Image Restoration

1. Image Restoration

2. What is Image Restoration • The purpose of image restoration is to restore a degraded/distorted image to its original content and quality. • Distinctions to Image Enhancement • Image restoration assumes a degradation model that is known or can be estimated. • Original content and quality ≠ Good looking

3. Image Degradation Model • Spatial variant degradation model • Spatial-invariant degradation model • Frequency domain representation

4. Most types of noise are modeled as known probability density functions 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. Parameters can be estimated based on histogram on small flat area of an image Noise Models

5. Mean filters Arithmetic mean filter Geometric mean filter Harmonic mean filter Contra-harmonic mean filter Order statistics filters Median filter Max and min filters Mid-point filter alpha-trimmed filters Adaptive filters Adaptive local noise reduction filter Adaptive median filter Noise Removal Restoration Method

6. Mean Filters

7. Contra-Harmonic Filters

8. Median Filter Effective for removing salt-and-paper (impulsive) noise.

9. Motion Blur Due to camera panning or fast motion Atmospheric turbulence blur Due to long exposure time through atmosphere Hufnagel and Stanley Uniform out-of-focus blur: Uniform 2D Blur LSI Degradation Models

10. Turbulence Blur Examples

11. Often due to camera panning or fast object motion. Linear along a specific direction. Motion Blur Blurdemo.m

12. 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! Inverse Filter Invfildemo.m

13. 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 Sf(u,v): Wiener Filtering

14. Given the degraded image g, the Wiener filter is an optimal filter hwinsuch that E{|| f – hwing||2} is minimized. Assume that f and  are uncorrelated zero mean stationary 2D random sequences with known power spectrum Sf and Sn. Thus, Derivation of Wiener Filters

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

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

17. To minimize CCLS, Set CCLS/ 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  = BL, increase , else if ||R||2 - ||N||2 > a, set  = Bu, decrease  , else stop iteration. 4. new = (Bu+BL)/2, go to step 2. Solution and Iterative Algorithm

18. CLS Demonstration