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In this corespondence, we study the extra-factor estimation problem with the assumption that the training image ensemble is expressed as an nth-order tensor with the nth-dimension characterizing all features for an image and other dimensions for different extra factors, such as illuminations, poses, and identities. To overcome the local minimum issue of conventional algorithms designed for this problem, we present a novel statistical learning framework called mode-kn Factor Analysis for obtaining a closed-form solution to estimating the extra factors of any test image. In the learning stage, for the kth (k ne = n) dimension of the data tensor, the mode-kn patterns are constructed by concatenating the feature dimension and the kth extra-factor dimension, and then a mode-kn factor analysis model is learnt based on the mode- kn patterns unfolded from the original data tensor. In the inference stage, for a test image, the mode classification of the kth dimension is performed within a probabilistic framework. The advantages of mode-kn factor analysis over conventional tensor analysis algorithms are twofold: (1) a closed-form solution, instead of iterative sub-optimal solution as conventionally, is derived for estimating the extra-factor mode of any test image; and (2) the classification capability is enhanced by interacting with the process of synthesizing data of all other modes in the k th dimension. Experiments on the Pointing'04 and CMU PIE databases for pose and illumination estimation both validate the superiority of the proposed algorithm over conventional algorithms for extra-factor estimation.
Date of Publication: March 2009