(HOOI) [11], and higher-order singular value decomposition (HOSVD). A good overview can be found in [11]. Returning to our main focus on the matrix-based case, below we preface the overview of CSA with a brief background on 2DPCA and B2DPCA to provide a basis for the development of the CSA algorithm in the next section. Background In the remainder of this article, we consider a set of M grayscale sample images with m rows and n columns of pixels. Let A k denote the kth sample image (k = 1, 2, f, M ) in the form of an m # n matrix, where the matrix entries are the pixel intensity values (typically 0-255 for 8-bit grayscale, or normalized values in [0, 1]). We form the set of M mean-adjusted, or centered, sample r, images X k where the kth-centered image X k = A k - A M r ^ h and A = 1/M R k = 1 A k , is the mean image. Preprocessing may include image registration, histogram equalization, and ensuring all images have an equal number of horizontal pixels and an equal number of vertical pixels. In the previous work considered in this section, various nomenclatures have been used. For consistency, we have modified the original nomenclature somewhat to provide consistency in the background we provide below. 2DPCA Yang et al. [3] proposed the row version of 2DPCA in 2004. Eigendecomposition is performed on the image covariance or scatter matrix S R defined as: 1 SR = M M |X T k X k (size n # n), (1) k=1 obtaining the n eigenvectors of S R as the columns of an orthogonal n # n matrix we denote as V, where the ith column (eigenvector) corresponds to the ith largest eigenvalue vR,i of S R . In [4], the number of eigenvectors, the " top p eigenvectors, " corresponding to a specified threshold proportion i R of the total scatter is the smallest integer p ( p # n) for which the following expression is true: p |v R, i i=1 n |v $ i R . (2) R, i i=1 Retaining only these top p eigenvectors, the size of V becomes n # p. The feature matrix Yk is then formed by projecting the rows of each centered image X k onto the top p eigenvectors of S R , where these eigenvectors are the p columns of V: Yk = X k V (size of Yk is m # p, p # n). (3) In the column version of 2DPCA proposed by Zhang and Zhou [4] in 2005, the m eigenvectors of the scatter matrix S C are first found (forming the columns of an m # m orthogonal matrix U), where S C is defined as: 1 SC = M M |X X k T k (size m # m). (4) k=1 The eigenvector of S C corresponding to the ith largest eigenvalue v C,i of S C is the ith column of U. These eigenvectors form a basis for a space onto which the columns of the centered image matrices X k are projected. The dimension of the space spanned by the m eigenvectors may be reduced to a value q ^q # m h, using an expression similar to (2) for a given proportion of the total scatter. The feature matrix Yk corresponding to each centered image is then given by Yk = U T X k (U is m # q, Yk is q # n, q # m). (5) B2DPCA While 2DPCA projects either the rows or the columns of the centered image matrices onto a single new subspace, B2DPCA projects both the rows and columns onto two new subspaces simultaneously. This approach was proposed by Zhang and Zhou [4] as a method to reduce the relatively large number of feature matrix entries (or coefficients) in the case of 2DPCA. In B2DPCA, eigendecomposition is performed on the two scatter matrices S R and S C defined in (1) and (4), leading to the two orthogonal matrices U (m # m) and V (n # n) already defined in the descriptions of column 2DPCA and row 2DPCA in the previous section. Then retaining only the top p eigenvectors (columns) in V and the top q eigenvectors (columns) in U, the B2DPCA feature vector Yk is obtained as Yk = U T X k V (size of Yk is q # p, q # m and p # n). (6) We emphasize that after dimension reduction, U and V sizes become m # q and n # p, respectively. Clearly, this B2DPCA feature matrix Yk has pq coefficients ( p # n and q # m). By comparison, the feature matrix for the row and column 2DPCA versions has mp ( p # n) coefficients and nq (q # m) coefficients, respectively. Typically in practical situations, pq % mp and pq % nq. CSA In CSA, as proposed by Xu et al. [5], the image matrix is projected onto two subspaces simultaneously just as with B2DPCA. The resulting feature matrix Yk is expressed exactly as in (6) for B2DPCA, where the columns of U are orthonormal basis vectors for one of the two subspaces and the columns of V are orthonormal basis vectors of the second subspace. However, in contrast to B2DPCA, the matrices U and V are found through an iterative process that seeks to minimize reconstruction error for a given number of retained basis vectors p and q (where q of the m columns of U and p of the n columns of V are retained). We note that the iterations converge [6]. In (6), if p = n and q = m , given a feature matrix Yk , the original centered image X k is perfectly reconstructed by the expression X k = UYk V T . However, for p 1 n and q 1 m , we have X k . UYk V T . Letting t k denote the centered reconstructed image (where X Ja nu a r y 2021 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE 27

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