A Riemannian Framework for Intrinsic Comparison of Closed Genus-Zero Shapes

Abstract

We present a framework for intrinsic comparison of surface metric structures and curvatures. This work parallels the work of Kurtek et al. on parameterization-invariant comparison of genus zero shapes. Here, instead of comparing the embedding of spherically parameterized surfaces in space, we focus on the first fundamental form. To ensure that the distance on spherical metric tensor fields is invariant to parameterization, we apply the conjugation-invariant metric arising from the L2 norm on symmetric positive definite matrices. As a reparameterization changes the metric tensor by a congruent Jacobian transform, this metric perfectly suits our purpose. The result is an intrinsic comparison of shape metric structure that does not depend on the specifics of a spherical mapping. Further, when restricted to tensors of fixed volume form, the manifold of metric tensor fields and its quotient of the group of unitary diffeomorphisms becomes a proper metric manifold that is geodesically complete. Exploiting this fact, and augmenting the metric with analogous metrics on curvatures, we derive a complete Riemannian framework for shape comparison and reconstruction. A by-product of our framework is a near-isometric and curvature-preserving mapping between surfaces. The correspondence is optimized using the fast spherical fluid algorithm. We validate our framework using several subcortical boundary surface models from the ADNI dataset.

Conformal invariants for multiply connected surfaces: Application to landmark curve-based brain morphometry analysis

“…In image analysis research, shape space was well studied for brain atlas estimation (Fletcher et al, 2009; Fletcher, 2013), shape analysis (Kurtek et al, 2011; Gutman et al, 2015; Su et al, 2015a), morphometry study (Younes et al, 2009; Boyer et al, 2011), and other applications. In a computational anatomy framework (Grenander and Miller, 1998), the space of diffeomorphisms was carefully studied (Miller et al, 2002; Miller and Younes, 2001; Trouvé, 1998; Younes, 2010).…”

“…Kurtek et al (2013) extended the work in (Kurtek et al, 2012) by adding landmark constraints. Jermyn et al (2012) simplified the RI metric computation and Gutman et al (2015) recently built a Riemannian framework for an intrinsic comparison of the RI metric structure. Lipman and Daubechies (2011) introduced a metric for shape comparison based on conformal uniformization and optimal mass transport.…”

Shape Analysis with Hyperbolic Wasserstein Distance

“…Shape space models, which usually measure similarities between two shapes by the deformation between them, may provide a suitable mathematical and computational description for shape analysis (as reviewed in [67]). In computer vision research, shape space has been well studied for brain atlas estimation [19, 18], shape analysis [33, 24, 56], morphometry study [69, 10], etc. Recently, the Wasserstein space is attracting more attention.…”

“…Currently, most of shape space work were developed on genus-0 surfaces, e.g. [34, 24], which cannot be directly applied to high-genus surfaces because of the difficulty in building a canonical space for them. Our framework, which adopts a hyperbolic harmonic map to build diffeomorphic mappings between general surfaces, may be used to generalize other shape space studies to general surfaces as well.…”

“…Kurtek et al [35] extended the work in [34] by adding landmark constraints. Jermyn et al [30] simplified the RI metric computation and Gutman et al [24] built a Riemannian framework for an intrinsic comparison of the RI metric structure. Lipman and Daubechies [37] introduced a metric for shape comparison based on conformal uniformization and OMT.…”