小波与滤波器组-中科院wavelets and filter banks

1、Adaptive wavelet selectionPeng Silong2007-06-01Happy Childrens day!ProblemMany wavelet.Good ! vs Bad!Different? Or same?How to choose?Energy Compaction filter bankx(n)H0H1y1(n)22v1y0(n)v0v0 (or y0)have some properties:Maximal energy with arbitraryLength.Which one?formulationDifficultNonlinear Real t

2、ime conputationMoulins methodConclusionIt is a global methodLinear programming can be computed easilyBut not real timeFactorization is not very easyExtension to filter approximationFilter approximationSee pdfExamples 1d（a）双正交小波bior5.5的低通滤波器。（b）与bior5.5等长的Coiflet2小波。（c）和 （d）分别是以补零的harr 和db3为初始值的全局优化结

3、果。（e）是只采用分部栅格优化得到的结果。（f）对结果（e）用随机优化得到的结果。 Examples 1d逼近滤波器的谱和脉冲响应（a）信号的功率谱 ，（b）理想的主分量滤波器组。（d）和 （f）分别是用全局优化得到的有限脉冲响应和频谱。（c）和（e）分别是用分步随机栅格优化得到的有限脉冲响应和频谱。 Example 2d Given texture with direction, we have method to use directional filter:Approximated waveletOne low-passThree high-pass filters Example 2d

4、Left: position using DbRight: position using approximate wavelet same length, with better sparse Adaptive partial pyramid (by Peng Huang, & Yan, 2007)Traditional pyramidss0s1h0h12f02f1S0S1+22New problem Find x with half length and a filter h such that the reconstructed signal S0 approximates S best.

5、S2hS0 x?minimalprobelmgiven a signal , to seek a signal and an adaptive filter h , such that the following reconstruction error is minimal: (1)Where is a low resolution approximation of , and is an up-sampling operator , “*” denotes convolution operator. For h, we hope it is a wavelet filter as its

6、prior information:For a wavelet filter with P vanishing moments,Equivalent:optimizationWe solve the following problem:Where D is the constrain on h of vanishing moments.Solving problemBy using alternative iterationGiven initial condition of both x and h, then update x and h alternatively:Example: er

7、ror is much smallerError Measurement (E)db10 proposedthe 1st level8.9562e-0042.2765e-004the 2nd level0.00195.8771e-004the1st level3.2958e-0049.3240e-005the 2nd level7.5391e-0042.3281e-004Extend to image positionMSEdb5 bior3.7proposedLena28.499027.710925.2563Barbara168.6360170.6712110.2073Boat53.876653.318849.1429Cameraman132.1388129.1666118.5879examplePa