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目标
- Haar、尺度和小波函数；
- 比拟函数wavefast 和函数wavedec2 的执行工夫；
- 小波的方向性和边缘检测。
步骤
Haar、尺度和小波函数
[Lo_D,Hi_D,Lo_R,Hi_R]=wfilters('haar')%Haar 变换的合成和重构滤波器的长度为2waveinfo('haar');[phi,psi,xval]=wavefun('haar',10);xaxis=zeros(size(xval));subplot(121);plot(xval,phi,'k',xval,xaxis,'--k');axis([0 1 -1.5 1.5]);axis square;title('haar scaling function');subplot(122);plot(xval,psi,'k',xval,xaxis,'--k');axis([0 1 -1.5 1.5]);axis square;title('haar wavelet function');应用haar 滤波器的一个简略FWT
f=magic(4)%应用Haar 的单尺度小波变换[c1,s1]=wavedec2(f,1,'haar')[c2,s2]=wavedec2(f,2,'haar')比拟函数wavefast 和函数wavedec2 的执行工夫
function [varargout]=wavefilter(wname,type)%Check the input and output argumentserror (nargchk(1,2,nargin));if (nargin==1&nargout~=4)|(nargin==2&nargout~=2) error('Invalid number of output arguments');endif nargin == 1 & ~ischar(wname) error('WNAME must be a string');endif nargin == 2 & ~ischar(type) error('TYPE must be a string');endswitch lower(wname) case {'haar','db1'} ld=[1 1]/sqrt(2); hd=[-1 1]/sqrt(2); lr=ld; hr=-hd; case 'db4' ld=[-1.059740178499728e-002 3.288301166698295e-002 ... 3.084138183598697e-002 -1.870348117188811e-001 ... -2.798376941698385e-002 6.308807679295904e-001 ... 7.14846575525415e-001 2.303778133088552e-001]; t=(0:7); hd=ld; hd(end:-1:1)=cos(pi*t).*ld; lr=ld; lr(end:-1:1)=ld; hr=cos(pi*t).*ld; case 'sym4' ld=[-7.576571478927333e-002 -2.963552764599851e-002 ... 4.976186676320155e-001 8.03738758059161e-001 ... 2.978577956052774e-001 -9.921954357684722e-002 ... -1.260396726203783e-002 3.222310060404270e-002]; t=(0:7); hd=ld; hd(end:-1:1)=cos(pi*t).*ld; lr=ld; lr(end:-1:1)=ld; hr=cos(pi*t).*ld; case 'bior6.8' ld=[0 1.908831736481291e-003 -1.914286129088767e-003 ... -1.699063986760234e-002 1.193456527972926e-002 ... 4.973490349094079e-002 -7.726317316720414e-002 ... -9.405920349573646e-002 4.207962846098268e-001 ... 8.259229974584023e-001 4.207962846098268e-001 ... -9.405920349573646e-002 -7.726317316720414e-002 ... 4.973490349094079e-002 1.193456527972926e-002 ... -1.699063986760234e-002 -1.914286129088767e-003 ... 1.908831736481291e-003]; hd=[0 0 0 1.442628250562444e-002 -1.446750489679015e-002 ... -7.872200106262882e-002 4.036797903033992e-002 ... 4.178491091502746e-001 -7.589077294536542e-002 ... 4.178491091502746e-001 4.036797903033992e-002 ... -7.872200106262882e-002 -1.446750489679015e-002 ... 1.442628250562444e-002 0 0 0 0]; t=(0:17); lr=cos(pi*(t+1)).*hd; hr=cos(pi*t).*ld; case 'jpep9.7' ld=[0 0.02674875741080976 -0.01686411844287495 ... -0.07822326652898785 0.266861184428732 ... 0.6029490182363579 0.266861184428732 ... -0.07822326652898785 -0.01686411844287495 ... 0.02674875741080976]; hd=[0 -0.09127176311424948 0.05754352622849957 ... 0.5912717631142470 -1.115087052456994 ... 0.5912717631142470 0.05754352622849957 ... -0.09127176311424948 0 0]; t=(0:9); lr=cos(pi*(t+1)).*hd; hr=cos(pi*t).*ld; otherwise error('Unrecognizable wavelet name (WNAME).');end%Output the requested filtersif(nargin == 1) varargout(1:4)={ld,hd,lr,hr};else switch lower(type(1)) case 'd' varargout={ld,hd}; case 'r' varargout={lr,hr}; otherwise error('Unrecognizable filter TYPE.') endendfunction [c,s]=wavefast(x,n,varargin)error(nargchk(3,4,nargin));if nargin == 3 if ischar(varargin{1}) [lp,hp]=wavefilter(varargin{1},'d'); else error('Missing wavelet name'); endelse lp=varargin{1};hp=varargin{2};endf1=length(lp);sx=size(x);if (ndims(x)~=2)|(min(sx<2))|~isreal(x)|~isnumeric(x) error('X must be a real, numeric matrix');endif (ndims(lp)~=2)|~isreal(lp)|~isnumeric(lp) ... |(ndims(hp)~=2)|~isreal(hp)|~isnumeric(hp) ... |(f1~=length(hp)) | rem(f1,2)~=0 error(['LP and HP must be even and equal length real ,'...'numeric filter vectors']);endif ~isreal(n)|~isnumeric(n)|(n<1)|(n>log2(max(sx))) error(['N must be a real scalar between 1 and' ...'log2(max(size(x)))']);endc=[];s=sx;app=double(x);%for each decomposition...for i=1:n [app,keep]=symextend(app,f1); rows=symconv(app,hp,'row',f1,keep); coefs=symconv(rows,hp,'col',f1,keep); c=[coefs(:)' c]; s=[size(coefs);s]; coefs=symconv(rows,lp,'col',f1,keep); c=[coefs(:)' c]; rows=symconv(app,lp,'row',f1,keep); coefs=symconv(rows,hp,'col',f1,keep); c=[coefs(:)' c]; app=symconv(rows,lp,'col',f1,keep);endc=[app(:)' c];s=[size(app);s];function [y,keep]=symextend(x,f1)keep=floor((f1+size(x)-1)/2);y=padarray(x,[(f1-1) (f1-1)],'symmetric','both');function [ratio, maxdiff]=fwtcompare(f,n,wname)tic;[c1,s1]=wavedec2(f,n,wname);reftime=toc;tic;[c2,s2]=wavefast(f,n,wname);t2=toc;%compare the resultsratio=t2/(reftime+eps);maxdiff=abs(max(c1-c2)); function y=symconv(x,h,type,f1,keep)if strcmp(type,'row') y=conv2(x,h); y=y(:,1:2:end); y=y(:,f1/2+1:f1/2+keep(2));else y=conv2(x,h'); y=y(1:2:end,:); y=y(f1/2+1:f1/2+keep(1),:);end %比拟两种办法下的速度clear allclcf=imread('D:\图像库\DIP3E_CH07_Original_Images\Fig0701.tif','tif');[ratio,maxdifference]=fwtcompare(f,5,'db4')小波的方向性和边缘检测
clear allclcf=imread('D:\图像库\DIP3E_CH07_Original_Images\Fig0701.tif');imshow(f);[c,s]=wavefast(f,1,'sym4');figure;wave2gray(c,s,-6);[nc,y]=wavecut('a',c,s);figure;wave2gray(nc,s,-6);edges=abs(waveback(nc,s,'sym4'));figure;imshow(mat2gray(edges));%子程序function w=wave2gray(c,s,scale,border)error(nargchk(2,4,nargin));if(ndims(c)~=2)|(size(c,1)~=1) error('c must be a row vector');endif(ndims(s)~=2)|~isreal(s)|~isnumeric(s)|(size(s,2)~=2) error('s must be a real,numeric two-column array');endelements=prod(s,2);if (length(c)<elements(end))|... ~(elements(1)+3*sum(elements(2:end-1))>=elements(end)) error(['[c s] must be a standard wavelet' ...'decomposition structure']);endif(nargin>2)&(~isreal(scale)|~isnumeric(scale)) error('scale must be a real,numeric scalar');endif(nargin>3)&(~ischar(border)) error('border must be character string');endif nargin==2 scale=1;%default scaleendif nargin<4 border='absord';%default borderendabsflag=scale<0;scale=abs(scale);if scale==0 scale=1;end[cd,w]=wavecut('a',c,s);w=mat2gray(w);cdx=max(abs(cd(:)))/scale;if absflag cd=mat2gray(abs(cd),[0,cdx]); fill=0;else cd=mat2gray(cd,[-cdx,cdx]); fill=0.5;end%build gray image one decomposition at a timefor i=size(s,1)-2:-1:1 ws=size(w); h=wavecopy('h',cd,s,i); pad=ws-size(h); frontporch=round(pad/2); h=padarray(h,frontporch,fill,'pre'); h=padarray(h,pad-frontporch,fill,'post'); v=wavecopy('v',cd,s,i); pad=ws-size(v); frontporch=round(pad/2); v=padarray(v,frontporch,fill,'pre'); h=padarray(v,pad-frontporch,fill,'post'); d=wavecopy('d',cd,s,i); pad=ws-size(d); frontporch=round(pad/2); d=padarray(d,frontporch,fill,'pre'); d=padarray(d,pad-frontporch,fill,'post'); %Add 1 pixel white border switch lower(border) case 'append' w=padarray(w,[1 1],1,'post'); h=padarray(h,[1 0],1,'post'); v=padarray(v,[0 1],1,'post'); case 'absord' w(:,end)=1; w(end,:)=1; h(end,:)=1; v(:,end)=1; otherwise error('Unrecognized border parameter'); end w=[w h;v d];endif nargout==0 imshow(w);%display resultendfunction [varargout]=waveback(c,s,varargin)error(nargchk(3,5,nargin));error(nargchk(1,2,nargout));if (ndims(c)~=2)|(size(c,1)~=1) error('c must be a row vector');endif(ndims(s)~=2)|~isreal(s)|~isnumeric(s)|(size(s,2)~=2) error('s must be a real,numeric two-column array');endelements=prod(s,2);if (length(c)<elements(end))|...~(elements(1)+3*sum(elements(2:end-1))>=elements(end)) error(['[c s] must be a standard wavelet' ... 'decomposition structure']);endnmax=size(s,1)-2;wname=varargin{1};filterchk=0;nchk=0;switch nargin case 3 if ischar(wname) [lp,hp]=wavefilter(wname,'r'); n=nmax; else error('updefined filter'); end if nargout~=1 error('wrong number of output arguments'); end case 4 if ischar(wname) [lp,hp]=wavefilter(wname,'r'); n=varargin{2}; nchk=1; else lp=varargin{1}; hp=varargin{2}; filterchk=1; n=nmax; if nargout~=1 error('wrong number of output arguments'); end endcase 5 lp=varargin{1}; hp=varargin{2}; filterchk=1; n=varargin{3}; nchk=1; otherwise error('improper number fo input arguments');endf1=length(lp);if filterchk if (ndims(lp)~=2)|~isreal(lp)|~isnumeric(lp) ... |(ndims(hp)~=2)|~isreal(hp)|~isnumeric(hp) ... |(f1~=length(hp)) | rem(f1,2)~=0 error(['LP and HP must be even and equal length real ,'... 'numeric filter vectors']); endendif nchk & (~isnumeric(n)|~isreal(n)) error('n must be a real numeric');endif (n>nmax)|(n<1) error('Invalid number (N) of reconstructions requested');endif (n~=nmax) & (nargout~=2) error('Not enough output arguments');endnc=c;ns=s;nnmax=nmax;for i=1:n a=symconvup(wavecopy('a',nc,ns),lp,lp,f1,ns(3,:))+ ... symconvup(wavecopy('h',nc,ns,nnmax),hp,lp,f1,ns(3,:))+ ... symconvup(wavecopy('v',nc,ns,nnmax),lp,hp,f1,ns(3,:))+ ... symconvup(wavecopy('d',nc,ns,nnmax),hp,hp,f1,ns(3,:)); nc=nc(4*prod(ns(1,:))+1:end); nc=[a(:)' nc]; ns=ns(3:end,:); ns=[ns(1,:);ns]; nnmax=size(ns,1)-2;endif nargout==1; a=nc; nc=repmat(0,ns(1,:)); nc(:)=a;endvarargout{1}=nc;if nargout==2 varargout{2}=ns;endfunction z=symconvup(x,f1,f2,fln,keep)y=zeros([2 1].*size(x));y(1:2:end,:)=x;y=conv2(y,f1');z=zeros([1 2].*size(y));z(:,1:2:end)=y;z=conv2(z,f2);z=z(fln-1:fln+keep(1)-2,fln-1:fln+keep(2)-2);function [nc,y]=wavecut(type,c,s,n)error(nargchk(3,4,nargin));if nargin==4 [nc,y]=wavework('cut',type,c,s,n);else [nc,y]=wavework('cut',type,c,s);Endfunction y=wavecopy(type,c,s,n)error(nargchk(3,4,nargin));if nargin==4 [nc,y]=wavework('cut',type,c,s,n);else [nc,y]=wavework('cut',type,c,s);Endfunction y=wavepaste(type,c,s,n)error(nargchk(5,5,nargin))nc=wavework('paste',type,c,s,n,x);function [varargout]=wavework(opcode,type,c,s,n,x)error(nargchk(4,6,nargin));if (ndims(c)~=2)|(size(c,1)~=1) error('c must be a row vector');endif(ndims(s)~=2)|~isreal(s)|~isnumeric(s)|(size(s,2)~=2) error('s must be a real,numeric two-column array');endelements=prod(s,2);if (length(c)<elements(end))|... ~(elements(1)+3*sum(elements(2:end-1))>=elements(end)) error(['[c s] must be a standard wavelet' ... 'decomposition structure']);endif strcmp (lower(opcode(1:3)),'pas') & nargin < 6 error('Not enough input arguments');endif nargin<5 n=1;endnmax=size(s,1)-2;aflag=(lower(type(1))=='a');if ~aflag & (n>nmax) error('N exceeds the decompositon in [c,s]');endswitch lower(type(1)) case 'a' nindex=1; start=1; stop=elements(1); ntst=nmax; case {'h','v','d'} switch type case 'h' offset=0; case 'v' offset=1; case 'd' offset=2; end nindex=size(s,1)-n; start=elements(1)+3*sum(elements(2:nmax-n+1))+ ... offset*elements(nindex)-1; stop=start+elements(nindex)-1; ntst=n; otherwise error('Type must begin with "a","h","v",or "d"');endswitch lower(opcode) case {'copy','cut'} y=repmat(0,s(nindex,:)); y(:)=c(start:stop); nc=c; if strcmp(lower(opcode(1:3)),'cut') nc(start:stop)=0; varargout={nc,y}; else varargout={y}; end case 'paste' if prod(size(x))~=elements(end-ntst) error('x is not sized for the requested paste'); else nc=c; nc(start:stop)=x(:); varargout={nc}; end otherwise error('unrecognized OPCODE');end参考文献：
[1] Rafael C. Gonzalez, Richard E. Woods, and Steven L. Eddins. 2003. Digital Image Processing Using MATLAB. Prentice-Hall, Inc., USA.
[2] [阮秋琦. 数字图像处理（MATLAB版）[M]. 北京：电子工业出版社, 2014.](https://github.com/timerring/digital-image-processing-matlab/blob/main/reference/Digital_Image_Processing_(MATLAB_version).pdf)
[3] [冈萨雷斯. 数字图像处理（第三版）[M]. 北京：电子工业出版社, 2011.](https://github.com/timerring/digital-image-processing-matlab/blob/main/reference/Digital_Image_Processing_(Third_Edition).pdf)