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

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.

Haar Wavelets Revisited The elements in the synthesis and analysis matrices are

Haar Revisited Synthesis Filter P3 Synthesis Filter Q3

In Other Words

How we got the numbers Orthonormal; also lead to energy conservation Averaging Orthogonality Differencing

How we got the numbers (cont) See Haar.ppt (p.30)

Daubechies Wavelets How they look like: Translated copy dilation Scaling functions Wavelets

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 Discrete sampling of the fractal curve; I.e., if you give enough Samples, you’ll see the complete “fractal” shape.

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

Daub4 Wavelets Similar “wrap-around” Obtained from natural basis Support/translation: Same as scaling functions Extends to lower-levels

Numbers of Scaling Function and Wavelets (Daub4)

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 2nd derivative: curvature of f(x)

Property of Daub4 (cont)

MRA

Example (Daub4)

More on Scaling Functions (Daub4, N=8) Synthesis Filter P3

Scaling Function (Daub4, N=16) Synthesis Filter P3

Scaling Functions (Daub4) Synthesis Filter P2 Synthesis Filter P1

More on Wavelets (Daub4) Synthesis Filter Q3 Synthesis Filter Q2 Synthesis Filter Q1

Summary Daub4 (N=32) j=5 j=4 j=3 j=2 In general N=2n support 1 4 10 22 ? translation 2 8

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 we’ll show that this is a property unique to orthogonal wavelets

Analysis and Synthesis f

MRA (Daub4)

Energy Compaction (Haar vs. Daub4)

How we got the numbers Orthonormal; also lead to energy conservation Orthogonality Averaging Differencing Constant Linear 4 unknowns; 4 eqns

Supplemental Here we re-interpret “difference” as “constant correlation”. We design D4 to capture both constant and linear correlation.

Conservation of Energy Define Therefore (Exercise: verify)

Energy Conservation By definition:

Orthogonal Wavelets By construction Haar is also orthogonal Not all wavelets are orthogonal! Semiorthogonal, Biorthogonal

Other Wavelets (Daub6) Wavelets are defined similarly.

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.

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.

Comparison (Daub20)

Supplemental on Daubechies Wavelets

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

Ex: Coif6