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CS 691 Computational Photography

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Presentation on theme: "CS 691 Computational Photography"— Presentation transcript:

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?

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

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