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Multiresolution Analysis

Multiresolution Analysis. CS474/674 – Prof. Bebis. Multiresolution Analysis. Use windows of different size (i.e., vary scale). Many signals or images contain information at different scales or levels of detail (e.g., people vs buildings)

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Multiresolution Analysis

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  1. Multiresolution Analysis CS474/674 – Prof. Bebis

  2. Multiresolution Analysis Use windows of different size (i.e., vary scale) Many signals or images contain information at different scales or levels of detail (e.g., people vs buildings) Analyzing information at the same scale will not be effective.

  3. Multiresolution Analysis (cont’d) Alternatively, use the same window size but analyze information at different resolutions: Small size objects should be examined at a high resolution High resolution Large size objects should be examined at a low resolution Low resolution

  4. Multiresolution Analysis (cont’d) • We will review two techniques for representing multiresolution information efficiently: • Pyramidal coding • Subband coding

  5. Image Pyramids A collection of decreasing resolution images arranged in the shape of a pyramid.

  6. Image Pyramids (cont’d) (high scale) low resolution j=0 Example: if N=256, there will be 8+1=9 levels (low scale) high resolution j = J

  7. Pyramidal coding • An efficient representation of image pyramids: averaging  mean pyramid Gaussian  Gaussian pyramid no filter  subsampling pyramid nearest neighbor biliner bicubic (details)

  8. Pyramidal coding (cont’d) Approximation pyramid (based on Gaussian filter) Prediction residual pyramid (based on bilinear interpolation) Note: the last level is the same as that of the approximation pyramid

  9. Pyramidal coding (cont’d) In the absence of quantization errors, the approximation pyramid can be re-constructed from the prediction residual pyramid. (details) (details) (details) • We only need to keep the prediction residual pyramid only! • - More efficient representation

  10. Pyramidal coding (cont’d) Just add the “details” or “residual error” back! Up-sample & Interpolate + (details)

  11. Subband coding Decompose an image (or signal) into different frequency bands (analysis step). Decomposition is performed so that the subbands can be re-assembled to reconstruct the original image without error(synthesis step) Need to choose appropriate filters (i.e., “filter bank”)

  12. Subband coding – 1D Example 2-band decomposition flp(n): approximation of f(n) low pass fhp(n): detail of f(n) high pass

  13. Subband coding (cont’d) • It can be shown that for perfect reconstruction, the synthesis filters (g0(n) and g1(n)) must be modulated versions of the analysis filters (h0(n), h1(n)): or original modulated

  14. Subband coding (cont’d) • An orthonormal filter bank can be designed from oneprototype filter: (non-orthonormal filters require two prototype filters)

  15. Subband coding (cont’d) Example: orthonornal filters prototype filter

  16. Subband coding – 2D Example 4-band decomposition (using separable filters) approximation LL low pass vertical detail LH horizontal detail high pass HL diagonal detail HH

  17. Subband coding (cont’d) horizontal detail approximation diagonal detail vertical detail

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