to-end trainable model that directly predicts implicit surface representations of arbitrary topology by optimising a novel geometric loss function. Specifically, we propose to represent the output as an oriented level set of a continuous embedding function, and incorporate this in a deep end-to-end learning framework by introducing a variational shape inference formulation. We investigate the benefits of our approach on the task of 3D surface prediction and demonstrate its ability to produce a more accurate reconstruction compared to voxel-based representations. We further show that our model is flexible and can be applied to a variety of shape inference problems.
1. Introduction
The introduction argues that existing learned shape representations struggle with object boundaries and motivates continuous level sets as a more expressive, geometrically informed representation compatible with deep networks.
1. Introduction
- Level sets use regular grids suited to CNNs while representing more geometric information and supporting quantities such as orientation, smoothness, and volume.Source: §1 · pp. 1–2
- Meshes capture detailed geometry compactly but are irregular and difficult to integrate into learning, while low-resolution voxel grids can yield coarse reconstructions.Point clouds are also sparse and share some drawbacks of meshes.Source: §1 · p. 1
- Existing deep reconstruction methods can misclassify boundary pixels, yielding fuzzy, coarse shapes that require meshing to obtain a 3D surface mesh.Source: §1 · p. 2
- The paper integrates level-set methods into deep convolutional networks to predict a continuous embedding whose zero level set represents the reconstructed object.Source: §1 · p. 2
- The proposed level-set representation produces more accurate reconstructions than voxel representations, with the improvement attributed to level-set incorporation rather than a more complex architecture.The evaluation uses ShapeNet and ShapeNet-Core with a simple encoder-decoder architecture.Source: Figure 1 · p. 1, §1 · p. 2
2. Related Work
Related work spans single- and multi-image reconstruction, learned voxel and point-cloud methods, and approaches using shape priors or adversarial models. The paper identifies representational limits in prior approaches and proposes continuous level sets integrated into CNNs.
2. Related Work
- Shape-prior methods address limited training data but rely on hand-crafted features that limit scalability.Source: §2 · p. 2
- Single-image CNN methods have used ground-truth supervision or weak supervision, while weakly supervised approaches do not scale well as input resolution increases.Source: §2 · p. 2
- GAN-based reconstruction methods can be computationally expensive and inaccurate on Chamfer distance, Earth Mover’s distance, or IoU, and may not support multiple reconstructions.Source: §2 · p. 3
- Prior reconstruction methods use voxel, point-cloud, or explicit shape representations, which the paper characterizes as limited in geometric expressiveness.Source: §2 · p. 3
- The paper proposes integrating level sets into CNNs to infer embedding functions from which a continuous 3D surface can be extracted.Source: §2 · p. 3
3. Preliminaries
The section formulates surfaces as zero level sets of embedding functions and develops an oriented fitting objective for implicit surfaces. It also describes level-set evolution, conditioning, and shape priors relevant to this formulation.
Level Set Surface Representations
- A level-set function implicitly represents an orientable surface as its zero isocontour, while manipulating the function captures isosurface motion.The function is a higher-dimensional scalar embedding function.Source: §3 · p. 3
- On a fixed Cartesian grid, the formulation supports computing normals, curvatures, and volumes, and can represent complex topologies and topology changes without explicit modeling.Parametric approaches typically require dedicated procedures to handle topological variations.Source: §3 · p. 3
Minimal Oriented Surface Models
- The fitting task minimizes both distance to sampled surface points and mismatch between estimated and target normals, whose signs may be ambiguous for open surfaces.The target surface need not be closed; its normals are defined only up to sign.Source: §3 · p. 3
- The proposed normal energy measures the Lp angular distance between estimated surface normals and desired normals, with target normals assigned from the nearest point.The point-set distance functional measures the Lp distance from the surface to the point set.Source: §3 · p. 3
- Surface-integral energies for distance and normals favor smaller-area surfaces, thereby introducing an implicit smoothing component.This follows from both functionals returning lower energies on surfaces with smaller area.Source: §3 · pp. 3–4
Shape Priors & Regularisation
- The section focuses on surface-area and volume priors, while noting that additional shape priors can presumably be incorporated into the framework.Generic priors concern geometric quantities; object-specific priors are computed from samples of the object of interest.Source: §3 · p. 4
Embedding functions and Ill-Conditioning
- Conventional level-set functions can become too steep or too flat near the zero level set, causing ill-conditioning and potentially unstable surface motion.Signed distance functions have unit gradient throughout their domain, and are commonly used to reinitialize the level-set function.Source: §3 · p. 4
- Because reinitialization is not directly applicable in learning-based shape inference, the method uses an energy functional to promote the unit-gradient property.The proposed approach is similar to.Source: §3 · p. 4
4. A Variational Loss Function for Shape Inference
The section introduces a variational loss for training a neural network to predict implicit surfaces from inputs. It evaluates a discretized approximation of the loss and remains independent of the isosurface extraction algorithm.
4. A Variational Loss Function for Shape Inference
- The proposed loss trains a neural network to predict a level-set function from each input by minimizing dissimilarity to corresponding ground-truth oriented shapes.Training examples pair inputs with shapes represented by discrete points and associated normals.Source: §4 · p. 4
- The predicted surface is the zero level set of the network's level-set function, and four weighting parameters govern the loss components.The loss is defined across the training examples.Source: §4 · p. 4
- In practice, the loss is evaluated on a fixed equidistant grid using continuous approximations of the Dirac delta and Heaviside functions.The approximated loss is expressed entirely in the predicted level-set function.Source: §4 · p. 5
- The framework is indifferent to the isosurface extraction algorithm, unlike work derived from a specific algorithm.Evaluating the derivative with respect to the predicted level-set function requires calculating its inferred zero-level surface.Source: §4 · p. 5
5. Experimental Validation
The experiments compare level-set and voxel representations using a shared, deliberately simple network. The reported results favor level sets overall, while noting smaller differences at higher resolution and a low-resolution challenge for chair legs.
5.1. Implementation Details
- The evaluation used bottles, cars, chairs, and sofas from ShapeNet, rendered into 20 views per model and divided into 80/20 train-test splits.Closed manifolds were generated as ground truth, with approximately 500 bottle models and 2,000 models in each other category.Source: §5.1 · p. 5
- Both representations used the same simple 64-dimensional autoencoder/CNN architecture and were trained with variational or voxel cross-entropy losses.Training used 3,000 epochs, batch size 64, and learning rate 10^-6; the two formulations had identical evaluation-time memory and computational costs.Source: §5.1 · pp. 5–6
5.2. Results
- Level sets produced more accurate, detailed shapes than voxels overall, though their advantage narrowed at higher resolution and chairs were an exception.The authors attribute the chair result to long, thin legs that are difficult for both representations to capture at low resolution.Source: §5.2 · p. 7
- Qualitative examples were reported to show higher-quality level-set predictions than volumetric predictions.The examples were selected as representative of typical network predictions.Source: §5.2 · p. 7
6. Conclusion and Future Work
The paper concludes that an oriented variational loss in an end-to-end framework can infer detailed 3D shapes, and proposes investigating higher-resolution inference and segmentation in future work.
6. Conclusion and Future Work
- An oriented variational loss in a deep end-to-end framework is proposed for flexible 3D surface prediction.The authors report that a unit-gradient regularizer addresses ill-conditioning and numerical and surface-motion instability in conventional level-set formulations.Source: §6 · p. 7
- Experiments showed more geometrical detail in inferred shapes than voxel-based representations.Source: §6 · p. 7
- Future work will investigate higher-resolution shape inference and segmentation.Source: §6 · p. 7