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Understanding Image Pyramids: A Comprehensive Guide to Gaussian and Laplacian Representations

This lecture explores the essential concepts of image pyramids, including Gaussian and Laplacian pyramids. Image pyramids provide a multi-scale representation of images, facilitating effective analysis and processing. We delve into the construction of Gaussian pyramids through filtering and subsampling, and we discuss the role of the Laplacian pyramid in capturing vital details for tasks such as image blending, denoising, and compression. This work emphasizes the significance of spatial frequency representation, aiding in both classic search strategies and texture mapping in graphics.

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Understanding Image Pyramids: A Comprehensive Guide to Gaussian and Laplacian Representations

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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 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

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

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

  19. Laplacian filter - ≅ Unit impulse Laplacian of Gaussian Gaussian

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

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

  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. 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 contributionskindly made available by the following authors • Alexei Efros • Svetlana Lazebnik • Frédo Durand • Bill Freeman • Steve Seitz • Derek Hoiem • David Forsyth

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