We propose the first algorithm for non-rigid 2D-to-3D shape matching, where
the input is a 2D shape represented as a planar curve and a 3D shape
represented as a surface; the output is a continuous curve on the surface. We
cast the problem as finding the shortest circular path on the product
3-manifold of the surface and the curve. We prove that the optimal matching can
be computed in polynomial time with a (worst-case) complexity of
$O(mn^2\log(n))$, where $m$ and $n$ denote the number of vertices on the
template curve and the 3D shape respectively. We also demonstrate that in
practice the runtime is essentially linear in $m\!\cdot\! n$ making it an
efficient method for shape analysis and shape retrieval. Quantitative
evaluation confirms that the method provides excellent results for sketch-based
deformable 3D shape retrieval.

本文提出了一种针对非刚性二维到三维形状匹配的算法，通过将问题表述为在曲线和形状的乘积三维流形上寻找最短圆形路径完成匹配，证明了可以在多项式时间内计算出最优匹配，具有较高的实用性和有效性。

高效全局最优的 2D-3D 可变形形状匹配

Efficient Globally Optimal 2D-to-3D Deformable Shape Matching

Corrected versions of the numerically invariant expressions for the affine
and Euclidean signature of a planar curve proposed by E.Calabi et. al are
presented. The new formulas are valid for fine but otherwise arbitrary
partitions of the curve. We also give numerically invariant expressions for the
four differential invariants parametrizing the three dimensional version of the
Euclidean signature curve, namely the curvature, the torsion and their
derivatives with respect to arc length.

介绍了 E.Calabi 等人提出的平面曲线仿射和欧几里德特征的校正版本的数值不变表达式，这些新公式适用于曲线的任意细分；我们还提供了描述欧几里德特征曲线的三维版本的四个微分不变量的数值不变表达式，即曲率、挠率及其相对弧长的导数。