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image warping image mapping image mapping examples

Image Warping Image Mapping Image Mapping - Examples Forward - PowerPoint PPT Presentation

Nov 28, 2022 •211 likes •717 views

Image Warping Image Mapping Image Mapping - Examples Forward Mapping Forward Mapping - Disadvantages Example Forward Mapping Original Rotated Zoom In Backward Mapping The Problem: (u,v) are not integers! "# Nearest Neighbor

  1. Image Warping

  2. Image Mapping

  3. Image Mapping - Examples

  4. Forward Mapping

  5. Forward Mapping - Disadvantages

  6. Example – Forward Mapping Original Rotated Zoom In

  7. Backward Mapping The Problem: (u,v) are not integers! 𝑔 "#

  10. Nearest Neighbor

  12. Example - Nearest Neighbor Original Rotated Zoom In

  13. Bi-linear Interpolation • Bi-linear interpolates four closest pixels. • The weight for each pixel is proportional to its distance from the sampling point (x,y)

  14. Bi-linear Interpolation

  15. Bi-linear Interpolation

  16. Bi-linear Interpolation

  17. Bi-linear Interpolation

  18. Example Bi-linear Original Rotated Zoom In

  19. Bi-cubic • Bicubic interpolates 16 closest neighbors (4x4 neighborhood) • The result is much more smooth (u,v)

  20. Bi-cubic Interpolation

  21. Bi-cubic Interpolation

  22. Bi-cubic Interpolation

  23. Example Bi-cubic Original Rotated Zoom In

  24. Nearest Neighbor

  25. Bi-Linear Interpolation

  26. Bi-Cubic Interpolation

  27. Comparison Nearest Neighbor Bi-cubic Bi-linear

  28. Direct Rotation

  29. Shear

  30. Shear

  31. Shear and Scale Operating line by line, faster and simpler filters

  32. Shear a a æ ö æ ö æ ö r x x cos - y sin ç ÷ = ç ÷ = ç ÷ T ç ÷ ç ÷ ç ÷ a + a s y x sin y cos è ø è ø è ø æ ö æ ö æ ö æ ö æ ö r u x x ç ÷ ç ÷ = ç ÷ = ç ÷ = ç ÷ B B A T ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ s v y y è ø è ø è ø è ø è ø æ ö æ ö u x ç ÷ ç ÷ = A ç ÷ ç ÷ v y è ø è ø

  33. Shear A preserve columns æ ö æ ö æ ö u x x ç ÷ ç ÷ ç ÷ = = A ç ÷ ç ÷ ç ÷ v y f(x, y) è ø è ø è ø B preserve rows æ ö æ ö æ ö r u g(u, v) ç ÷ ç ÷ ç ÷ = = B ç ÷ ç ÷ ç ÷ s v v è ø è ø è ø

  34. Shear a a æ ö æ ö æ ö r x x cos - y sin ç ÷ ç ÷ ç ÷ = = T ç ÷ ç ÷ ç ÷ a + a s y x sin y cos è ø è ø è ø æ ö æ ö æ ö æ ö æ ö æ ö r u x x g(x, f(x, y)) ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ = = = = B B A B ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ s v y f(x, y) f(x, y) è ø è ø è ø è ø è ø è ø We get = = a + a f(x, y) s x sin y cos = a - a g(u, v) x cos y sin

  35. Shear = a - a g(u, v) x cos y sin We need to express it in terms of u,v We know that x=u, and = = a + a v f(x, y) x sin y cos We get a a v - x sin v - u sin = = y a a cos cos

  36. Shear We put it all together and get a v - u sin = a a = a - a g(u, v) u cos - sin u sec v tan a cos

  37. Shear At last we get æ ö æ ö x x ç ÷ ç ÷ = A ç ÷ ç ÷ a + a y x sin y cos è ø è ø a - a æ ö æ ö u u sec v tan ç ÷ ç ÷ = B ç ÷ ç ÷ v v è ø è ø

  38. Shear Using a large angle (80 degree)

  39. Shear

  40. Shear

  41. Shear Second pass with Backward Mapping

  42. Shear • Rotate in 90 degree, then use shear with a small angle • We still have a scale factor in the shear which create holes; one solution is to use filter

  43. Shear The other solution is by using three shear transformations a - a - a - a æ ö æ ö æ ö æ ö cos sin 1 tan / 2 1 0 1 tan / 2 ç ÷ ç ÷ ç ÷ ç ÷ = ç ÷ ç ÷ ç ÷ ç ÷ a a a sin cos 0 1 sin 1 0 1 è ø è ø è ø è ø We need Three passes instead of Two. But no scale! Just shift lines!

  44. Shear Two first shears

  45. Shear

  46. Shear

  47. Shear

  48. Shear

  49. Shear

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