1 / 85

Advanced Multimedia

Advanced Multimedia. Warping & Morphing Tamara Berg. Image Warping. http://www.jeffrey-martin.com. Slides from: Alexei Efros & Steve Seitz. Image Warping in Biology. D'Arcy Thompson http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/darcy.html

aneko
Download Presentation

Advanced Multimedia

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Advanced Multimedia Warping & Morphing Tamara Berg

  2. Image Warping http://www.jeffrey-martin.com Slides from: Alexei Efros & Steve Seitz

  3. Image Warping in Biology • D'Arcy Thompson • http://www-groups.dcs.st-and.ac.uk/~history/Miscellaneous/darcy.html • http://en.wikipedia.org/wiki/D'Arcy_Thompson • Importance of shape and structure in evolution Slide by Durand and Freeman

  4. f f f T T x x x f x Image Transformations • image filtering: change range of image • g(x) = T(f(x)) image warping: change domain of image g(x) = f(T(x))

  5. T T Image Transformations • image filtering: change range of image • g(x) = T(f(x)) f g image warping: change domain of image g(x) = f(T(x)) f g

  6. Parametric (global) warping • Examples of parametric warps: aspect rotation translation perspective cylindrical affine

  7. T Parametric (global) warping • Transformation T is a coordinate-changing machine: • p’ = T(p) • What does it mean that T is global? • Is the same for any point p • can be described by just a few numbers (parameters) • Let’s represent T as a matrix: • p’ = Mp p = (x,y) p’ = (x’,y’)

  8. Scaling • Scaling a coordinate means multiplying each of its components by a scalar • Uniform scaling means this scalar is the same for all components: 2

  9. X  2,Y  0.5 Scaling • Non-uniform scaling: different scalars per component:

  10. Scaling • Scaling operation: • Or, in matrix form: scaling matrix S What’s inverse of S?

  11. (x’, y’) (x, y)  2-D Rotation x’ = x cos() - y sin() y’ = x sin() + y cos()

  12. (x’, y’) (x, y)  2-D Rotation x = r cos (f) y = r sin (f) x’ = r cos (f + ) y’ = r sin (f + ) f

  13. (x’, y’) (x, y)  2-D Rotation x = r cos (f) y = r sin (f) x’ = r cos (f + ) y’ = r sin (f + ) Trig Identity… x’ = r cos(f) cos() – r sin(f) sin() y’ = r sin(f) cos() + r cos(f) sin() f

  14. (x’, y’) (x, y)  2-D Rotation x = r cos (f) y = r sin (f) x’ = r cos (f + ) y’ = r sin (f + ) Trig Identity… x’ = r cos(f) cos() – r sin(f) sin() y’ = r sin(f) cos() + r cos(f) sin() Substitute… x’ = x cos() - y sin() y’ = x sin() + y cos() f

  15. 2-D Rotation • This is easy to capture in matrix form: • What is the inverse transformation? R

  16. 2-D Rotation • This is easy to capture in matrix form: • What is the inverse transformation? • Rotation by –q • For rotation matrices R

  17. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Identity?

  18. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Identity?

  19. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Identity? 2D Scale around (0,0)?

  20. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Identity? 2D Scale around (0,0)?

  21. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Rotate around (0,0)? 2D Shear?

  22. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis?

  23. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis?

  24. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis? 2D Mirror over (0,0)?

  25. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis? 2D Mirror over (0,0)?

  26. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Translation?

  27. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Translation? NO! Only linear 2D transformations can be represented with a 2x2 matrix

  28. All 2D Linear Transformations • Linear transformations are combinations of … • Scale, • Rotation, • Shear, and • Mirror • Properties of linear transformations: • Origin maps to origin • Lines map to lines • Parallel lines remain parallel • Ratios are preserved • Closed under composition

  29. Homogeneous Coordinates • Q: How can we represent translation as a 3x3 matrix?

  30. Homogeneous Coordinates • Homogeneous coordinates • represent coordinates in 2 dimensions with a 3-vector

  31. y 2 (2,1,1) or (4,2,2) or (6,3,3) 1 x 2 1 Homogeneous Coordinates • Add a 3rd coordinate to every 2D point • (x, y, w) represents a point at location (x/w, y/w) • (x, y, 0) represents a point at infinity • (0, 0, 0) is not allowed Convenient coordinate system to represent many useful transformations

  32. Homogeneous Coordinates • Q: How can we represent translation as a 3x3 matrix?

  33. Homogeneous Coordinates • Q: How can we represent translation as a 3x3 matrix? • A: Using the rightmost column:

  34. Translation • Example of translation Homogeneous Coordinates tx = 2ty= 1

  35. Basic 2D Transformations • Basic 2D transformations as 3x3 matrices Translate Scale Rotate Shear

  36. Matrix Composition • Transformations can be combined by matrix multiplication p’ = T(tx,ty) R(Q) S(sx,sy) p

  37. Affine Transformations • Affine transformations are combinations of … • Linear transformations, and • Translations • Properties of affine transformations: • Origin does not necessarily map to origin • Lines map to lines • Parallel lines remain parallel • Ratios are preserved • Closed under composition • Models change of basis

  38. Projective Transformations • Projective transformations … • Affine transformations, and • Projective warps • Properties of projective transformations: • Origin does not necessarily map to origin • Lines map to lines • Parallel lines do not necessarily remain parallel • Ratios are not preserved • Closed under composition • Models change of basis

  39. 2D image transformations • These transformations are a nested set of groups • Closed under composition and inverse is a member

  40. Recovering Transformations • What if we know f and g and want to recover the transform T? • willing to let user provide correspondences • How many do we need? ? T(x,y) y y’ x x’ f(x,y) g(x’,y’)

  41. Translation: # correspondences? • How many correspondences needed for translation? ? T(x,y) y y’ x x’

  42. Translation: # correspondences? • How many correspondences needed for translation? • How many Degrees of Freedom? ? T(x,y) y y’ x x’

  43. Translation: # correspondences? • How many correspondences needed for translation? • How many Degrees of Freedom? • What is the transformation matrix? ? T(x,y) y y’ x x’

  44. Euclidian: # correspondences? • How many correspondences needed for translation+rotation? • How many DOF? ? T(x,y) y y’ x x’

  45. Affine: # correspondences? • How many correspondences needed for affine? • How many DOF? ? T(x,y) y y’ x x’

  46. Projective: # correspondences? • How many correspondences needed for projective? • How many DOF? ? T(x,y) y y’ x x’

  47. Example: warping triangles B’ • Given two triangles: ABC and A’B’C’ in 2D (12 numbers) • Need to find transform T to transfer all pixels from one to the other. • What kind of transformation is T? • How can we compute the transformation matrix: B ? T(x,y) C’ A C A’ Source Destination

  48. Image warping • Given a coordinate transform (x’,y’)=T(x,y) and a source image f(x,y), how do we compute a transformed image g(x’,y’)=f(T(x,y))? T(x,y) y y’ x x’ f(x,y) g(x’,y’)

  49. Forward warping • Send each pixel f(x,y) to its corresponding location • (x’,y’)=T(x,y) in the second image T(x,y) y y’ x x’ f(x,y) g(x’,y’) Q: what if pixel lands “between” two pixels?

  50. Forward warping • Send each pixel f(x,y) to its corresponding location • (x’,y’)=T(x,y) in the second image T(x,y) y y’ x x’ f(x,y) g(x’,y’) Q: what if pixel lands “between” two pixels? • A: distribute color among neighboring pixels (x’,y’) • Known as “splatting” • Check out griddata in Matlab

More Related