1. Fourier Transform and its Application in Image Processing
Md Shiplu Hawlader
Roll: SH-224

2. Overview Fourier Series Theorem Fourier Transform
Discrete Fourier Transform
Fast Fourier Transform

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

4. Fourier Series

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

6. Discrete Fourier Transform (DFT)

7. Discrete Fourier Transform (DFT)
Forward DFT
Inverse DFT

8. 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

9. Magnitude and Phase of DFT (1/2)

10. 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!)

11. Why is FT Useful? Easier to remove undesirable frequencies.
Faster perform certain operations in the frequency domain than in the spatial domain.

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

13. 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

14. Example of noise reduction using FT

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

16. 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

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

18. 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.

19. 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

20. Thank You