CS4670: Computer Vision

Transcription

1 CS467: Computer Vision Noah Snavely Lecture 8: Geometric transformations

2 Szeliski: Chapter 3.6 Reading

3 Announcements Project 2 out today, due Oct. 4 (demo at end of class today)

4 Image alignment Why don t these image line up eactly?

5 What is the geometric relationship between these two images?? Answer: Similarity transformation (translation, rotation, uniform scale)

6 What is the geometric relationship between these two images??

7 What is the geometric relationship between these two images? Very important for creating mosaics!

8 Image Warping image filtering: change range of image g() = h(f()) f image warping: change domain of image f h g() = f(h()) h g g Richard Szeliski Image Stitching 8

9 Image Warping image filtering: change range of image g() = h(f()) f h g image warping: change domain of image f g() = f(h()) h g Richard Szeliski Image Stitching 9

10 Parametric (global) warping Eamples of parametric warps: translation rotation aspect Richard Szeliski Image Stitching 1

11 Parametric (global) warping T p = (,y) p = (,y ) 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 consider linear forms (can be represented by a 2D matri):

12 Common linear transformations Uniform scaling by s: (,) (,) What is the inverse?

13 Common linear transformations Rotation by angle θ (about the origin) (,) (,) θ What is the inverse? For rotations:

14 22 Matrices What types of transformations can be represented with a 22 matri? 2D mirror about Y ais? 2D mirror across line y =?

15 22 Matrices What types of transformations can be represented with a 22 matri? 2D Translation? NO! Translation is not a linear operation on 2D coordinates

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

17 Homogeneous coordinates Trick: add one more coordinate: w (, y, w) homogeneous plane homogeneous image coordinates y w = 1 (/w, y/w, 1) Converting from homogeneous coordinates

18 Translation Solution: homogeneous coordinates to the rescue

19 Affine transformations any transformation with last row [ 1 ] we call an affine transformation

20 Basic affine transformations 1 1 cos sin sin cos 1 ' ' y y ' ' y t t y y ' ' y sh sh y y Translate 2D in-plane rotation Shear ' ' y s s y y Scale

21 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 w y f e d c b a w y 1 ' '

22 Is this an affine transformation?

23 Where do we go from here? affine transformation what happens when we mess with this row?

24 Projective Transformations aka Homographies aka Planar Perspective Maps Called a homography (or planar perspective map)

25 Eample on board Homographies

26 Image warping with homographies image plane in front black area where no piel maps to image plane below

27 Homographies

28 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 w y i h g f e d c b a w y ' ' '

29 2D image transformations These transformations are a nested set of groups Closed under composition and inverse is a member

30 Image Warping Given a coordinate form (,y ) = T(,y) and a source image f(,y), how do we compute an formed image g(,y ) = f(t(,y))? T(,y) y y f(,y) g(,y )

31 Forward Warping Send each piel f() to its corresponding location (,y ) = T(,y) in g(,y ) What if piel lands between two piels? T(,y) y y f(,y) g(,y )

32 Forward Warping Send each piel f(,y) to its corresponding location = h(,y) in g(,y ) What if piel lands between two piels? Answer: add contribution to several piels, normalize later (splatting) Can still result in holes T(,y) y y f(,y) g(,y )

33 Inverse Warping Get each piel g(,y ) from its corresponding location (,y) = T -1 (,y) in f(,y) Requires taking the inverse of the transform What if piel comes from between two piels? T -1 (,y) y y f(,y) g(,y )

34 Inverse Warping Get each piel g( ) from its corresponding location = h() in f() What if piel comes from between two piels? Answer: resample color value from interpolated (prefiltered) source image T -1 (,y) y y f(,y) g(,y )

35 Interpolation Possible interpolation filters: nearest neighbor bilinear bicubic (interpolating) sinc Needed to prevent jaggies and teture crawl (with prefiltering)