Surface Matching and Registration by Landmark Curve-Driven Canonical Quasiconformal Mapping

This work presents a novel surface matching and registration method based on the landmark curve-driven canonical surface quasiconformal mapping, where an open genus zero surface decorated with landmark curves is mapped to a canonical domain with horizontal or vertical straight segments and the local shapes are preserved as much as possible. The key idea of the canonical mapping is to minimize the harmonic energy with the landmark curve straightening constraints and generate a quasi-holomorphic 1-form which is zero in one parameter along landmark and results in a quasiconformal mapping. The mapping exists and is unique and intrinsic to surface and landmark geometry. The novel shape representation provides a conformal invariant shape signature. We use it as Teichmüller coordinates to construct a subspace of the conventional Teichmüller space which considers geometry feature details and therefore increases the discriminative ability for matching. Furthermore, we present a novel and efficient registration method for surfaces with landmark curve constraints by computing an optimal mapping over the canonical domains with straight segments, where the curve constraints become linear forms. Due to the linearity of 1-form and harmonic map, the algorithms are easy to compute, efficient and practical. Experiments on human face and brain surfaces demonstrate the efficiency and efficacy and the potential for broader shape analysis applications.