Analysis of Discrete Shapes Using Lie Groups

Abstract

Discrete shapes can be described and analyzed using Lie groups, which are mathematical structures having both algebraic and geometrical properties. These structures, borrowed from mathematical physics, are both algebraic groups and smooth manifolds. A key property of a Lie group is that a curved space can be studied, using linear algebra, by local linearization with an exponential map.

Here, a discrete shape was a Euclidean-invariant computer representation of an object. Highly variable shapes are known to exist in non-linear spaces where linear analysis tools, such as Pearson's decomposition of principal components, are inadequate. The novel method proposed herein represented a shape as an ensemble of homogenous matrix transforms. The Lie group of homogenous transforms has elements that both represented a local shape and acted as matrix operators on other local shapes. For the manifold, a matrix transform was found to be equivalent to a vector transform in a linear space. This combination of representation and linearization gave a simple implementation for solving a computationally expensive problem.

Two medical datasets were analyzed: 2D contours of femoral head-neck cross-sections and 3D surfaces of proximal femurs. The Lie-group method outperformed the established principal-component analysis by capturing higher variability with fewer components. Lie groups are promising tools for medical imaging and data analysis.