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Lec 07 image enhancement in frequency domain i

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    Lec 07 image enhancement in frequency domain i Lec 07 image enhancement in frequency domain i Presentation 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 "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
    • 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