We consider the problem of finding a continuous and non-rigid matching between a 2D contour and a 3D mesh. While such problems can be solved to global optimality by finding a shortest path in the product graph between both shapes, existing solutions heavily rely on unrealistic prior assumptions to avoid degenerate solutions (e.g. knowledge to which region of the 3D shape each point of the 2D contour is matched). To address this, we propose a novel 2D-3D shape matching formalism based on the conjugate product graph between the 2D contour and the 3D shape. Doing so allows us for the first time to consider higher-order costs, i.e. defined for edge chains, as opposed to costs defined for single edges. This offers substantially more flexibility, which we utilise to incorporate a local rigidity prior. By doing so, we effectively circumvent degenerate solutions and thereby obtain smoother and more realistic matchings, even when using only a one-dimensional feature descriptor. Overall, our method finds globally optimal and continuous 2D-3D matchings, has the same asymptotic complexity as previous solutions, produces state-of-the-art results for shape matching and is even capable of matching partial shapes. Our code is publicly available (https://github.com/paul0noah/sm-2D3D).

翻译：我们考虑找到2D轮廓和3D mesh之间的连续和非刚性匹配的问题。虽然可以通过在两种形状之间的乘积图中找到最短路径来解决这些问题，但现有解决方案严重依赖于不现实的先验假设，以避免退化解决方案（例如，了解2D轮廓的每个点匹配到3D形状的哪个区域）。为解决这个问题，我们提出了一种基于2D轮廓和3D形状之间的共轭乘积图的新型2D-3D形状匹配形式。这样做允许我们首次考虑更高阶的成本，即为边链定义的成本，而不是为单个边定义的成本。这提供了更大的灵活性，我们利用它来合并本地刚度先验。通过这样做，我们有效地规避了退化的解决方案，从而获得更平滑和更现实的匹配，即使只使用一维特征描述符。总的来说，我们的方法找到全局最优和连续的2D-3D匹配，具有与先前解决方案相同的渐近复杂度，为形状匹配提供了最先进的结果，甚至能够匹配部分形状。我们的代码是公开可用的（https://github.com/paul0noah/sm-2D3D）。