1. © 2002-2003 by Yu Hen Hu 1 ECE533 Digital Image Processing Image Restoration

2. © 2002-2003 by Yu Hen Hu 2 ECE533 Digital Image Processing What is Image Restoration l The purpose of image restoration is to restore a degraded/distorted image to its original content and quality. l Distinctions to Image Enhancement »Image restoration assumes a degradation model that is known or can be estimated. »Original content and quality ≠ Good looking

3. © 2002-2003 by Yu Hen Hu 3 ECE533 Digital Image Processing Image Degradation Model l Spatial variant degradation model l Spatial-invariant degradation model »Frequency domain representation

4. © 2002-2003 by Yu Hen Hu 4 ECE533 Digital Image Processing Noise Models l Most types of noise are modeled as known probability density functions l Noise model is decided based on understanding of the physics of the sources of noise. »Gaussian: poor illumination »Rayleigh: range image »Gamma, exp: laser imaging »Impulse: faulty switch during imaging, »Uniform is least used. l Parameters can be estimated based on histogram on small flat area of an image

5. © 2002-2003 by Yu Hen Hu 5 ECE533 Digital Image Processing Noise Removal Restoration Method l Mean filters »Arithmetic mean filter »Geometric mean filter »Harmonic mean filter »Contra-harmonic mean filter l Order statistics filters »Median filter »Max and min filters »Mid-point filter »alpha-trimmed filters l Adaptive filters »Adaptive local noise reduction filter »Adaptive median filter

6. © 2002-2003 by Yu Hen Hu 6 ECE533 Digital Image Processing Mean Filters

7. © 2002-2003 by Yu Hen Hu 7 ECE533 Digital Image Processing Contra-Harmonic Filters

8. © 2002-2003 by Yu Hen Hu 8 ECE533 Digital Image Processing Median Filter Effective for removing salt-and- paper (impulsive) noise.

9. © 2002-2003 by Yu Hen Hu 9 ECE533 Digital Image Processing LSI Degradation Models l Motion Blur »Due to camera panning or fast motion l Atmospheric turbulence blur »Due to long exposure time through atmosphere »Hufnagel and Stanley l Uniform out-of-focus blur: l Uniform 2D Blur

10. © 2002-2003 by Yu Hen Hu 10 ECE533 Digital Image Processing Turbulence Blur Examples

11. © 2002-2003 by Yu Hen Hu 11 ECE533 Digital Image Processing Motion Blur l Often due to camera panning or fast object motion. l Linear along a specific direction. Blurdemo.m

12. © 2002-2003 by Yu Hen Hu 12 ECE533 Digital Image Processing Inverse Filter l Recall the degradation model: Given H(u,v), one may directly estimate the original image by At (u,v) where H(u,v)  0, the noise N(u,v) term will be amplified! Invfildemo.m

13. © 2002-2003 by Yu Hen Hu 13 ECE533 Digital Image Processing Wiener Filtering l Minimum mean-square error filter »Assume f and  are both 2D random sequences, uncorrelated to each other. »Goal: to minimize »Solution: Frequency selective scaling of inverse filter solution! »White noise, unknown S f (u,v):

14. © 2002-2003 by Yu Hen Hu 14 ECE533 Digital Image Processing Derivation of Wiener Filters l Given the degraded image g, the Wiener filter is an optimal filter h win such that E{|| f – h win g|| 2 } is minimized. l Assume that f and  are uncorrelated zero mean stationary 2D random sequences with known power spectrum S f and S n. Thus,

15. © 2002-2003 by Yu Hen Hu 15 ECE533 Digital Image Processing Constrained Least Square (CLS) Filter l For each pixel, assume the noise  has a Gaussian distribution. This leads to a likelihood function: l A constraint representing prior distribution of f will be imposed: the exponential form of pdf of f is known as the Gibbs’ distribution. l Since L(f)  p(g|f), use Bayes rule, since g is given, to maximize the posterior probability, one should minimize l q is an operator based on prior knowledge about f. For example, it may be the Laplacian operator!

16. © 2002-2003 by Yu Hen Hu 16 ECE533 Digital Image Processing Intuitive Interpretation of CLS l Prior knowledge: Most images are smooth  ||q**f|| should be minimized l However, the restored image, after going through the same degradation process h, should be close to the given degraded image g. l The difference between g and is bounded by the amount of the additive noise: l In practice, ||  || is unknown and needs to be estimated with the variance of the noise

17. © 2002-2003 by Yu Hen Hu 17 ECE533 Digital Image Processing Solution and Iterative Algorithm To minimize C CLS, Set  C CLS /  F = 0. This yields The value of  however, has to be determined iteratively! It should be chosen such that Iterative algorithm (Hunt) 1.Set initial value of , 2.Find, and compute R(u,v). 3.If ||R|| 2 - ||N|| 2 < - a, set  = B L, increase , else if ||R|| 2 - ||N|| 2 > a, set  = B u, decrease , else stop iteration. 4.  new = (B u +B L )/2, go to step 2.

18. © 2002-2003 by Yu Hen Hu 18 ECE533 Digital Image Processing CLS Demonstration