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**Fourier Transform and its Application in Image Processing**

Md Shiplu Hawlader Roll: SH-224

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**Overview Fourier Series Theorem Fourier Transform**

Discrete Fourier Transform Fast Fourier Transform

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**Fourier Series Theorem**

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

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Fourier Series

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

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**Discrete Fourier Transform (DFT)**

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**Discrete Fourier Transform (DFT)**

Forward DFT Inverse DFT

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**Visualizing DFT 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 scaling after scaling

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**Magnitude and Phase of DFT (1/2)**

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**Magnitude and Phase of DFT (2/2)**

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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**Why is FT Useful? Easier to remove undesirable frequencies.**

Faster perform certain operations in the frequency domain than in the spatial domain.

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**Removing undesirable frequencies**

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

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**How do frequencies show up in an image?**

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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**Example of noise reduction using FT**

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**Frequency Filtering Steps**

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

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**Fast Fourier Transform (FFT)**

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 FFT Algorithm The running time is O(n log n)

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Conclusion 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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References 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

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Thank You

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