Daubechies Wavelets

1
Daubechies Wavelets

• A first look
• Ref Walker (Ch.2)
• Jyun-Ming Chen, Spring 2001

2
Introduction

• A family of wavelet transforms discovered by
  Ingrid Daubechies
• Concepts similar to Haar (trend and fluctuation)
• Differs in how scaling functions and wavelets are
  defined
• longer supports

Wavelets are building blocks that can quickly
decorrelate data.
3
Haar Wavelets Revisited

• The elements in the synthesis and analysis
  matrices are

4
Haar Revisited
Synthesis Filter P3
Synthesis Filter Q3
5
In Other Words
6
How we got the numbers

• Orthonormal also lead to energy conservation
• Averaging

• Orthogonality
• Differencing

7
How we got the numbers (cont)
8
Daubechies Wavelets

• How they look like
• Translated copy
• dilation

Scaling functions
Wavelets
9
Daub4 Scaling Functions (n-1 level)

• Obtained from natural basis
• (n-1) level Scaling functions
• wrap around at end due to periodicity
• Each (n-1) level function
• Support 4
• Translation 2
• Trend average of 4 values

10
Daub4 Scaling Function (n-2 level)

• Obtained from n-1 level scaling functions
• Each (n-2) scaling function
• Support 10
• Translation 4
• Trend average of 10 values

• This extends to lower levels

11
Daub4 Wavelets

• Similar wrap-around
• Obtained from natural basis
• Support/translation
• Same as scaling functions
• Extends to lower-levels

12
Numbers of Scaling Function and Wavelets (Daub4)
13
Property of Daub4

• If a signal f is (approximately) linear over the
  support of a Daub4 wavelet, then the
  corresponding fluctuation value is
  (approximately) zero.
• True for functions that have a continuous 2nd
  derivative

14
Property of Daub4 (cont)
15
MRA
16
Example (Daub4)
17
More on Scaling Functions (Daub4, N8)
Synthesis Filter P3
18
Scaling Function (Daub4, N16)
Synthesis Filter P3
19
Scaling Functions (Daub4)
20
More on Wavelets (Daub4)
Synthesis Filter Q3
21
Summary
22
Analysis and Synthesis

• There is another set of matrices that are related
  to the computation of analysis/decomposition
  coefficient
• In the Daubechies case, they are also the
  transpose of each other
• Later well show that this is a property unique
  to orthogonal wavelets

23
Analysis and Synthesis
24
MRA (Daub4)
25
Energy Compaction (Haar vs. Daub4)
26
How we got the numbers

• Orthonormal also lead to energy conservation
• Orthogonality

• Averaging
• Differencing
• Constant
• Linear

4 unknowns 4 eqns
27
Supplemental
28
Conservation of Energy

• Define
• Therefore (Exercise verify)

29
Energy Conservation

• By definition

30
Orthogonal Wavelets

• By construction

• Haar is also orthogonal
• Not all wavelets are orthogonal!
• Semiorthogonal, Biorthogonal

31
Other Wavelets (Daub6)
32
Daub6 (cont)

• Constraints
• If a signal f is (approximately) quadratic over
  the support of a Daub6 wavelet, then the
  corresponding fluctuation value is
  (approximately) zero.

33
DaubJ

• Constraints
• If a signal f is (approximately) equal to a
  polynomial of degree less than J/2 over the
  support of a DaubJ wavelet, then the
  corresponding fluctuation value is
  (approximately) zero.

34
Comparison (Daub20)
35
Supplemental on Daubechies Wavelets
36
(No Transcript)
37
Coiflets

• Designed for maintaining a close match between
  the trend value and the original signal
• Named after the inventor R. R. Coifman

38
Ex Coif6