HYPERCOMPLEX ALGEBRAS FOR DICTIONARY LEARNING Srd ¯an Lazendi´ c a , Aleksandra Piˇ zurica b and Hendrik De Bie a a Department of Mathematical Analysis, Clifford Research Group b Department of Telecommunications and Information Processing, UGent-IPI-imec Faculty of Engineering and Architecture, Ghent University, Belgium Srdan.Lazendic@UGent.be [presenter, corresponding] Aleksandra.Pizurica@UGent.be Hendrik.DeBie@UGent.be ABSTRACT. This paper presents an application of hypercomplex algebras combined with dicti- onary learning for sparse representation of multichannel images. Two main representatives of hypercomplex algebras, Clifford algebras and algebras generated by the Cayley-Dickson proce- dure are considered. Related works reported quaternion methods (for color images) and octo- nion methods, which are applicable to images with up to 7 channels. We show that the current constructions cannot be generalized to dimensions above eight. 1. I NTRODUCTION The complex (C) and quaternion (H) algebras are special cases of more general hypercomplex algebras [18]. Rooted in Hamilton’s seminal paper about quaternions published in 1843, the theory of hypercomplex algebras evolved, finding applications in many disciplines, ranging from theoretical physics [6] to robotics [17] and signal processing [2, 3, 7, 16]. In general, a hypercomplex algebra is defined as unital, distributive algebra, not necessarily associative, over the field of real or complex numbers with n generators (e 1 ,..., e n ). Usually e 2 k ∈ {−1, 0, 1} and different multiplication rules between the basis elements generate different algebras [18]. Basis elements are referred to as the imaginary units. We address here a recent application of hypercomplex algebras in sparse image representation, with some new insights, regarding applicability to general multichannel images. Dictionary learning techniques provide the most succinct representation of signals and images. Initiated by the classical work of Olsahusen and Field on sparse coding [22], many dictionary learning methods emerged, K-SVD [1] being among the best known ones, especially in image and video processing. An overview of different models for color image processing by using dictionary learning techniques has been given in [5]. State-of-the-art methods for sparse repre- sentation of color images typically concatenate pixel values from collocated image patches in the three channels and treat them then by the standard K-SVD or by a variant thereof [20]. Re- cently reported quaternion-based K-SVD known as K-QSVD [28, 29], introduced quaternions into dictionary construction, where the three color channels are assigned to the three imaginary units. The quaternion model demonstrated improvements compared to the classical K-SVD model, especially in the terms of color fidelity as reported in [28,30]. A limitation of the model is that it cannot treat more than three spectral channels. Recently, in [19] a new model was introduced, called octonion dictionary learning (ODL), which is a generalization of the K-QSVD, in the sense that it can handle up to 7 spectral channels. This is of interest e.g., for multispectral imaging. To the best of our knowledge, the quaternion and the octonion algebras are the only examples of hypercomplex algebra that have been used for multichannel image processing in the combination with dictionary learning techniques. From a different perspective, the approach This research has been partially supported by the institutional grant no. BOF15/24J/078 of Ghent University.