1. A first look Ref: Walker (ch1) Jyun-Ming Chen, Spring 2001
Haar Wavelets
A first look
Ref: Walker (ch1)
Jyun-Ming Chen, Spring 2001

2. Introduction Simplest; hand calculation suffice
A prototype for studying more sophisticated wavelets
Related to Haar transform, a mathematical operation

3. Haar Transform
Assume discrete signal (analog function occurring at discrete instants)
Assume equally spaced samples (number of samples 2n)
Decompose the signal into two sub-signals of half its length
Running average (trend)
Running difference (fluctuation)

4. Haar transform, 1-level Running average
Multiplication by is needed to ensure energy conservation (see later)
Running difference
Denoted by:
Meaning of superscript explained later

5. Example

6. Inverse Transform

7. Small Fluctuation Feature
Magnitudes of the fluctuation subsignal (d) are often significantly smaller than those of the original signal
Logical: samples are from continuous analog signal with very short time increment
Has application to signal compression
Mallat order

8. Energy Concerns Energy of signals
The 1-level Haar transform conserves energy

9. Proof of Energy Conservation

10. Haar Transform, multi-level

11. Compaction of Energy Compare with 1-level
Can be seen more clearly by cumulative energy profile

12. Cumulative Energy Profile
Definition

13. Algebraic Operations Addition & subtraction Constant multiple
Scalar product

14. Haar Wavelets 1-level Haar wavelets “wavelet”: plus/minus wavy nature
Translated copy of mother wavelet
support of wavelet =2
The interval where function is nonzero
Significance of “compact” support:
Able to detect very sort-lived, transient, fluctuations in the signal (p. 35)…. What differs from FFT
Property 1. If a signal f is (approximately) constant over the support of a Haar wavelet, then the fluctuation value is (approximately) zero.

15. Haar Scaling Functions
1-level scaling functions
Graph: translated copy of father scaling function
Support = 2

16. Haar Wavelets (cont) 2-level Haar scaling functions support = 4
2-level Haar wavelets
support = 4
Regard these as “definitions”. (“defined as the following …”)
Put these in matrix form …

17. Multiresolution Analysis (MRA)
Natural basis:
Therefore:
The dimension of f (C3) is N.

18. MRA Note: the coefficient vectors
1. The dimension of C2 and D2 should be N/2 (seen from the number of basis functions, and number of coefficients to represent a function: c1, …, c(n/2)).
2. As to why the coefficients are determined by inner product is due to orthogonality of V’s and W’s:
Write f=c2 +d2 then you’ll see.

19. MRA If do it all the way through, representing the average of all data
Similarly, dimension of c1 and d1 is N/4
If do it all the way through,
representing the average of all data

20. Example

21. Example (cont)
Decomposition coefficients obtained by inner product with basis function
In fact, all V and W are functions (V(x), W(x)). Here we use 8 numbers to represent sampling of such functions over the designated interval (as f(x)).
Note: The coefficient of the basis represent some kind of time/frequency resolution …
Q: What if the scaling fns and wavelets are not orthogonal? How would we obtain the decomposition coeff?
How would this affect L2 norm of compression?

22. Haar MRA

23. More on Scaling Functions (Haar)
They are in fact related
Pj is called the synthesis filter (more later)
Up to now, we haven’t mentioned how the scaling functions and wavelets are obtained.

24. Ex: Haar Scaling Functions
Synthesis
Filter P3

25. Ex: Haar Scaling Functions
Synthesis
Filter P2
Synthesis
Filter P1

26. More on Wavelets (Haar)
They are in fact related
Qj is called the synthesis filter (more later)

27. Ex: Haar Wavelets
Synthesis
Filter Q3

28. Ex: Haar Wavelets
Synthesis
Filter Q1
Synthesis
Filter Q2

29. Analysis Filters
There is another set of matrices that are related to the computation of analysis/decomposition coefficient
In the Haar case, they are the transpose of each other
Later we’ll show that this is a property unique to orthogonal wavelets

30. Analysis/Decomposition (Haar)B2
Analysis
Filter Aj
Analysis
Filter Bj B3 A1 B1

31. Synthesis Filters
On the other hand, synthesis filters have to do with reconstructing the signal from MRA results

32. Synthesis/Reconstruction (Haar)Q2 P2 Q1 P1 Q3 P3
Synthesis
Filter Pj
Synthesis
Filter Qj

33. Conclusion/Exercise Haar (N=8) j=3 j=2 j=1 j=0 In general N=2n support4 8
2n-j
translation