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CS559: Computer Graphics

CS559: Computer Graphics. Lecture 3: Digital Image Representation Li Zhang Spring 2008. Today. Eyes Cameras. Represented by a matrix. Image as a discreet function. 10. 5. 3. 4. 5. 7. 1. 1. 1. 7. Some function. Local image data. Modified image data. What is image filtering?.

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CS559: Computer Graphics

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  1. CS559: Computer Graphics Lecture 3: Digital Image Representation Li Zhang Spring 2008

  2. Today • Eyes • Cameras

  3. Represented by a matrix Image as a discreet function

  4. 10 5 3 4 5 7 1 1 1 7 Some function Local image data Modified image data What is image filtering? • Modify the pixels in an image based on some function of a local neighborhood of the pixels.

  5. 0 0 0 0 0.5 0 0 1 0.5 10 5 3 4 7 5 1 1 1 7 Local image data kernel Modified image data 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”.

  6. I Convolution

  7. coefficient 0 Pixel offset Linear filtering (warm-up slide) ? 1.0 original

  8. coefficient 0 Pixel offset Linear filtering (warm-up slide) 1.0 original Filtered (no change)

  9. coefficient 0 Pixel offset Linear filtering 1.0 ? original

  10. coefficient 0 Pixel offset shift 1.0 original shifted

  11. coefficient 0 Pixel offset Linear filtering ? 0.3 original

  12. coefficient 0 Pixel offset Blurring 0.3 original Blurred (filter applied in both dimensions).

  13. 8 coefficient 0 original Pixel offset Blur Examples 2.4 impulse 0.3 filtered

  14. 8 coefficient coefficient 0 0 original Pixel offset Pixel offset Blur Examples 2.4 impulse 0.3 filtered 8 8 edge 4 4 0.3 filtered original

  15. Linear filtering (warm-up slide) 2.0 1.0 ? 0 0 original

  16. Linear Filtering (no change) 2.0 1.0 0 0 Filtered (no change) original

  17. 0.33 0 Linear Filtering 2.0 ? 0 original

  18. coefficient 0 Pixel offset (remember blurring) 0.3 original Blurred (filter applied in both dimensions).

  19. 0.33 0 Sharpening 2.0 0 Sharpened original original

  20. Sharpening example 1.7 11.2 8 8 coefficient -0.25 -0.3 original Sharpened (differences are accentuated; constant areas are left untouched).

  21. Sharpening before after

  22. Spatial resolution and color R G B original

  23. R G B Blurring the G component processed original

  24. R G B Blurring the R component processed original

  25. R G B processed Blurring the B component original

  26. Lab Color Component L A rotation of the color coordinates into directions that are more perceptually meaningful: L: luminance, a: red-green, b: blue-yellow a b

  27. L a b processed Bluring L original

  28. L a b processed Bluring a original

  29. L a b processed Bluring b original

  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. Look for peaks in Solution: smooth first • Where is the edge?

  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: • Find local edge normal directions for each pixel • Along this direction, compute image gradient • 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

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