1. Image Analysis
Image Restoration

2. Image Restoration
Image enhancement tries to improve subjective image quality.
Image restoration tries to recover the original image.
Bahadir K. Gunturk
EE4780

3. Noise Models
Noise may arise during image due to sensors, digitization, transmission, etc.
Most of the time, it is assumed that noise is independent of spatial coordinates, and that there is no correlation between noise component and pixel value.
Noise may be considered as a random variable, its statistical behavior is characterized by a probability density function (PDF).
Gaussian noise
Bahadir K. Gunturk
EE4780

4. Noise Models Uniform noise Impulse (salt-and-pepper) noise
Bahadir K. Gunturk
EE4780

5. Noise Models Original Noisy images and their histograms
Bahadir K. Gunturk
EE4780

6. Noise Models How to estimate noise parameters?
If imaging device is available
Take a picture of a flat surface.
See the shape of the histogram; decide on the noise model.
Estimate the parameters. (e.g., find mean and standard deviation.)
When only images already generated are available
Get a small patch of image with constant gray level
Inspect histogram
Estimate the parameters
Bahadir K. Gunturk
EE4780

7. Restoration When There is Only Noise
Low-Pass Filters:
Smoothes local variations in an image.
Noise is reduced as a result of blurring.
For example, Arithmetic Mean Filter is
 Convolve with a uniform filter of size m-by-n.
Bahadir K. Gunturk
EE4780

8. Restoration When There is Only Noise
Adaptive, local noise reduction filter
Let be the noise variance at (x,y).
Let be the local variance of pixels around (x,y).
Let be the local mean of pixels around (x,y).
We want a filter such that
If noise variance is zero, it should return g(x,y).
If local variance is high relative to noise variance, the filter should return a value close to g(x,y). (Therefore, edges are preserved!)
If two variances are equal, the filter should return the average of the pixels within the neighborhood.
Bahadir K. Gunturk
EE4780

9. Restoration When There is Only Noise
Bahadir K. Gunturk
EE4780

10. Restoration When There is Only Noise
Median Filter
Replaces the value of a pixel by the median of intensities in the neighborhood of that pixel.
Is very effective against the salt-and-pepper noise.
Bahadir K. Gunturk
EE4780

11. Restoration When There is Only Noise
Adaptive Median Filter: The basic idea is to avoid extreme values
Let
z_min: minimum gray level value in a neigborhood of a pixel at (x,y).
z_max: maximum gray level value…
z_med: median…
z(x,y): gray level at (x,y).
Is z_med=z_min or z_med=z_max? (That is, is z_med an extreme value?)
No:
Is z(x,y) an extreme value? (Is z(x,y)=z_min or z(x,y)=z_max?)
No: Output is z(x,y)
Yes: Output is z_med.
Yes: Increase window size (to find a non-extreme z_med) and go to the first step. (When a maximum allowed window size is reached, stop and output z(x,y).)
Bahadir K. Gunturk
EE4780

12. Restoration When There is Only Noise
Bahadir K. Gunturk
EE4780

13. Restoration When There is Only Noise
Removing Periodic Noise with Band-Reject Filters
Spikes are due to noise
Periodic Noise
Periodic noise arises typically from electrical or electromechanical interference during image acquisition. This is the only type of spatially dependent noise (considered in the book).
Inspect the Fourier characteristics (look for spikes) to estimate the parameters of periodic noise.
Band-reject filter
Bahadir K. Gunturk
EE4780

14. Restoration When There is Only Noise
Finding Periodic Noise from the Spectrum and Using Notch Filters
Filter out these spikes
Periodic noise arises typically from electrical or electromechanical interference during image acquisition. This is the only type of spatially dependent noise (considered in the book).
Inspect the Fourier characteristics (look for spikes) to estimate the parameters of periodic noise.
Noise due to interference
Bahadir K. Gunturk
EE4780

15. Image Restoration Spatial domain: Frequency domain: Bahadir K. Gunturk
EE4780

16. Image Restoration Inverse Filtering This could dominate signal.
Bahadir K. Gunturk
EE4780

17. Image Restoration
Bahadir K. Gunturk
EE4780

18. Image Restoration
Cut off the inverse filter for large frequencies. (Signal-to-noise ratio is typically low for large frequencies.)
Bahadir K. Gunturk
EE4780

19. Image Restoration Minimum Mean Square Error (Wiener) Filtering:
Find such that the expected value of error is minimized:
Solution is
Investigate this equation for different signal-noise ratios.
Bahadir K. Gunturk
EE4780

20. Original image
Bahadir K. Gunturk
EE4780

21. Image Restoration
Bahadir K. Gunturk
EE4780

22. Least Squares Filtering
Find F(u,v) that minimizes the following cost function:
The solution is
(Unconstrained solution)
(See the derivations in the classroom)
Bahadir K. Gunturk
EE4780

23. Least Squares Filtering
Find F(u,v) that minimizes the following cost function:
Choose a P(u,v) to have a smooth solution. (A high-pass filter would do the trick.)
Bahadir K. Gunturk
EE4780

24. Least Squares Filtering
The frequency domain solution to this optimization problem is
(Constrained solution)
where P(u,v) is the Fourier Transform of p(x,y), which is typically chosen as a high-pass filter.
Example:
Bahadir K. Gunturk
EE4780

25. Least Squares Filtering
Bahadir K. Gunturk
EE4780