Difference Subspace and Its Generalization for Subspace-Based Methods

Subspace-based methods are known to provide a practical solution for image set-based object recognition. Based on the insight that local shape differences between objects offer a sensitive cue for recognition, this paper addresses the problem of extracting a subspace representing the difference components between class subspaces generated from each set of object images independently of each other. We first introduce the difference subspace (DS), a novel geometric concept between two subspaces as an extension of a difference vector between two vectors, and describe its effectiveness in analyzing shape differences. We then generalize it to the generalized difference subspace (GDS) for multi-class subspaces, and show the benefit of applying this to subspace and mutual subspace methods, in terms of recognition capability. Furthermore, we extend these methods to kernel DS (KDS) and kernel GDS (KGDS) by a nonlinear kernel mapping to deal with cases involving larger changes in viewing direction. In summary, the contributions of this paper are as follows: 1) a DS/KDS between two class subspaces characterizes shape differences between the two respectively corresponding objects, 2) the projection of an input vector onto a DS/KDS realizes selective visualization of shape differences between objects, and 3) the projection of an input vector or subspace onto a GDS/KGDS is extremely effective at extracting differences between multiple subspaces, and therefore improves object recognition performance. We demonstrate validity through shape analysis on synthetic and real images of 3D objects as well as extensive comparison of performance on classification tests with several related methods; we study the performance in face image classification on the Yale face database B+ and the CMU Multi-PIE database, and hand shape classification of multi-view images.