这篇文章详细解析了FcaNet中基于2DDiscreteCosineTransform(DCT)的频率通道注意力网络，讨论了DCT的计算逻辑，特别强调了GAP操作与DCT低频成分的关系，并通过实验研究了不同频率分量对性能的影响。
ICCV 2021 | FcaNet: Frequency Channel Attention Networks 中的频率分析
文章是围绕 2D 的 DCT 进行展开的，本文针对具体的计算逻辑进行梳理和解析。
f ( u , v ) = α u α v H W ∑ i = 0 H − 1 ∑ j = 0 W − 1 f ( i , j ) cos ( 2 i + 1 ) u π 2 H cos ( 2 j + 1 ) v π 2 W = ∑ i = 0 H − 1 [ α u H cos ( 2 i + 1 ) u π 2 H ] ∑ j = 0 W − 1 [ α v W cos ( 2 j + 1 ) v π 2 W ] x ( i , j ) = ∑ i = 0 H − 1 A u i ∑ j = 0 W − 1 A v j x ( i , j ) = ∑ i = 0 H − 1 ∑ j = 0 W − 1 x ( i , j ) B u , v i , j , u ∈ { 0 , 1 , … , H − 1 } , v ∈ { 0 , 1 , … , W − 1 } α u = { 1 u = 0 2 u ≠ 0 , α v = { 1 v = 0 2 v ≠ 0 , x = ∑ u = 0 H − 1 ∑ v = 0 W − 1 f ( u , v ) B u , v i , j \begin{align} \\ f(u,v) &= \sqrt{\frac{\alpha_{u}\alpha_{v}}{HW }} \sum^{H-1}_{i=0} \sum^{W-1}_{j=0} f(i,j) \cos\frac{(2i+1)u\pi}{2H} \cos\frac{(2j+1)v\pi}{2W} \\ & = \sum^{H-1}_{i=0} \left[ \sqrt{ \frac{\alpha_{u}}{H} }\cos\frac{(2i+1)u\pi}{2H}\right] \sum^{W-1}_{j=0} \left[ \sqrt{ \frac{\alpha_{v}}{W} }\cos\frac{(2j+1)v\pi}{2W} \right] x(i,j) \\ & = \sum^{H-1}_{i=0} A^{i}_{u} \sum^{W-1}_{j=0} A^{j}_{v} x(i,j) \\ & = \sum^{H-1}_{i=0} \sum^{W-1}_{j=0} x(i,j) B^{i,j}_{u,v}, \, u \in \{0, 1, \dots, H-1\}, \, v \in \{0, 1, \dots, W-1\} \\ \alpha_{u} & = \left\{ \begin{matrix} 1 & u = 0 \\ 2 & u \ne 0, \end{matrix} \right. \quad \alpha_{v} = \left\{ \begin{matrix} 1 & v = 0 \\ 2 & v \ne 0, \end{matrix} \right. \\ x & = \sum^{H-1}_{u=0} \sum^{W-1}_{v=0} f(u,v) B^{i,j}_{u,v} \end{align} f(u,v)αux=HWαuαvi=0∑H−1j=0∑W−1f(i,j)cos2H
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