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Linear algebra, the algebra of vectors and matrices, has traditionally been a veritable workhorse in image processing. Linear algebraic methods such as principal components analysis (PCA) and its refinement known as independent components analysis (ICA) model single-factor linear variation in image formation or the linear combination of multiple sources. In this exploratory signal processing article, we review a novel, multilinear (tensor) algebraic framework for image processing, particularly for the synthesis, analysis, and recognition of images. In particular, we will discuss multilinear generalizations of PCA and ICA and present new applications of these tensorial methods to image-based rendering and the analysis and recognition of facial image ensembles.