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Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord

Machine Vision. Lecture 07 – Pyramids. Dr. J. Shanbehzadeh Shanbehzadeh@gmail.com M.HosseinKord. Science and Research Branch of Islamic Azad University. 1/49 slides. Table of Contents. 7-1-1) Reduce 7-1-2) Expand. 7-4) Interpolation. 7-3-1) Image compression

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Dr. J. Shanbehzadeh Shanbehzadeh@gmail M.HosseinKord

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  1. Machine Vision Lecture 07 – Pyramids Dr. J. Shanbehzadeh Shanbehzadeh@gmail.com M.HosseinKord Science and Research Branch of Islamic Azad University 1/49 slides

  2. Table of Contents 7-1-1) Reduce 7-1-2) Expand 7-4) Interpolation 7-3-1) Image compression 7-3-2) Image composting

  3. 7-1)Gaussian Pyramids Lowest Resolution 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids . . . Original Img Highest Resolution

  4. 7-1-1) Reduce 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids Level l Level l-1

  5. 7-1-1) Reduce- Convolution 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  6. 7-1-1) Reduce (1D):Example Convolution Mask: [w(-2), w(-1), w(0), w(1), w(2)] 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids i = 2

  7. Convolution Mask: [w(-2), w(-1), w(0), w(1), w(2)] [ c , b , a , b , c ] 7-1-1) Reduce (1D) 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids gl = REDUCE (gl-1)

  8. 7-1-2) Expand 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids n=1 n=2 Notice:

  9. 7-1-2) Expand(1D) [w(-2), w(-1), w(0), w(1), w(2)] [ c , b , a , b , c ] 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids i = 4 Involved weights  [c , a , c]

  10. 7-1-2) Expand(1D) [w(-2), w(-1), w(0), w(1), w(2)] [ c , b , a , b , c ] 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids i = 3 Involved weights  [b , b]

  11. Expand 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  12. Convolution Mask 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids • Separable • •Symmetric

  13. Convolution Mask 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids • The sum of mask should be 1. •All nodes at a given level must contribute the same total weight to the nodes at the next higher level.

  14. Convolution Mask 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids c a b c b a + 2b + 2c = 1 a + 2c = 2b b= ¼

  15. Convolution Mask 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids a= 0.4 GAUSSIAN a= 0.5 TRINGULAR

  16. Gaussian Mask 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  17. Gaussian Pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  18. Gaussian Pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  19. Laplacian Pyramids

  20. Laplacian Pyramids 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids • Similar to edge detected images. • Most pixels are zero. • Can be used for image compression. L1 = g1 – EXPAND[g2] L2 = g2 – EXPAND[g3] L3 = g3 – EXPAND[g4] • L4= g4

  21. Laplacian Pyramids L1 = g1 – EXPAND[g2] L2 = g2 – EXPAND[g3] L3 = g3 – EXPAND[g4] • L4= g4 Lower in size and resolution 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids Gaussian Pyramid Laplacian Pyramid

  22. Applications of Laplacian pyramids

  23. Image compression •Compute Gaussian pyramid •Compute Laplacian pyramid •Code Laplacian pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  24. Image compression 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids • Decode Laplacian pyramid. • Compute Gaussian pyramid from Laplacian pyramid. • g1 is reconstructed image.

  25. Image Compression (Entropy) 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids 7.6 4.4 0.77 5.0 1.9 5.6 3.3 6.2 4.2

  26. Image Compression 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids 1.58 0.73

  27. Combining Apple & Orange 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  28. Algorithm • • Generate Laplacian pyramid Lo of orange image. • • Generate Laplacian pyramid La of apple • image. • • Generate Laplacian pyramid Lc by • – copying left half of nodes at each level from apple • and • – right half of nodes from orange pyramids. • • Reconstruct combined image from Lc. 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  29. Interpolation

  30. Interpolation 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  31. 1‐D Interpolation 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids 1=< x =<2

  32. 2‐D Interpolation 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids Bilinear

  33. Bi‐linear Interpolation 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  34. Optical flow using Laplacian Pyramid

  35. Why Lucas Kanade with Pyramids? • Horn-Schunck and Lucas-Kanade optical method works only for small motion. • If object moves faster, the brightness changes rapidly, 2x2 or 3x3 masks fail to estimate spatiotemporal derivatives. • Pyramids can be used to compute large optical flow vectors. 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  36. Lucas Kanade with Pyramids Lucas Kanade Lucas Kanade 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids Interpolation LK for highest level of Laplacian pyramid  ui , vi Do the interpolation  u*i-1 , v*i-1 Multiply by 2  u*i-1 , v*i-1 Calculate ft according to displacement of u*i-1 , v*i-1 LK for level l-1 of Laplacian pyramid  u’i-1 , v’i-1 Accurate value of Optical flow is  ui-1 = u*i-1 + u’i-1 vi-1 = v*i-1 + v’i-1

  37. Laplacian Pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  38. Laplacian Pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  39. Laplacian Pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  40. Laplacian Pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

  41. Laplacian Pyramid 7-1) Gaussian Pyramids 7-1-1) Reduce 7-1-2) Expand 7-1-3) Convolution Mask 7-2) Laplacian Pyramids 7-3) Applications of Laplacian pyramids 7-3-1) Image compression 7-3-2) Image composting 7-4) interpolation 7-5) Optical flow using Pyramids

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