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CS 691 Computational Photography Instructor: Gianfranco Doretto Image Pyramids

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

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

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

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

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

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

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Overview Gaussian Pyramid Laplacian Pyramid

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

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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

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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

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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

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

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Overview Gaussian Pyramid Laplacian Pyramid

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What does blurring take away? original

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

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High-Pass filter smoothed – original

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

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Laplacian filter Gaussian Unit impulseLaplacian of Gaussian - ≅

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

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Can we reconstruct the original? interpolate Need this!

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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

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Laplacian pyramid

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

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

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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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