Presentation on theme: "A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001"— Presentation transcript:
1A first look Ref: Walker (Ch.2) Jyun-Ming Chen, Spring 2001 Daubechies WaveletsA first lookRef: Walker (Ch.2)Jyun-Ming Chen, Spring 2001
2IntroductionA family of wavelet transforms discovered by Ingrid DaubechiesConcepts similar to Haar (trend and fluctuation)Differs in how scaling functions and wavelets are definedlonger supportsWavelets are building blocks that can quickly decorrelate data.
3Haar Wavelets Revisited The elements in the synthesis and analysis matrices are
6How we got the numbers Orthonormal; also lead to energy conservation AveragingOrthogonalityDifferencing
7How we got the numbers (cont) See Haar.ppt (p.30)
8Daubechies Wavelets How they look like: Translated copy dilation Scaling functionsWavelets
9Daub4 Scaling Functions (n-1 level) Obtained from natural basis(n-1) level Scaling functionswrap around at end due to periodicityEach (n-1) level functionSupport: 4Translation: 2Trend: average of 4 valuesDiscrete sampling of the fractal curve; I.e., if you give enoughSamples, you’ll see the complete “fractal” shape.
10Daub4 Scaling Function (n-2 level) Obtained from n-1 level scaling functionsEach (n-2) scaling functionSupport: 10Translation: 4Trend: average of 10 valuesThis extends to lower levels
11Daub4 Wavelets Similar “wrap-around” Obtained from natural basis Support/translation:Same as scaling functionsExtends to lower-levels
12Numbers of Scaling Function and Wavelets (Daub4)
13Property of Daub4If 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 derivative2nd derivative: curvature of f(x)
20More on Wavelets (Daub4) SynthesisFilter Q3SynthesisFilter Q2SynthesisFilter Q1
21Summary Daub4 (N=32) j=5 j=4 j=3 j=2 In general N=2n support 1 4 10 22 ?translation28
22Analysis and Synthesis There is another set of matrices that are related to the computation of analysis/decomposition coefficientIn the Daubechies case, they are also the transpose of each otherLater we’ll show that this is a property unique to orthogonal wavelets