## Deconvolution

30 min.
Objectives:
At the end of this practical you should be able to:
- perform basic image arithmetic
- perform Fourier Deconvolution

## Terminology:

See:
Complex Images
Difference of Gaussians
Fourier Transform
Gaussian Smoothing
 Download the two images above as .pgm files. For this practical you should use any matlab routines developed in previous practicals. Use MATLAB to investigate the use of Gaussian smoothing at various scales (set by the width of the convolution kernal). In particular observe the effect at different scales on discontinuities. Construct a difference of Gaussian operator to enhance edges. Now investigate the consequences of the convolution theorem. Attempt to repeat a large scale DOG convolution in the Fourier domain. Can you see any appreciable difference between the resulting images? Attempt Fourier deconvolution of the resulting image (essentially a division of two complex images in the Fourier Domain). You may well have problems with numerical stability, due to division by small numbers. Try to correct this. Finally attept fourier deconvolution on the 4 way symmetric image. This has continuity at the image boundaries which helps eliminate the "Gibbs" osscillations generated at discontinuities by deconvolution. This trick is essentially equivalent to using the Discrete Cosine Transform rather than the FFT for generation of the "frequency domain" representation.

 (c) Imaging Science and Biomedical Engineering 2000 [paul.bromiley@man.ac.uk]