Abstract

We introduce Localized Components Analysis (LoCA) for describing surface shape variation in an ensemble of biomedical objects using a linear subspace of spatially localized shape components. In contrast to earlier methods, LoCA optimizes explicitly for localized components and allows a flexible trade-off between localized and concise representations. Experiments comparing LoCA to a variety of competing shape representation methods on 2D and 3D shape ensembles establish the superior ability of LoCA to modulate the locality-conciseness tradeoff and generate shape components corresponding to intuitive modes of shape variation. Our formulation of locality in terms of compatibility between pairs of surface points is shown to be flexible enough to enable spatially-localized shape descriptions with attractive higher-order properties such as spatial symmetry.

Results

LoCA was performed on three data sets: a set of corpora callosa, a set humeri scans, and a set of monkey crania. The summary images show the 16 basis vectors minimizing the reconstruction error, where each vector is represented by an object/graph pair. The object represents the average item of the data set, and the arrows indicate the degree that the object changes as the corresponding shape parameter is varied.

The graphs plot the magnitude of basis vector's effect on a point against the point's distance to the vector's "center point". The center point of the region is defined as the point that minimizes our cost function.

The example on the right corresponds to the animation above. One can see that PCA generates a global deformation, bending the corpus callosum in half. In contrast, LoCA generates a localized deformation that expands the genu and leaves everything else alone.

The accompanying movies show how the shape parameters alter objects in the space. Each parameter is controlled by a slider, and is initially set to 0. The ends of each slider are capped at 4 standard deviations away from 0. Each slider doubles as a histogram showing how many objects from the original data set have a particular value for the corresponding parameter.

Corpora Callosa

55 2D tracings of corpora callosa were used, each consisting of 103 points.

Principal Components Analysis 7/54 vectors required to get 90% variation	Localized Components Analysis Parameters set so 26/54 vectors required to get 90% variation
First basis vectors (Large image) (Movie)	First basis vectors (Large image) (Movie)

Humeri

This data set consisted of 71 3D scans of humeri from both humans and various species of monkeys. Each humerus was represented by approximately 140 points. Blue locations of the humeri are affected by a larger amount than the white areas.

Principal Components Analysis 20/70 vectors required to get 90% variation	Localized Components Analysis Parameters set so 50/70 vectors required to get 90% variation
Front view of the first basis vectors (Large image) (Movie)	Front view of the first basis vectors (Large image) (Movie)
Back view of the first basis vectors (Large image) (Movie)	Back view of the first basis vectors (Large image) (Movie)

Monkey crania

A set of 239 monkey crania from different species were used. Each cranium was represented by a set of 45 points.

Principal Components Analysis 26/135 vectors required to get 90% variation	Localized Components Analysis Parameters set so that 82/135 vectors were required to get 90% variation
Front view of the first vectors (Large image) (Movie)	Front view of the first vectors (Large image) (Movie)
Side view of the first vectors (Large image) (Movie)	Side view of the first vectors (Large image) (Movie)
Symmetric Localized Components Analysis In addition to distance, symmetry is taken into account to determine compatibility. Parameters were set so that 64/135 vectors were required for 90% variation
First basis vectors (front view) (Large image) (Movie)	First 16 basis vectors (side view) (Large image) (Movie)
First 16 basis vectors (under) (Large image) (Movie)