Continuous-Domain Linear (Spline) Projection Operators and Projector PCA for Shape Space Representation
Daniel Schmitter, EPFL STI LIB

Seminar • 08 July 2014 • BM 4.233

Abstract
The standard approach to define shape spaces is to consider shapes that are described by an ordered set of points or landmarks. They are geometrically normalized by aligning them to acommon reference in order to remove some effects of rigid-body transforms. A standard PCA is then applied. By normalizing, a bias is introduced in the model, because computing distances between normalized shapes generally does not yield the same result as for non-normalized shapes. Our alternative proposal is to define a finite-dimensional vector space that contains all possible shapes w.r.t. a given linear transformation of a reference shape. The idea is to generically characterize a shape space as a subspace containing all shapes that are related to a reference shape by a specific transformation. Thereby, the shape space itself is implicitly characterized by the orthogonal projection onto the vector space. This allows us to compute the "best match" among curves defined by a subspace w.r.t. an arbitrary shape. Our method does not include any normalization step prior to the shape space definition and hence, no bias is introduced when comparing shapes. Describing shape spaces by projection operators onto vector spaces provides new possibilities to compare arbitrary shapes with each other. The concept can be extended to compute eigenshapes via “projector” PCA, as well as other shape statistics. The computation of such measures without the need of normalizing implies that they are invariant to the transformation that defines the shape space.