| Recently, two manifold-based dimensionality reduction algorithms were proposed to recover a low-dimensional representation of very high-dimensional data libraries. Isomap and Locally Linear Embedding (LLE) both construct a low-dimensional embedding using the geometry of a data sample assumed to lie on a nonlinear manifold.; Implicit in the Isomap algorithm is the assumption that the data lie on a nonlinear submanifold of Euclidean space that is globally isometric to an underlying low-dimensional Euclidean parameter space. We consider a specific kind of data—families of images generated by the articulation of an object—in an idealization where the images are functions on the continuum plane. Using the ambient L2-distances between images defines a continuous articulation family as a nonlinear manifold. For images with edges, the continuum viewpoint suffers from various infinities, but we demonstrate a natural renormalization of geodesic distance that is well-defined. Global isometry requires both that the data manifold be locally isometric and that its associated parameter space be convex. There exists a set of interesting image-articulation manifolds where the strict global isometry criterion is exactly fulfilled, including articulations of a horizon, translations or rotations of closed figures, and independent multiple object articulations such as expressions of a cartoon face.; The introduction of imaging examples also produces a number of natural image families that are clearly not globally isometric to Euclidean space, but satisfy instead a local isometry condition, where the manifold is infinitesimally isometric and connected, rather than convex. We introduce a new methodology, Hessian Locally Linear Embedding (HLLE), derived from the original LLE methodology within a theoretical framework similar to that of Laplacian Eigenmaps, where a quadratic form based on the Hessian is substituted in place of the Laplacian operator. Theoretical and experimental results demonstrate that HLLE perfectly recovers underlying parameter spaces in a significantly broader class of imaging examples than those permitted by the original isometry criterion. |
