Hallucinating optimal high-dimensional subspaces

• Matching appearance subspaces of different scales not addressed in the literature.
• Naïve basis re-projection inadequate.
• Proposed refinement using constrained rotation.
• Closed-form solution for optimal subspace reconstruction.
• Class separation increased up to an order of magnitude.

Linear subspace representations of appearance variation are pervasive in computer vision. This paper addresses the problem of robustly matching such subspaces (computing the similarity between them) when they are used to describe the scope of variations within sets of images of different (possibly greatly so) scales. A naïve solution of projecting the low-scale subspace into the high-scale image space is described first and subsequently shown to be inadequate, especially at large scale discrepancies. A successful approach is proposed instead. It consists of (i) an interpolated projection of the low-scale subspace into the high-scale space, which is followed by (ii) a rotation of this initial estimate within the bounds of the imposed “downsampling constraint”. The optimal rotation is found in the closed-form which best aligns the high-scale reconstruction of the low-scale subspace with the reference it is compared to. The method is evaluated on the problem of matching sets of (i) face appearances under varying illumination and (ii) object appearances under varying viewpoint, using two large data sets. In comparison to the naïve matching, the proposed algorithm is shown to greatly increase the separation of between-class and within-class similarities, as well as produce far more meaningful modes of common appearance on which the match score is based.