Image Warping. Many slides from Alyosha Efros + Steve Seitz. Photo by Sean Carroll

Transcription

1 Image Warping Man slides from Alosha Efros + Steve Seitz Photo b Sean Carroll

2 Morphing Blend from one object to other with a series of local transformations

3 Image Transformations image filtering: change range of image g() T(f()) f T gf image warping: change domain of image f g() f(t()) T gf

4 Image Transformations image filtering: change range of image g() T(f()) f T g image warping: change domain of image f g() f(t()) T g

5 Parametric (global) warping T p (,) p (, ) Transformation T is a coordinate-changing machine: p T(p) What does it mean that T is global? Is the same for an point p can be described b just a few numbers (parameters) For linear transformations, we can represent T as a matri p Mp M

6 Parametric (global) warping Eamples of parametric warps: translation rotation aspect affine perspective clindrical

7 Scaling Scaling a coordinate means multipling each of its components b a scalar Uniform scaling means this scalar is the same for all components: 2

8 Scaling Non-uniform scaling: different scalars per component: X 2, Y.5

9 Scaling Scaling operation: Or, in matri form: b a b a scaling matri S What is the transformation from (, ) to (, )?

10 2-D Rotation (, ) (, ) θ cos(θ) - sin(θ) sin(θ) + cos(θ)

11 2-D Rotation θ φ (, ) (, ) Polar coordinates 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) cos(θ) + r cos(φ) sin(θ) Substitute cos(θ) - sin(θ) sin(θ) + cos(θ)

12 2-D Rotation This is eas to capture in matri form: cos sin θ ( θ) sin( θ) ( ) ( ) R cos θ 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 T R R

13 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Identit? 2D Scale around (,)? s s * * s s

14 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Rotate around (,)? * cos * sin * sin * cos Θ + Θ Θ Θ Θ Θ Θ Θ cos sin sin cos 2D Shear? k k + + * * k k

15 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Mirror about Y ais? 2D Mirror over (,)?

16 22 Matrices What tpes of transformations can be represented with a 22 matri? 2D Translation? + t + t NO!

17 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 a c b d a c be d g f i h k j l

18 Homogeneous Coordinates Q: How can we represent translation in matri form? + + t t

19 Homogeneous Coordinates Homogeneous coordinates represent coordinates in 2 dimensions with a 3-vector homogeneous coords

20 Homogeneous Coordinates 2D Points! Homogeneous Coordinates Append to ever 2D point: ( )! ( ) Homogeneous coordinates! 2D Points Divide b third coordinate ( w)! (/w /w) Special properties Scale invariant: ( w) k * ( w) (,, ) represents a point at infinit (,, ) is not allowed 2 Scale Invariance (2,,) or (4,2,2) or (6,3,3) 2

21 Homogeneous Coordinates Q: How can we represent translation in matri form? A: Using the rightmost column: t t Translation t t + +

22 Translation Eample + + t t t t t 2 t

23 Basic 2D transformations as 33 matrices Θ Θ Θ Θ cos sin sin cos t t β β Translate Rotate Shear s s Scale

24 Matri Composition Transformations can be combined b matri multiplication Θ Θ Θ Θ w s s t t w cos sin sin cos p T(t,t ) R(Θ) S(s,s ) p Does the order of multiplication matter?

25 Affine Transformations f e d c b a Affine transformations are combinations of Linear transformations, and Translations Properties of affine transformations: Origin does not necessaril map to origin Lines map to lines Parallel lines remain parallel Ratios are preserved Closed under composition

26 Recovering Transformations? T(,) f(,) g(, ) What if we know f and g and want to recover the transform T? willing to let user provide correspondences How man do we need?

27 Translation: # correspondences? How man Degrees of Freedom? How man correspondences needed for translation? What is the transformation matri? T(,)? p p p p M

28 Robust Motion Recover Given feature matches and a motion model ou can estimate motion parameters

29 How to handle outliers... What do we do about the bad matches? Richard Szeliski CSE 576 (Spring 25): Computer Vision 9

30 Random Sample Consensus (RANSAC) 3

31 Least squares: Robustness to noise Least squares fit to the red points: Slide: S. Lazebnik

32 Least squares: Robustness to noise Least squares fit with an outlier: Problem: squared error heavil penalizes outliers Slide: S. Lazebnik

33 RANSAC 33

34 RANSAC How man samples to fit a line? 34

35 RANSAC How man samples to fit a line? 2! 35

36 RANSAC How man samples to fit a line? 2! 36

37 RANSAC How man samples to fit a line? 2! 37

38 RANSAC How man samples to fit a line? 2! 38

39 RANSAC How man samples to fit a line? 2! 39

40 RANSAC How man samples to fit a line? 2! 4

41 RANSAC Best: Most number of inliers 4

42 RANSAC Do global fitting from all the inliers 42

43 RAndom SAmple Consensus Select one match, count inliers (in this case, onl one) Richard Szeliski CSE 576 (Spring 25): Computer Vision 2

44 RAndom SAmple Consensus Select one match, count inliers (4 inliers) Richard Szeliski CSE 576 (Spring 25): Computer Vision 2

45 Least squares fit Find average translation vector for largest set of inliers Richard Szeliski CSE 576 (Spring 25): Computer Vision 22

46 Euclidian: # correspondences?? T(,) How man DOF? How man correspondences needed for translation+rotation?

47 Affine: # correspondences?? T(,) How man DOF? How man correspondences needed for affine?

48 Affine transformations Matri form 2n 6 6 2n

49 Least squares Find t that minimizes To solve, form the normal equations

50 Projective Transformations Projective transformations are combos of Affine transformations, and Projective warps Properties of projective transformations: Origin does not necessaril map to origin Lines map to lines w Parallel lines do not necessaril remain parallel Ratios are not preserved Closed under composition Models change of basis Projective matri is defined up to a scale (8 DOF) a d g b e h c f i w

51 2D image transformations These transformations are a nested set of groups Closed under composition and inverse is a member

52 RANSAC motion model 52

53 RANSAC motion model (pick a motion model) 53

54 RANSAC motion model 54

55 Recognizing Panoramas [Brown & Lowe, ICCV 3] 55

56 Finding the panoramas 56

57 Finding the panoramas 57

58 Finding the panoramas 58

59 Finding the panoramas 59

60 Gigapiel