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Wavelets

Wavelets Fast Multiresolution Image Querying Jacobs et.al. SIGGRAPH95 Outline Overview / Background Wavelets 2D Image matching L1, L2 metrics Wavelet metric Evaluation Use in 3D Image matching 2D analogue of 3D shape matching Raster instead of XYZ What are we trying to match?

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Wavelets

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  1. Wavelets Fast Multiresolution Image QueryingJacobs et.al. SIGGRAPH95

  2. Outline • Overview / Background • Wavelets • 2D Image matching • L1, L2 metrics • Wavelet metric • Evaluation • Use in 3D

  3. Image matching • 2D analogue of 3D shape matching • Raster instead of XYZ

  4. What are we trying to match? • Looking for different images of the same things? • Different projections • Different colors • Looking for images that look the same? • Look for similar shapes and colors • Metric to discern like human eye

  5. Wavelets • Decompose a signal into component parts • Fourier analysis: a signal can be represented as a (possibly infinite) sum of sine and cosine functions • Signal becomes a set of wavelet coefficients • Coefficients represent features of signal

  6. Wavelets II • Signal can be completely reconstructed from all the coefficients • Signal can be partially reconstructed from some coefficients

  7. Wavelets for 2D Images • Each color plane in image is signal • Coefficients will represent visual features in the image • Store as many coefficients as needed • Image compression (see next slide) • c.f. Statistical shape descriptors

  8. Wavelet Reconstruction SIGGRAPH 96 Course Notes: Wavelets in Computer Graphics

  9. Comparing Images • Develop a metric that describes how closely two images match • Smaller difference in metric = images more similar

  10. Image metrics • Comparing images Q and T, with dimensions i,j • L1-Norm • For each pixel in Q, calculate the difference between Q[i,j] and T[i,j] • Add absolute value of differences of all i,j to form metric

  11. Image metrics II • L2-Norm • For each pixel in Q, calculate the square of the difference between Q[i,j] and T[i,j] • Add for all i,j, and take square root • Better than L1?

  12. Image metrics III • Problems with L1 and L2 • Expensive to compute / compare: O(i*j) • Not discriminating in cases with • Color Shift • Misregistration • Noise / Dithering • In general, not good descriptors • c.f. D1, D2 in 3D

  13. Wavelets as image metrics • Capture features of images e.g. edges in coefficients • Use Haar wavelets • Square basis functions • Easy to implement and compute • Calculate coefficients, truncate, quantize

  14. Truncation • 128x128 image has 1282 coefficients • Truncation = only storing largest ‘n’ coefficients • ‘n’ ~ 40-60 depending on exact use • Discarding smaller coefficients discards high frequency information i.e. detail • Loss of that information is desirable

  15. Quantization • Reduce precision of wavelet magnitude • Large +ve  +1 • Large –ve  -1 • Else  0 • Turns out this works well for matching

  16. Wavelet metric • Q,T are query and target image coefficients • w is weighting function

  17. Weighting function • Weighting function applied to give particular pairs of coefficients different significance in comparison • Gives ability to statistically tune the metric • Determined experimentally from dataset (Appendix A) • Weights expensive to calculate • Compute fewer • Bins to map range of i,j onto a weight

  18. Wavelet metric II • For i=0, j=0 value in Q and T is proportional to the average overall color • From quantization, use ≠

  19. Calculating coefficients • Standard two-dimensional Haar wavelet decomposition • Decompose each row, then decompose each column of the result • Trivial to implement

  20. Wavelet metric III • Final metric is • T[0,0] • Sign, i and j of n largest coefficients in T

  21. Faster Matching • To speed up matching, use 6 arrays • One for each combination of R,G,B,+,- • DR+,DR-, … • Each i,j in Dx is a list of all images with a metric coefficient in that color range, with that sign

  22. Evaluation • Better than L1 and L2 • Matches to ~1% of database • Compact • Fast to compare: similar complexity to an 8x8 pixel image L1/L2 for any resolution • More robust • Misregistration • Color shifting • Dithering • Different resolutions

  23. Evaluation II

  24. Evaluation III • Limits • Scaling ~1.5 times • Rotation ~20 degrees • Translation ~ 15% of width

  25. Matching in 3D • Can this be used in 3D as well • Compares image rather than geometry • Render 3D into voxels? • Projection of 3D object into 2D? • In general, other 3D specific methods probably much better

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