CoE4TN4

Image Processing
Chapter 4
Filtering in the Frequency
Domain
 Fourier Transform
• Sections 4.1 to 4.5 will be done on the board

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2D Fourier Transform

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2D Sampling and Aliasing

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 2D Sampling and Aliasing
• Perfect reconstruction of a bank-limited image from a set of
its samples requires 2-D convolution in the spatial domain
with a sinc function.
• To reduce aliasing it is a good idea to blur an image before
shrinking or sampling

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2D Sampling and Aliasing

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 2D Sampling and Aliasing
• In images with strong edge content, the effects of aliasing are
called jaggies

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 2D Sampling and Aliasing
• Moire patterns happens in sampling scenes with periodic or
nearly perodic components
• Scanning of media prints such as newspaper

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 2D Sampling and Aliasing
• Newspapers and other printed materials use halftone dots
• Halftone: black dots or ellipses whose sizes are used to
simulate gray tones

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2D Sampling and Aliasing

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2D DFT

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2D DFT

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 Fourier Transform
• Fourier Transform (FT) of a 2-D signal:

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 Fourier Transform
• Properties of Fourier Transform (FT) of a 2-D real, signal:

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 Fourier Transform
• It is common to multiply input image by (-1)x+y prior to
computing the Fourier Transform.
• This shift the center of the FT to (M/2,N/2)

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Fourier Transform

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2D DFT

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2D DFT

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 Discrete Fourier transform
• When working with discrete Fourier transform, we have to
keep in mind the periodicity of functions involved.
• This periodicity is a byproduct of the way in which discrete
FT (DFT) is defined
• Using DFT allows us to perform convolution in the frequency
domain but the functions are treated as periodic.

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2D DFT

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2D DFT

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2D DFT

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 Frequency domain filtering
1. Multiply the input image by (-1)x+y to center the transform
Compute the DFT of the image from step 1
2. Multiply the result by the transfer function of the filter
(centered)
3. Take the inverse transform.
4. Multiply the result by (-1)x+y

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• H(u,v)is 0 at the center of the transform and 1 elsewhere

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• Low frequencies: slowly varying components in an image
• High frequencies: caused by sharp transitions in intensity such
as edges and noise
 Steps for filtering in frequency domain
• For input image f(x,y) of size MxN obtain the padding
parameters P and Q
• Form a padded image fp(x,y) of size PxQ by appending the
necessary number of zeros to f(x,y)
• Multiply fp(x,y) by (-1)x+y to center the Fourier transform
• Compute DFT of f, F(u,v)
• Generate a real, symmetric filter H(u,v) of size PxQ. Form
product G(u,v)=H(u,v)F(u,v)
• Obtain the processed image: gp (x, y) = {real(F 1 (G(u, v)))}( 1)x+y
• Obtain the final processed result by extracting the MxN
region from the top left quadrant of gp(x,y)

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 Frequency domain filtering
• G(u,v)=F(u,v)H(u,v)
• Based on convolution theorem: g(x,y)=h(x,y)*f(x,y)
– f(x,y) is the input image
– g(x,y) is the processed image
– h(x,y): impulse response or point spread function
• Based on the form of H(u,v), the output image exhibits some
features of f(x,y)

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 Frequency domain filtering
Convolution Implementing a mask
w(x,y)
h(x,y) f(x,y)
f(x,y)

1) Flip 2) Shift, multiply, add

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 Frequency domain filtering
• Convolution with a filter and implementing a mask are very
similar
• The only difference is the flipping operation
• If the impulse response of the filter is symmetric about the
origin the two operations are the same.
• Instead of filtering in the frequency domain, we can
approximate the impulse response of the filter by a mask, and
use the mask in the spatial domain.

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 Frequency domain filtering
• Edges and sharp transitions (e.g., noise) in the gray levels of
an image contribute significantly to high-frequency content of
FT.
• Low frequency in the Fourier transform of an image are
responsible for the general gray-level appearance of the image
over smooth areas.
• Blurring (smoothing) is achieved by attenuating range of high
frequency components of FT.
• We consider 3 types of lowpass filters: ideal, Butterworth and
Gaussian

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 Ideal low-pass filter
• Ideal: all the frequencies inside a circle of radius D0 are
passed and all the frequencies outside this circle are
completely removed.

|H| |H|

v
u
u
M/2-D0 M/2+D0

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 Ideal low-pass filter
• Effects of ideal low-pass filtering: blurring and ringing

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 Ideal low-pass filter

• The radii of the concentric rings in h(x,y) are proportional to 1/D0

D0 1/D0 Less blurring,

Less ringing (amplitude
of rings drop)

1/D0 More blurring,

D0
More ringing
(amplitude of rings
increase)

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 Butterworth Lowpass Filters
• Smooth transfer function, no sharp discontinuity, no clear
cutoff frequency.

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Butterworth Lowpass Filters

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Butterworth Lowpass Filters

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 Gaussian Lowpass Filters
• Smooth transfer function, smooth impulse response, no
ringing

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Gaussian Lowpass Filters

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Applications of Lowpass Filters

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Applications of Lowpass Filters

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 High-pass filtering
• Image sharpening can be achieved by a high-pass filtering
process.
• Hhp(u,v)=1-Hlp(u,v)

• Ideal:

• Butterworth:

• Gaussian:

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High-pass filtering

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High-pass filtering

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High-pass filtering

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High-pass filtering

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High-pass filtering

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Laplacian in Frequency Domain

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Laplacian in Frequency Domain

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 Unsharp Masking, High-boost Filtering
• Unsharp masking: fhp(x,y)=f(x,y)-flp(x,y)
• Hhp(u,v)=1-Hlp(u,v)

• High-boost filtering: fhb(x,y)=Af(x,y)-flp(x,y)

• fhb(x,y)=(A-1)f(x,y)+fhp(x,y)
• Hhb(u,v)=(A-1)+Hhp(u,v)

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 Homomorphic filtering
• In some images, the quality of the image has reduced because
of non-uniform illumination
• Homomorphic filtering can be used to perform illumination
correction
• We can view an image f(x,y) as a product of two components:

• This equation cannot be used directly in order to operate

separately on the frequency components of illumination and
reflectance

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Homomorphic filtering

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 Homomorphic filtering
• The key to the approach is that separation of the illumination
and reflectance components is achieved. The homomorphic
filter can then operate on these components separately
• Illumination component of an image generally has slow
variations, while the reflectance component vary abruptly
• By removing the low frequencies (highpass filtering) the
effects of illumination can be removed

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Homomorphic filtering

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 Selective Filtering
• Bandreject filters remove or attenuate a band of frequencies

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Selective Filtering

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Selective Filtering

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 Bandpass and band reject filters
• A bandpass filter performs the opposite of a bandreject filter.
– Hbp(u,v)=1-Hbr(u,v)
• A notch filter rejects (or passes) frequencies in predefined
neighborhood about a center frequency
• Due to symmetry of the FT, notch filters must appear in
symmetric pairs about the origin in order to obtain meaningful
results

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Periodic noise reduction

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Periodic noise reduction

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