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CS 691 Computational Photography Instructor: Gianfranco Doretto Image Pyramids.

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Presentation on theme: "CS 691 Computational Photography Instructor: Gianfranco Doretto Image Pyramids."— 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. Basis Vectorized image transformed image

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

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

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

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

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 –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 image Overcomplete representation. Low-pass filters, sampled appropriately for their blur. Gaussian pyramid

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) Created from Gaussian pyramid by subtraction Gaussian Pyramid (low-pass images)

19 Laplacian filter Gaussian Unit impulseLaplacian of Gaussian - ≅

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

21 Can we reconstruct the original? interpolate Need this!

22 The Laplacian Pyramid Synthesis –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 = * Pixel domain image Overcomplete representation. Transformed pixels represent bandpassed image information. Laplacian pyramid

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

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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