  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).
