1. Digital Image Processing
Image Restoration
Digital Image Processing

2. Content Introduction Image degradation/restoration model Noise models
Restoration by spatial filtering
Estimation of degradation functions
Inverse filtering
Wiener filtering
Geometric transformation

3. Introduction Objective of image restoration The distortion is due to
to recover a distorted image to the original form based on idealized models.
The distortion is due to
Image degradation in sensing environment e.g. random atmospheric turbulence
Noisy degradation from sensor noise.
Blurring degradation due to sensors
e.g. camera motion or out-of-focus
Geometric distortion
e.g. earth photos taken by a camera in a satellite

4. Introduction Enhancement Restoration
Concerning the extraction of image features
Difficult to quantify performance
Subjective; making an image “look better”
Restoration
Concerning the restoration of degradation
Performance can be quantified
Objective; recovering the original image

5. Image degradation / restoration model

6. Noise models Assuming degradation only due to additive noise (H = 1)
Noise from sensors
Electronic circuits
Light level
Sensor temperature
Noise from environment
Lightening
Atmospheric disturbance
Other strong electric/magnetic signals

7. Noise models Assuming that noise is
independent of spatial coordinates, and
uncorrelated with respect to the image content

8. Noise models

9. Noise models

10. Noise models Other common noise models Rayleigh noise Gamma noise
Exponential noise
Uniform noise

11. Noise Models
Rayleigh Noise
Gamma(Erlang) Noise
Exponential Noise

12. Noise models paper salt -3-levels -simple constant areas
(spans from black to white)
paper
salt

13. Additive Noise
Histograms

14. Additive Noise
Histograms

15. Periodic Noise Noise components
Are generated due to electrical or electromechanical interference during image acquisition
Periodic noise can be reduced in via frequency domain

16. Restoration by spatial filtering
Noise is unknown
Spatial filtering is appropriate when only additive noise is present

17. Restoration by spatial filtering

18. Restoration by spatial filtering

19. Restoration by spatial filtering

20. Restoration by spatial filtering
Q is the order of filter

21. Restoration by spatial filtering
Noise level is
Mean =0
Variance = 400

22. Restoration by spatial filtering
Mean filters (noise reduced by blurring)
Arithmetic mean filter and geometric mean filter are well suited for random noise such as Gaussian noise
Contraharmonic mean filter is well suited for impulse noise
Disadvantage: must know pepper noise or salt noise in advance

23. Restoration by spatial filtering

24. Restoration by spatial filtering
wrong

25. Restoration by spatial filtering
-- Repeated passes of median filter tend to blur the image.
-- Keep the number of passes as low as possible.

26. Restoration by spatial filtering
Fig. 8 next page

27. Restoration by spatial filtering
Pepper noise
Salt noise

28. Median and alpha-trimmed filter performed better
High level of noise  large filter
Median and alpha-trimmed filter performed better
Alpha-trimmed did better than median filter

29. Restoration by spatial filtering
Filters discussed so far
Do not consider image characteristics
Adaptive filters to be discussed
Behaviors based on statistical characteristics of the subimage under a filter window
Better performance
More complicated
Adaptive, local noise reduction filter
Adaptive median filter

30. Restoration by spatial filtering

31. Restoration by spatial filtering

32. Restoration by spatial filtering

33. Restoration by spatial filtering
Adaptive filtering

34. Restoration by spatial filtering

35. Restoration by spatial filtering
Is Z_med impulse?
Is Z_xy impulse?

36. Restoration by spatial filtering

37. Periodic Noise Reduction (Frequency Domain Filtering)
Band-Reject Filters
Ideal Band-reject Filter
-D(u,v) =distance from the origin of the centered freq. rectangle
-W =width of the band
-D0=Radial center of the band.

38. Periodic Noise Reduction (Frequency Domain Filtering)
Butterworth Band-Reject Filter of order n
Gaussian Band-Reject Filter

39. Periodic Noise Reduction (Frequency Domain Filtering)

40. Periodic Noise Reduction (Frequency Domain Filtering)
Band-Pass Filters
Opposite operation of a band-reject fiter

41. Periodic Noise Reduction (Frequency Domain Filtering)
Notch Filters
Rejects (or passes) frequencies in predefined neighborhoods about a center frequency
Ideal
Must appear in symmetric pairs about the origin.
Butterworth
Gaussian

42. Periodic Noise Reduction (Frequency Domain Filtering)
Notch Filters
Ideal
Center frequency components
Shift with respect to the center

43. Notch pass filter
Horizontal lines of the noise pattern I can be seen

44. Optimum Notch Filtering
Several pairs of components are present  more than just one sinusoidal component

45. Optimum Notch Filtering

46. Estimation of degradation functions

47. Estimation of degradation functions

48. Estimation of degradation functions

49. Estimation of degradation functions

50. Estimation of degradation functions

51. Estimation of degradation functions (model bases)
Mathematical model for uniform linear motion between the image and the sensor during image acquisition
Let x0(t) and y0(t) denote time varying components of motion in the x- and y-directions
Degradation model

52. Estimation of degradation functions (model bases)
Uniform Linear motion in the x and y direction x0(t)=at/T and y0(t)=bt/T where the image has been displaced by a total distance a in the x-direction and b in the y-direction

53. Estimation of degradation functions (model bases)

54. Inverse filtering

55. Inverse filtering

56. Inverse filtering Degradation function Curtain of noise
Cutting off values of the ratio outside a radius of 40, 70,85.

57. Wiener filtering

58. Wiener filtering

59. Wiener filtering
White noinse

60. Wiener filtering

61. Geometric transformations
Objective: to eliminate geometric distortion that occurs when an image is captured
Examples of geometric distortion
Pincushion distortion (associated with zoom lenses)

62. Geometric transformations

63. Geometric transformations
Two steps in geometric transformation
Spatial transformation: rearrangement of pixels on the image plane
Gray-level interpolation: assignment of gray levels to pixels in the spatially transformed image

64. Geometric transformations

65. Geometric transformations
• Solution
To formulate the spatial relocation of pixels by the use of the corresponding tiepoints
Tiepoints: a subset of pixels whose locations in the input (distorted) and output (restored) images are known.

66. Geometric transformations

67. Geometric transformations

68. Geometric transformations

69. Geometric transformations

70. Geometric transformations