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Low-rank approximation of images via singular value decomposition iswell-received in the era of big data. However, singular value decomposition(SVD) is only for order-two data, i.e., matrices. It is necessary to flatten ahigher order input into a matrix or break it into a series of order-two slicesto tackle higher order data such as multispectral images and videos with theSVD. Higher order singular value decomposition (HOSVD) extends the SVD and canapproximate higher order data using sums of a few rank-one components. Weconsider the problem of generalizing HOSVD over a finite dimensionalcommutative algebra. This algebra, referred to as a t-algebra, generalizes thefield of complex numbers. The elements of the algebra, called t-scalars, arefix-sized arrays of complex numbers. One can generalize matrices and tensorsover t-scalars and then extend many canonical matrix and tensor algorithms,including HOSVD, to obtain higher-performance versions. The generalization ofHOSVD is called THOSVD. Its performance of approximating multi-way data can befurther improved by an alternating algorithm. THOSVD also unifies a wide rangeof principal component analysis algorithms. To exploit the potential ofgeneralized algorithms using t-scalars for approximating images, we use a pixelneighborhood strategy to convert each pixel to "deeper-order" t-scalar.Experiments on publicly available images show that the generalized algorithmover t-scalars, namely THOSVD, compares favorably with its canonicalcounterparts.

Approximation of Images via Generalized Higher Order Singular Value Decomposition over Finite-dimensional Commutative Semisimple Algebra