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# Lec 07 image enhancement in frequency domain i

## on Oct 24, 2010

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## Lec 07 image enhancement in frequency domain iPresentation Transcript

• Digital Image Processing Lecture 7: Image Enhancement in Frequency Domain-I Naveed Ejaz
• 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
• What is the difference between fourier series and fourier transform? The Fourier series is an expression of a pattern (such as an electrical waveform or signal) in terms of a group of sine or cosine waves of different frequencies and amplitude. This is the frequency  domain . The Fourier transform is the process or function used to convert from time domain (example: voltage samples over time, as you see on an oscilloscope) to the frequency domain, which you see on a graphic equalizer or spectrum analyzer)
•
• Background (Fourier Transform)
• Even functions that are not periodic (but whose area under the curve is 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
• 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 &quot;transforms&quot; 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
• 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