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# Computer Vision - Image Filters

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Basic intro and hands on exercises im image filtering

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### Computer Vision - Image Filters

1. 1. 1 Yossi CohenIntro to computer Vision
2. 2. 2
3. 3. 3 How to filter 2d Correlation h=filter2(g,f); or h=imfilter(f,g); 2d Convolution h=conv2(g,f); ],[],[],[ , lnkmflkgnmh lk −−= ∑ f=image g=filter ],[],[],[ , lnkmflkgnmh lk ++= ∑
4. 4. 4 Correlation filtering Say the averaging window size is 2k+1 x 2k+1: Loop over all pixels in neighborhood around image pixel F[i,j] Attribute uniform weight to each pixel Now generalize to allow different weights depending on neighboring pixel’s relative position: Non-uniform weights
5. 5. 5 Correlation filtering Filtering an image: replace each pixel with a linear combination of its neighbors. The filter “kernel” or “mask” H[u,v] is the prescription for the weights in the linear combination. This is called cross-correlation, denoted
6. 6. 6 Properties of smoothing filters Smoothing  Values positive  Sum to 1  constant regions same as input  Amount of smoothing proportional to mask size  Remove “high-frequency” components; “low-pass” filter
7. 7. 7 Filtering an impulse signal 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c d e f g h i What is the result of filtering the impulse signal (image) F with the arbitrary kernel H? ?
8. 8. 8 Filtering an impulse signal 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a b c d e f g h i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i h g 0 0 0 0 f e d 0 0 0 0 c b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 What is the result of filtering the impulse signal (image) F with the arbitrary kernel H?
9. 9. 9 Averaging filter  What values belong in the kernel H for the moving average example? 0 10 20 30 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 90 0 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 111 111 111 “box filter” ?
10. 10. 10 Smoothing by averaging depicts box filter: white = high value, black = low value original filtered
11. 11. 11 Gaussian filter 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 90 0 90 90 90 0 0 0 0 0 90 90 90 90 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 2 4 2 1 2 1 This kernel is an approximation of a 2d Gaussian function:
12. 12. 12 Smoothing with a Gaussian
13. 13. 13 Smoothing with a Box
14. 14. 14  Weight contributions of neighboring pixels by nearness 0.003 0.013 0.022 0.013 0.003 0.013 0.059 0.097 0.059 0.013 0.022 0.097 0.159 0.097 0.022 0.013 0.059 0.097 0.059 0.013 0.003 0.013 0.022 0.013 0.003 5 x 5, σ = 1 Slide credit: Christopher Rasmussen Gaussian
15. 15. 15 Gaussian filters What parameters matter here? Size of kernel or mask Note, Gaussian function has infinite support, but discrete filters use finite kernels σ = 5 with 10 x 10 kernel σ = 5 with 30 x 30 kernel
16. 16. 16 Gaussian filters What parameters matter here? Variance of Gaussian: determines extent of smoothing σ = 2 with 30 x 30 kernel σ = 5 with 30 x 30 kernel
17. 17. 17 Matlab >> hsize = 10; >> sigma = 5; >> h = fspecial(‘gaussian’ hsize, sigma); >> mesh(h); >> imagesc(h); >> outim = imfilter(im, h); % correlation >> imshow(outim); outim
18. 18. 18 Smoothing with a Gaussian for sigma=1:3:10 h = fspecial('gaussian‘, fsize, sigma); out = imfilter(im, h); imshow(out); pause; end … Parameter σ is the “scale” / “width” / “spread” of the Gaussian kernel, and controls the amount of smoothing.
19. 19. 19 Gaussian filters • Remove “high-frequency” components from the image (low-pass filter) Images become more smooth • Convolution with self is another Gaussian –So can smooth with small-width kernel, repeat, and get same result as larger-width kernel would have –Convolving two times with Gaussian kernel of width σ is same as convolving once with kernel of width σ√2 • Separable kernel –Factors into product of two 1D Gaussians Source: K. Grauman
20. 20. 20 Separability of the Gaussian filter Source: D. Lowe
21. 21. 21 Separability example * * = = 2D convolution (center location only) Source: K. Grauman The filter factors into a product of 1D filters: Perform convolution along rows: Followed by convolution along the remaining column:
22. 22. 22 Convolution  Convolution:  Flip the filter in both dimensions (bottom to top, right to left)  Then apply cross-correlation Notation for convolution operator F H
23. 23. 23 Convolution vs. correlation Convolution Cross-correlation
24. 24. 24 Key properties of linear filters Linearity: filter(f1 + f2) = filter(f1) + filter(f2) Shift invariance: same behavior regardless of pixel location filter(shift(f)) = shift(filter(f)) Any linear, shift-invariant operator can be represented as a convolution Source: S. Lazebnik
25. 25. 25 More properties • Commutative: a * b = b * a  Conceptually no difference between filter and signal  But particular filtering implementations might break this equality • Associative: a * (b * c) = (a * b) * c  Often apply several filters one after another: (((a * b1) * b2) * b3)  This is equivalent to applying one filter: a * (b1 * b2 * b3) • Distributes over addition: a * (b + c) = (a * b) + (a * c) • Scalars factor out: ka * b = a * kb = k (a * b) • Identity: unit impulse e = [0, 0, 1, 0, 0], a * e = a Source: S. Lazebnik
26. 26. 26 Matlab Lab 1 – Smooth/Blur Filters
27. 27. 27 Lets try to blur filter %%basic image filters imrgb = imread('peppers.png'); imshow(imrgb); a = [1 1 1; 1 1 1; 1 1 1] Corroutimg = filter2(a, imgray); Convoutimg = conv2(imgray,a); figure; imshow(Corroutimg) figure imshow(Convoutimg)
28. 28. 28 Fix it Can we conv2 an RGB image?? Use rgb2gray Whats wrong now? Try preserving the image power Convert image to double
29. 29. 29 Results %%basic image filters imrgb = imread('peppers.png'); imgray = im2double(rgb2gray(imrgb)); imshow(imgray); a = 1/16*ones(4) Corroutimg = filter2(a, imgray); Convoutimg = conv2(a, imgray); figure; imshow(Corroutimg) figure imshow(Convoutimg)
30. 30. 30 Sharp %%basic sharp filter imrgb = imread('peppers.png'); imgray = im2double(rgb2gray(imrgb)); imshow(imgray); a = [ 0 0 0; 0 2 0; 0 0 0] - 1/9*ones(3) Corroutimg = filter2(a, imgray); figure; imshow(Corroutimg)
31. 31. 31 Practical matters What happens near the edge? the filter window falls off the edge of the image need to extrapolate methods: clip filter (black) wrap around copy edge reflect across edge Source: S. Marschner
32. 32. 32 Matlab edge aware filtering methods (MATLAB): clip filter (black): imfilter(f, g, 0) wrap around: imfilter(f, g, ‘circular’) copy edge: imfilter(f, g, ‘replicate’) reflect across edge: imfilter(f, g, ‘symmetric’)
33. 33. 33 Practical matters What is the size of the output? • MATLAB: filter2(g, f, shape) shape = ‘full’: output size is sum of sizes of f and g shape = ‘same’: output size is same as f shape = ‘valid’: output size is difference of sizes of f and g f gg gg f gg gg f gg gg full same valid
34. 34. 34 Median filters A Median Filter operates over a window by selecting the median intensity in the window. What advantage does a median filter have over a mean filter? Is a median filter a kind of convolution?
35. 35. 35 Matlab Lab 2 – Median Filter
36. 36. 36 Comparison: salt and pepper noise
37. 37. 37 Median Filter Example %%Median imrgb = imread('peppers.png'); imgray = im2double(rgb2gray(imrgb)); imnoise = imnoise(imgray,'salt & pepper',0.1); imshow(imnoise) figure imshow(medfilt2(imnoise))
38. 38. 38 Basic Filters summary Linear filtering is sum of dot product at each position Can smooth, sharpen, translate (among many other uses) Be aware of details for filter size, extrapolation, cropping 111 111 111