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

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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.

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

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Haar Revisited Synthesis Filter P 3 Synthesis Filter Q 3

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In Other Words

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

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How we got the numbers (cont)

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

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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

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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

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

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Numbers of Scaling Function and Wavelets (Daub4)

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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 2 nd derivative

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Property of Daub4 (cont)

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MRA

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Example (Daub4)

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More on Scaling Functions (Daub4, N=8) Synthesis Filter P 3

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Scaling Function (Daub4, N=16) Synthesis Filter P 3

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Scaling Functions (Daub4) Synthesis Filter P 2 Synthesis Filter P 1

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More on Wavelets (Daub4) Synthesis Filter Q 2 Synthesis Filter Q 1 Synthesis Filter Q 3

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Summary Daub4 (N=32) j=5j=4j=3j=2 In general N=2 n support141022? translation1248?

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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

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Analysis and Synthesis f

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MRA (Daub4)

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Energy Compaction (Haar vs. Daub4)

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

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Supplemental

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Conservation of Energy Define Therefore (Exercise: verify)

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Energy Conservation By definition:

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Orthogonal Wavelets By constructionHaar is also orthogonal Not all wavelets are orthogonal! –Semiorthogonal, Biorthogonal

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Other Wavelets (Daub6)

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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.

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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.

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Comparison (Daub20)

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Supplemental on Daubechies Wavelets

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

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Ex: Coif6

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