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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. transformed image Vectorized image Basis

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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**

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

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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 Fourier transform**

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

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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 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

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**= * Gaussian pyramid Pixel domain Gaussian pyramid image**

Overcomplete representation. Low-pass filters, sampled appropriately for their blur.

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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)**

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

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Laplacian filter - ≅ Unit impulse Gaussian Laplacian of Gaussian

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**Laplacian pyramid algorithm**

subsample blur _ blur blur _ subsample subsample _

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**Can we reconstruct the original?**

Need this! interpolate interpolate interpolate

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**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

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Laplacian pyramid

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**= * Laplacian pyramid Laplacian pyramid Pixel domain image**

Overcomplete representation. Transformed pixels represent bandpassed image information.

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**Hybrid Image in Laplacian Pyramid**

Extra points for project 1 High frequency Low frequency

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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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