Interactive Computer Graphics. Warping and morphing. Warping and Morphing. Warping and Morphing. Lecture 14+15: Warping and Morphing. What is.

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1 Interactive Computer Graphics Warping and morphing Lecture 14+15: Warping and Morphing Lecture 14: Warping and Morphing: Slide 1 Lecture 14: Warping and Morphing: Slide 2 Warping and Morphing What is Warping and Morphing What is warping? morphing? warping? morphing?? Lecture 14: Warping and Morphing: Slide 3 Lecture 14: Warping and Morphing: Slide 4 1

2 Warping The term warping refers to the geometric transformation of graphical objects (images, surfaces or volumes) from one coordinate sstem to another coordinate sstem. Warping does not affect the attributes of the underling graphical objects. Attributes ma be color (RGB, HSV) teture maps and coordinates normals, etc. Morphing The term morphing stands for metamorphosing and refers to an animation technique in which one graphical object is graduall turned into another. Morphing can affect both the shape and attributes of the graphical objects. Lecture 14: Warping and Morphing: Slide 5 Lecture 14: Warping and Morphing: Slide 6 Morphing = Object Averaging Averaging The aim is to find an average between two objects Not an average of two images of objects but an image of the average object! How can we make a smooth transition in time? Do a weighted average over time t How do we know what the average object looks like? Need an algorithm to compute the average geometr and appearance What s the average of P and Q? Linear Interpolation (Affine Combination): New point ap + bq, defined onl when a+b = 1 So ap+bq = ap+(1-a)q v = Q - P P P + 0.5v = P + 0.5(Q P) = 0.5P Q Q P + 1.5v = P + 1.5(Q P) = -0.5P Q (etrapolation) Lecture 14: Warping and Morphing: Slide 7 Lecture 14: Warping and Morphing: Slide 8 2

3 Morphing using cross-dissolve Morphing using warping and cross-dissolve Interpolate whole images: I(t) = t*i1 + (1-t)*I2 This is called cross-dissolve But what is the images are not aligned? Align first, then cross-dissolve Lecture 14: Warping and Morphing: Slide 9 Lecture 14: Warping and Morphing: Slide 10 Image warping Image warping image filtering: change range of image g() = T(f()) f image filtering: change range of image g() = h(t()) f f T image warping: change domain of image image warping: change domain of image g() = f(t()) f f g T f g() = f(t()) T Lecture 14: Warping and Morphing: Slide 11 T g Lecture 14: Warping and Morphing: Slide 12 3

4 Parametric (global) warping Eamples of parametric warps: Parametric (global) warping T p = (,) p = (, ) translation rotation aspect Transformation T can be epressed as a mapping: p = T(p) Transformation T can be epressed as a matri: p = M*p affine Lecture 14: Warping and Morphing: Slide 13 perspective clindrical Lecture 14: Warping and Morphing: Slide 14 Scaling Scaling Scaling a coordinate means multipling each of its components b a scalar Uniform scaling means this scalar is the same for all components: Non-uniform scaling: different scalars per component: X 2, Y Lecture 14: Warping and Morphing: Slide 15 Lecture 14: Warping and Morphing: Slide 16 4

5 Scaling Scaling operation: 2-D Rotation (, ) Or, in matri form: (, ) scaling matri S θ = cos(θ) - sin(θ) = sin(θ) + cos(θ) What is the inverse of S? Lecture 14: Warping and Morphing: Slide 17 Lecture 14: Warping and Morphing: Slide 18 2-D Rotation 2-D Rotation θ φ (, ) (, ) = r cos (φ) = r sin (φ) = r cos (φ + θ) = r sin (φ + θ) Trig Identit = r cos(φ) cos(θ) r sin(φ) sin(θ) = r sin(φ) sin(θ) + r cos(φ) cos(θ) Substitute = cos(θ) - sin(θ) = sin(θ) + cos(θ) This is eas to capture in matri form: R Even though sin(θ) and cos(θ) are nonlinear functions of θ, is a linear combination of and is a linear combination of and What is the inverse transformation? Rotation b θ For rotation matrices, det(r) = 1 so Lecture 14: Warping and Morphing: Slide 19 Lecture 14: Warping and Morphing: Slide 20 5

6 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Identit? 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Rotate around (0,0)? 2D Scale around (0,0)? 2D Shear? Lecture 14: Warping and Morphing: Slide 21 Lecture 14: Warping and Morphing: Slide Matrices What tpes of transformations can be represented with a 22 matri? 2D Mirror about Y ais? 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Translation? NO! 2D Mirror over (0,0)? Onl linear 2D transformations can be represented with a 22 matri Lecture 14: Warping and Morphing: Slide 23 Lecture 14: Warping and Morphing: Slide 24 6

7 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 Homogeneous Coordinates Q: How can we represent translation as a matri transformation? A: Using the translation parameters as the rightmost column: Lecture 14: Warping and Morphing: Slide 25 Lecture 14: Warping and Morphing: Slide 26 Basic 2D Transformations 2D image transformations Basic 2D transformations as 33 matrices Translate Scale Rotate Shear Lecture 14: Warping and Morphing: Slide 27 Lecture 14: Warping and Morphing: Slide 28 7

8 Transformations Dimensions of transformation 1D: curves 2D: images 3D: volumes Tpes of transformations rigid affine polnomial quadratic cubic splines Transformations in 3D: Rigid Rigid transformation (6 degrees of freedom) t, t, t z describe the 3 translations in, and z r 11,..., r 33 describe the 3 rotations around,, z Lecture 14: Warping and Morphing: Slide 29 Lecture 14: Warping and Morphing: Slide 30 Transformations in 3D: Rigid Transformations in 3D: Affine Affine transformations (12 degrees of freedom) Lecture 14: Warping and Morphing: Slide 31 Lecture 14: Warping and Morphing: Slide 32 8

9 Non-rigid transformations Quadratic transformation (30 degrees of freedom) Non-rigid transformations Can be etended to other higher-order polnomials: 3 rd order (60 DOF) 4 th order (105 DOF) 5 th order (168 DOF) Problems: can model onl global shape changes, not local shape changes higher order polnomials introduce artifacts such as oscillations Lecture 14: Warping and Morphing: Slide 33 Lecture 14: Warping and Morphing: Slide 34 Image warping Forward warping T(,) f(,) g(, ) T(,) f(,) g(, ) Given a coordinate transform (, ) = T(,) and a source image f(,), how do we compute a transformed image g(, ) = f(t(,))? Send each piel f(,) to its corresponding location (, ) = T(,) in the second image Lecture 14: Warping and Morphing: Slide 35 Lecture 14: Warping and Morphing: Slide 36 9

10 Forward warping Forward warping T(,) f(,) g(, ) T(,) f(,) g(, ) Send each piel f(,) to its corresponding location (, ) = T(,) in the second image Q: what if piel lands between two piels? Send each piel f(,) to its corresponding location (, ) = T(,) in the second image Q: what if piel lands between two piels? A: distribute color among neighboring piels (, ) known as splatting Lecture 14: Warping and Morphing: Slide 37 Lecture 14: Warping and Morphing: Slide 38 Inverse warping Inverse warping T -1 (,) f(,) g(, ) T -1 (,) f(,) g(, ) Get each piel g(, ) from its corresponding location (,) = T -1 (, ) in the first image Get each piel g(, ) from its corresponding location (,) = T -1 (, ) in the first image Q: what if piel comes from between two piels? Lecture 14: Warping and Morphing: Slide 39 Lecture 14: Warping and Morphing: Slide 40 10

11 Inverse warping Interpolation T -1 (,) f(,) g(, ) Get each piel g(, ) from its corresponding location (,) = T -1 (, ) in the first image Q: what if piel comes from between two piels? A: Interpolate color value from neighbors nearest neighbor, bilinear, Gaussian, bicubic Lecture 14: Warping and Morphing: Slide 41 Lecture 14: Warping and Morphing: Slide 42 Interpolation Interpolation: Linear, 2D p 2 p 3 s p 0 r p 1 Lecture 14: Warping and Morphing: Slide 43 Lecture 14: Warping and Morphing: Slide 44 11

12 Interpolation: Linear, 3D Non-rigid transformations t p 6 p 7 p 4 p 5 s z p 2 p 3 r p 0 p 1 Lecture 14: Warping and Morphing: Slide 45 Lecture 14: Warping and Morphing: Slide 46 Non-rigid transformations: Correspondences Non-rigid transformations: Correspondences Lecture 14: Warping and Morphing: Slide 47 Lecture 14: Warping and Morphing: Slide 48 12

13 Feature-Based Warping: Beier-Neele Feature-Based Warping: Beier-Neele Beier & Neele use pairs of lines to specif warp Given p in destination image, where is p in source image? Lecture 14: Warping and Morphing: Slide 49 Lecture 14: Warping and Morphing: Slide 50 Feature-Based Warping: Beier-Neele For each piel p in the destination image find the corresponding u,v find the p in the source image for that u,v destination(p) = source(p ) Warping with One Line Pair: Beier-Neele What happens to the F? Translation! Lecture 14: Warping and Morphing: Slide 51 Lecture 14: Warping and Morphing: Slide 52 13

14 Warping with One Line Pair (cont.): Beier-Neele What happens to the F? Warping with One Line Pair (cont.): Beier-Neele What happens to the F? Scale! Rotation! Lecture 14: Warping and Morphing: Slide 53 Lecture 14: Warping and Morphing: Slide 54 Warping with One Line Pair (cont.): Beier-Neele What happens to the F? Warping with Multiple Line Pairs: Beier-Neele Use weighted combination of points defined each pair of corresponding lines In general, similarit transformations Lecture 14: Warping and Morphing: Slide 55 Lecture 14: Warping and Morphing: Slide 56 14

15 Warping with Multiple Line Pairs: Beier-Neele Use weighted combination of points defined b each pair corresponding lines Weighting Effect of Each Line Pair: Beier-Neele To weight the contribution of each line pair where length[i] is the length of L[i] dist[i] is the distance from X to L[i] a, b, p are constants that control the warp p is a weighted average Lecture 14: Warping and Morphing: Slide 57 Lecture 14: Warping and Morphing: Slide 58 Warping Pseudocode: Beier-Neele foreach destination piel p do psum = (0, 0) wsum = (0, 0) foreach line L[i] in destination do p [i] = p transformed b (L[i], L [i]) psum = psum + p [i] * weight[i] wsum += weight[i] end p = psum / wsum destination(p) = source(p ) end Lecture 14: Warping and Morphing: Slide 59 Lecture 14: Warping and Morphing: Slide 60 15