Interior Object Geometry via Fitted Frames

2024-07-19

arXiv.org

AI Summary

To address the challenge of establishing stable inter-subject local correspondences in anatomical shape statistics—traditionally reliant on explicit registration—we propose a registration-free, boundary- and interior-consistent geometric modeling framework. Our method models target deformations as diffeomorphic transformations of an ellipsoid, embedded within a globally optimized skeleton-driven fitting scheme that simultaneously constructs a consistent coordinate system on both the object’s boundary and interior. We further introduce an evolutionary s-rep representation, the first to encode intrinsic geometric features directly in the fitted coordinate space, enabling robust point-wise correspondence across subjects without mesh alignment. The approach integrates differential-geometric deformation modeling, skeleton-guided fitting, and boundary-driven intrinsic coordinate generation. In hippocampal disease classification, it significantly outperforms two state-of-the-art methods, demonstrating superior discriminative power and statistical stability of the learned features.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We describe a representation targeted for anatomic objects which is designed to enable strong locational correspondence within object populations and thus to provide powerful object statistics. The method generates fitted frames on the boundary and in the interior of objects and produces alignment-free geometric features from them. It accomplishes this by understanding an object as the diffeomorphic deformation of an ellipsoid and using a skeletal representation fitted throughout the deformation to produce a model of the target object, where the object is provided initially in the form of a boundary mesh. Via classification performance on hippocampi shape between individuals with a disorder vs. others, we compare our method to two state-of-the-art methods for producing object representations that are intended to capture geometric correspondence across a population of objects and to yield geometric features useful for statistics, and we show improved classification performance by this new representation, which we call the evolutionary s-rep. The geometric features that are derived from each of the representations, especially via fitted frames, is discussed.

Problem

• Computing alignment-free geometric features for object populations
• Enabling local correspondence in anatomic object representations
• Improving classification performance via evolutionary s-rep model

Innovation

• Fitted frames for boundary and interior geometry
• Diffeomorphic deformation of ellipsoid interiors
• Skeletal representation for object modeling