Frequency domain filters for image enhancement
Low Pass
filter
Smoothing Filters
High Pass
filter
Sharpening Filters
Band Pass Filter
Band Reject Filter
Ideal
Ideal
Ideal
Ideal
Butterworth
Butterworth
Butterworth
Butterworth
Gaussian
Gaussian
Gaussian
Gaussian
Frequency domain representation of an image
FFT
Original Image
FFT shift
Standard Representation
(Frequency domain)
Optical Representation
(Frequency domain)
Image Smoothening (Ideal Low Pass Filter)
Ideal LPF
H(u,v)=1 if D(u,v) ≤ Do
H(u,v)=0 if D(u,v) > Do
𝐷 𝑢, 𝑣 = (𝑢 − 𝑀/2)2 +(𝑣 − 𝑁/2)2
1/2
Response of ILPF
Optical Representation
(Frequency domain)
Inverse Fourier
transform
Image Sharpening (Ideal High Pass Filter)
Ideal HPF
H(u,v)=0 if D(u,v) ≤ Do
H(u,v)=1 if D(u,v) > Do
𝐷 𝑢, 𝑣 = (𝑢 − 𝑀/2)2 +(𝑣 − 𝑁/2)2
1/2
Response of IHPF
Optical Representation
(Frequency domain)
Inverse Fourier
transform
Image Smoothening (Butterworth Low Pass Filter)
1
𝐷(𝑢, 𝑣) 2𝑛
1+
𝐷0
2
𝐷 𝑢, 𝑣 = (𝑢 − 𝑀/2) +(𝑣 − 𝑁/2)2
Butterworth LPF
𝐻(𝑢, 𝑣) =
1/2
Response of BLPF
Optical Representation
(Frequency domain)
Inverse Fourier
transform
Image Sharpening (Butterworth High Pass Filter)
Butterworth HPF
𝐻(𝑢, 𝑣) =
1
2𝑛
𝐷0
1+
𝐷(𝑢, 𝑣)
𝐷 𝑢, 𝑣 = (𝑢 − 𝑀/2)2 +(𝑣 − 𝑁/2)2
1/2
Response of BLPF
Optical Representation
(Frequency domain)
Inverse Fourier
transform
Image Smoothening (Gaussian Low Pass Filter)
𝐻 𝑢, 𝑣 =
𝐷 𝑢, 𝑣 = (𝑢 − 𝑀/2)2 +(𝑣 − 𝑁/2)2
Gaussian LPF
−𝐷(𝑢,𝑣)2
2
𝑒 2𝐷0
1/2
Response of GLPF
Optical Representation
(Frequency domain)
Inverse Fourier
transform
Image Sharpening (Gaussian High Pass Filter)
Gaussian HPF
−𝐷(𝑢,𝑣)2
2
𝑒 2𝐷0
𝐻 𝑢, 𝑣 = 1 −
𝐷 𝑢, 𝑣 = (𝑢 − 𝑀/2)2 +(𝑣 − 𝑁/2)2
1/2
Response of GHPF
Optical Representation
(Frequency domain)
Inverse Fourier
transform
Thank you