Low Pass Filters pass low frequency components up to cutoff frequency and attenuate or eliminate high frequency components present in the image.

1. Ideal LPF

   Transfer function of Ideal LPF is given by,

   H(u,v)= $1 \ \ \ \ if D(u,v)≤ D0 \\ 0 \ \ \ \ D(u,v)› D0$

   Where,

   D0: is a non-negative quantity (cut-off frequency)

   D(u,v): is the distance from point (u,v) to the origin of the frequency plane and D(u,v) and $D(u,v)=\sqrt{(u^2+v^2 )}$

1. Butterworth LPF

   Transfer function of Butterwoth LPF is given by, $H(u,v)=\dfrac{1}{(1+[D(u,v)/D0]2N)}$

   Where

   D0 is the cutoff frequency

   $D(u,v)= \sqrt{(u^2+v^2 )}$

   N is the order of the filter

   Butterworth LPF produces smooth transition in blurring as a function of increases cutoff frequency.

   A Butterworth Filter of order 1 has no ringing. Ringing is imperceptive in filters of order 2 but can become a significant factor in filters of higher order.

1. Gaussian Low-Pass Filter

   Transfer function of Gaussian LPF is given by, $H(u,v) = \dfrac{e(-D(u,v))}{2σ2}$

   Where

   D(u,v): is the distance from the origin of the Fourier Transform

   σ : is a measure of the spread of the Guassian Curve

   Let σ = D0, then $H(u,v)=e\dfrac{(-D(u,v))}{2σ2}$ where D0 is cutoff frequency.

   GLPF produces smooth transition I blurring as a function of increasing cutoff frequency. GLPF does not produce ringing effect. This is an important characteristic in practice.