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upper W Super Superscript l plus 1 Superscript upper D Super Superscript l plus 1 Superscript right-parenthesis times left-parenthesis upper H Super Superscript l Superscript upper W Super Superscript l Superscript upper D Super Superscript l Superscript right-parenthesis"/>.

      One triplet of indexes (il + 1, jl + 1, dl + 1) specifies a row in S, while (il, jl, dl) specifies a column. These two triplets together pinpoint one element in (xl). We set that element to 1 if dl + 1 = dl and x Subscript i Sub Superscript l Subscript comma j Sub Superscript l Subscript comma d Sub Superscript l Subscript Superscript l Baseline greater-than-or-equal-to y Subscript i plus i Sub Superscript l plus 1 Subscript times upper H comma j plus j Sub Superscript l plus 1 Subscript times upper W comma d Sub Superscript l Subscript Baseline comma for-all 0 less-than-or-equal-to i less-than upper H comma 0 less-than-or-equal-to j less-than upper W, are simultaneously satisfied, and 0 otherwise. One row of S(xl) corresponds to one element in y, and one column corresponds to one element in xl. By using this indicator matrix, we have vec(y) = S(xl)vec(xl) and

StartFraction partial-differential v e c left-parenthesis y right-parenthesis Over partial-differential left-parenthesis v e c left-parenthesis x Superscript l Baseline right-parenthesis Superscript upper T Baseline right-parenthesis EndFraction equals upper S left-parenthesis x Superscript l Baseline right-parenthesis comma StartFraction partial-differential z Over partial-differential left-parenthesis v e c left-parenthesis x Superscript l Baseline right-parenthesis Superscript upper T Baseline right-parenthesis EndFraction equals StartFraction partial-differential z Over partial-differential left-parenthesis v e c left-parenthesis y right-parenthesis Superscript upper T Baseline right-parenthesis EndFraction upper S left-parenthesis x Superscript l Baseline right-parenthesis Schematic illustration of preprocessing in a cooperative neural network (CoNN)-based image classifier.

      (3.104)StartFraction partial-differential z Over partial-differential v e c left-parenthesis x Superscript l Baseline right-parenthesis EndFraction equals upper S left-parenthesis x Superscript l Baseline right-parenthesis Superscript upper T Baseline StartFraction partial-differential z Over partial-differential v e c left-parenthesis y right-parenthesis EndFraction period

      For further readings on CoNNs, the reader is referred to [30].

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      17 17 Pan, C. and Naeemi, A. (2016). A proposal for energy‐efficient cellular neural network based on spintronic devices. IEEE Trans. Nanotechnol. 15 (5): 820–827.

      18 18 Wang, L. et al. (1998). Time multiplexed color image processing based on a CNN with cell‐state outputs. IEEE Trans. VLSI Syst. 6 (2): 314–322.

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      28 28

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