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Fourier Transform and its Application in Image Processing. Md Shiplu Hawlader Roll: SH-224. Overview. Fourier Series Theorem Fourier Transform Discrete Fourier Transform Fast Fourier Transform. Fourier Series Theorem.

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## Fourier Transform and its Application in Image Processing

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**Fourier Transform and its Application in Image Processing**Md Shiplu Hawlader Roll: SH-224**Overview**• Fourier Series Theorem • Fourier Transform • Discrete Fourier Transform • Fast Fourier Transform**Fourier Series Theorem**Any periodic function can be expressed as a weighted sum (infinite) of sine and cosine functions of varying frequency**Fourier Transform**• Transforms a signal (i.e., function) from the spatialdomain to the frequency domain. where**Discrete Fourier Transform (DFT)**• Forward DFT • Inverse DFT**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**Magnitude and Phase of DFT (1/2)**magnitude phase**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!)**Why is FT Useful?**• Easier to remove undesirable frequencies. • Faster perform certain operations in the frequencydomain than in the spatialdomain.**Removing undesirable frequencies**frAequencies noisy signal To remove certain frequencies, set their corresponding F(u) coefficients to zero! remove high frequencies reconstructed signal**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**Frequency Filtering Steps**1. Take the FT of f(x): 2. Remove undesired frequencies: 3. Convert back to a signal:**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**The FFT Algorithm**The running time is O(n log n)**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.**References**• Image Processing, Analysis and Machine Vision, chapter 6.2.3. Chapman and Hall, 1993 • The Image Processing Handbook, chapter 4. CRC Press, 1992 • Fundamentals of Electronic Image Processing, chapter 8.4. IEEE Press, 1996 • http://en.wikipedia.org/wiki/Fourier_transform

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