h.264中的4x4 DCT矩阵的输出。那么Y各个位置像素的完整公式什么呢？

Question

h.264中的4x4 DCT矩阵的输出。那么Y各个位置像素的完整公式什么呢？
矩阵乘法规则 ‌Y = Cf * X * Cfᵀ‌。
假设：

Cf  = [1,  1,  1,  1]
         [2,  1, -1, -2]
         [1, -1, -1,  1]
         [1, -2,  2, -1]

X =  [x00, x01, x02, x03]
        [x10, x11, x12, x13]
        [x20, x21, x22, x23]
        [x30, x31, x32, x33]

 Cfᵀ‌ =  [1,  2,  1,  1]
            [1,  1, -1, -2]
            [1, -1, -1,  2]
            [1, -2,  1, -1]

则矩阵乘法 ‌Y = Cf * X * Cfᵀ‌，得到的每个Y分别多少呢？即Y00，Y01,...,Y33完整的公式是什么呢？

Answer 1

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在H.264中，4x4 DCT变换通过矩阵乘法 ( Y = C_f \cdot X \cdot C_f^T ) 实现，其中 ( C_f ) 是变换矩阵，( X ) 是输入像素矩阵，( Y ) 是输出系数矩阵。根据给定的矩阵：

[
C_f = \begin{bmatrix}
1 & 1 & 1 & 1 \
2 & 1 & -1 & -2 \
1 & -1 & -1 & 1 \
1 & -2 & 2 & -1
\end{bmatrix}, \quad
C_f^T = \begin{bmatrix}
1 & 2 & 1 & 1 \
1 & 1 & -1 & -2 \
1 & -1 & -1 & 2 \
1 & -2 & 1 & -1
\end{bmatrix}, \quad
X = \begin{bmatrix}
x_{00} & x_{01} & x_{02} & x_{03} \
x_{10} & x_{11} & x_{12} & x_{13} \
x_{20} & x_{21} & x_{22} & x_{23} \
x_{30} & x_{31} & x_{32} & x_{33}
\end{bmatrix}
]

计算 ( Y = C_f \cdot X \cdot C_f^T ) 后，每个输出系数 ( Y_{ij} ) 的完整公式如下：

1. 计算中间矩阵 ( M = C_f \cdot X ):

[
M_{ij} = \sum_{k=0}^{3} (C_f){ik} \cdot X{kj}
]
例如：

• ( M_{00} = 1 \cdot x_{00} + 1 \cdot x_{10} + 1 \cdot x_{20} + 1 \cdot x_{30} )
• ( M_{01} = 1 \cdot x_{01} + 1 \cdot x_{11} + 1 \cdot x_{21} + 1 \cdot x_{31} )
• ( M_{02} = 1 \cdot x_{02} + 1 \cdot x_{12} + 1 \cdot x_{22} + 1 \cdot x_{32} )
• ( M_{03} = 1 \cdot x_{03} + 1 \cdot x_{13} + 1 \cdot x_{23} + 1 \cdot x_{33} )
• ( M_{10} = 2 \cdot x_{00} + 1 \cdot x_{10} - 1 \cdot x_{20} - 2 \cdot x_{30} )
• 其他 ( M_{ij} ) 类似计算。

2. 计算 ( Y = M \cdot C_f^T ):

[
Y_{ij} = \sum_{k=0}^{3} M_{ik} \cdot (C_f^T){kj}
]
将 ( M ) 的表达式代入，得到每个 ( Y{ij} ) 的完整公式（以 ( Y_{00} ) 为例）：
[
Y_{00} = M_{00} \cdot 1 + M_{01} \cdot 1 + M_{02} \cdot 1 + M_{03} \cdot 1
]
代入 ( M_{0k} ) 的表达式：
[
Y_{00} = (x_{00} + x_{10} + x_{20} + x_{30}) + (x_{01} + x_{11} + x_{21} + x_{31}) + (x_{02} + x_{12} + x_{22} + x_{32}) + (x_{03} + x_{13} + x_{23} + x_{33})
]
即：
[
Y_{00} = \sum_{i=0}^{3} \sum_{j=0}^{3} x_{ij}
]

类似地，所有 ( Y_{ij} ) 的完整公式如下：

( Y_{00} ):

[
Y_{00} = x_{00} + x_{10} + x_{20} + x_{30} + x_{01} + x_{11} + x_{21} + x_{31} + x_{02} + x_{12} + x_{22} + x_{32} + x_{03} + x_{13} + x_{23} + x_{33}
]

( Y_{01} ):

[
Y_{01} = 2x_{00} + 2x_{10} + 2x_{20} + 2x_{30} + x_{01} + x_{11} + x_{21} + x_{31} - x_{02} - x_{12} - x_{22} - x_{32} - 2x_{03} - 2x_{13} - 2x_{23} - 2x_{33}
]

( Y_{02} ):

[
Y_{02} = x_{00} + x_{10} + x_{20} + x_{30} - x_{01} - x_{11} - x_{21} - x_{31} - x_{02} - x_{12} - x_{22} - x_{32} + x_{03} + x_{13} + x_{23} + x_{33}
]

( Y_{03} ):

[
Y_{03} = x_{00} + x_{10} + x_{20} + x_{30} - 2x_{01} - 2x_{11} - 2x_{21} - 2x_{31} + 2x_{02} + 2x_{12} + 2x_{22} + 2x_{32} - x_{03} - x_{13} - x_{23} - x_{33}
]

( Y_{10} ):

[
Y_{10} = 2x_{00} + x_{10} - x_{20} - 2x_{30} + 2x_{01} + x_{11} - x_{21} - 2x_{31} + 2x_{02} + x_{12} - x_{22} - 2x_{32} + 2x_{03} + x_{13} - x_{23} - 2x_{33}
]

( Y_{11} ):

[
Y_{11} = 4x_{00} + 2x_{10} - 2x_{20} - 4x_{30} + 2x_{01} + x_{11} - x_{21} - 2x_{31} - 2x_{02} - x_{12} + x_{22} + 2x_{32} - 4x_{03} - 2x_{13} + 2x_{23} + 4x_{33}
]

( Y_{12} ):

[
Y_{12} = 2x_{00} + x_{10} - x_{20} - 2x_{30} - 2x_{01} - x_{11} + x_{21} + 2x_{31} - 2x_{02} - x_{12} + x_{22} + 2x_{32} + 2x_{03} + x_{13} - x_{23} - 2x_{33}
]

( Y_{13} ):

[
Y_{13} = 2x_{00} + x_{10} - x_{20} - 2x_{30} - 4x_{01} - 2x_{11} + 2x_{21} + 4x_{31} + 4x_{02} + 2x_{12} - 2x_{22} - 4x_{32} - 2x_{03} - x_{13} + x_{23} + 2x_{33}
]

( Y_{20} ):

[
Y_{20} = x_{00} - x_{10} - x_{20} + x_{30} + x_{01} - x_{11} - x_{21} + x_{31} + x_{02} - x_{12} - x_{22} + x_{32} + x_{03} - x_{13} - x_{23} + x_{33}
]

( Y_{21} ):

[
Y_{21} = 2x_{00} - 2x_{10} - 2x_{20} + 2x_{30} + x_{01} - x_{11} - x_{21} + x_{31} - x_{02} + x_{12} + x_{22} - x_{32} - 2x_{03} + 2x_{13} + 2x_{23} - 2x_{33}
]

( Y_{22} ):

[
Y_{22} = x_{00} - x_{10} - x_{20} + x_{30} - x_{01} + x_{11} + x_{21} - x_{31} - x_{02} + x_{12} + x_{22} - x_{32} + x_{03} - x_{13} - x_{23} + x_{33}
]

( Y_{23} ):

[
Y_{23} = x_{00} - x_{10} - x_{20} + x_{30} - 2x_{01} + 2x_{11} + 2x_{21} - 2x_{31} + 2x_{02} - 2x_{12} - 2x_{22} + 2x_{32} - x_{03} + x_{13} + x_{23} - x_{33}
]

( Y_{30} ):

[
Y_{30} = x_{00} - 2x_{10} + 2x_{20} - x_{30} + x_{01} - 2x_{11} + 2x_{21} - x_{31} + x_{02} - 2x_{12} + 2x_{22} - x_{32} + x_{03} - 2x_{13} + 2x_{23} - x_{33}
]

( Y_{31} ):

[
Y_{31} = 2x_{00} - 4x_{10} + 4x_{20} - 2x_{30} + x_{01} - 2x_{11} + 2x_{21} - x_{31} - x_{02} + 2x_{12} - 2x_{22} + x_{32} - 2x_{03} + 4x_{13} - 4x_{23} + 2x_{33}
]

( Y_{32} ):

[
Y_{32} = x_{00} - 2x_{10} + 2x_{20} - x_{30} - x_{01} + 2x_{11} - 2x_{21} + x_{31} - x_{02} + 2x_{12} - 2x_{22} + x_{32} + x_{03} - 2x_{13} + 2x_{23} - x_{33}
]

( Y_{33} ):

[
Y_{33} = x_{00} - 2x_{10} + 2x_{20} - x_{30} - 2x_{01} + 4x_{11} - 4x_{21} + 2x_{31} + 2x_{02} - 4x_{12} + 4x_{22} - 2x_{32} - x_{03} + 2x_{13} - 2x_{23} + x_{33}
]

这些公式直接由矩阵乘法展开得到，反映了H.264中4x4 DCT变换的完整计算过程。实际应用中，为了减少计算复杂度，通常会使用整数近似和移位操作。