Feature Patch Illumination Spaces and Karcher Compression for Face Recognition via Grassmannians

Abstract

Recent work has established that digital images of a human face, when collected with a fixed pose but under a variety of illumination conditions, possess discriminatory information that can be used in classification. In this paper we perform classification on Grassmannians to demonstrate that sufficient discriminatory information persists in feature patch (e.g., nose or eye patch) illumination spaces. We further employ the use of Karcher mean on the Grassmannians to demonstrate that this compressed representation can accelerate computations with relatively minor sacrifice on performance. The combination of these two ideas introduces a novel perspective in performing face recognition.

Interpolation on Symmetric Spaces Via the Generalized Polar Decomposition

“…Not surprisingly, this task has received much attention in the literature; see [2,9] and the references therein. Our constructions in this paper recover some of the well-known interpolation schemes on the Grassmannian, including those that appear in [4,9,13].…”

“…More sophisticated approaches with better numerical properties can be found in [3,22]. The identity (13) can be leveraged to derive formulas for higher-order derivatives of Iu(x), provided of course that G is a matrix group. As shown in Appendix A, we have…”

“…Manifold-valued data and manifold-valued functions play an important role in a wide variety of applications, including mechanics [14,20,32], computer vision and graphics [13,16,17,24,26,35], medical imaging [8], and numerical relativity [6]. By their very nature, such applications demand that care be taken when performing computations that would otherwise be routine, such as averaging, interpolation, extrapolation, and the numerical solution of differential equations.…”

“…Another symmetric space which we address in this paper is the Grassmannian Gr(p, n), which consists of all p-dimensional linear subspaces of R n . Interpolation on the Grassmannian is a task of importance in a variety of contexts, including reduced-order modeling [4,36] and computer vision [13,16,24,35]. Not surprisingly, this task has received much attention in the literature; see [2,9] and the references therein.…”

“…Since O(n) is compact, Lemma 3.3 shows that Algorithm 4 produces the weighted Riemannian mean on Gr(p, n). This interpolant has been considered previously by several authors, including [9,13,18].…”