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

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

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

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

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

Results Original ImageFourier Transformed Image

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 o e.g. linear, gaussian, etc.

Blurred Image

Linear Blur

Gaussian Blur

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

Deblurred Image ÷

Problems Division by very small Fourier values creates amplified noise

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

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

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

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

Matlab Implementation 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 No RegularizationRegularizationRichardson-Lucy

Conclusion 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

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", 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", bDeblur/OGO3.2b_2004_Deblur.pdf Y. Fan. ``Deblurring Images: Python, CGI, and Web", 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 pp Z. Zhao and R. Blahut. ``The Richardson-Lucy Algorithm Based Demodulation Algorithms for the Two-dimensional Intersymbol Interference Channel",