$($A$)$ Before we study Gaussian filtering using MATLAB, we will investigate some of its

analytical properties.

We model a one$-$dimensional image as a sequence of five pixels $x=(a_1,$dots,$a_5)$ where

$a_i,i=1,$dots,$5,$ are constants. The noise is modeled as $Z=(Z_1,$dots,$Z_5),$ where all the

$Z_i\text{'}$s are independent, $Z_{i}N(0,{}^2),i=1,$dots,$5.$ Hence the noisy image is

$Y=x+Z.$

The Gaussian filter replaces $Y_3$ with its filtered version hat$(Y{)}_3$ :

hat$(Y{)}_3={\displaystyle \underset{i=1}{\overset{5}{}}}{}_i{Y}_i,$

where we assume ${}_1={}_5=0\mathrm{.}1,{}_2={}_4=0\mathrm{.}2$ and ${}_3=0\mathrm{.}4.$ Note that hat$(Y{)}_3$ has two

components, one due to the image and one due to noise.

$($a$)$ Consider a smooth image where all pixels are equal, that is $x=(a,$dots,$a).$

i$.$ What is the distribution of the noise component in hat$(Y{)}_3?$

ii$.$ What is image component in hat$(Y{)}_3?$

iii. Comment on the effect of Gaussian filtering on the noise and the image.

$($b$)$ Now we consider an image with a edge, modeled as $x=(a,a,a,b,b),$ where $ab.$

i$.$ What is the distribution of the noise component in hat$(Y{)}_3?$

ii$.$ What is image component in hat$(Y{)}_3?$

iii. Comment on the effect of Gaussian filtering on the noise and the image. Do

you see the effect of blurring on the image?