1. September 19, 2013Computer Vision Lecture 6: Image Filtering 1 Image Filtering Many basic image processing techniques are based on convolution. In a convolution, a convolution filter W is applied to every pixel of an image I to create a filtered image I*. The filter W itself is a 2D matrix of real values. To simplify the mathematics, we could consider W to have a center [0, 0] and extend from –m to m vertically and –n to n horizontally. This means that W is of size (2m + 1)×(2n + 1).

2. September 19, 2013Computer Vision Lecture 6: Image Filtering 2 j iConvolution Example: Averaging filter: 1/91/91/9 1/91/91/9 1/91/91/9

3. September 19, 2013Computer Vision Lecture 6: Image Filtering 3Convolution Grayscale Image: 16329 2113100 510697 31028 4429101/91/91/91/91/91/9 1/91/91/9 Averaging Filter:

4. September 19, 2013Computer Vision Lecture 6: Image Filtering 4Convolution Original Image: 16329 2113100 510697 31028 442910 Filtered Image:0000000 00 00 00000 1/91/91/91/91/91/9 1/91/91/9 value = 1  1/9 + 6  1/9 + 3  1/9 + 2  1/9 + 11  1/9 + 3  1/9 + 5  1/9 + 10  1/9 + 6  1/9 = 47/9 = 5.222 5

5. September 19, 2013Computer Vision Lecture 6: Image Filtering 5Convolution Original Image: 16329 2113100 510697 31028 442910 Filtered Image:0000000 00 00 00000 1/91/91/91/91/91/9 1/91/91/9 value = 6  1/9 + 3  1/9 + 2  1/9 + 11  1/9 + 3  1/9 + 10  1/9 + 10  1/9 + 6  1/9 + 9  1/9 = 60/9 = 6.667 57

6. September 19, 2013Computer Vision Lecture 6: Image Filtering 6Convolution Original Image: 16329 2113100 510697 31028 442910 Filtered Image:000000 0 0 0 0 0 00000 57 5 5 5 56 64 Now you can see the averaging (smoothing) effect of the 3  3 filter that we applied.

7. September 19, 2013Computer Vision Lecture 6: Image Filtering 7Convolution It needs to be noted that for convolution the filter needs to be rotated by 180  before starting the computations (otherwise it’s a correlation). An intuitive explanation is that we would like convolution to be like a “local multiplication of patterns.” For example, if our image contains a few 1s and otherwise 0s, we would expect the convolution result to contain a copy of the filter pattern centered at each 1. Let us look at an image with one 1-pixel:

8. September 19, 2013Computer Vision Lecture 6: Image Filtering 8Convolution Image: 00000 00000 00100 00000 00000 123456 789 Filter: Result:0000009870 06540 03210 00000 Oops! The copy of the filter is rotated by 180  !

9. September 19, 2013Computer Vision Lecture 6: Image Filtering 9Convolution If we rotate the filter by 180  beforehand, we get the desired result. This leads to the following definition of convolution for image I, filter W, and result I * : This formula needs to be applied to all coordinates [i, j] in I in order to create the complete image I *. Convolution is commutative, i.e., W and I are exchangeable.

10. September 19, 2013Computer Vision Lecture 6: Image Filtering 10 Image Filtering More common: Gaussian Filters 14741 41626164 72641267 4162616414741 Discrete version: 1/273 implement decreasing influence by more distant pixels Continuous version:

11. September 19, 2013Computer Vision Lecture 6: Image Filtering 11 Image Filtering original 33333333 99999999 15  15 Effect of Gaussian smoothing:

12. September 19, 2013Computer Vision Lecture 6: Image Filtering 12 Properties of Gaussian Filters The application of Gaussian convolution filters can be made more efficient. This is important, for example, if we want to apply different Gaussian filters to a large number of big input images. The basic idea is to separate the convolution with the 2D Gaussian filter into two successive convolutions with 1D Gaussian filters. One of these filters is vertical, the other one horizontal.

13. September 19, 2013Computer Vision Lecture 6: Image Filtering 13 Properties of Gaussian Filters The general form of the Gaussian filter, without a normalizing factor, is given by: The convolution of an image F with a Gaussian filter G of size (2m + 1)  (2n + 1) is given by: This formula needs to be applied to all coordinates [i, j] in F in order to create the convoluted image.

14. September 19, 2013Computer Vision Lecture 6: Image Filtering 14 Properties of Gaussian Filters Then we have:

15. September 19, 2013Computer Vision Lecture 6: Image Filtering 15 Properties of Gaussian Filters The formula in the curly braces describes the convolution of F[i, j] with a horizontal one- dimensional Gaussian filter. The remainder of the formula takes the result of this first convolution and performs a convolution with a vertical one-dimensional Gaussian filter on it. So instead of applying an (2m + 1)  (2n + 1) Gaussian convolution filter, we can successively apply a 1  (2n + 1) filter and an (2m + 1)  1 filter. This increases the efficiency of the computation.

16. September 19, 2013Computer Vision Lecture 6: Image Filtering 16 Different Types of Filters Smoothing can reduce noise in the image.Smoothing can reduce noise in the image. This can be useful, for example, if you want to find regions of similar color or texture in an image.This can be useful, for example, if you want to find regions of similar color or texture in an image. However, there are different types of noise.However, there are different types of noise. For so-called “salt-and-pepper” noise, for example, a median filter can be more effective.For so-called “salt-and-pepper” noise, for example, a median filter can be more effective. Note that it is not a convolution filter, but it works similarly.Note that it is not a convolution filter, but it works similarly.

17. September 19, 2013Computer Vision Lecture 6: Image Filtering 17 Median Filter Use, for example, a 3  3 filter and move it across the image like we did before.Use, for example, a 3  3 filter and move it across the image like we did before. For each position, compute the median of the brightness values of the nine pixels in question.For each position, compute the median of the brightness values of the nine pixels in question. –To compute the median, sort the nine values in ascending order. –The value in the center of the list (here, the fifth value) is the median. Use the median as the new value for the center pixel.Use the median as the new value for the center pixel.

18. September 19, 2013Computer Vision Lecture 6: Image Filtering 18 Median Filter Advantage of the median filter: Capable of eliminating outliers such as the extreme brightness values in salt-and-pepper noise.Advantage of the median filter: Capable of eliminating outliers such as the extreme brightness values in salt-and-pepper noise. Disadvantage: The median filter may change the contours of objects in the image.Disadvantage: The median filter may change the contours of objects in the image.

19. September 19, 2013Computer Vision Lecture 6: Image Filtering 19 Median Filter original image 3  3 median 7  7 median