Note on Implementation of “Fast Noise Variance Estimation” by J. Immerkær

Noise variance estimation is a fundamental task of image processing. Among various approaches of noise variance estimate, fast noise variance estimation method proposed by J. Immerkær in [1] stands out because of its good balance between complexity and accuracy.

Based on J. Immerkær's method, to estimate the standard deviation of image noise \(\sigma\), we can use equations below:

\(F=
\begin{bmatrix}
1 & -2 & 1\\
-2 & 4 & -2\\
1 & -2 & 1\\
\end{bmatrix}
\)

\(\sigma=\frac{\sqrt{\pi/2}}{6(W-2)(H-2)}\sum_{I}|I(x,y)*F|\)

\(F\) is the high-pass filter. To estimate noise variance, filter \(F\) will be applied on the whole image \(I(x,y)\) with convolution. The sum of absolute value of convolution results is normalized by width of height of the image, which is \((W-2)(H-2)\). "-2" counts for the boundary effect. The summation is also scaled by \(\frac{\sqrt{\pi/2}}{6}\), which is explained next.

Assume a 3x3 block of \(I(x,y)\) with each element of \(x_{i}\) to be a Gaussian random variable with mean \(\mu\) and standard deviation \(\sigma\)
\(I=
\begin{bmatrix}
x_{1} & x_{2} & x_{3}\\
x_{4} & x_{5} & x_{6}\\
x_{7} & x_{8} & x_{9}\\
\end{bmatrix}
\)
Then \(|I*F|=|x_{1}-2*x_{2}+x_{3}-2*x_{4}+4*x_{5}-2*x_{6}+x_{7}-2*x_{8}+x_{9}|\)
Since \(E(|I*F|^2)=36\sigma^2\), \(|I*F|=6|s|\) where s is Gaussian random variable with mean 0 and standard deviation \(\sigma\). We can derive the distribution of \(y=|s|\) and it is \(\frac{2}{\sqrt{2\pi}\sigma}e^{-\frac{y^2}{2\sigma^2}}\)  for \(y>=0\) and 0 for \(y < 0\). <0 .="" br=""> 

Since \(E(|s|)=\sqrt{\frac{2}{\pi}}\sigma\), it explains why the summation is also scaled by \(\frac{\sqrt{\pi/2}}{6}\).


[1] J. Immerkær, “Fast Noise Variance Estimation”, Computer Vision and Image Understanding, Vol. 64, No. 2, pp. 300-302, Sep. 1996