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An Integer Linear Programming Approach to Geometrically Consistent Partial-Partial Shape Matching

Viktoria Ehm, Paul Roetzer, Florian Bernard, Daniel Cremers

TL;DR

This work tackles partial-partial 3D shape matching, where both shapes are incompletely observed, by introducing the first integer linear programming (ILP) formulation that enforces geometric consistency to infer the unknown overlap and recover neighbourhood-preserving correspondences. It builds on surface-cycle representations and product graphs to encode per-triangle matching as coupled subproblems, augmented with injectivity and surjectivity constraints to handle overlap. The ILP objective combines per-edge matching costs with overlap probabilities via a weighting parameter $\lambda$, and a coarse-to-fine strategy scales the approach to higher resolutions. Empirical results on CP2P24 and PSMAL show superior overlap prediction and geodesic accuracy compared to learning-based baselines, with competitive smoothness and improved scalability over prior non-linear methods. The approach offers a principled, scalable pathway for robust partial-partial matching in realistic 3D scanning scenarios while enabling overlap-aware post-processing and downstream tasks.

Abstract

The task of establishing correspondences between two 3D shapes is a long-standing challenge in computer vision. While numerous studies address full-full and partial-full 3D shape matching, only a limited number of works have explored the partial-partial setting, very likely due to its unique challenges: we must compute accurate correspondences while at the same time find the unknown overlapping region. Nevertheless, partial-partial 3D shape matching reflects the most realistic setting, as in many real-world cases, such as 3D scanning, shapes are only partially observable. In this work, we introduce the first integer linear programming approach specifically designed to address the distinctive challenges of partial-partial shape matching. Our method leverages geometric consistency as a strong prior, enabling both robust estimation of the overlapping region and computation of neighbourhood-preserving correspondences. We empirically demonstrate that our approach achieves high-quality matching results both in terms of matching error and smoothness. Moreover, we show that our method is more scalable than previous formalisms.

An Integer Linear Programming Approach to Geometrically Consistent Partial-Partial Shape Matching

TL;DR

This work tackles partial-partial 3D shape matching, where both shapes are incompletely observed, by introducing the first integer linear programming (ILP) formulation that enforces geometric consistency to infer the unknown overlap and recover neighbourhood-preserving correspondences. It builds on surface-cycle representations and product graphs to encode per-triangle matching as coupled subproblems, augmented with injectivity and surjectivity constraints to handle overlap. The ILP objective combines per-edge matching costs with overlap probabilities via a weighting parameter , and a coarse-to-fine strategy scales the approach to higher resolutions. Empirical results on CP2P24 and PSMAL show superior overlap prediction and geodesic accuracy compared to learning-based baselines, with competitive smoothness and improved scalability over prior non-linear methods. The approach offers a principled, scalable pathway for robust partial-partial matching in realistic 3D scanning scenarios while enabling overlap-aware post-processing and downstream tasks.

Abstract

The task of establishing correspondences between two 3D shapes is a long-standing challenge in computer vision. While numerous studies address full-full and partial-full 3D shape matching, only a limited number of works have explored the partial-partial setting, very likely due to its unique challenges: we must compute accurate correspondences while at the same time find the unknown overlapping region. Nevertheless, partial-partial 3D shape matching reflects the most realistic setting, as in many real-world cases, such as 3D scanning, shapes are only partially observable. In this work, we introduce the first integer linear programming approach specifically designed to address the distinctive challenges of partial-partial shape matching. Our method leverages geometric consistency as a strong prior, enabling both robust estimation of the overlapping region and computation of neighbourhood-preserving correspondences. We empirically demonstrate that our approach achieves high-quality matching results both in terms of matching error and smoothness. Moreover, we show that our method is more scalable than previous formalisms.
Paper Structure (22 sections, 10 equations, 4 figures, 4 tables)

This paper contains 22 sections, 10 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. Background
  4. 3D Shapes and Shape Representations
  5. Shape Matching with Product Graphs
  6. Partial-Partial Geometric Consistency
  7. Partial-Partial Shape Matching
  8. Coupled Surface Cycle Matching Problems
  9. Injectivity and Surjectivity of Overlap
  10. ILP for Partial-Partial Shape Matching
  11. Scaling to Higher Resolutions
  12. Experiments
  13. Dataset & Competing Methods & Setup
  14. Metrics
  15. Results
  16. ...and 7 more sections

Figures (4)

  • Figure 1: (Left) We show matching results for pairs of partial shapes computed with our method using colour transfer. (Right) Runtime comparison to the recent (geometrically consistent) partial-partial shape matching method GC-PPSM ehm2024partial which approaches the task using integer non-linear programming. In contrast, we propose the first integer linear programming approach explicitly tailored to partial-partial shape matching with geometric consistency, making our method more scalable in comparison.
  • Figure 2: Overview over our partial-partial 3D shape matching approach. We match a partial source 3D shape ${\boldsymbol{X}}$ to a non-rigidly deformed partial target 3D shape ${\boldsymbol{Y}}$. $(i)$ We represent the source shape using surface cycles (cf. \ref{['def:surface-cycles']}). $(ii)$ For each surface cycle $\boldsymbol{C}_i$, we formulate an independent subproblem as the product graphs ${\boldsymbol{P}}_i$ between a $\boldsymbol{C}_i$ and the whole 3D shape ${\boldsymbol{Y}}$ (cf. \ref{['def:prodgraph']}). $(iii)$ By employing coupling constraints (cf. \ref{['sec:coupled-matching-problems']}) as well as injectivity and surjectivity constraints of the overlap (cf. \ref{['sec:inj-surj']}) we obtain a formalism for geometrically consistent partial-partial 3D shape matching (cf. \ref{['sec:resulting-ilp']}).
  • Figure 3: We prune instances of \ref{['eq:partial-partial-ILP']} for pairs of higher resolution shapes by using computed matching of (the same pair of) lower resolution shapes. In particular, on higher resolution shapes, we only allow for pairs of vertices which map to a matching on lower resolutions shapes within their $N$-ring neighbourhood (we show $N=1$ in the figure), i.e. in this example, all coloured vertices on higher resolution shape ${\boldsymbol{X}}$ can be matched to all coloured vertices higher resolution shape ${\boldsymbol{Y}}$.
  • Figure 4: We show qualitative results and the ground-truth (GT) of shape pairs of CP2P24 (first two rows) and PSMAL (last two rows) using methods GC-PPSM ehm2024partial, DPFM attaiki2021dpfm, Echo xie2025echomatch and ours. We can see that our method leads to overall smoother matching results.

Theorems & Definitions (5)

  • Definition 1: 3D Shape
  • Definition 2: Surface Cycle Shape Representation
  • Definition 3: Product graph
  • Definition 4: Partial-Partial Geometric Consistency
  • Definition 5: Collection of Product Graphs