Periodic Noise

Periodic noise is generally due to electrical or electromechanical interference during image acquisition. It is a spatially dependent noise.

We will have to tackle in the frequency domain. This is because its energy is concentrated at a few predictable spectral locations instead of being spread across the spectrum.

How we can reduce the specific frequencies is by using bandreject, bandpass, or notch filters to restore the images in the frequency domain.

Ideal Bandreject Filter

$$H(u,v)=\{\begin{array}{ll}1,& D(u,v)<D_0-\frac{W}{2}\\ 0,& D_0-\frac{W}{2}\le D(u,v)\le D_0+\frac{W}{2}\\ 1,& D(u,v)>D_0+\frac{W}{2}\end{array}$$

where $D(u,v)$ is the distance from the origin, $W$ is the width of the band, and $D_0$ is its radial centre, which is the distance from center (DC) to the middle of the rejected ring

Example:

Ideal Bandpass Filter

$$H_{bp}(u,v)=1-H_{br}(u,v)$$

When we perform the same process but using a bandpass filter, we are getting the noise estimate as the output (instead of the image with noise removed). In otherwords, it maintains the selected annulus (ring) and discards the rest.

Notch Reject Filter

This filter suppresses only small neighbourhoods around the selected spectral peaks while passing the remaining frequencies.

$$H(u,v)=\{\begin{array}{ll}0,& D_1(u,v)\le D_0\text{ or }{D}_2(u,v)\le D_0\\ 1,& \text{otherwise}\end{array}$$ $$D_1(u,v)=\sqrt{(u-u_0{)}^2+(v-v_0{)}^2}{D}_2(u,v)=\sqrt{(u+u_0{)}^2+(v+v_0{)}^2}$$