Low Pass Frequency Domain Filters:

The basic formula for any kind of filtering is based on the convolution integral.

i.e. f(x,y)*h(x,y) = FT =F(u,v) x H(u,v)

f(x,y) - original image

h(x,y) - Filtering mask

F(u,v) – Fourier transform of the original image.

H(u,v) – Fourier transform of the filtering mask

Hence fro the filtering we use the formula,

G(u,v) = F(u,v) x H(u,v)

In this cases three types of filtering is possible:

Ideal low pass filter (ILPF):

These are the simplest of the three filters. This filters cut off all the high frequency components of the fourier transform that are at a distance greater than a specified distance D0.

H(u,v) =1 ; if D(u,v) ≤ D0 =0 ; if D(u,v) > D0

Butterworth Low Pass filters (BLPS):

The ringing effects due to the sharp cut-offs in the ideal filter and to get rid of ringing effects, elimination of sharp cut-offs is necessary. This exactly happens in butterworth low pass filters. The transfer function of the butterworth low pass filter of order n and the cut off frequency at a distance D0 from the origin is defined as

$H(u, v) = \frac{1}{1+[D(u, v) / D0]^{2n}}........ (1)$

D(u,v) is the distance from the point (u,v) to the origin of the frequency rectangle for an M x N image. H(u,v) is the Fourier transform to the filtering mask.

Unlike the ILPF, the BLPF does not have sharp discontinuities and hence there are no ringing effects present when a BLPF is used. But as the order of the filter goes on increasing, a small amount of ringing effects does not creep in because the butterworth low pass filter tends to be an ideal filter.

Gaussian Low Pass Filter (GLPF):

Gaussian filter LPF is given by,

$H(u, v) = e^{-D^2(u, v)/2σ^2}$

Here σ is the standard deviation and is a measure of spread of the Gaussian curve. If we put σ = D0 we get,

$H(u, v) = e^{-D^2(u, v)/2σ^2}$