Multi-Class Boundary Extraction from Implicit Representations

Jash Vira; Andrew Myers; Simon Ratcliffe

cs.LGFeb 18, 2026

Multi-Class Boundary Extraction from Implicit Representations

Authors: Jash Vira, Andrew Myers, Simon Ratcliffe

Abstract

Surface extraction from implicit neural representations modelling a single class surface is a well-known task. However, there exist no surface extraction methods from an implicit representation of multiple classes that guarantee topological correctness and no holes. In this work, we lay the groundwork by introducing a 2D boundary extraction algorithm for the multi-class case focusing on topological consistency and water-tightness, which also allows for setting minimum detail restraint on the approximation. Finally, we evaluate our algorithm using geological modelling data, showcasing its adaptiveness and ability to honour complex topology.

Figures & tables

Figure 1: The top part shows the output of a Neural Network for all in

\{C_{1},C_{2},\dots,C_{k}\}

, where

k=14

, densely sampled across

\mathbb{R}^{1}

. The bottom figure displays the results of the 1D Root-Finder, i.e. the precise intervals where each class with the maximum probability exists.

Figure 2: The graph depicts the behaviour of classes

C_{1}

C_{2}

, and

C_{3}

on a 1D axis in the interval

[-4,4]

. The domain transition occurs in the interval

[0,2]

, which the 1D algorithm eventually localises using the topological bracketing method.

Figure 3: Illustration of the gradient-based localisation approach for root finding. The figure demonstrates the scenario where

g(x)

exhibits multiple domain transitions, with tangents plotted at

x_{1}

and

x_{2}

. The regions

[x_{1}-\delta,x_{1}]

and

[x_{2},x_{2}+\delta]

are shown in yellow, while the interval

[x_{1},x_{2}]

is in green, indicating the interval under consideration for root localisation. The plot captures a potential root classification since the signs of

g(x)

change at

x_{1}

and

x_{2}

, and not both projected intersections

x_{1}^{*}

and

x_{2}^{*}

lie in the green area.

Panel of Figure 4 — Figure 4: This figure showcases all possible permutations of the Polygonisable, Three Domains Meeting and Ambiguous cases of a 2D rectangle.

Figure 5: Illustration of method imposing geometric criteria on a naive edge represented in yellow, which is trying to approximate the boundary in purple.

Panel of Figure 6 — Figure 6: The top row shows slices from a geological model, selected for their complexity, sampled at a resolution 1000x1000 from the neural network. The bottom row shows the 2D polygoniser’s result.

Panel of Figure 7 — Figure 7: The first column showcases the output of the 2D polygoniser when no geometric criteria were set. The second column shows the output when the geometric criteria threshold was set to 1e-3.

Implicit Neural Representations

Implicit neural representations encode signals and fields as continuous functions that can be queried at arbitrary coordinates and resolutions.

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M-plicits: Neural Implicit Surfaces via Nested Multiscale Residuals

Vinícius da Silva, Isabelle Melo, Matheus Bessa, Guilherme Schardong+7

PUC-Rio · Universidade Federal de Santa Maria · Google DeepMind+2

Implicit Neural Representations 3D Reconstruction Residual Learning Neural Rendering

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Encoding input coordinates with sinusoidal functions into multi-layer perceptrons (MLPs) has proven effective for implicit neural representations (INRs) of surfaces defined as zero-level sets. However, existing methods often struggle to balance training efficiency, rendering speed, and noise robustness: single-MLP approaches are expensive at inference, grid-based representations are fast but can limit surface smoothness and overfit input noise, and previous multiscale approaches frequently capture noise and produce artifacts due to hard spectral truncation. To address these limitations, we propose M-plicits, a multiscale framework that models surfaces as a residual sum of MLPs trained via a sequence of nested neighborhoods. Unlike existing residual approaches that rely on standard domain-wide sampling and require costly mesh extraction for visualization, our method strictly localizes supervision to narrow bands around the previous zero-level sets. This nested design naturally provides robustness against noisy input data: the coarse network acts as a low-pass filter that establishes a clean geometric prior, while subsequent residuals progressively refine the geometry without fitting to high-frequency artifacts. We further introduce a multiscale sphere-tracing algorithm and a GEMM-based analytical normal computation that bypasses auto-differentiation entirely, yielding high-fidelity real-time rendering. On Stanford and Thingi32, M-plicits achieves the best mean Chamfer distance in the coarse configuration and the best median Chamfer distance and IoU in the fine configuration, with substantially better noise robustness than iNGP, BACON, and IDF, while using an order of magnitude fewer parameters than grid-based baselines. Code, models, and data are available at https://github.com/dsilvavinicius/m-plicits.

Abstract

Implicit Neural Representations

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