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1、Haar Wavelet Analysis,吳育德 陽明大學(xué)放射醫(yī)學(xué)科學(xué)研究所 臺北榮總整合性腦功能實(shí)驗(yàn)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 Decomp

2、osition 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) Subsequen

3、t 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 us

4、ed 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 (father wavelet) Wavelet (mother wavelet) These two function

5、s 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

6、of the signal. Disadvantages of Harr wavelet: discontinuous and does not approximate continuous signals very well.,Figure 2,Daubechies 8,Dubieties 3,Daubechies 4,Chap 6,4.2.2 Basic Properties of the Haar Scaling Function,The Haar Scaling function is defined as,(x-k) : same graph but translated by to

7、 the right (if k0) 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 discontinuities at x=0,1,3, and 4,Let V1 be th

8、e 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

9、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,A function f(x) belongs to V0 iff f(2jx) belong

10、s to Vj,A function f(x) belongs to Vj iff f(2-jx) belongs to V0,How to decompose a signal into its Vj-components,When j is large, the graph of (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 pro

11、duct,Theorem:,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 , we

12、 need:,Harr wavelet,Theorem 4.8 (extend to Vj),Decomposing Vj,Theorem:,4.3 Haar Decomposition and Reconstruction Algorithms,Implementation,Step 1 : Approximate the original signal f by a step function of the form,Example 4.11,General decomposition scheme,Wj-1-component,,,Vj-1-component,,,Theorem 4.1

13、2 (Haar Decomposition),Example 4.13,V8-component,V7-component,V6-component,V4-component,W7-component,4.3.2 Reconstruction,General reconstruction scheme,General reconstruction scheme,Theorem 4.14 (Haar Reconstruction),Example 4.15,80% compression,90% compression,sample signal,4.3.3 Filters and Diagrams,,,k=-1,0,k=-1,0,Decomposition algorithm,downsampling operator,Reconstruction,,,k=0,1,k=0,1,,upsampling operator,Summary,,,,,