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Haar Wavelet Analysis. 吳育德 陽明大學放射醫學科學研究所 台北榮總整合性腦功能實驗室. A First Course in Wavelets with Fourier Analysis Albert Boggess Francis J. Narcowich Prentice-Hall, Inc., 2001. Outlines. Why Wavelet Haar Wavelets The Haar Scaling Function Basic Properties of the Haar Scaling Function
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Haar Wavelet Analysis 吳育德 陽明大學放射醫學科學研究所 台北榮總整合性腦功能實驗室
A First Course in Wavelets with Fourier Analysis Albert Boggess Francis J. Narcowich Prentice-Hall, Inc., 2001
Outlines • Why Wavelet • Haar Wavelets • The Haar Scaling Function • Basic Properties of the Haar Scaling Function • The Haar Wavelet • Haar Decomposition and Reconstruction Algorithms • Decomposition • Reconstruction • Filters and Diagrams • Summary
4.1 Why Wavelet • Wavelets were first applied in geophysics to analyze data from seismic surveys. • Seismic survey geophones seismic trace Sesimic trace Direct wave (along the surface) Subsequent waves (rock layers below ground)
Fourier Transform (FT) is not a good tool – gives no direct information about when an oscillation occurred. • Short-time FT : equal time interval, high- frequency bursts occur are hard to detect. • Wavelets can keep track of time and frequency information. They can be used to “zoom in” on the short bursts, or to “zoom out” to detect long, slow oscillations
frequency frequency + time (equal time intervals) frequency + time
4.2 Haar Wavelets 4.2.1 The Haar Scaling Function • Wavelet functions • Scaling function Φ (father wavelet) • Wavelet Ψ (mother wavelet) • These two functions generate a family of functions that can be used to break up or reconstruct a signal • The Haar Scaling Function • Translation • Dilation
Using Haar blocks to approximate a signal • High-frequency noise shows up as tall, thin blocks. • Needs an algorithm that eliminates the noise and not distribute the rest of the signal. • Disadvantages of Harr wavelet: discontinuous and does not approximate continuous signals very well. Figure 2
Chap 6 Dubieties 3 Daubechies 4 Daubechies 8
4.2.2 Basic Properties of the Haar Scaling Function • The Haar Scaling function is defined as • Φ(x-k) : same graph but translated by to the right (if k>0) by k units • Let V0 be the space of all functions of the form
V0 consists of all piecewise constant functions whose discontinuities are contained in the set of integers • V0 has compact support. Typical element in V0 Figure 5 Figure 6 has discontinuitiesat x=0,1,3, and 4
Let V1 be the space of piecewise constant functions of finite support with discontinuities at the half integers has discontinuities at x=0,1/2,3/2, and 2
Suppose j is any nonnegative integer. The space of step functions at level j, denoted by Vj , , is defined to be the space spanned by the set over the real numbers. • Vj is the space of piecewise constant functions of finite support whose discontinuities are contained in the set • means no information is lost as the resolution gets finer. Vj contains all relevant information up to a resolution scale order 2-j
How to decompose a signal into its Vj-components • When j is large, the graph of Φ(2j x) is similar to one of the spikes of a signal that we may wish to filter out. • One way is to construct an orthonormal basis for Vj using the L2 inner product
4.2.4 The Haar Wavelet • We want to isolate the spikes that belong to Vj, but that are not members of Vj-1 • The way is to decompose Vj as an orthonormal sum of Vj-1 and its complement. • Start with V1, assume the orthonormal complement of Vo is generated by translates of some functions Ψ, we need:
Implementation • Step 1 : Approximate the original signal f by a step function of the form
General decomposition scheme Wj-1-component Vj-1-component
Example 4.13 V8-component V7-component V6-component V4-component W7-component
Example 4.15 sample signal 80% compression 90% compression
4.3.3 Filters and Diagrams • Decomposition algorithm • k=-1,0 • k=-1,0
Reconstruction • k=0,1 • k=0,1