1. Image Pyramids and Blending
© Kenneth Kwan
Slides Modified from Alexei Efros, CMU,

2. Image Pyramids Known as a Gaussian Pyramid [Burt and Adelson, 1983]
In computer graphics, a mip map [Williams, 1983]
A precursor to wavelet transform

3. A bar in the big images is a hair on the zebra’s nose; in smaller images, a stripe; in the smallest, the animal’s nose
Figure from David Forsyth

4. Gaussian Pyramid for encoding
[Burt & Adelson, 1983]
Prediction using weighted local Gaussian average
Encode the difference as the Laplacian
Both Laplacian and the Averaged image is easy to encode

5. Gaussian pyramid

6. Image sub-sampling
1/8
1/4
Throw away every other row and column to create a 1/2 size image
- called image sub-sampling

7. Image sub-sampling 1/2 1/4 (2x zoom) 1/8 (4x zoom)
Why does this look so bad?

8. Sampling Good sampling: Bad sampling: Sample often or, Sample wisely
see aliasing in action!

9. Gaussian pre-filtering
Solution: filter the image, then subsample
Filter size should double for each ½ size reduction. Why?

10. Subsampling with Gaussian pre-filtering
Solution: filter the image, then subsample
Filter size should double for each ½ size reduction. Why?
How can we speed this up?

11. Compare with...
1/2
1/4 (2x zoom)
1/8 (4x zoom)

12. What does blurring take away?
original

13. What does blurring take away?
smoothed (5x5 Gaussian)

14. High-Pass filter
smoothed – original

15. Gaussian pyramid is smooth=> can be subsampled
Laplacian pyramid has narrow band of frequency=> compressed

17. Image Blending

18. + = Feathering Encoding transparency I(x,y) = (aR, aG, aB, a)1 + =
Encoding transparency
I(x,y) = (aR, aG, aB, a)
Iblend = Ileft + Iright

19. Affect of Window Size 1
left 1
right

20. Affect of Window Size 1 1

21. Good Window Size 1
“Optimal” Window: smooth but not ghosted

22. What is the Optimal Window?
To avoid seams
window >= size of largest prominent feature
To avoid ghosting
window <= 2*size of smallest prominent feature

23. Pyramid Blending 1 1 1
Left pyramid
blend
Right pyramid

24. Pyramid Blending

25. laplacian
level 4
laplacian
level 2
laplacian
level
left pyramid
right pyramid
blended pyramid

26. Laplacian Pyramid: Blending
General Approach:
Build Laplacian pyramids LA and LB from images A and B
Build a Gaussian pyramid GR from selected region R
Form a combined pyramid LS from LA and LB using nodes of GR as weights:
LS(i,j) = GR(I,j,)*LA(I,j) + (1-GR(I,j))*LB(I,j)
Collapse the LS pyramid to get the final blended image

27. Blending Regions

28. Horror Photo
© prof. dmartin

29. Simplification: Two-band Blending
Brown & Lowe, 2003
Only use two bands: high freq. and low freq.
Blends low freq. smoothly
Blend high freq. with no smoothing: use binary mask

30. 2-band Blending Low frequency (l > 2 pixels)
High frequency (l < 2 pixels)

31. Linear Blending

32. 2-band Blending

33. Gradient Domain
In Pyramid Blending, we decomposed our image into 2nd derivatives (Laplacian) and a low-res image
Let us now look at 1st derivatives (gradients):
No need for low-res image
captures everything (up to a constant)
Idea:
Differentiate
Blend
Reintegrate

34. Gradient Domain blending (1D)
bright
Two
signals
dark
Regular
blending
Blending
derivatives

35. Gradient Domain Blending (2D)
Trickier in 2D:
Take partial derivatives dx and dy (the gradient field)
Fidle around with them (smooth, blend, feather, etc)
Reintegrate
But now integral(dx) might not equal integral(dy)
Find the most agreeable solution
Equivalent to solving Poisson equation
Can use FFT, deconvolution, multigrid solvers, etc.

36. Comparisons: Levin et al, 2004

37. Perez et al., 2003

38. Perez et al, 2003 Limitations: editing
Can’t do contrast reversal (gray on black -> gray on white)
Colored backgrounds “bleed through”
Images need to be very well aligned

39. Don’t blend, CUT!
Moving objects become ghosts
So far we only tried to blend between two images. What about finding an optimal seam?

40. Davis, 1998 Segment the mosaic Single source image per segment
Avoid artifacts along boundries
Dijkstra’s algorithm

41. constrained by overlap
Efros & Freeman, 2001
block
Input texture B1 B2
Neighboring blocks
constrained by overlap B1 B2
Minimal error
boundary cut B1 B2
Random placement
of blocks

42. Minimal error boundary
overlapping blocks
vertical boundary _ = 2
overlap error
min. error boundary

43. Kwatra et al, 2003
Actually, for this example, DP will work just as well…

44. Lazy Snapping (Li el al., 2004)
Interactive segmentation using graphcuts