Last Lecture. photomatix.com
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1 Last Lecture photomatix.com
2 Today Image Processing: from basic concepts to latest techniques Filtering Edge detection Re-sampling and aliasing Image Pyramids (Gaussian and Laplacian) Removing handshake blur from a single image
3 Image as a discreet function Represented by a matrix
4 What is image filtering? Modify the pixels in an image based on some function of a local neighborhood of the pixels Local image data Some function 7 Modified image data
5 Linear functions Simplest: linear filtering. Replace each pixel by a linear combination of its neighbors. The prescription for the linear combination is called the convolution kernel Local image data kernel 7 Modified image data
6 Convolution f [ m, n] I g I[ m k, n l] g[ k, l] k, l I
7 coefficient Linear filtering (warm-up slide) 1.0? original 0 Pixel offset
8 coefficient Linear filtering (warm-up slide) 1.0 original 0 Pixel offset Filtered (no change)
9 coefficient Linear filtering 1.0? original 0 Pixel offset
10 coefficient shift 1.0 original 0 Pixel offset shifted
11 coefficient Linear filtering 0.3? original 0 Pixel offset
12 coefficient Blurring 0.3 original 0 Pixel offset Blurred (filter applied in both dimensions).
13 coefficient Blur Examples impulse original 0 Pixel offset filtered
14 coefficient coefficient Blur Examples impulse 0.3 original 0 Pixel offset filtered 8 8 edge original 0 Pixel offset filtered
15 Linear filtering (warm-up slide) ? 0 0 original
16 Linear Filtering (no change) original Filtered (no change)
17 Linear Filtering ? 0 0 original
18 coefficient (remember blurring) 0.3 original 0 Pixel offset Blurred (filter applied in both dimensions).
19 Sharpening original Sharpened original
20 coefficient Sharpening example original Sharpened (differences are accentuated; constant areas are left untouched).
21 Sharpening before after
22 Spatial resolution and color R G original B
23 Blurring the G component R G original processed B
24 Blurring the R component R G original processed B
25 Blurring the B component R G original processed B
26 Lab Color Component L a b A rotation of the color coordinates into directions that are more perceptually meaningful: L: luminance, a: red-green, b: blue-yellow
27 Bluring L L a original processed b
28 Bluring a L a original processed b
29 Bluring b L a original processed b
30 Application to image compression (compression is about hiding differences from the true image where you can t see them).
31 Edge Detection Convert a 2D image into a set of curves Extracts salient features of the scene More compact than pixels
32 How can you tell that a pixel is on an edge?
33 Image gradient The gradient of an image: The gradient points in the direction of most rapid change in intensity The gradient direction is given by: how does the gradient relate to the direction of the edge? The edge strength is given by the gradient magnitude
34 Effects of noise Consider a single row or column of the image Plotting intensity as a function of position gives a signal How to compute a derivative? Where is the edge?
35 Solution: smooth first Where is the edge? Look for peaks in
36 Derivative theorem of convolution This saves us one operation:
37 Laplacian of Gaussian Consider Laplacian of Gaussian operator Where is the edge? Zero-crossings of bottom graph
38 Canny Edge Detector Smooth image I with 2D Gaussian: G I Find local edge normal directions for each pixel Along this direction, compute image gradient G arctan I I I y x Locate edges by finding max gradient magnitude (Non-maximum suppression)
39 Non-maximum Suppression Check if pixel is local maximum along gradient direction requires checking interpolated pixels p and r
40 The Canny Edge Detector original image (Lena)
41 The Canny Edge Detector magnitude of the gradient
42 The Canny Edge Detector After non-maximum suppression
43 Canny Edge Detector original Canny with Canny with The choice of depends on desired behavior large detects large scale edges small detects fine features
44 Image Scaling This image is too big to fit on the screen. How can we reduce it? How to generate a halfsized version?
45 Image sub-sampling 1/8 1/4 Throw away every other row and column to create a 1/2 size image - called image sub-sampling
46 Image sub-sampling 1/2 1/4 (2x zoom) 1/8 (4x zoom) Why does this look so crufty?
47 Even worse for synthetic images
48 Really bad in video
49 Alias: n., an assumed name Input signal: Picket fence receding Into the distance will produce aliasing WHY? Matlab output: x = 0:.05:5; imagesc(sin((2.^x).*x)) Aj-aj-aj: Alias! Not enough samples
50 Aliasing occurs when your sampling rate is not high enough to capture the amount of detail in your image Can give you the wrong signal/image an alias Where can it happen in images? During image synthesis: sampling continous singal into discrete signal e.g. ray tracing, line drawing, function plotting, etc. During image processing: resampling discrete signal at a different rate e.g. Image warping, zooming in, zooming out, etc. To do sampling right, need to understand the structure of your signal/image Enter Monsieur Fourier
51 Antialiasing What can be done? 1. Raise sampling rate by oversampling Sample at k times the resolution
52 Antialiasing What can be done? 1. Raise sampling rate by oversampling Sample at k times the resolution
53 Antialiasing What can be done? 1. Raise sampling rate by oversampling Sample at k times the resolution 2. Lower the max frequency by prefiltering Smooth the signal enough Works on discrete signals
54 Antialiasing What can be done? 1. Raise sampling rate by oversampling Sample at k times the resolution 2. Lower the max frequency by prefiltering Smooth the signal enough Works on discrete signals 3. Improve sampling quality with better sampling (CS559)
55 The Gaussian Pyramid Low resolution G 4 ( G 3 * gaussian) 2 G3 ( G2 * gaussian blur ) 2 blur G 2 ( G 1 * gaussian) 2 blur G 1 ( G 0 * gaussian) 2 G 0 blur Image High resolution
56 Gaussian pre-filtering G 1/8 G 1/4 Gaussian 1/2 Solution: filter the image, then subsample
57 Subsampling with Gaussian pre-filtering Gaussian 1/2 G 1/4 G 1/8 Solution: filter the image, then subsample
58 Compare with... 1/2 1/4 (2x zoom) 1/8 (4x zoom)
59 Pyramids at Same Resolution
60 Gaussian Pyramid The Laplacian Pyramid L i G i G L i i expand( G 1 ) expand( G 1 ) i i Laplacian Pyramid G 2 G n L n G n - = L2 G 1 - = L 1 G 0 L 0 - =
61
62 Recap Image Processing: from basic concepts to latest techniques Filtering Edge detection Re-sampling and aliasing Image Pyramids (Gaussian and Laplacian) Next
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