DiscoMatch: Fast Discrete Optimisation for Geometrically Consistent 3D Shape Matching

TL;DR

A novel combinatorial solver that combines a unique set of favourable properties that is initialisation free, is massively parallelisable and powered by a quasi-Newton method, provides optimality gaps, and delivers improved matching quality with decreased runtime and globally optimal results for many instances.

Abstract

In this work we propose to combine the advantages of learningbased and combinatorial formalisms for 3D shape matching. While learningbased methods lead to state-of-the-art matching performance, they do not ensure geometric consistency, so that obtained matchings are locally non-smooth. On the contrary, axiomatic, optimisation-based methods allow to take geometric consistency into account by explicitly constraining the space of valid matchings. However, existing axiomatic formalisms do not scale to practically relevant problem sizes, and require user input for the initialisation of non-convex optimisation problems. We work towards closing this gap by proposing a novel combinatorial solver that combines a unique set of favourable properties: our approach (i) is initialisation free, (ii) is massively parallelisable and powered by a quasi-Newton method, (iii) provides optimality gaps, and (iv) delivers improved matching quality with decreased runtime and globally optimal results for many instances.

Figures (15)

• Figure 1: We efficiently solve a large integer linear program (ILP) to guarantee geometric consistency of matchings between pairs of 3D shapes. Our matchings are smooth and more accurate than the recent state-of-the-art shape matching approach URSSMcao2023unsupervised. Improved solution quality can be seen when transferring triangulation between shapes.
• Figure 2: Left: Geometric consistency is an essential property of correspondences between shapes, meaning that neighbouring elements in $\mathcal{M}$ are consistently matched to neighbouring elements in $\mathcal{N}$. Right: Matching elements of the approach by windheuser2011geometrically: triangle-triangle, triangle-edge and triangle-vertex between shapes $\mathcal{M}$ and $\mathcal{N}$. At least one triangle is involved in every matching element.
• Figure 3: Illustration of our shape matching pipeline. We use a pretrained feature extractor to define the cost function of ILP \ref{['eq:ilp-sm']}, which we solve using our combinatorial solver to find a geometrically consistent matching between shapes. The dual solver performs second-order quasi-Newton updates on top of the first-order min-marginal averaging scheme (MMA) of abbas2022fastdog. For primal recovery we adapt the approach of abbas2022fastdog by replacing their dual optimisation procedure by ours leading to faster and better-quality solutions.
• Figure 4: Top: Percentage of correct keypoints w.r.t. geodesic error thresholds on FAUST, SMAL, DT4D-Intra and DT4D-Inter datasets. The numbers in the legends are mean geodesic errors ($\downarrow$). Middle: Conformal distortion errors on FAUST, SMAL, DT4D-Intra and DT4D-Inter datasets. Bottom: Statistics of runtime of all methods.
• Figure 5: Qualitative Comparison on DT4D-Inter ( 1- 3), DT4D-Intra ( 4, 5), SHREC'20 ( 6, 7), SMAL ( 8, 9) and FAUST ( 10, 11). To better visualise the quality of matchings we transfer triangulation and the color from source to target shapes.