1. General Orthonormal MRA Ref: Rao & Bopardikar, Ch. 3

2. Outline MRA characteristics –Nestedness, translation, dilation, … Properties of scaling functions Properties of wavelets Digital filter implementations

3. Recall Formal Definition of an MRA An MRA consists of the nested linear vector space such that There exists a function  (t) (called scaling function) such that is a basis for V 0 If and vice versa Remarks: –Does not require the set of  (t) and its integer translates to be orthogonal (in general) –No mention of wavelet

4. Properties of Scaling Functions Explained using Haar basis

5. Dilation of Scaling Functions

6. Nested Spaces Every vector in V 0 belongs to V 1 as well –In particular  (t) Possible to express  (t) as a linear combination of the basis for V 1

7. Haar may be misleading … One can translate an arbitrary function by integers and compress it by 2; BUT there is no reason to think that the spaces V j created by the function and its translates and dilates will necessarily be nested in each other V0V0 V1V1 Remark

8. Two-Scale Relations (Scaling Fns)

9. Constraints on c(n)

10. Orthogonal Projection in Subspaces Finer approx Coarser approx See next page

11. From previous page Finer coefficients and coarser ones are related by c(n)

12. Properties of Wavelets Orthogonality

13. Two-Scale Relations (wavelet)

14. We showed : Similarly : Constraints on c(n) and d(n)

15. Function Reconstruction See next page

16. Detail coefficients and finer representation are related by d(n)

17. Nested Space VNVN V N-1 W N-1 V N-2 W N-2 V N-3 W N-3

18. Digital Filter Implementation Use existing methodology in signal processing for discrete wavelet computation

19. Digital Filter Implementation Recall Then

22. Similarly, …

23. Signal Reconstruction

24. Subdivision … getting a(1,n): Zero insertion (upsampling) and convolve with 2H n=0

25. Detail part: … getting a(1,n): upsampling and convolve with 2G n=0 Similarly, …

26. Notations of Digital Filters

27. Interpolator and Decimator

28. analysis filter bank perfect reconstruction pair: Whatever goes into analysis bank is recovered perfectly by the synthesis bank synthesis filter bank

29. Haar Revisited Analysis Filters 0 9 7 3 5 0 8 5 4 2.5 2 0 1 2 2.5 2 0 h(-n) 0.5 0 g(-n) 0.5 -0.5 Haar:

30. Haar Revisited Synthesis Filters 01 2 h(n) 1 0 2 g(n) 0 9 7 3 5 0 8 4 2 0 1 2 1 11