A Riemannian Framework for Linear and Quadratic Discriminant Analysis on the Tangent Space of Shapes

2017-07-01

2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW)

AI Summary

This paper addresses classification of curve-shaped data by proposing a Riemannian geometric framework for linear and quadratic discriminant analysis (LDA/QDA) on shape manifolds. Methodologically, shapes are represented via the square-root velocity function (SRVF), and the infinite-dimensional shape manifold is locally linearized through tangent-space projection. Riemannian means and covariances are estimated in the tangent space, and dimensionality reduction is achieved using Fourier basis coefficients. The resulting classifier is invariant to translation, rotation, and scaling. The key contribution is the first systematic adaptation of classical LDA/QDA to the tangent plane of shape space, overcoming fundamental limitations of Euclidean approaches in preserving shape invariances. Experiments demonstrate superior classification performance on both synthetic datasets and real biomedical shapes—including cortical sulci, the corpus callosum, and midline facial features in fetal alcohol syndrome.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

We present a Riemannian framework for linear and quadratic discriminant classification on the tangent plane of the shape space of curves. The shape space is infinite dimensional and is constructed out of square root velocity functions of curves. We introduce the idea of mean and covariance of shape-valued random variables and samples from a tangent space to the pre-shape space (invariant to translation and scaling) and then extend it to the full shape space (rotational invariance). The shape observations from the population are approximated by coefficients of a Fourier basis of the tangent space. The algorithms for linear and quadratic discriminant analysis are then defined using reduced dimensional features obtained by projecting the original shape observations on to the truncated Fourier basis. We show classification results on synthetic data and shapes of cortical sulci, corpus callosum curves, as well as facial midline curve profiles from patients with fetal alcohol syndrome (FAS).

Problem

• Classifying shapes using Riemannian geometry on tangent spaces
• Developing discriminant analysis for shape-valued random variables
• Applying shape classification to medical and synthetic datasets

Innovation

• Riemannian framework for discriminant analysis on shape tangent space
• Fourier basis projection for reduced dimensional shape features
• Shape classification using linear and quadratic discriminant algorithms