文章和代碼以及樣例圖片等相關(guān)資源，已經(jīng)歸檔至【Github倉庫：digital-image-processing-matlab】或者公眾號(hào)【AIShareLab】回復(fù) 數(shù)字圖像處理 也可獲取。

目的

1. Haar、尺度和小波函數(shù)；

2. 比較函數(shù)wavefast 和函數(shù)wavedec2 的執(zhí)行時(shí)間；

3. 小波的方向性和邊緣檢測(cè)。

步驟

Haar、尺度和小波函數(shù)

[Lo_D,Hi_D,Lo_R,Hi_R]=wfilters('haar')
%Haar 變換的分解和重構(gòu)濾波器的長(zhǎng)度為2
waveinfo('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');

使用haar 濾波器的一個(gè)簡(jiǎn)單FWT

f=magic(4)%使用Haar 的單尺度小波變換
[c1,s1]=wavedec2(f,1,'haar')
[c2,s2]=wavedec2(f,2,'haar')

比較函數(shù)wavefast 和函數(shù)wavedec2 的執(zhí)行時(shí)間

function [varargout]=wavefilter(wname,type)%Check the input and output arguments
error (nargchk(1,2,nargin));
if (nargin==1&nargout~=4)|(nargin==2&nargout~=2)error('Invalid number of output arguments');
end
if 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;otherwiseerror('Unrecognizable wavelet name (WNAME).');
end%Output the requested filters
if(nargin == 1)varargout(1:4)={ld,hd,lr,hr};
elseswitch lower(type(1))case 'd'varargout={ld,hd};case 'r'varargout={lr,hr};otherwiseerror('Unrecognizable filter TYPE.')end
endfunction [c,s]=wavefast(x,n,varargin)error(nargchk(3,4,nargin));
if nargin == 3if ischar(varargin{1})[lp,hp]=wavefilter(varargin{1},'d');elseerror('Missing wavelet name');end
elselp=varargin{1};hp=varargin{2};
end
f1=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)~=0error(['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)))']);
end
c=[];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);
end
c=[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 results
ratio=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));
elsey=conv2(x,h');y=y(1:2:end,:);y=y(f1/2+1:f1/2+keep(1),:);
end %比較兩種方法下的速度
clear all
clc
f=imread('D:\圖像庫\DIP3E_CH07_Original_Images\Fig0701.tif','tif');
[ratio,maxdifference]=fwtcompare(f,5,'db4')

小波的方向性和邊緣檢測(cè)

clear all
clc
f=imread('D:\圖像庫\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');
end
if(ndims(s)~=2)|~isreal(s)|~isnumeric(s)|(size(s,2)~=2)error('s must be a real,numeric two-column array');
end
elements=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']);
end
if(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==2scale=1;%default scale
end
if nargin<4border='absord';%default border
endabsflag=scale<0;
scale=abs(scale);
if scale==0scale=1;
end[cd,w]=wavecut('a',c,s);
w=mat2gray(w);
cdx=max(abs(cd(:)))/scale;
if absflagcd=mat2gray(abs(cd),[0,cdx]);fill=0;
elsecd=mat2gray(cd,[-cdx,cdx]);fill=0.5;
end
%build gray image one decomposition at a time
for i=size(s,1)-2:-1:1ws=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 borderswitch 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;otherwiseerror('Unrecognized border parameter');endw=[w h;v d];
endif nargout==0imshow(w);%display result
endfunction [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');
end
if(ndims(s)~=2)|~isreal(s)|~isnumeric(s)|(size(s,2)~=2)error('s must be a real,numeric two-column array');
end
elements=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 nargincase 3if ischar(wname)[lp,hp]=wavefilter(wname,'r');n=nmax;elseerror('updefined filter');endif nargout~=1error('wrong number of output arguments');endcase 4if ischar(wname)[lp,hp]=wavefilter(wname,'r');n=varargin{2};nchk=1;elselp=varargin{1};hp=varargin{2};filterchk=1;n=nmax;if nargout~=1error('wrong number of output arguments');endend
case 5lp=varargin{1};hp=varargin{2};filterchk=1;n=varargin{3};nchk=1;otherwiseerror('improper number fo input arguments');
endf1=length(lp);
if filterchkif (ndims(lp)~=2)|~isreal(lp)|~isnumeric(lp) ...|(ndims(hp)~=2)|~isreal(hp)|~isnumeric(hp) ...|(f1~=length(hp)) | rem(f1,2)~=0error(['LP and HP must be even and equal length real ,'...'numeric filter vectors']);end
endif nchk & (~isnumeric(n)|~isreal(n))error('n must be a real numeric');
end
if (n>nmax)|(n<1)error('Invalid number (N) of reconstructions requested');
end
if (n~=nmax) & (nargout~=2)error('Not enough output arguments');
endnc=c;
ns=s;
nnmax=nmax;
for i=1:na=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;
end
varargout{1}=nc;
if nargout==2varargout{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');
end
if(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 < 6error('Not enough input arguments');
endif nargin<5n=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 typecase 'h'offset=0;case 'v'offset=1;case 'd'offset=2;endnindex=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;otherwiseerror('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};elsevarargout={y};endcase 'paste'if prod(size(x))~=elements(end-ntst)error('x is not sized for the requested paste');elsenc=c;nc(start:stop)=x(:);varargout={nc};endotherwiseerror('unrecognized OPCODE');
end

參考文獻(xiàn)：

[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. 數(shù)字圖像處理（MATLAB版）[M]. 北京：電子工業(yè)出版社, 2014.

[3] 岡薩雷斯. 數(shù)字圖像處理（第三版）[M]. 北京：電子工業(yè)出版社, 2011.