MRA (from subdivision viewpoint)

1. MRA(from subdivision viewpoint) Jyun-Ming Chen Spring 2001

2. : record the combined effect of splitting and averaging done to the initial control pointsto achieve the limit f(x) The same limit curve can be defined from each iteration Using matrix notation : some yet undetermined functions; (later we’ll show they are the scaling functions) Nested Spaces We will show that every subdivision scheme gives rise to refinable scaling functions and, hence, to nested spaces

3. Subdivision (refinement) matrix Pj Represent the combined effect of splitting and averaging (both are linear operations) refinement relation for scaling functions Observe the similar relation for Haar and Daub4 Nested Spaces (cont)

4. Nested Space (cont) • The refinement relation states that each of the coarser scaling functions can be written as a linear combination of the finer scaling functions. • This linear combination depends on the subdivision(refinement) scheme used.

5. Define space Vj that includes all linear combinations of scaling functions (of j), whose dimension denoted by v(j) Then From Pj is a v(j) by v(j-1) matrix Nested Space

6. Wavelet Space • Define wavelet space Wj to be the complement of Vj in Vj+1 ; implying • Any function in Vj+1 can be written as the sum of a unique function in Vj and a unique function in Wj • The dimensions of these spaces are related • The basis for Wj are called wavelets • The corresponding scaling function space

7. Also write wavelet space Wj as linear combination using basis of next space Vj+1 From before, Combining, this is called the two-scale relation. Wavelet Space (cont)

8. VN WN-1 VN-1 WN-3 VN-3 VN-2 WN-2 Splitting of MRA Subspaces

9. Example: Haar

10. Two-Scale Relations (graphically)

11. Consider the approxi-mation of a function in some subspace Vj Assume the function is described in some scaling function basis Write these coefficients as a column matrix Suppose we wish to create a lower resolution version with a smaller number of coefficients v(j-1), this can be done by Aj is a constant matrix of dimension v(j-1) by v(j) Analysis Filters

12. To capture the lost details as another column matrix dj-1 Bj is a constant matrix of dimension w(j-1) by v(j), relating to Aj This process is called analysis or decomposition The pair of matrices Aj and Bj are called analysis filters Analysis/Decomposition

13. If analysis matrices are chosen appropriately, original signal can be recovered using subdivision matrices This process is called synthesis or reconstruction The pair of matrices Pj and Qj are called synthesis filters Synthesis/Reconstruction

14. Performing the splitting and averaging to bring cj-1 to a finer scale A perturbation by interpolating the wavelets Closer Look at Synthesis

15. Doing the aforementioned task repeatedly Recall Haar (next page) Filter Bank

16. A2 B2 Haar (Analysis)

17. Q2 P2 Haar (Synthesis)

18. Relation Between Analysis and Synthesis Filters • In general, analysis filters are not necessarily transposed multiples of the synthesis filters (as in the Haar case)

19. Dimension of filter matrices Aj: v(j-1) by v(j); Bj: w(j-1) by v(j) Pj: v(j) by v(j-1) Qj: v(j) by w(j-1) Hence are both square … and should be invertible Combining: We get: Analysis & Synthesis Filters

20. Orthogonal Wavelets

21. Orthogonal Wavelets • Scaling function orthogonal to one another in the same level • Wavelets orthogonal to one another in the same level and in all scales • Each wavelet orthogonal to every coarser scaling function • Haar and Daubechies are both orthogonal wavelets

22. Two row matrices of functions Define matrix Has these properties: Implication of Orthogonality

23. Implication of Orthogonality Changing subscript to j-1

24. Scaling functions Scaling functions are orthonormal only w.r.t. translations in a given scale Not w.r.t. the scale (because of the nested nature of MRA) Wavelets The wavelets are orthonormal w.r.t. scale as well as w.r.t. translation in a given scale Orthonormal …