# A Combinatorial Solution to Non-Rigid 3D Shape-to-Image Matching

arXiv: https://arxiv.org/abs/1611.05241 · HTML: https://mlsanity.com/paper/1611.05241/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Nov 2016.

## Built on by (most cited first, 5 of 5)

- [Parameterized Temperature Scaling for Boosting the Expressive Power in Post-Hoc Uncertainty Calibration](https://mlsanity.com/paper/2102.12182/) (2021)
- [Conjugate Product Graphs for Globally Optimal 2D-3D Shape Matching](https://mlsanity.com/paper/2211.11589/) (2022)
- [SIGMA: Scale-Invariant Global Sparse Shape Matching](https://mlsanity.com/paper/2308.08393/) (2023)
- [Lifting the Convex Conjugate in Lagrangian Relaxations: A Tractable Approach for Continuous Markov Random Fields](https://mlsanity.com/paper/2107.06028/) (2021)
- [Efficient and Flexible Sublabel-Accurate Energy Minimization](https://mlsanity.com/paper/2206.09596/) (2022)

## Abstract (arXiv)

We propose a combinatorial solution for the problem of non-rigidly matching a 3D shape to 3D image data. To this end, we model the shape as a triangular mesh and allow each triangle of this mesh to be rigidly transformed to achieve a suitable matching to the image. By penalising the distance and the relative rotation between neighbouring triangles our matching compromises between image and shape information. In this paper, we resolve two major challenges: Firstly, we address the resulting large and NP-hard combinatorial problem with a suitable graph-theoretic approach. Secondly, we propose an efficient discretisation of the unbounded 6-dimensional Lie group SE(3). To our knowledge this is the first combinatorial formulation for non-rigid 3D shape-to-image matching. In contrast to existing local (gradient descent) optimisation methods, we obtain solutions that do not require a good initialisation and that are within a bound of the optimal solution. We evaluate the proposed method on the two problems of non-rigid 3D shape-to-shape and non-rigid 3D shape-to-image registration and demonstrate that it provides promising results.