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Multiresolution Analysis (Section 7.1) CS474/674 – Prof. Bebis
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Multiresolution Analysis Many signals or images contain features at different scales or level of detail (e.g., people vs buildings) Analyzing information at the same resolution or at a single scale will not be sufficient!
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Multiresolution Analysis (cont’d) Many signals or images contain features at different scales or level of detail (e.g., people vs buildings) Small size objects should be examined at a high resolution Large size objects should be examined at a low resolution Equivalent to using windows of multiple size! High resolution (high frequencies) Low resolution (low frequencies) Intermediate resolution
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Multiresolution Analysis (cont’d) We will review two techniques for representing multiresolution information efficiently: –Pyramidal coding –Subband coding
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high resolution j = J low resolution j=0 A collection of decreasing resolution images arranged in the shape of a pyramid. Image Pyramid (low scale) (high scale) If N=256, the pyramid will have 8+1=9 levels
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Pyramidal coding Two pyramids: approximation and prediction residual averaging mean pyramid Gaussian Gaussian pyramid no filter subsampling pyramid nearest neighbor biliner bicubic
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Pyramidal coding (cont’d) Approximation pyramid (based on Gaussian filter) Prediction residual pyramid (based on bilinear interpolation) Last level same as that in the approximation pyramid
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Pyramidal coding (cont’d) In the absence of quantization errors, the approximation pyramid can be re-constructed from the prediction residual pyramid. Keep the prediction residual pyramid only! Much more efficient to represent!
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Subband coding Decompose an image (or signal) into a set of 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) Analysis/Synthesis are performed using appropriate filters.
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Subband coding – 1D Example f lp (n): approximation of f(n) f hp (n): detail of f(n) 2-band decomposition
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Subband coding (cont’d) For perfect reconstruction, the synthesis filters (g 0 (n) and g 1 (n)) must be modulated versions of the analysis filters (h 0 (n), h 1 (n)): or original modulated
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Subband coding (cont’d) Of special interest, are filters satisfying orthonormality conditions: An orthonormal filter bank can be designed from a single prototype filter: (non-orthonormal filters require two prototypes)
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Subband coding (cont’d) Example: orthonornal filters
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Subband coding – 2D Example 4-band decomposition (using separable filters) approximation horizontal detail vertical detail diagonal detail LL LH HL HH
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Subband coding (cont’d) approximation vertical detail horizontal detail diagonal detail
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