A brief talk on compression perception (14): MATLAB implementation of Fourier matrix and wavelet transform matrix

Main content:

1. Fourier matrix and its MATLAB implementation
2. wavelet transform Matrix and its MATLAB implementation

Fourier matrix and its MATLAB implementation

Definition of the Fourier matrix: (Source: http://mathworld.wolfram.com/FourierMatrix.html)

The MATLAB implementation of the Fourier matrix:

Dftmtx (N) is the n-by-n complex matrix of values around  the unit-circle whose inner Product with a column vector  of Length N yields the discrete Fourier transform of the  vector. If X is a column vector of length N, then  dftmtx (N) *x yields the same result as FFT (X);   however,  FFT (X) is more efficient.

The inverse discrete Fourier transform matrix is Conj (DFTMTX (N))/N.

%clc;clear; N= -; X= Randn (N,1);d FT_RESULT1= DFTMTX (N) *X;DFT_RESULT2=FFT (X);% IsEqual = ALL (DFT_RESULT1 = =dft_result2); Err= Norm (Dft_result1 (:)-dft_result2 (:));ifErr <0.01fprintf ('dftmtx (N) *x yields the same result as FFT (X)');Elsefprintf ('dftmtx (N) *x does not yield the same result as FFT (X)'); end

Wavelet transform matrix and its MATLAB implementation

The concept of a wavelet transform matrix:

Reference: http://blog.csdn.net/jbb0523/article/details/42470103

MATLAB implementation of wavelet transform matrix:

function [WW] =dwtmtx (N,wtype,wlev)%dwtmtx discrete wavelet transform matrix%This function generates the transform matrix WW according to input%parameters N,wtype,wlev.%detailed explanation goes here% N isThe dimension of WW% Wtype isThe wavelet type% Wlev isThe number of decomposition level%note:the extension mode must be periodization ('per') [H,g]= Wfilters (Wtype,'D'); %decomposition low&High pass Filterl=length (h); %Filter lengthh_1= FLIPLR (h); %Flip Matrix left to rightg_1=FLIPLR (g); Loop_max=log2 (N); Loop_min=Double(Int8 (log2 (L))) +1;ifwlev>loop_max-loop_min+1fprintf ('\nwaring:wlev is too big\n'); fprintf ('The biggest Wlev is%d\n', loop_max-loop_min+1); Wlev= loop_max-loop_min+1; ENDWW=1; forloop = loop_max-wlev+1: Loop_max Nii=2^Loop; P1_0= [h_1 Zeros (1, nii-L)]; P2_0= [G_1 Zeros (1, nii-L)]; P1= Zeros (nii/2, NII); P2= Zeros (nii/2, NII);  forIi=1: nii/2P1 (ii,:)=circshift (P1_0', (ii-1) +1-(L-1) +l/2-1)'; P2 (ii,:)=circshift (p2_0', (ii-1) +1-(L-1) +l/2-1)'; End W1=[P1;P2]; MM=2^loop_max-Length (W1); W=[W1,zeros (Length (W1), mm), zeros (Mm,length (W1)), Eye (mm,mm)]; WW=ww*W; Clear P1;clear P2;end%the END!!!End

Verify that the results of the function wavedec with MATLAB are consistent:

%Verify the correctness of the function dwtmtx clear all;close ALL;CLC; N=2^7;%'DB1'Or'Haar','DB2', ... ,'DB10', ... ,'Db45'%'COIF1', ... ,'Coif5'%'sym2', ... ,'SYM8', ... ,'Sym45'%'bior1.1','bior1.3','bior1.5'%'bior2.2','bior2.4','bior2.6','bior2.8'%'bior3.1','bior3.3','bior3.5','bior3.7'%'bior3.9','bior4.4','bior5.5','bior6.8'%'rbio1.1','rbio1.3','rbio1.5'%'rbio2.2','rbio2.4','rbio2.6','rbio2.8'%'rbio3.1','rbio3.3','rbio3.5','rbio3.7'%'rbio3.9','rbio4.4','rbio5.5','rbio6.8'wtype='rbio6.8'; Wlev_max=Wmaxlev (n,wtype);ifWlev_max = =0fprintf ('\nthe parameter N and wtype does not match!\n'); Enddwtmode ('per'); forWlev =1: Wlev_max ww=dwtmtx (N,wtype,wlev); X= Randn (1, N); Y1= (ww*x')'; [Y2,Y2L]=Wavedec (X,wlev,wtype); Y_err= SUM ((y1-y2). * (y1-y2)); fprintf ('Wlev =%d:y_err =%f\n', Wlev,y_err); end

A brief talk on compression perception (14): MATLAB implementation of Fourier matrix and wavelet transform matrix