Contraction-Expansion Analysis Module

Overview

The Contraction-Expansion module provides three distinct low-level metrics for quantifying the spatial spread of body configurations in 3D (or 2D) space:

1. Bounding Box Filled Area
2. Ellipsoid Sphericity
3. Points Density

These shape metrics are used to track structural changes relative to a movement baseline—for example, distinguishing a tightly curled posture from a sprawling, open posture.

1. Bounding Box Filled Area

Theoretical Interpretation

• Input Requirements: A static frame of points in space.
• Value Interpretation: This computes the "Contraction Index" based on surface area compactness. A value closer to 1.0 typically indicates the posture fully occupies its volumetric bounds (highly expanded and flush), whereas smaller values indicate a tightly clustered or irregular shape relative to its global bounds.
• Robustness: Relies on Convex Hull computation which gracefully handles outlier limbs but can fail mathematically if all points form a flat plane (degeneracy).

Algorithm Details & Mathematics

The index is the ratio of the squared surface area of the enclosed convex hull to the surface area of the axis-aligned bounding box (AABB).

where \(Area_{hull}\) is the internal surface area derived purely from the outermost perimeter of points, and \(Area_{bbox}\) is the theoretical bounding rectangle/prism containing all movement.

2. Ellipsoid Sphericity

Theoretical Interpretation

• Input Requirements: A static frame of spatial coordinates.
• Value Interpretation: Sphericity quantifies how closely the current body posture resembles a perfect sphere versus an elongated line or flat disc. A perfectly spherical posture yields 1.0, while a highly stretched limb posture approaches 0.0.

Algorithm Details & Mathematics

The algorithm fits a 3D ellipsoid to the skeletal joints using PCA on the mean-centered point cloud. The sorted eigenvalues yield the primary radii \((a, b, c)\) where \(a\) is the major axis and \(c\) is the minor axis.

Sphericity \(S\) is defined as the ratio of the smallest to the largest radius:

where \(c\) is the length of the shortest semi-axis and \(a\) is the length of the longest semi-axis of the fitted ellipsoid.

3. Points Density

Theoretical Interpretation

• Input Requirements: A static snippet of spatial coordinates.
• Value Interpretation: Evaluates the dispersion of the body. Smaller values mean all points are tightly clustered near the center of mass (high density / contracted). Larger values indicate the body is widely spread out (low density / expanded).

Algorithm Details & Mathematics

Computes the average Euclidean distance of all points from their shared barycenter.

where \(N\) is the total number of points, \(X_i\) is the position of the \(i\)-th point, and \(\bar{X}\) is the exact spatial mean of the point cloud.

References