An Optimal Transport Approach to Robust Reconstruction and Simplification of 2D Shapes

, 2011

Category: Computational Geometry

Overall Rating

3.1/5 (22/35 pts)

While the original paper's greedy, heuristic-driven implementation for computing the transport plan between input points and the output complex limits its practical resurrection, the specific measure-theoretic error metric (W2 distance between input Dirac measures and output uniform measures on simplices) remains a conceptually interesting "gem."
This metric could potentially be repurposed as a novel, principled loss function within modern geometric deep learning frameworks for learning simplified shapes directly from point clouds, offering an alternative to heuristic pipelines and common point-wise error metrics.

The core concept of using Optimal Transport (specifically the W2 distance between a discrete measure representing the input points and a mixed measure representing the simplified simplicial complex) as the primary metric for shape reconstruction and simplification is quite novel.
The method's inherent robustness to noise/outliers and feature preservation capabilities, derived directly from the OT cost structure (normal vs. tangential components), offer a principled alternative to common heuristic-driven or feature-detection-dependent methods.
The fundamental idea of approximating a complex, discrete distribution of "mass" (points) with a simpler, structured geometric entity (a simplicial complex acting as a measure) based on Optimal Transport principles has broad potential.
The OT cost defined in the thesis could serve as a principled, differentiable loss function for training networks to directly output simplified geometric structures from raw point data, potentially bypassing traditional geometric algorithms or offering a novel training signal.

The core idea applies Optimal Transport (OT) to 2D geometric data... its specific application here feels constrained by outdated assumptions about the target representation and the nature of geometric data challenges.
Firstly, the heuristic point-to-simplex assignment (closest edge/vertex, local flip-based optimization) is a major theoretical weakness... likely introduces brittleness...
Secondly, the greedy decimation strategy, while standard in simplification, is inherently suboptimal.
The paper's primary technical limitation is the reliance on a heuristic for the fundamental transport plan computation.

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