CS 691 Computational Photography Instructor: Gianfranco Doretto Image Pyramids

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

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

= * Identity transform Pixel domain image 1000… 01000… 00100… 000100… = * …0001000… …00010 …0001 Pixel domain image Spatial bases are local: each transform coefficient depends on one pixel location. Pixel domain image

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

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

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

Overview Gaussian Pyramid Laplacian Pyramid

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

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

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

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

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

Overview Gaussian Pyramid Laplacian Pyramid

What does blurring take away? original

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

High-Pass filter smoothed – original

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

Laplacian filter - ≅ Unit impulse Gaussian Laplacian of Gaussian

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

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

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

Laplacian pyramid

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

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

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