【QA】什麼是轉置卷積（Transposed Convolution）？ - 【教材專區】學習AI有困難？讓Cupoy助教來幫你！

目錄 Convolution 簡介 Transposed Convolution 簡介用數學方式理解 Convolution、Transpose Convolution 的運作實際案例卷積（Convolution）簡介 Convolution 是一個「將原始資料縮透過遮罩（Mask）縮減成少量關鍵特徵」的技術，用數學角度想就是將原始資料與 Mask 進行「線性轉換」。圖像辨識：透過 Convolution 將原始圖片搭配 Mask 運算後，得出有用的特徵圖（Feature Map）。 Output Size = 1 + \frac{size\ of\ image\ – kernel\ size\ + 2padding}{stride}1+size of image –kernel size +2∗paddingstride1 + \frac{size\ of\ image\ – kernel\ size\ + 2padding}{stride}1+stridesize of image –kernel size +2∗padding 訊號處理：透過 1D Convolution 將原始訊號轉換成「特徵訊號」。 Transposed Convolution 簡介剛好跟 Convolution 的功能「完全相反」。將某個 Pixel 或資料點以「線性轉換」方式轉換成多個資料用數學方式理解 Convolution、Transpose Convolution 的運作已知有一張 44 的 Image、一張 33 的 Kernel，以及透過 Kernel 將 Image 線性轉換後的 Output，Image 中每個 Pixel 為 $I={I_{ij} | i = 1, 2, 3, 4; j = 1, 2, 3, 4}$，Kernel 中每個 Pixel 為 K={K_{ij} | i = 1, 2, 3; j = 1, 2, 3}K={Kij∣i=1,2,3;j=1,2,3}K={K_{ij} | i = 1, 2, 3; j = 1, 2, 3}K={Kij∣i=1,2,3;j=1,2,3}，Output 中每個 Pixel 為 O={O_{ij} | i = 1, 2; j = 1, 2}O={Oij∣i=1,2;j=1,2}O={O_{ij} | i = 1, 2; j = 1, 2}O={Oij∣i=1,2;j=1,2}。O_{11}=I_{11}K_{11}+I_{12}K_{12}+I_{13}K_{13}+I_{21}K_{21}+I_{22}K_{22}+I_{23}K_{23}+I_{31}K_{31}+I_{32}K_{32}+I_{33}K_{33} \
O_{12}= I_{12}K_{11}+I_{13}K_{12}+I_{14}K_{13}+I_{22}K_{21}+I_{23}K_{22}+I_{24}K_{23}+I_{32}K_{31}+I_{33}K_{32}+I_{34}K_{33}\
O_{21}= I_{21}K_{11}+I_{22}K_{12}+I_{23}K_{13}+I_{31}K_{21}+I_{32}K_{22}+I_{33}K_{23}+I_{41}K_{31}+I_{42}K_{32}+I_{43}K_{33}\
O_{22}= I_{22}K_{11}+I_{23}K_{12}+I_{24}K_{13}+I_{32}K_{21}+I_{33}K_{22}+I_{34}K_{23}+I_{42}K_{31}+I_{43}K_{32}+I_{44}K_{33}\O11=I11K11+I12K12+I13K13+I21K21+I22K22+I23K23+I31K31+I32K32+I33K33O12=I12K11+I13K12+I14K13+I22K21+I23K22+I24K23+I32K31+I33K32+I34K33O21=I21K11+I22K12+I23K13+I31K21+I32K22+I33K23+I41K31+I42K32+I43K33O22=I22K11+I23K12+I24K13+I32K21+I33K22+I34K23+I42K31+I43K32+I44K33O_{11}=I_{11}K_{11}+I_{12}K_{12}+I_{13}K_{13}+I_{21}K_{21}+I_{22}K_{22}+I_{23}K_{23}+I_{31}K_{31}+I_{32}K_{32}+I_{33}K_{33} \O_{12}= I_{12}K_{11}+I_{13}K_{12}+I_{14}K_{13}+I_{22}K_{21}+I_{23}K_{22}+I_{24}K_{23}+I_{32}K_{31}+I_{33}K_{32}+I_{34}K_{33}\O_{21}= I_{21}K_{11}+I_{22}K_{12}+I_{23}K_{13}+I_{31}K_{21}+I_{32}K_{22}+I_{33}K_{23}+I_{41}K_{31}+I_{42}K_{32}+I_{43}K_{33}\O_{22}= I_{22}K_{11}+I_{23}K_{12}+I_{24}K_{13}+I_{32}K_{21}+I_{33}K_{22}+I_{34}K_{23}+I_{42}K_{31}+I_{43}K_{32}+I_{44}K_{33}\O11=I11K11+I12K12+I13K13+I21K21+I22K22+I23K23+I31K31+I32K32+I33K33O12=I12K11+I13K12+I14K13+I22K21+I23K22+I24K23+I32K31+I33K32+I34K33O21=I21K11+I22K12+I23K13+I31K21+I32K22+I33K23+I41K31+I42K32+I43K33O22=I22K11+I23K12+I24K13+I32K21+I33K22+I34K23+I42K31+I43K32+I44K33 若將 Convolution 以 Matrix Multiplication 計算，則：I= \begin{bmatrix}
I_{11} \
I_{12} \
I_{13} \
I_{14} \
I_{21} \
I_{22} \
I_{23} \
I_{24} \
I_{31} \
I_{32} \
I_{33} \
I_{34} \
I_{41} \
I_{42} \
I_{43} \
I_{44}
\end{bmatrix}{1*16}
\
K= \begin{bmatrix}
K{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 & 0 \
0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 \
0 & 0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33}
\end{bmatrix}{4*16}
\
O= K{416} \cdot I_{161}=\begin{bmatrix}
O_{11} \\ O_{12} \\ O_{21} \\ O_{22} \\ \end{bmatrix} \\textarea><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Imi><mo>=mo><msub><mrow><mo fence="true">[mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Imi><mn>11mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Imi><mn>12mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Imi><mn>13mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Imi><mn>14mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Imi><mn>21mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Imi><mn>22mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Imi><mn>23mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" 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displaystyle="false"><msub><mi>Kmi><mn>33mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>11mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>12mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>13mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>21mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>22mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>23mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>31mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>32mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>33mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>11mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>12mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>13mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>21mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>22mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>23mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>31mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>32mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>33mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>11mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>12mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>13mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>21mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>22mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>23mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0mn>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>31mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>32mn>msub>mstyle>mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Kmi><mn>33mn>msub>mstyle>mtd>mtr>mtable><mo fence="true">]mo>mrow><mrow><mn>4mn><mo>∗mo><mn>16mn>mrow>msub><mspace linebreak="newline">mspace><mi>Omi><mo>=mo><msub><mi>Kmi><mrow><mn>4mn><mo>∗mo><mn>16mn>mrow>msub><mo>⋅mo><msub><mi>Imi><mrow><mn>16mn><mo>∗mo><mn>1mn>mrow>msub><mo>=mo><mrow><mo fence="true">[mo><mtable rowspacing="0.15999999999999992em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Omi><mn>11mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Omi><mn>12mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Omi><mn>21mn>msub>mstyle>mtd>mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>Omi><mn>22mn>msub>mstyle>mtd>mtr>mtable><mo fence="true">]mo>mrow><mspace linebreak="newline">mspace>mrow><annotation encoding="application/x-tex">I= \begin{bmatrix}
I_{11} \
I_{12} \
I_{13} \
I_{14} \
I_{21} \
I_{22} \
I_{23} \
I_{24} \
I_{31} \
I_{32} \
I_{33} \
I_{34} \
I_{41} \
I_{42} \
I_{43} \
I_{44}
\end{bmatrix}{1*16}
\
K= \begin{bmatrix}
K{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 & 0 \
0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 \
0 & 0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33}
\end{bmatrix}{4*16}
\
O= K{416} \cdot I_{161}=\begin{bmatrix}
O_{11} \\ O_{12} \\ O_{21} \\ O_{22} \\ \end{bmatrix} \\annotation>semantics>math>span>span>Ispan>span>=span>span>span>span>span>⎣span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎡span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>1span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>1span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>1span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>1span>4span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>2span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>2span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>2span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>2span>4span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>3span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>3span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>3span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>3span>4span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>4span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>4span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>4span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ispan>span>4span>4span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>⎦span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎤span>span>span>span>span>span>span>span>span>span>span>span>span>span>1span>∗span>1span>6span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>=span>span>span>span>span>⎣span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎡span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>0span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>Kspan>span>1span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>Kspan>span>1span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>1span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>Kspan>span>2span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>Kspan>span>1span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>3span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>Kspan>span>2span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>3span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>3span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>3span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>3span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>2span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>Kspan>span>3span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>Kspan>span>2span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>Kspan>span>3span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>Kspan>span>3span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>3span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>Kspan>span>3span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Kspan>span>3span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>0span>span>span>span>0span>span>span>span>0span>span>span>span>Kspan>span>3span>3span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>⎦span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎤span>span>span>span>span>span>span>span>span>span>span>span>span>span>4span>∗span>1span>6span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ospan>span>=span>span>span>span>Kspan>span>4span>∗span>1span>6span>span>span>span>span>span>span>span>span>span>span>span>span>span>⋅span>span>span>span>Ispan>span>1span>6span>∗span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>=span>span>span>span>span>⎣span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎢span>span>span>span>⎡span>span>span>span>span>span>span>span>span>span>span>span>span>Ospan>span>1span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ospan>span>1span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ospan>span>2span>1span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>Ospan>span>2span>2span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>⎦span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎥span>span>span>span>⎤span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>span>

Transposed Convolution 就是從轉成矩陣的 Convolution 進行轉換的

K^{T}=\begin{bmatrix} K_{11} & 0 & 0 & 0 \ K_{12} & K_{11} & 0 & 0 \ K_{13} & K_{12} & 0 & 0 \ 0 & K_{13} & 0 & 0 \ K_{21} & 0 & K_{11} & 0 \ K_{22} & K_{21} & K_{12} & K_{11} \ K_{23} & K_{22} & K_{13} & K_{12} \ 0 & K_{23} & 0 & K_{13} \ K_{31} & 0 & K_{21} & 0 \ K_{32} & K_{31} & K_{22} & K_{21} \ K_{33} & K_{32} & K_{23} & K_{22} \ 0 & K_{33} & 0 & K_{23} \ 0 & 0 & K_{31} & 0 \ 0 & 0 & K_{32} & K_{31} \ 0 & 0 & K_{33} & K_{32} \ 0 & 0 & 0 & K_{33} \end{bmatrix}{4*16} O=\begin{bmatrix} O{11} \ O_{12} \ O_{21} \ O_{22} \end{bmatrix}{1*4} I= O{14}\cdot K_{416} \KT=[K11000K12K1100K13K12000K1300K210K110K22K21K12K11K23K22K13K120K230K13K310K210K32K31K22K21K33K32K23K220K330K2300K31000K32K3100K33K32000K33]4∗16O=[O11O12O21O22]1∗4I=O1∗4⋅K4∗16K^{T}=\begin{bmatrix} K_{11} & 0 & 0 & 0 \ K_{12} & K_{11} & 0 & 0 \ K_{13} & K_{12} & 0 & 0 \ 0 & K_{13} & 0 & 0 \ K_{21} & 0 & K_{11} & 0 \ K_{22} & K_{21} & K_{12} & K_{11} \ K_{23} & K_{22} & K_{13} & K_{12} \ 0 & K_{23} & 0 & K_{13} \ K_{31} & 0 & K_{21} & 0 \ K_{32} & K_{31} & K_{22} & K_{21} \ K_{33} & K_{32} & K_{23} & K_{22} \ 0 & K_{33} & 0 & K_{23} \ 0 & 0 & K_{31} & 0 \ 0 & 0 & K_{32} & K_{31} \ 0 & 0 & K_{33} & K_{32} \ 0 & 0 & 0 & K_{33} \end{bmatrix}{4*16}O=\begin{bmatrix} O{11} \ O_{12} \ O_{21} \ O_{22} \end{bmatrix}{1*4}I= O{14}\cdot K_{416} \KT=⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡K11K12K130K21K22K230K31K32K33000000K11K12K130K21K22K230K31K32K3300000000K11K12K130K21K22K230K31K32K33000000K11K12K130K21K22K230K31K32K33⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤4∗16O=⎣⎢⎢⎢⎡O11O12O21O22⎦⎥⎥⎥⎤1∗4I=O1∗4⋅K4∗16 實際案例 Generative Adversarial Network 一種影像生成的深度學習演算法 Discriminator Network：判斷生成出來的圖片與真實圖片是否相同或相似。 Generator Network：藉由 D-dimensional noise vector 產生出一張偽圖片，其中就會大量使用到 Transposed Convolution。 U-Net 影像切割案例中最常使用的深度學習架構在 Encoder 階段，將圖片以 CNN 方式萃取出關鍵特徵在 Decoder 階段，除了以 Transposed Convolution 將 Feature Map 中每個特徵轉換成多個特徵外，部分 Encoder 階段得到的 Feature Map 也會在此階段搭配運算。 Reference Mozaffari M.H. and Tay L.L. (2020). 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