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