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：我们研究了在连续且非刚性条件下，实现二维轮廓与三维网格之间匹配的问题。尽管此类问题可通过在两个形状间的积图中寻找最短路径以获得全局最优解，但现有方法严重依赖不切实际的先验假设来避免退化解（例如，预先知道二维轮廓上每个点对应三维形状的哪个区域）。为此，我们提出了一种基于二维轮廓与三维形状间共轭积图的新型二维-三维形状匹配形式化框架。该方法首次允许我们考虑高阶代价（即定义为边链的代价），而非局限于单一边的代价。这提供了极大的灵活性，我们利用该特性引入了局部刚性先验。通过此方式，我们有效规避了退化解，即使在仅使用一维特征描述符的情况下，也能获得更平滑、更真实的匹配结果。总体而言，我们的方法可找到全局最优且连续的二维-三维匹配，渐近复杂度与现有方案相同，在形状匹配中达到了当前最优性能，甚至能够实现部分形状的匹配。我们的代码已公开（https://github.com/paul0noah/sm-2D3D）。