Multiresolution Analysis (Section 7.1)

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

2. Multiresolution Analysis Small size objects should be examined at a high resolution. Large size objects should be examined at a low resolution. Many signals or images contain features at various levels of detail (i.e., scales).

3. Multiresolution Analysis (cont’d) Local image statistics are quite different from global image statistics. Modeling entire image is difficult or impossible. Need to analyze images at multiple levels of detail.

4. Multiresolution Analysis (cont’d) • Do you remember the issue of choosing the “right” window size in Short-Time Fourier Transform (STFT)? • Multiresolution methods analyze a signal or image using windows of different size! • As a result, multiresolution analysis provides information simultaneously in the spatial and frequency (i.e., scale) domains.

5. Multiresolution Analysis and Wavelets • Mallat (1987) showed that wavelets unify a number of multiresolution techniques, including: • Pyramidal coding (image processing). • Subband coding (signal processing)

6. Image Pyramids A collection of decreasing resolution images arranged in the shape of a pyramid. low resolution j=0 high resolution j = J

7. Pyramidal coding Approximation and prediction residual pyramids: Downsampling: half resolution averaging  mean pyramid Gaussian  Gaussian pyramid no filter  subsampling pyramid nearest neighbor biliner bicubic Upsampling: double resolution A variety of approximation/interpolation filters can be used!

8. Pyramidal coding (cont’d) Approximation pyramid (based on Gaussian filter) Prediction residual pyramid (based on bilinear interpolation) Top levels are the same

9. Pyramidal coding (cont’d) In the absence of quantization errors, the approximation pyramid can be constructed from the prediction residual pyramid.

10. Subband coding • Decompose an image (or signal) into a set of different frequency bands (i.e., subbands). • Decomposition is performed so that the subbands can be reassembled to reconstruct the original image without error. • Decomposition/Reconstruction are performed using digital filters.

11. Digital Filter - Example main components: unit delays, multipliers, adders K-order filter h(n): impulse response. FIR filter (finite impulse response)

12. Digital Filter – Example (cont’d) order reversal sign reversed original order reversal modulation order reversal and modulation

13. Subband coding (cont’d) Example: 1D, 2-band decomposition flp(n): approximation of f(n) 1D fhp(n): detail of f(n)

14. Subband coding (cont’d) • There are many 2-band, real coefficient, FIR perfect reconstruction filter banks. • In all of them, the synthesis filters (g0(n) and g1(n)) are modulated versions of the analysis filters (h0(n), h1(n)).

15. Subband coding (cont’d) • For perfect reconstruction, h0(n), h1(n), g0(n) and g1(n) must be related in one of the following two ways: • Also, they are bio-orthogonal: or

16. Subband coding (cont’d) • Of special interest, are filters satisfying orthonormality conditions: • An orthonormal filter bank can be designed from a single prototype filter; all other filters are computed from the prototype (biorthogonal filters require two prototypes).

17. Subband coding (cont’d) Example: orthonornal filters

18. Subband coding (cont’d) Example: 2D, 4-band decomposition (using separable filters) approximation vertical detail 2D horizontal detail diagonal detail

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