The document discusses image enhancement in the frequency domain using Fourier transforms, explaining both one-dimensional and two-dimensional transforms. It outlines various applications such as image restoration and enhancement, as well as filtering techniques like low-pass, high-pass, and band-pass filters. The document emphasizes the significance of frequency components in image processing, including effects on image clarity and detail enhancement.

1. Image Enhancement in Frequency Domain
Image Enhancement in Frequency Domain
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2. Introduction
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3. Background (Fourier Series)
 Any function that periodically repeats itself can be
expressed as the sum of sines and cosines of
different frequencies each multiplied by a different
coefficient
 This sum is known as Fourier Series
 It does not matter how complicated the function is;
as long as it is periodic and meet some mild
conditions it can be represented by such as a sum
 It was a revolutionary discovery
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5. Background (Fourier Transform)
 Even functions that are not periodic but Finite can be
expressed as the integrals of sines and cosines multiplied
by a weighing function
 This is known as Fourier Transform
 A function expressed in either a Fourier Series or
transform can be reconstructed completely via an inverse
process with no loss of information
 This is one of the important characteristics of these
representations because they allow us to work in the
Fourier Domain and then return to the original domain of
the function
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6. Fourier Transform
• ‘Fourier Transform’ transforms one function into
another domain , which is called the frequency
domain representation of the original function
• The original function is often a function in the
Time domain
• In image Processing the original function is in the
Spatial Domain
• The term Fourier transform can refer to either the
Frequency domain representation of a function or
to the process/formula that "transforms" one
function into the other.
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7. Our Interest in Fourier Transform
• We will be dealing only with functions (images) of
finite duration so we will be interested only in Fourier
Transform
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8. Applications of Fourier Transforms
 1-D Fourier transforms are used in Signal Processing
 2-D Fourier transforms are used in Image Processing
 3-D Fourier transforms are used in Computer Vision
 Applications of Fourier transforms in Image processing: –
– Image enhancement,
– Image restoration,
– Image encoding / decoding,
– Image description
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10. One Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u) of a single variable, continuous
function f (x) is
 Given F(u) we can obtain f (x) by means of the Inverse
Fourier Transform
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11. Discrete Fourier Transforms (DFT)
1-D DFT for M samples is given as
The Inverse Fourier transform in 1-D is given as
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12. Discrete Fourier Transforms (DFT)
1-D DFT for M samples is given as
The inverse Fourier transform in 1-D is given as
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13. Two Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u,v) of a two variable, continuous
function f (x,y) is
 Given F(u,v) we can obtain f (x,y) by means of the Inverse
Fourier Transform
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14. 2-D DFT
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15. Fourier Transform
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16. 2-D DFT
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Shifting the Origin to the Center

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Shifting the Origin to the Center

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Properties of Fourier Transform
 As we move away from the origin in F(u,v) the lower
frequencies corresponding to slow gray level changes
 Higher frequencies correspond to the fast changes in gray
levels (smaller details such edges of objects and noise)
 The direction of amplitude change in spatial domain and the
amplitude change in the frequency domain are orthogonal
(see the examples)

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Properties of Fourier Transform
The Fourier Transform pair has the following translation property

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Properties of Fourier Transform

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Properties of Fourier Transform

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DFT Examples

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DFT Examples

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Properties of Fourier Transform

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Properties of Fourier Transform

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Properties of Fourier Transform
 The lower frequencies corresponds to slow
gray level changes
 Higher frequencies correspond to the fast
changes in gray levels (smaller details such
edges of objects and noise)

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Filtering using Fourier Transforms

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Example of Gaussian LPF and HPF

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Filters to be Discussed

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Low Pass Filtering
A low-pass filter attenuates high frequencies and retains low
frequencies unchanged. The result in the spatial domain is
equivalent to that of a smoothing filter; as the blocked high
frequencies correspond to sharp intensity changes, i.e. to the
fine-scale details and noise in the spatial domain image.

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High Pass Filtering
A high pass filter, on the other hand, yields edge enhancement
or edge detection in the spatial domain, because edges contain
many high frequencies. Areas of rather constant gray level
consist of mainly low frequencies and are therefore
suppressed.

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Band Pass Filtering
A bandpass attenuates very low and very high frequencies, but
retains a middle range band of frequencies. Bandpass filtering
can be used to enhance edges (suppressing low frequencies)
while reducing the noise at the same time (attenuating high
frequencies).
Bandpass filters are a combination of both lowpass and
highpass filters. They attenuate all frequencies smaller than a
frequency Do and higher than a frequency D1 , while the
frequencies between the two cut-offs remain in the resulting
output image.

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Ideal Low Pass Filter
D(u,v)= distance between a point (u,v) in the frequency
domain and the center of the frequency rectangle

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Ideal Low Pass Filter

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Ideal Low Pass Filter (example)

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Why Ringing Effect

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Butterworth Low Pass Filter

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Butterworth Low Pass Filter

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Butterworth Low Pass Filter (example)

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Gaussian Low Pass Filters

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Gaussian Low Pass Filters

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Gaussian Low Pass Filters (example)

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Gaussian Low Pass Filters (example)

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Sharpening Fourier Domain Filters

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Sharpening Spatial Domain Representations

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Sharpening Fourier Domain Filters (Examples)

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Sharpening Fourier Domain Filters (Examples)

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Sharpening Fourier Domain Filters (Examples)

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Laplacian in Frequency Domain

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Unsharp Masking, High Boost Filtering

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Example of Modified High Pass Filtering

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Homomorphic Filtering

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Homomorphic Filtering

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Homomorphic Filtering

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Homomorphic Filtering

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Homomorphic Filtering (Example)

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Basic Filters
And scaling rest of values.

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Example (Notch Function)

63. Any question