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  1. Image Warping http://www.jeffrey-martin.com 15-463: Computational Photography Alexei Efros, CMU, Fall 2008 Some slides from Steve Seitz

  2. 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))

  3. 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

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

  5. 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’)

  6. 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

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

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

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

  10. (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

  11. 2-D Rotation • This is easy to capture in matrix form: • Even though sin(q) and cos(q) are nonlinear functions of q, • x’ is a linear combination of x and y • y’ is a linear combination of x and y • What is the inverse transformation? • Rotation by –q • For rotation matrices R

  12. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Identity? 2D Scale around (0,0)?

  13. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Rotate around (0,0)? 2D Shear?

  14. 2x2 Matrices • What types of transformations can be represented with a 2x2 matrix? 2D Mirror about Y axis? 2D Mirror over (0,0)?

  15. 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

  16. 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

  17. v =(vx,vy) p u=(ux,uy) Consider a different Basis j =(0,1) q i =(1,0) q=4i+3j = (4,3) p=4u+3v

  18. é ù é ù u v u v é ù 4 x x x x = = pij puv ê ú ê ú ê ú u v u v 3 ë û ë û ë û y y y y Linear Transformations as Change of Basis • Any linear transformation is a basis!!! v =(vx,vy) j =(0,1) puv pij u=(ux,uy) i =(1,0) puv = (4,3) pij= 4u+3v px=4ux+3vx py=4uy+3vy

  19. -1 -1 é ù é ù u v u v é ù 5 x x x x = = puv pij ê ú ê ú ê ú u v u v 4 ë û ë û ë û y y y y What’s the inverse transform? • How can we change from any basis to any basis? • What if the basis are orthogonal? j =(0,1) v =(vx,vy) pij puv u=(ux,uy) i =(1,0) pij= (5,4) = pxu + pyv puv = (px,py) = ?

  20. Projection onto orthogonal basis j =(0,1) v =(vx,vy) pij puv u=(ux,uy) i =(1,0) pij= (5,4) puv = (u·pij, v·pij) é ù é ù u u u u é ù 5 x x x y = = puv pij ê ú ê ú ê ú v v v v 4 ë û ë û ë û y y x y

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

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

  23. 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

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

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

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

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

  28. 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 • Will the last coordinate w always be 1?

  29. 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

  30. 2D image transformations • These transformations are a nested set of groups • Closed under composition and inverse is a member