文章和代码以及样例图片等相关资源，已经归档至【Github仓库：digital-image-processing-matlab】或者公众号【AIShareLab】回复 数字图像处理 也可获取。
目的
- Haar、尺度和小波函数；
- 比较函数wavefast 和函数wavedec2 的执行时间；
- 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 变换的分解和重构滤波器的长度为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 滤波器的一个简单FWT
f=magic(4)%使用Haar 的单尺度小波变换
[c1,s1]=wavedec2(f,1,'haar')
[c2,s2]=wavedec2(f,2,'haar')
比较函数wavefast 和函数wavedec2 的执行时间
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');
end
if nargin == 2 & ~ischar(type)
error('TYPE must be a string');
end
switch 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 filters
if(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.')
end
end
function [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');
end
else
lp=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');
end
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']);
end
if ~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));
else
y=conv2(x,h');
y=y(1:2:end,:);
y=y(f1/2+1:f1/2+keep(1),:);
end
%比较两种方法下的速度
clear all
clc
f=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 all
clc
f=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');
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');
end
if(nargin>3)&(~ischar(border))
error('border must be character string');
end
if nargin==2
scale=1;%default scale
end
if nargin<4
border='absord';%default border
end
absflag=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 time
for 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];
end
if nargout==0
imshow(w);%display result
end
function [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']);
end
nmax=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
end
case 5
lp=varargin{1};
hp=varargin{2};
filterchk=1;
n=varargin{3};
nchk=1;
otherwise
error('improper number fo input arguments');
end
f1=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']);
end
end
if 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');
end
nc=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;
end
if nargout==1;
a=nc;
nc=repmat(0,ns(1,:));
nc(:)=a;
end
varargout{1}=nc;
if nargout==2
varargout{2}=ns;
end
function 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);
End
function 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);
End
function 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');
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 strcmp (lower(opcode(1:3)),'pas') & nargin < 6
error('Not enough input arguments');
end
if nargin<5
n=1;
end
nmax=size(s,1)-2;
aflag=(lower(type(1))=='a');
if ~aflag & (n>nmax)
error('N exceeds the decompositon in [c,s]');
end
switch 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"');
end
switch 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：
[2] 阮秋琦. 数字图像处理（MATLAB版）[M]. 北京：电子工业出版社, 2014.
[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.
标签：...,end,002,变换,nc,f1,实现,Matlab,error From： https://blog.51cto.com/u_15736437/6102292