Statistical analysis of locally parameterized shapes

TL;DR

The paper addresses the biases introduced by global Procrustes alignment in statistical shape analysis and proposes a locally parameterized ds-rep (LP-ds-rep) that uses hierarchical local frames to achieve translation and rotation invariance. By separating local frame definitions from global coordinates, LP-ds-rep enables accurate mean estimation and hypothesis testing without alignment, and provides improved interpretability of local deformations. The authors develop mean definitions for LP-ds-rep, formalize LP-ds-rep–to–GP-ds-rep conversion, and implement a permutation-based testing framework with BH and Bonferroni adjustments. Through simulations and a Parkinson’s disease hippocampus study, LP-ds-rep demonstrates reduced false positives and more medically plausible localization of differences, with main effects localized to the spinal region of the hippocampal skeleton.

Abstract

The alignment of shapes has been a crucial step in statistical shape analysis, for example, in calculating mean shape, detecting locational differences between two shape populations, and classification. Procrustes alignment is the most commonly used method and state of the art. In this work, we uncover that alignment might seriously affect the statistical analysis. For example, alignment can induce false shape differences and lead to misleading results and interpretations. We propose a novel hierarchical shape parameterization based on local coordinate systems. The local parameterized shapes are translation and rotation invariant. Thus, the inherent alignment problems from the commonly used global coordinate system for shape representation can be avoided using this parameterization. The new parameterization is also superior for shape deformation and simulation. The method's power is demonstrated on the hypothesis testing of simulated data as well as the left hippocampi of patients with Parkinson's disease and controls.

Figures (12)

• Figure 1: Problem of false positives due to alignment. (a) Red and blue indicate two populations of PDMs. Small crosses are the mean centroids. (b,c) Separation of corresponding local distributions.
• Figure 2: Skeletal structure. (a) 2D m-reps of two ellipsoidal objects. $\bm{s}$ and $\bm{s}'$ are corresponding spokes with unit directions $\bm{u}$ and $\bm{u}'$. (b) A fitted ds-rep to a left hippocampus's mesh. Green, cyan, magenta, and yellow respectively indicate skeletal sheet, up spokes, down spokes, and crest spokes.
• Figure 3: Ellipsoid's Medial axis deformation. Left: Medial axis of an eccentric 3D ellipsoid. Right: Object's s-rep skeletal sheet.
• Figure 4: Illustration of a local frames. $\bm{n}$ is normal of tangent planes $T_{\bm{p}}(M)$ and $T_{\bm{p}}(M_\Omega)$. (a) $\bm{s}_1$ and $\bm{s}_2$ are equal-length spokes with unit directions $\bm{u}_1$ and $\bm{u}_2$, and $\bm{b}=\frac{\bm{u}_1+\bm{u}_2}{\|\bm{u}_1+\bm{u}_2\|}$ (b) $c$ is a smooth curve on $M$. $-\bm{p}'_1$ and $\bm{p}'_2$ are the projection of $\bm{p}_1$ and $\bm{p}_2$ on $T_{\bm{p}}(M)$. $\hat{\bm{v}}'_1=\frac{\bm{p}-\bm{p}'_1}{\|\bm{p}-\bm{p}'_1\|}$, $\hat{\bm{v}}'_2=\frac{\bm{p}'_2-\bm{p}}{\|\bm{p}'_2-\bm{p}\|}$, and $\bm{b}=\frac{\hat{\bm{v}}'_2+\hat{\bm{v}}'_1}{\|\hat{\bm{v}}'_2+\hat{\bm{v}}'_1\|}$.
• Figure 5: LP-ds-rep. Top: Hierarchical structure of the ellipsoid's medial axis. Black and blue arrows are connections on the medial line and m-rep spokes. Red dot is the ellipsoid's centroid. Bottom: A fitted LP-ds-rep to a hippocampus. Red arrows indicate spokes. Black and blue arrows are connections on the spine and veins. Orange arrows depict orthogonal local frames. Red dot is the s-centroid.