Statistic based smoothing

Statistical thresholding

First find the mean $\mu$ and the variance $\sigma^2$ of all pixels in the neighborhood $W$ of a given pixel $x[m,n]$ of the input image:

$$\mu=\frac{1}{N} \sum_{(i,j) \in W} x[i,j]$$

$$\sigma^2=\frac{1}{N} \sum_{(i,j) \in W} [x[i,j]-\mu]^2$$

where $N$ is the number of pixels in the neighborhood $W$.

Then find the corresponding pixel for the output image:

$$y[m,n]=\left\{ \begin{array}{ll} x[m,n] & \mbox{if $\vert x[... ...mu\vert < \sigma T$} \mu & \mbox{else} \end{array} \right.$$

where $T$ is a user specified threshold value.

Nagao's algorithm

• Define a set of K neighborhoods around the pixel $x[m,n]$ under consideration, for example, the eight 3x3 neighborhoods in N, NE, E, SE, S, SW, W, and NW directions. The pixel $x[m,n]$ should be included as a corner in each of these neighborhoods.

• Find the mean $\mu_i$ and variance $\sigma_i^2$ for each of the K neighborhoods.

• Replace pixel$x[m,n]$ by one of the means corresponding to the smallest $\sigma^2$:
  
  $$y[m,n]=\mu_i, \;\;\;\;\;\mbox{iff}\;\;\;\;\; \sigma_i^2=min\{\sigma_1^2,\cdots,\sigma_K^2\}$$
  
  

In this example, the pixel in question $x[m,n]=5$ is replaced by the mean 7.1 from the northeast neighborhood with minimum variance 105.