1. Course 2 Image Filtering

2. Image filtering is often required prior any other vision processes to remove image noise, overcome image corruption and change distribution in an image. ——linear filter ——nonlinear filter

3. 1. Linear system delta function impulse response Continuous case: Discrete case: Linear Space Invariant System

4. LSI h(x, y) input f (x, y) output g(x, y) —— convolution ( 卷积 )

5. For discrete case: In Fourier domain: then:

6. 2. Mean Filter —— average intensity values of the neighbor pixels at a pixel. e.g. window neighbors. where 111 111 111

7. The larger window size, the efficient in reducing noise, but the more blurring original image (lose image details) —— you must make a trade-off ! To reduce windows boundary effect and for a smooth filtering, filter elements are often weighted.

8. Mean Filter on Gaussian noise corrupted image

9. 3. Gaussian Filter: impulse response of Gaussian filter: —— linear system —— rotational symmetric —— Fourier Transformation is still a Gaussian

10. For 1-D case: It is still a Gaussian!

11. ——Separatibility It can easily be seen hat the Gaussian core can be separated in both space domain and Fourier domain, i.e. So,

12. In Fourier Domain, ——Cascading operation: Let If a Gaussian filter has a large parameter, it requires a large operation mask and thus, it will take long time to process.

13. We can separate the large Gaussian of into two smaller Gaussian filters of parameter and perform the filtering processes in cascade. 4. Discrete Guassian Filter Where window size

14.  To form a Gaussian Filter: (1) Choose a proper size N of mask window. (2) Calculate the value of each mask element. e.g. (3) Scale all mask elements to integer with the same scaling factor.

15. Scale (4) find the normalizing factor k Therefore, for an input image output of Gaussian smooth is:

16. By the way, A two-dimenssional Gaussian filter can be extended by two one – dimentional Gaussian (how? Homework).

17. 5. Median Filter (nonlinear) ------ Very efficient to remove salt-and – pepper noise. (1)In a window of of filter mask, order the intensity value of points. set (2) Set the image pixel with intensity … ………… ………… ………

18. 759936 384910 199822 For example,

19. Other nonlinear filters: —— minimum filter(remove salt noise): —— maximum filter (remove pepper noise) —— Midpoint filter ( remove Gaussian noise and uniform noise)

20. Median filter on salt-and-pepper noise corrupted image

21. Minimum filter on salt noise corrupted image

22. Morphological Filters (1)Review of “ set ” : Intersection: Union: Complement: Translation: where are position, and is a vector operation.  A B

23. (2) Dilation: given an image and structuring element dilatation of A by B is defined as Dilatation operation tends to inflate the original image.

24. (3) Erosion: Erosion of A by B is defined by all pixel that makes inside the image A, i.e. or, equivalently Erosion operation intends to shrink original image.

25. (4)Opening: Open operation to an image is erosion followed by dilation with the same structuring element. It will remove the image regions that a too small to contain the structuring element, leaving the resulted image approximately unchanged. (5)Closing: Closing operation to an image is dilation followed by erosion. It will fill in holes and concavities smaller than structuring element, leaving the resulted image approximately unchanged.

26. 6.Histogram Modification (Image Enhancement) Uniform intensity distribution of an image can fit human interpretation of the image, it also meets the requirements of some image operations, such as image thresholding. original intensity distribution uniform intensity distribution Let original image has intensity and with probability. The transformed image has intensity with probability. Then, the transformation

27. will makes resultant image have uniform intensity distribution, i.e. Prove: For discrete case, let the original image has intensity is the number of

28. pixels at level in its histogram, we want to map the image to a new one with the same of number of intensity level but different histogram, where the histogram is pre- designed (e.g., uniform): 1)find a intensity level K, which that: then, map pixels of levels to level in the new image.

29. 2) find intensity : then, map pixels of level to level in new image. 3) repeat until all intensity levels of original image have been included.

30. Original image Enhanced image