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

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

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)

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 Image Fourier 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 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 Filter With Inverse Filter Not a large difference, but still helpful at a low cost of efficiency.

Inverse Filter Without Inverse Filter With 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 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()

Further Work Test Matlab functions and collect results