1. Recap from Friday
Image Completion Synthesis Order Graph Cut Scene Completion

2. Image Warping (Sz 3.6.1) cs129: Computational Photography
cs129: Computational Photography
James Hays, Brown, Spring 2011
Slides from Alexei Efros and Steve Seitz

3. Image Transformations
image filtering: change range of image
g(x) = T(f(x)) f x f x T

4. For example: Gamma Correction
Is an instance of these power law intensity transformations. Typically, gamma = 2.2 for a display device, and 1/2.2 for encoding. The result is a more perceptually uniform encoding.

5. Gamma Correction Image encoded for various display gammas.
Source: Wikipedia

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

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

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

9. Parametric (global) warping
p = (x,y)
p’ = (x’,y’)
Transformation T is a coordinate-changing machine:
p’ = T(p)
What does it mean that T is global and parametric?
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

10. 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

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

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

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

14. 2-D Rotation R 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

15. 2x2 Matrices 2D Identity? 2D Scale around (0,0)?
What types of transformations can be represented with a 2x2 matrix?
2D Identity?
2D Scale around (0,0)?

16. 2x2 Matrices 2D Rotate around (0,0)? 2D Shear?
What types of transformations can be represented with a 2x2 matrix?
2D Rotate around (0,0)?
2D Shear?

17. 2x2 Matrices 2D Mirror about Y axis? 2D Mirror over (0,0)?
What types of transformations can be represented with a 2x2 matrix?
2D Mirror about Y axis?
2D Mirror over (0,0)?

18. 2x2 Matrices 2D Translation? Only linear 2D transformations
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

19. 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

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

21. Homogeneous Coordinates
represent coordinates in 2 dimensions with a 3-vector
homogenous coords

22. 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 1 2
(2,1,1)
or (4,2,2)
or (6,3,3) x y
Convenient coordinate system to represent many useful transformations

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

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

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

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

27. 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?

28. 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

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

30. Matlab Demo
“help imtransform”

31. Recovering Transformations?
T(x,y) y y’ x x’
f(x,y)
g(x’,y’)
What if we know f and g and want to recover the transform T?
e.g. better align images from Project 1
willing to let user provide correspondences
How many do we need?

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

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

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

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

36. Image warping T(x,y) y y’ x x’ f(x,y) g(x’,y’)
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))?

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

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

39. Inverse warping T-1(x,y) y y’ x x x’ f(x,y) g(x’,y’)
Get each pixel g(x’,y’) from its corresponding location
(x,y) = T-1(x’,y’) in the first image
Q: what if pixel comes from “between” two pixels?

40. Inverse warping T-1(x,y) y y’ x x x’ f(x,y) g(x’,y’)
Get each pixel g(x’,y’) from its corresponding location
(x,y) = T-1(x’,y’) in the first image
Q: what if pixel comes from “between” two pixels?
A: Interpolate color value from neighbors
nearest neighbor, bilinear, Gaussian, bicubic
Check out interp2 in Matlab

41. Forward vs. inverse warping
Q: which is better?
A: usually inverse—eliminates holes
however, it requires an invertible warp function—not always possible...