1. CS 691 Computational Photography
Instructor: Gianfranco Doretto
Image Pyramids

2. Linear image transformations
In analyzing images, it’s often useful to make a change of basis.
transformed image
Vectorized image
Basis

3. Spatial Domain Basis Basis functions: Tells you where things are….
… but no concept of what it is
…………..

4. = * Identity transform Pixel domain image
1000…
01000…
00100…
000100… = *
… …
…00010
…0001
Pixel domain image
Spatial bases are local: each transform coefficient depends on one pixel location.
Pixel domain image

5. Fourier Domain Basis Basis functions:
Tells you what (frequency) is in the image….
… but not where it is
………
………

6. = * Fourier transform Fourier transform
Fourier bases are global: each transform coefficient depends on all pixel locations.
Pixel domain image

7. Image Analysis Want representation that combines what and where.
 Image Pyramids

8. Overview
Gaussian Pyramid
Laplacian Pyramid

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

10. Gaussian pyramid construction
filter mask
Repeat
Filter (keep filter the same size)
Subsample (by a factor of two)
Until minimum resolution reached
can specify desired number of levels (e.g., 3-level pyramid)
Total number of pixels in pyramid?
1 + ¼ + 1/16 + 1/32…….. = 4/3  Over-complete representation

11. 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

12. What are they good for? Improve Search Pre-computation Compression
Search over translations
Classic coarse-to-fine strategy
Search over scale
Template matching
E.g. find a face at different scales
Pre-computation
Need to access image at different blur levels
Useful for texture mapping at different resolutions (called mip-mapping)
Compression
Capture important structures with fewer bytes
Denoising
Model statistics of pyramid sub-bands
Image blending

13. = * Gaussian pyramid Pixel domain Gaussian pyramid image
Overcomplete representation. Low-pass filters, sampled appropriately for their blur.

14. Overview
Gaussian Pyramid
Laplacian Pyramid

15. What does blurring take away?
original

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

17. High-Pass filter
smoothed – original

18. Band-pass filtering Laplacian Pyramid (subband images)
Gaussian Pyramid (low-pass images)
Laplacian Pyramid (subband images)
Created from Gaussian pyramid by subtraction

19. Laplacian filter - ≅
Unit impulse
Gaussian
Laplacian of Gaussian

20. Laplacian pyramid algorithm
subsample
blur _
blur
blur _
subsample
subsample _

21. Can we reconstruct the original?
Need this!
interpolate
interpolate
interpolate

22. The Laplacian Pyramid Synthesis Analysis
preserve difference between upsampled Gaussian pyramid level and Gaussian pyramid level
band pass filter - each level represents spatial frequencies (largely) unrepresented at other levels
Analysis
reconstruct Gaussian pyramid, take top layer

23. Laplacian pyramid

24. = * Laplacian pyramid Laplacian pyramid Pixel domain image
Overcomplete representation. Transformed pixels represent bandpassed image information.

25. Hybrid Image in Laplacian Pyramid
Extra points for project 1
High frequency  Low frequency

26. Slide Credits
This set of sides also contains contributions kindly made available by the following authors
Alexei Efros
Svetlana Lazebnik
Frédo Durand
Bill Freeman
Steve Seitz
Derek Hoiem
David Forsyth