By Soraya Mofty

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After the rearrangement of the components of each vector ~ L1s , it is transformed into the vector t 1 1 1 1 ~ Ls ðrÞ ¼ ½L1s ðrÞ; L2s ðrÞ; . ; LNs ðrÞ . The decision to continue with the next (second) HAPCA is based on the analysis of the covariance matrix ½K1L ðrÞ of the rearranged vectors ~ L1s ðrÞ for s = 1, 2, …, S, from which could be calculated the achieved decorrelation in the ﬁrst level. In case that full decorrelation is achieved, the matrix ½K1L ðrÞ is diagonal. The HAPCA algorithm could be stopped before the second level even if the decorrelation is not full, provided that the relation below is satisﬁed: ( N X N X i¼1 j¼1 , ½ki;j ðrÞ2jði6¼jÞ N X N X ) ½ki;j ðrÞ2jði¼jÞ d: ð1:60Þ i¼1 j¼1 Here ki;j ðrÞ is the element (i, j) of the matrix ½KL1 ðrÞ, and δ is a threshold with preliminary set small value.

39). As a result, it is decomposed into two components: 1 New Approaches for Hierarchical Image Decomposition … 21 Fig. 5 Flowgraph of the HSVD algorithm represented through the vector-radix (2 × 2) for a matrix of size 4 × 4 ½Xk ð2Þ ¼ r1;k ½T1;k ð2Þ þ r2;k ½T2;k ð2Þ ¼ ½C1;k ð2Þ þ ½C2;k ð2Þ for k ¼ 1; 2; 3; 4; where r1;k ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x1;k þ A1;k ; 2 r2;k ¼ ð1:41Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ x2;k ÀA2;k t t ~1;k ~ ~2;k ~ V1;k V2;k ; ½T2;k ð2Þ ¼ U : ; ½T1;k ð2Þ ¼ U 2 22 R. Kountchev and R. Kountcheva Using the matrices ½Cm;k ð2Þ of size 2 × 2 for k = 1, 2, 3, 4 and m = 1, 2, are composed the matrices ½Cm ð4Þ of size 4 × 4: ½Cm;1 ð2Þ ½Cm;2 ð2Þ ½Cm ð4Þ ¼ ½C ð2Þ ½Cm;4 ð2Þ 3 2 m;3 c11 ðm; 2Þ c12 ðm; 2Þ c11 ðm; 1Þ c12 ðm; 1Þ 7 6 c13 ðm; 1Þ c14 ðm; 1Þ c13 ðm; 2Þ c14 ðm; 2Þ 7 for m ¼ 1; 2: ¼ 6 4 c11 ðm; 3Þ c12 ðm; 3Þ c11 ðm; 4Þ c12 ðm; 4Þ 5 c13 ðm; 3Þ c14 ðm; 3Þ c13 ðm; 4Þ c14 ðm; 4Þ ð1:42Þ Hence, the SVD decomposition of the matrix [X] in the ﬁrst level is represented by two components: ½Xð4Þ¼ ½C1 ð4Þ þ ½C2 ð4Þ ¼ ð½C1;1 ð2Þ þ ½C2;1 ð2ÞÞ ð½C1;3 ð2Þ þ ½C2;3 ð2ÞÞ ð½C1;2 ð2Þ þ ½C2;2 ð2ÞÞ : ð½C1;4 ð2Þ þ ½C2;4 ð2ÞÞ ð1:43Þ In the second level (r = 2) of the HSVD, on each matrix ½Cm ð4Þ of size 4 × 4 is applied four times the SVD2×2.

They both use iterative learning algorithms, for which the number of needed operations can reach several hundreds. The third approach is based on the Sequential KLT/SVD [29], already commented in the preceding section. In [28, 29] is presented one more approach, based on the recursive calculation of the covariance matrix of the vectors, its eigen values and eigen vectors. In the papers [57, 58] is introduced hierarchical recursive block processing of matrices. The next approach is based on the so-called Distributed KLT [59, 60], where each vector is divided into sub-vectors and on each is applied Partial KLT.