Approximation of Images via Generalized Higher Order Singular Value Decomposition over Finite-dimensional Commutative Semisimple Algebra
This work addresses the limitation of traditional SVD and HOSVD in handling higher-order data for image approximation, offering a unified approach that could benefit fields like computer vision and data compression, though it appears incremental as it builds on existing HOSVD with algebraic generalization.
The paper tackles the problem of low-rank approximation for higher-order data like images by generalizing Higher Order Singular Value Decomposition (HOSVD) over a finite-dimensional commutative algebra, called t-algebra, resulting in THOSVD, which shows favorable performance compared to canonical methods in experiments on publicly available images.
Low-rank approximation of images via singular value decomposition is well-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 a higher order input into a matrix or break it into a series of order-two slices to tackle higher order data such as multispectral images and videos with the SVD. Higher order singular value decomposition (HOSVD) extends the SVD and can approximate higher order data using sums of a few rank-one components. We consider the problem of generalizing HOSVD over a finite dimensional commutative algebra. This algebra, referred to as a t-algebra, generalizes the field of complex numbers. The elements of the algebra, called t-scalars, are fix-sized arrays of complex numbers. One can generalize matrices and tensors over t-scalars and then extend many canonical matrix and tensor algorithms, including HOSVD, to obtain higher-performance versions. The generalization of HOSVD is called THOSVD. Its performance of approximating multi-way data can be further improved by an alternating algorithm. THOSVD also unifies a wide range of principal component analysis algorithms. To exploit the potential of generalized algorithms using t-scalars for approximating images, we use a pixel neighborhood strategy to convert each pixel to "deeper-order" t-scalar. Experiments on publicly available images show that the generalized algorithm over t-scalars, namely THOSVD, compares favorably with its canonical counterparts.