1. Warping and Morphing

2. Warping
There are two ways to warp an image. The first, called forward mapping, scans through the source image pixel by pixel, and copies them to the appropriate place in the destination image. The second, reverse mapping, goes through the destination image pixel by pixel, and samples the correct pixel from the source image. The most important feature of inverse mapping is that every pixel in the destination image gets set to something appropriate. In the forward mapping case, some pixels in the destination might not get painted, and would have to be interpolated. We calculate the image deformation as a reverse mapping. The problem can be stated "Which pixel coordinate in the source image do we sample for each pixel in the destination image?"

5. Image Warping
Move pixels of image
Mapping
Resampling

6. Mapping Define transformation
Describe the destination (x, y) for every location (u, v) in the source image(or vice-versa, if invertible)

7. Example Mappings
Scale by factor
x = factor * u
y = factor * v

8. Example Mappings Rotate by  degrees x = ucos - vsin
y = usin + vcos

9. Other Mappings
Any function of u and v
x = fx(u, v)
y = fy(u, v)

10. Forward Mapping
Traverse input pixels
Miss/overlap output pixels W

11. Forward Mapping Iterate over source image
Some destination pixels may not be covered
Many source pixels can map to the same destination pixel

12. Inverse Mapping
Traverse output pixels
Does not waste work
W-1

13. Reverse Mapping Iterate over destination image Must resample source
May oversample, but much simpler!

14. Resampling Evaluate source image at arbitrary (u, v)
(u, v) does not usually have integer coordinates
Some kinds of resampling
Point resampling
Triangle filter
Gaussian filter
Source Image
Destination Image

15. Point Sampling Take value at closest pixel Simple, but causes aliasing
int iu = trunc(u + 0.5);
int iv = trunc(v + 0.5);
dst(x, y) = src(iu, iv);
Simple, but causes aliasing

16. Triangle Filter
Convolve with triangle filter

17. Triangle Filter Bilinearly interpolate four closest pixels
a = linear interpolation of src(u1, v2) and src(u2, v2)
b = linear interpolation of src(u1, v1) and src(u2, v1)
dst(x, y) = linear interpolation of ‘a’ and ‘b’

18. Gaussian Filter Convolve with Gaussian filter
Width of Gaussian kernel affects bluriness

19. Filtering Method Comparison
Trade-offs
Aliasing versus blurring
Computation speed

20. Image Warping Implementation
Reverse mapping
for (int x = 0; x < xmax; x++) {
for (int y = 0; y < ymax; y++) {
float u = fx-1(x, y);
float v = fy-1(x, y);
dst(x, y) = resample_src(u, v, w); }
Source Image
Destination Image

21. Polynomial Transformation
Geometric correction requires a spatial transformation to invert an unknown distortion function. The mapping functions, U and V , have been universally chosen to be global bivariate polynomial transformations of the form:
(20)
where aij and bij are constant polynomial coefficients
A first degree ( N=1) bivariate polynomial defines those mapping functions that are exactly given by a general 3 x 3 affine transformation matrix .

22. Polynomial Warping
In the remote sensing, the polynomial coefficients are not given directly. Instead spatial information is supplied by means of control points, corresponding positions in the input and output images whose coordinates can be defined precisely. In these cases, the central task of the spatial transformation stage is to infer the coefficients of the polynomial that models the unknown distortion. Once these coefficients are known, Eq.20 is fully specified and it is used to map the observed (x,y) points onto the reference ( u,v) coordinate system. This is also called polynomial warping. It is practical to use polynomials up to the fifth order for the transformation function.

24. Least – Squares with Ordinary Polynomials
From Equation (20) with N=2, coefficients aij can be determined by minimizing
This is achieved by determining the partial derivatives of E with respect to coefficients aij, and equating them to zero. For each coefficient aij, we have :
By considering the partial derivative of E with respect to all six coefficients, we obtain the system of linear equations.

25. Weighted Least Squares
As you see the least-squares formulation is global error measure - distance between control points ( xk, yk) and approximation points ( x,y).
The least –squares method may be localized by introducing a weighting function Wk that represents the contribution of control point (xk,yk) on point (x,y)
where  determines the influence of distant control points and approximating points

26. Pseudoinverse Solution
Let a correspondence be established between M points in the observed and reference images. The spatial transformation that approximates this correspondence is chosen to be a polynomial of degree N. In two variables ( x and y) , such a polynomial has K coefficients where
For example, a second-degree approximation requires only six coefficients to be solved. In this case , N=2 and K=6.
In matrix form:
U=WA ; V=WB
 WTU = WTWA A = (WTW)-1WTU ; B =(WTW)-1WTV

27. Warping Summary Reverse mapping is simpler to implement
Different filters trade-off speed and aliasing/blurring
Fun and creative warps are easy to implement

28. Image Morphing
Animate transition between two images

29. Cross-Dissoving Blend images blend(i, j) = (1-t)src(i, j) + tdst(i, j)

30. Image Morphing
Combines warping and cross-dissolving

31. Image Morphing Warping step is the hard one
Aim to align features in images

32. Image Morphing
There are two necessary components to morphing algorithms:
the user must have a mechanism to establish correspondence between the two images - perhaps as simple as silhouettes
from this correspondence, the algorithm must determine a dense pixel correspondence which is the warp algorithm
A common technique often used for facial morphing is named feature-based morphing. Important feature are chosen rather than a grid of points that must cover "unimportant" areas. Typical features might be ears, nose, and eyebrows. Lines are drawn in each image and are associated with each other. If there is only one pair of lines, the algorithm produces a purely local change. Therefore, a high spatial density is achieved in areas of interest.

33. Feature-based Morphing
Beier & Neeley use pairs of lines to specify warp
Given p in dst image, where is p’ in source image?
u is a fraction
v is a length(in pixels)

34. Feature-based Morphing
The mapping function is simple. An inverse mapping function is used.                                                                   

35. Feature-based Morphing
If more than one pair of lines is used, the transformations must be blended. Beier and Neeley of Pacific Data Images who developed the technique suggested the use of weight. (NOTE: Beier-92.pdf work).

36. Warping with One Line Pair
What happens to the ‘F’?

37. Warping with Multiple Line Pairs
Use weighted combination of points defined by each pair of corresponding lines
p’ is a weighted average

38. Weighting Effect of Each Line Pair
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

39. Warping Pseudocode WarpImage(Image, L’[], L[]) begin
foreach destination pixel p do
psum = (0, 0)
wsum = (0, 0)
foreach line L[i] in destination do
p’[i] = p transformed by (L[i], L’[i])
psum = psum + p’[i] * weight[i]
wsum += weight[i]
end
p’ = psum / wsum
Result(p) = Image(p’)

40. Morphing Pseudocode GenerateAnimation(Image0, L0[], Image1, L1[])
begin
foreach intermediate frame time t do
for i = 1 to number of line pairs do
L[i] = line t-th of the way from L0[i] to L1[i]
end
Warp0 = WarpImage(Image0, L0, L)
Warp1 = WarpImage(Image1, L1, L)
foreach pixel p in FinalImage do
Result(p) = (1-t)Warp0 + tWarp1

41. Morphing Summary
Specifying correspondences
Warping
Blending

42. Graphical Objects and Metamorphosis
Object O1
Object O2
Metamorphosis
Geometric
Data Set
Geometric
Data Set
Geometry Alignment
Attributes
Attributes
Attribute Interpolation

43. Warping Pipeline Generation of attributes Deform Adjust
Transformation of geometry
Generation of attributes

44. Morphing Pipeline
Deform
Adjust
Blend
Adjust
Deform

45. Warping Morphing Warping and Morphing Single object
Specification of original and deformed states
Morphing
Two objects
Specification of initial and final states

46. Warping and Morphing http://www-2.cs.cmu.edu/~seitz/vmorph/vmorph.html