1. Applications of Fourier Analysis in Image Recovery
Kang Guo
TJHSST Computer Systems Lab

2. Image Blur Common cause of image quality degradation in photography
Removing blur from images is a practical application

3. Fourier Transform Spatial domain to Frequency domain N, N^2 transforms
Imaginary numbers, displayed with magnitude
Given a+bi, √(a^2+b^2)
e^(iӨ) = cos(Ө)+isin(Ө)

4. Fast Fourier Transform
Speed improvement
2N, N transforms
2 components

5. Logarithmic Transform
Fit all values within the range of 8-bit color values
Image without the logarithmic transform

6. Results
Original Image
Fourier Transformed Image

7. Applying a Blur
Fourier Transform of Blur Filter must be multiplied with Fourier Transform of the Original Image
A Point Spread Function (PSF) is simply the Fourier Transform of the blur filter
Different kinds of blur can be modeled with a PSF
e.g. linear, gaussian, etc.

8. Blurred Image

9. Linear Blur

10. Gaussian Blur

11. Deblur the Image
If the PSF is known, then the reverse process can be done to remove the blur
That is, instead of multiplying, dividing will result in the inverse process

12. Deblurred Image ÷

13. Problems
Division by very small Fourier values creates amplified noise

14. Inverse Filter Goal to reduce/remove noise
All values that are determined to be too small are set to a common value, gamma

15. Inverse Filter
Without Inverse Filter
With Inverse Filter
Not a large difference, but still helpful at a low cost of efficiency.

16. Inverse Filter
Without Inverse Filter
With Inverse Filter
However, in the case of a Gaussian blur, the Inverse Filter is not as helpful.

18. Regularization
Introducing additional information into a problem to allow a proper solution
The inverse process (deconvolution), division of blurred image by PSF produces loss of data
Regularization → some sort of restriction, forcing final solution to fall in a set boundary of answers

19. Matlab Image Processing Toolbox
Due to the difficulty of these image processing algorithms and the limited scope of this project, I will be using Matlab to implement regularization, filters, and blind deconvolution
deconvblind(), deconvlucy(), deconvreg(), deconvwnr()

20. Further Work
Test Matlab functions and collect results