Digital Image Processing Chapter # 4 Image Enhancement in Frequency Domain Digital Image Processing Chapter # 4 Image Enhancement in Frequency Domain.

Digital Image ProcessingDigital Image Processing

Chapter # 4 Chapter # 4

Image Enhancement in Frequency DomainImage Enhancement in Frequency Domain

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Introduction

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

Background (Fourier Transform)

Even functions that are not periodic 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

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.

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

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

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

Discrete Fourier Transforms (DFT)

1-D DFT for M samples is given as

The Inverse Fourier transform in 1-D is given as

Discrete Fourier Transforms (DFT)

1-D DFT for M samples is given as

The inverse Fourier transform in 1-D is given as

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

2-D DFT

Fourier Transform

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

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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DFT Examples

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DFT Examples

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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 highpass 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

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Ideal Low Pass Filter

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Ideal Low Pass Filter (example)

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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 and High 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)