Spatial Filtering

1. Spatial Filtering

2. output image Spatial Filtering Methods (or Mask Processing Methods)

3. Spatial Filtering • The word “filtering” has been borrowed from the frequency domain. • Filters are classified as: • Low-pass (i.e., preserve low frequencies) • High-pass (i.e., preserve high frequencies) • Band-pass (i.e., preserve frequencies within a band) • Band-reject (i.e., reject frequencies within a band)

4. output image Spatial Filtering (cont’d) • Spatial filtering is defined by: (1) A neighborhood (2) An operation that is performed on the pixels inside the neighborhood

5. Spatial Filtering - Neighborhood • Typically, the neighborhood is rectangular and its size • is much smaller than that of f(x,y) • - e.g., 3x3 or 5x5

6. Spatial filtering - Operation Assume the origin of the mask is the center of the mask. for a 3 x 3 mask: for a K x K mask:

7. output image Spatial filtering - Operation • A filtered image is generated as the center of the mask moves to every pixel in the input image.

8. Handling Pixels Close to Boundaries pad with zeroes 0 0 0 ……………………….0 or 0 0 0 ……………………….0

9. Linear vs Non-LinearSpatial Filtering Methods • A filtering method is linear when the output is a weighted sum of the input pixels. • Methods that do not satisfy the above property are called non-linear. • e.g.,

10. Linear Spatial Filtering Methods • Two main linear spatial filtering methods: • Correlation • Convolution

11. w(i,j) Output Image f(i,j) Correlation g(i,j)

12. Correlation (cont’d) Often used in applications where we need to measure the similarity between images or parts of images (e.g., pattern matching).

13. Convolution • Similar to correlation except that the mask is first flipped both horizontally and vertically. Note: if w(x,y) is symmetric, that is w(x,y)=w(-x,-y), then convolution is equivalent to correlation!

14. Example Correlation: Convolution:

15. 2nd derivative of Gaussian 1st derivative of Gaussian Gaussian How do we choose the elements of a mask? • Typically, by sampling certain functions.

16. Gaussian Filters • Smoothing (i.e., low-pass filters) • Reduce noise and eliminate small details. • The elements of the mask must be positive. • Sum of mask elements is 1 (after normalization)

17. 2nd derivative of Gaussian 1st derivative of Gaussian Filters (cont’d) • Sharpening (i.e., high-pass filters) • Highlight fine detail or enhance detail that has been blurred. • The elements of the mask contain both positive and negative weights. • Sum of the mask weights is 0 (after normalization)

18. Smoothing Filters: Averaging(Low-pass filtering)

19. Smoothing Filters: Averaging (cont’d) • Mask size determines the degree of smoothing and loss of detail. original 3x3 5x5 7x7 15x15 25x25

20. Smoothing Filters: Averaging (cont’d) Example: extract, largest, brightest objects 15 x 15 averaging image thresholding

21. Smoothing filters: Gaussian • The weights are samples of the Gaussian function σ = 1.4 mask size is a function of σ :

22. Smoothing filters: Gaussian (cont’d) • σ controls the amount of smoothing • As σ increases, more samples must be obtained to represent • the Gaussian function accurately. σ = 3

23. Smoothing filters: Gaussian (cont’d)

24. Averaging vs Gaussian Smoothing Averaging Gaussian

25. Smoothing Filters: Median Filtering(non-linear) • Very effective for removing “salt and pepper” noise (i.e., random occurrences of black and white pixels). median filtering averaging

26. Smoothing Filters: Median Filtering (cont’d) • Replace each pixel by the median in a neighborhood around the pixel.

27. Sharpening Filters (High Pass filtering) • Useful for emphasizing transitions in image intensity (e.g., edges).

28. Sharpening Filters (cont’d) • Note that the response of high-pass filtering might be negative. • Values must be re-mapped to [0, 255] sharpened images original image

29. Sharpening Filters: Unsharp Masking • Obtain a sharp image by subtracting a lowpass filtered (i.e., smoothed) image from the original image: - =

30. Sharpening Filters: High Boost • Image sharpening emphasizes edges but details (i.e., low frequency components) might be lost. • High boost filter: amplify input image, then subtract a lowpass image. (A-1) + =

31. Sharpening Filters: High Boost (cont’d) • If A=1, we get a high pass filter • If A>1, part of the original image is added back to the high pass filtered image.

32. Sharpening Filters: High Boost (cont’d) A=1.9 A=1.4

33. Sharpening Filters: Derivatives • Taking the derivative of an image results in sharpening the image. • The derivative of an image can be computed using the gradient.

34. Sharpening Filters: Derivatives (cont’d) • The gradient is a vector which has magnitude and direction: or (approximation)

35. Sharpening Filters: Derivatives (cont’d) • Magnitude: provides information about edge strength. • Direction: perpendicular to the direction of the edge.

36. sensitive to vertical edges Δx sensitive to horizontal edges Sharpening Filters: Gradient Computation • Approximate gradient using finite differences:

37. Sharpening Filters: Gradient Computation (cont’d)

38. Example

39. Sharpening Filters: Gradient Computation (cont’d) • We can implement and using masks: (x+1/2,y) good approximation at (x+1/2,y) (x,y+1/2) * * good approximation at (x,y+1/2) • Example: approximate gradient at z5

40. Sharpening Filters: Gradient Computation (cont’d) • A different approximation of the gradient: good approximation (x+1/2,y+1/2) * • We can implement and using the following masks:

41. Sharpening Filters: Gradient Computation (cont’d) • Example: approximate gradient at z5 (ROBERT CROSS) • Other approximations Sobel

42. Example

43. Sharpening Filters: Laplacian The Laplacian (2nd derivative) is defined as: (dot product) Approximate derivatives:

44. Sharpening Filters: Laplacian (cont’d) Laplacian Mask detect zero-crossings

45. Sharpening Filters: Laplacian (cont’d) Sobel Laplacian