Multiresolution Analysis (Section 7.1) CS474/674 – Prof. Bebis

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!

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

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

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

Pyramidal coding Two pyramids: approximation and prediction residual averaging  mean pyramid Gaussian  Gaussian pyramid no filter  subsampling pyramid nearest neighbor biliner bicubic

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

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!

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.

Subband coding – 1D Example f lp (n): approximation of f(n) f hp (n): detail of f(n) 2-band decomposition

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

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)

Subband coding (cont’d) Example: orthonornal filters

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

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