Statistical Shape Analysis of Shape Graphs with Applications to Retinal Blood-Vessel Networks

TL;DR

This paper develops a mathematical framework for statistical shape analysis of shape graphs, applied to retinal blood-vessel networks, enabling characterization, comparison, and modeling of complex vascular structures.

Contribution

It introduces elastic Riemannian shape metrics and a multi-scale representation to compare and analyze shape graphs with varying complexities, specifically applied to retinal blood-vessel networks.

Findings

Effective resistance clustering aids in reducing shape graph complexity.

Shape graph registration and geodesic computation facilitate visualization of deformations.

The methods successfully analyze retinal blood-vessel networks from databases.

Abstract

This paper provides theoretical and computational developments in statistical shape analysis of shape graphs, and demonstrates them using analysis of complex data from retinal blood-vessel (RBV) networks. The shape graphs are represented by a set of nodes and edges (planar articulated curves) connecting some of these nodes. The goals are to utilize shapes of edges and connectivities and locations of nodes to: (1) characterize full shapes, (2) quantify shape differences, and (3) model statistical variability. We develop a mathematical representation, elastic Riemannian shape metrics, and associated tools for such statistical analysis. Specifically, we derive tools for shape graph registration, geodesics, summaries, and shape modeling. Geodesics are convenient for visualizing optimal deformations, and PCA helps in dimension reduction and statistical modeling. One key challenge here is comparisons of shape graphs with vastly different complexities (in number of nodes and edges). This paper introduces a novel multi-scale representation of shape graphs to handle this challenge. Using the notions of (1) ``effective resistance" to cluster nodes and (2) elastic shape averaging of edge curves, one can reduce shape graph complexity while maintaining overall structures. This way, we can compare shape graphs by bringing them to similar complexity. We demonstrate these ideas on Retinal Blood Vessel (RBV) networks taken from the STARE and DRIVE databases.