Wavelets and Filter Banks小波与滤波器组,中国科学院自动化研究所,参考书 Wavelet and filter banks, G. Strang, T. Nguyen, Wellesley-Cambridge Press, 1997 （据说有翻译版，也有MIT的ppt 中文有翻译，也可参考瑞士联邦工学院M. Vetterlli的ppt） 多抽样率信号处理，宗孔德，清华大学出版社，1996。 Multirate systems and filter banks, Vaidyanathan, PP., Englewood Cliffs, New Jersey, Prentice Hall Inc. 1993. A wavelet tour of signal processing, S. Mallat, Academic Press. NY, 1998 Ten Lectures on Wavelets, Ingrid Daubechies, 1992 Matlab 6.5, M.,Background needed,Mathematics: Linear algebra （线性代数） Polynomial Theory（多项式理论） Mathematical analysis （数学分析） Functional analysis （泛函分析） Signal processing （信号处理） Image processing （图像处理） Matlab programming,课程名称解释,Filter banks（滤波器组）=a set of filters, filter is widely used in many fields of engineering and science for a long time. Wavelet （小波）, an old and new tool to produce filter banks, have been thoroughly studied in past 20 years. Here we use wavelets to indicate many kinds of wavelets with different properties.,Contents （课程目录）,Signal processing basic （信号处理基础） Filter bank （滤波器组） Mathematical basic MRA (multiresolution analysis) （多尺度分析） Wavelet lifting scheme （小波提升算法） Two dimensional wavelet （二维小波） The application of wavelet （小波应用） Wavelet domain denoising （小波域去噪） Fast object searching （快速目标搜索）,Contents (cont.),Wavelet domain image deconvolution （小波域图像反卷积） Wavelet domain image super-resolution （小波域图像超分辨率） Wavelet domain image compression and post-processing （小波域图像压缩和后处理） Wavelet domain image fusion and mosaicing （小波域图像融合和拼接） Filter approximation （滤波器逼近） Adaptive wavelet (pyramid) （自适应小波）,Contents (cont.),Advances of wavelet now: （小波前沿） Nonlinear signal transform: （非线性信号变换） Empirical Mode Decomposition (Hilbert-Huang transform) （经验模式分解） Local narrow band signal based decomposition（基于局部窄带信号分解） Geometry wavelet in 2D (optional) （几何小波） Image decomposition (optional) （图像分解）,Contents (cont.),Some ideas in life and research （杂谈） How to win before forty ,信号处理基础,Signal （信号）: x(t) or x(n) Filter（滤波器）: a vector, h=h(n), for a given signal x(n), the process of filtering: y=h*x, where * is the convolution operator: FIR=Finite Impulse Response=finite length（有限脉冲响应滤波器） IIR=Infinite Impulse Response=infinite length （无限脉冲响应滤波器） Example of filtering:x=sin(-4:0.08:4)+0.1*randn(1,101);h=1 1 1 1/4;y=x*h,Continuous Fourier Transform（连续Fourier 变换） Some basic properties: Linearity （线性性） Parseval Identity: （Parseval 等式）,Inner product （内积） F_1 and F_2 are two functions in L_2, the inner product of these two functions is defined as: Orthogonality （正交性） If =0, we say they are orthogonal.,Biorthogonality （双正交性） Two sets of function F_j and G_j, if =1 if j=k and 0 otherwise. We call the two set of functions are biorthogonal. Compact support （紧支撑） For a given function f, supp(f)=x|f(x) is not 0 If measure(supp(f) is finite, we say f is compactly supported or f has compact support. Corresponding to FIR,Basis and frames （基和框架） Basis: unique representation; linear independence（线性独立） and completeness（完备） Frame: linear dependence and completeness but stable Riesz basis: stable basis,Z transform （Z-变换）,Lowpass Filter（低通滤波器）=moving average=passing low frequency h=h(n), if sum of h is not zero, we call it a lowpass filter, in most time, the sum of h is 1. H(z), H(1)=1 Example: Simplest: H=1 1/2; (average) Spline: 1 2 1/4 General: h(n) Previous figure Highpass Filter（高通滤波器）=moving difference=passing high frequency,h=h(n), is called highpass filter is the sum of h is zero. H(z), H(1)=0 Examples: Simplest: h=1 1/2, difference Dual spline: -1 2 1/4; General:h(n) sum of h is 0. Figure,x as before, h=-1 2 1/4, y=x*h;,Magnitude response（幅频响应） of 1 1/2,Magnitude response of 1 2 1/4,Magnitude response of -1 2 -1/4,Phase（相位），Magnitude（幅度） is called the magnitude response of H. is called the phase of H If , we say H has linear phase （线性相位） H has linear phase is equivalent to say H is symmetric or antisymmetric （反对称）, Previous filters are symmetric, have linear phase,Invertibility （可逆性） or noninvertibility （不可逆性） Y(z)=H(z)X(z), If we want to reconstruct X, H can not be 0 at any point z. If H does not equal to 0 at any |z|=1, we say H is invertible, that is to say, we can reconstruct X by: X(z)=Y(z)/H(z), which is a inverse filtering, the filter is 1/H(z). But in most cases, H equals to 0 at some points, we can not reconstruct X exactly. Example: H(z)=1+0.5z is invertible, but H(z)=(1+z)/2 is not invertible. How to reconstruct a signal? We can use filter banks,Filter banks （滤波器组）=Lowpass + Highpass (inter-complement （互补）), Simplest idea: H0 and H1, where H0 is a lowpass filter, and H1 is a highpass filter, the lost information in the process of lowpass filtering can be fund in the output of the highpass filter. Some problems: How to reconstruct the signal? How to find such filter bank? How to reduce the computation and/or storage? Any more properties beside reconstruction?,Signal sampling 信号采样,Signal Sequence of numbers, x(-1), x(0), x(1), Unit impulse: x(n)=1 if n=0 and 0 otherwise Usually use Dir