1. Image Restoration
Chapter 5

2. Image Restoration
As in image enhancement, the principal goal of restoration techniques is to improve an image in some predefined sense.
Restoration attempts to recover an image that has been degraded by using a priori knowledge of the degradation phenomenon.
Thus, restoration techniques are oriented toward modeling the degradation and applying the inverse process in order to recover the original image.
By contrast, enhancement techniques basically are procedures designed to manipulate an image in order to take advantage of aspects of the human visual system. For example, contrast stretching is considered an enhancement technique because it is based primarily on the pleasing aspects it might present to the viewer. Whereas removal of image blur by applying deblurring function is considered a restoration technique.

3. Image Restoration
We consider in this chapter the restoration problem only from the point where a degraded digital image is given; thus we consider topics dealing with sensor, digitizer, and display degradations only superficially.
Some restoration techniques are best suited in the spatial domain while others are better formulated in the frequency domain.
For example: spatial processing is applicable when the only degradation is additive noise while degradations such as image blur are difficult to approach in the spatial domain using small filter masks. In this case frequency domain is best suited.

4. Image Restoration A Model of the Image Degradation/Restoration Process
Noise Models
Some Important Noise Probability Density Functions
Gaussian noise
Rayleigh noise
Erlang(gamma) noise
Exponential noise
Uniform noise
Impulse(salt-and-pepper) noise

5. Image Restoration
Restoration in the presence of Noise Only – spatial Filtering
Mean Filters
Arithmetic mean filter
Harmonic mean filter
Contraharmonic mean filter
Order-Statistic Filters
Median filter
Max and Min filters
Mid point filters

6. Image Restoration
Periodic Noise Reduction by Frequency Domain Filtering.
Bandreject filters
Bandpass filters
Notch Filters
Optimum Notch Filtering
Estimating the degradation function
Estimation by Image Observation
Estimation by Experimentation
Estimation by Modeling

7. A Model of the Image Degradation/Restoration Process
The degradation process is modeled as a degradation function that, together with an additive noise term, operates on an input image f(x,y) to produce a degraded image g(x,y). Given g(x,y), some knowledge about the degradation function H, and some knowledge about the additive noise term, the objective of restoration is to obtain an estimate of the original image.

8. Noise Models
The principal sources of noise in digital images arise during image acquisition and/or transmission.
The performance of imaging sensors is affected by a variety of factors, such as:
Environmental conditions during image acquisition
Quality of the sensing elements themselves.

9. Noise Models
Some Important Noise Probability Density Functions

10. Noise Models

11. Noise Models

12. Restoration in the presence of Noise Only – spatial Filtering
When the only degradation present in an image is noise.
Spatial filtering is the method of choice in situations when only additive random noise is present. Spatial filtering was discussed in details in Ch3.
Mean Filters:
Arithmetic mean filter
This is the simplest of the mean filters, the arithmetic mean filter computes the average value of the corrupted image g(x,y). This operation can be implemented using a spatial filter of size m x n in which all coefficients have value 1/mn. A mean filter smoothes local variations in an image, and noise is reduced as a result of blurring.
Geometric mean filter:
Here each restored pixel is given by the product of the pixels in the subimage window, raised to the power 1/mn. A geometric mean filter achieves smoothing comparable to the arithmetic mean filter, but it tends to lose less image detail in the process.

13. Mean Filters: Harmonic mean filter
The harmonic mean filter works well for salt noise, but fails for pepper noise. It does well also with other types of noise like Gaussian moise.
Contraharmonic mean filter:
It has order of the filter Q.This filter is well suited for reducing or eleminating the effects of salt-and-pepper noise. For positive values of Q , the filter eleminates pepper noise. For negative valuse of Q it eleminates salt noise. It cannot do both simultaneously. Note that contraharmonic filter reduces to the arithmetic filter mean filter if Q=0, and to the harmonic mean filter if Q=-1.

14. Mean Filters:

15. Mean Filters:

16. Order-Statistic Filters
Order-statistic filters are spatial filters whose response is based on ordering(ranking) the values of the pixels contained in the image area encompassed by the filter.
Median filter
The best known order-statistic filter is the median filter, which replaces the value of a pixel by the median of the intensity levels in the neighborhood of the pixel. They provide excellent noise-reduction capabilities, with less blurring than linear smoothing filters if similar size.
Max and min filters
The median filter represents the 50th percentile of a ranked set of numbers. Using the 100th percentile results in so called max filter.
This filter is useful for finding the brightest points in an image. Also, because pepper noise had very low values, it is reduced by this filter as a result of the max selection process in the subimage area.
The 0th percentile filter is the min filter. This filter is useful for finding the darkest points in an image. Also, it reduces salt noise as a result of the min operation.

17. Order-Statistic Filters
Midpoint filter
the midpoint filter simply computes the midpoint between the maximum and minimum values in the area encompassed by the filter:
(x,y) = ½(max + min)
note that this filter combines order statistics and averaging. It works best for randomly distributed noise, like Gaussian or uniform noise.

18. Order-Statistic Filters

19. Order-Statistic Filters

20. Order-Statistic Filters

21. Periodic Noise Reduction by Frequency Domain Filtering.
Periodic Noise can be analyzed and filtered effectively using frequency domain techniques.
The approach is to use a selective filter to isolate the noise.
The three types of selective filters (bandreject, bandpass, and notch) are used for basic periodic noise reduction. Also an optimum notch filtering approach is obtained.
Bandreject filters
Bandpass filters
Notch Filters
Optimum Notch Filtering

22. Estimating the degradation function
There are three principal ways to estimate the degradation function for use in image restoration: (1) observation, (2) experimentation, and (3) mathematical modeling. The process of restoring an image by using a degradation function that had been estimated in some way sometimes is called blind deconvolution due to the fact that the true degradation function is seldom known completely.
Estimation by Image Observation
Estimation by Experimentation
Estimation by Modeling

23. Estimation by Image Observation
Suppose that we are given a degraded image without any knowledge about the degradation function H. one way to estimate H is to gather information from the image itself. For example, if the image is blurred, we can look at a small rectangular section of the image containing sample structures. Like part of an object and the background. In order to reduce the effect of noise, we would look for an area in which the signal content is strong (e.g., an area of high contrast). The next step would be to process the subimage to arrive at a result that is as unblurred as possible, for example, we can do this by sharpening the subimage with a sharpening filter.

24. Estimation by Experimentation
If equipment similar to the equipment used to acquire the degraded image is available, it is possible to obtain an accurate estimate of the degradation.
Images similar to the degraded image can be acquired with various system settings until they are degraded as closely as possible to the image we wish to restore.

25. Estimation by Modeling
Degradation modeling has been used for many years.
In some cases, the model can take into account environmental conditions that cause degradations.
Another approach in modeling is to derive a mathematical model starting from basic principles.