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Applications of Fourier Analysis in Image Recovery

Applications of Fourier Analysis in Image Recovery. Kang Guo TJHSST Computer Systems Lab 2009-2010. Common cause of image quality degradation in photography Removing blur from images is a practical application. Image Blur. http://web.media.mit.edu/~raskar/deblur/Whitecar.htm.

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Applications of Fourier Analysis in Image Recovery

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  1. Applications of Fourier Analysis in Image Recovery Kang Guo TJHSST Computer Systems Lab 2009-2010

  2. Common cause of image quality degradation in photography Removing blur from images is a practical application Image Blur http://web.media.mit.edu/~raskar/deblur/Whitecar.htm

  3. 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) Fourier Transform

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

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

  6. Results Original Image Fourier Transformed Image

  7. 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. Applying a Blur

  8. Blurred Image

  9. Linear Blur

  10. Gaussian Blur

  11. 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 Deblur the Image

  12. Deblurred Image ÷

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

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

  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.

  17. 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 Regularization

  18. 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() Matlab Implementation

  19. Matlab Implementation No Regularization Richardson-Lucy Regularization

  20. Non-blind deconvolution is relatively simple, but still not perfect Blind deconvolution is much more difficult However, if the blind deconvolution problem is ever solved, it will be a huge step forward for image processing and the general community of photographers Conclusion

  21. References • Q. Shan, J. Jia, and A. Agarwala. ``High-quality Motion Deblurring from a Single Image" ACM Trans. Graph. 27, 3, Article 73 (August 2008). • L. Yuan, J. Sun, L. Quan, and H. Shum. ``Image Deblurring With Blurry/Noisy Image Pairs" ACM Trans. Graph. 26, 3, Article 1 (July 2007). • R. Fisher, S. Perkins, A. Walker, and E. Wolfart. ``Image Transforms - Fourier Transform", 2003. http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm • T. Chan and J. Shen. ``Theory and Computation of Variational Image Deblurring" WSPC/Lecture Notes Series (November 2005). • S. Allon, M. Debertrand, and B. Sleutjes. ``Fast Deblurring Algorithms", 2004. http://www.bmi2.bmttue.nl/image-analysis/Education/OGO/0504-3.2bDeblur/OGO3.2b_2004_Deblur.pdf • Y. Fan. ``Deblurring Images: Python, CGI, and Web", 2006. www.mathcs.emory.edu/~yfan/SSProject.pdf • G. Kempen and L. Vliet. (2000). ``The influence of the regularization parameter and the first estimate on the performance of Tikhonov regularized non-linear image restoration algorithms", Journal of Microscopy. 198. pp. 63-75. • Z. Zhao and R. Blahut. ``The Richardson-Lucy Algorithm Based Demodulation Algorithms for the Two-dimensional Intersymbol Interference Channel", 2005. www.ifp.illinois.edu/~zzhao/publications/slides_wustl23Mar05.pdf

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