The Proof of the ssim_end1() in FFmpeg

这篇文章是对FFmpeg中计算SSIM算法中用到的公式的证明。其中的$fs1$，$fs2$，$fs12$，$fss$的含义和FFmpeg保持一致，具体可以参考FFmpeg如何计算图像的SSIM中的说明。

$\begin{aligned} SSIM(a,b)&=\frac{(2\mu_a\mu_b+C_1)(2\sigma_{ab}+C_2)}{(\mu_a^2+\mu_b^2+C_1)(\sigma_a^2+\sigma_b^2+C_2)} \\ \mu_a&=\frac{1}{64}fs1 \\ \mu_b&=\frac{1}{64}fs2 \\ \sigma_a^2+\sigma_b^2&=\frac{1}{63}(\sum_{i,j}(a(i,j)-\mu_a)^2 + \sum_{i,j}(b(i,j)-\mu_b)^2) \\ &=\frac{1}{63}\sum_{i,j}(a(i,j)^2+\mu_a^2-2 \cdot a(i,j) \cdot \mu_a+b(i,j)^2+\mu_b^2-2 \cdot b(i,j) \cdot \mu_b) \\ &=\frac{1}{63}(\sum_{i,j}(a(i,j)^2+b(i,j)^2)-2(\mu_a\sum_{i,j}a(i,j)+\mu_b\sum_{i,j}b(i,j))+\sum_{i,j} (\mu_a^2+\mu_b^2)) \\ &=\frac{1}{63}(fss-2 \cdot \frac{1}{64}(fs1^2+fs2^2)+\frac{1}{64}(fs1^2+fs2^2) \\ &=\frac{1}{63}(fss-\frac{1}{64}(fs1^2+fs2^2)) \\ &=\frac{1}{63}\frac{1}{64}(64fss-fs1^2-fs2^2) \\ \sigma_{ab}&=\frac{1}{63}\sum_{i,j}((a(i,j)-\mu_a)(b(i,j)-\mu_b)) \\ &=\frac{1}{63}\sum_{i,j}(a(i,j) \cdot b(i,j)-a(i,j)\mu_b-b(i,j)\mu_a+\mu_a \cdot \mu_b) \\ &=\frac{1}{63}(fs12-fs1 \cdot \mu_b-fs2 \cdot \mu_a+fs1 \cdot \mu_b) \\ &=\frac{1}{63}(fs12-\frac{1}{64}fs1 \cdot fs2) \\ &=\frac{1}{63}\frac{1}{64}(64fs12-fs1 \cdot fs2) \\ \therefore SSIM(a,b)&=\frac{(2fs1 \cdot fs2+64^2C_1)(2 \cdot 64fs12-2fs1fs2+63 \cdot 64C_2)}{(fs1^2+fs2^2+64^2C_1)(64fss-fs1^2-fs2^2+63 \cdot 64C_2)} \end{aligned}$