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Md Shiplu Hawlader Roll: SH-224

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Fourier Series Theorem Fourier Transform Discrete Fourier Transform Fast Fourier Transform

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Any periodic function can be expressed as a weighted sum (infinite) of sine and cosine functions of varying frequency

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Transforms a signal (i.e., function) from the spatial domain to the frequency domain. where

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Forward DFT Inverse DFT

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Typically, we visualize |F(u,v)| The dynamic range of |F(u,v)| is typically very large Apply stretching: (c is const) original image before scalingafter scaling

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magnitude phase

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Reconstructed image using magnitude only (i.e., magnitude determines the contribution of each component!) Reconstructed image using phase only (i.e., phase determines which components are present!)

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Easier to remove undesirable frequencies. Faster perform certain operations in the frequency domain than in the spatial domain.

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noisy signal frAequencies remove high frequencies reconstructed signal To remove certain frequencies, set their corresponding F(u) coefficients to zero!

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Low frequencies correspond to slowly varying information (e.g., continuous surface). High frequencies correspond to quickly varying information (e.g., edges) Original Image Low-passed

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1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal:

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The FFT is an efficient algorithm for computing the DFT The FFT is based on the divide-and-conquer paradigm: If n is even, we can divide a polynomial into two polynomials and we can write

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The running time is O(n log n)

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Fourier Transform has multitude of applications in all the field of engineering but has a tremendous contribution in image processing fields like image enhancement and restoration.

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Image Processing, Analysis and Machine Vision, chapter Chapman and Hall, 1993 The Image Processing Handbook, chapter 4. CRC Press, 1992 Fundamentals of Electronic Image Processing, chapter 8.4. IEEE Press, 1996 rm rm

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