This document summarizes Haar wavelet transforms and Daubechies wavelet transforms, including their forward and inverse transforms. It provides MATLAB code implementations for 1D and 2D transforms using Haar, Daub4, Daub6, Daub5/3 and Daub4 wavelets. The transforms can be applied for multiple levels (levels 1 through k) to decompose signals into approximation and detail coefficients.

1. 以下是小波变换的相关程序 2.Daub4 变换
function y = Daub4(f)
1.Haar 变换
% Level 1 Daub4 transform
function h = Haar(f)
r = f(2:2:end)-sqrt(3)*f(1:2:end);
s = f(1:2:end)+sqrt(3)/4*r+(sqrt(3)-
% N = length(f);
2)/4*[r(2:end), r(1)];
% a = zeros(1,N/2);
a = (1+sqrt(3))/sqrt(2)*s;
% d = zeros(1,N/2);
d = (1-sqrt(3))/sqrt(2)*(s+[r(2:end), r(1)]);
% for m = 1:N/2
y = [a, d];
% a(m) = (f(2*m-1)+f(2*m))/sqrt(2);
% d(m) = (f(2*m-1)-f(2*m))/sqrt(2);
% end Daub4 逆变换
function f = Daub4I(y)
% a = (f(1:2:end-1)+f(2:2:end))/sqrt(2); % Level 1 Inverse Daub4 Transform
% d = (f(1:2:end-1)-f(2:2:end))/sqrt(2); s = sqrt(2)/(1+sqrt(3))*y(1:end/2);
% h = [a d]; r = sqrt(2)/(1-sqrt(3))*[y(end),y(end/2+1:end-
1)]...
h = [(f(1:2:end-1)+f(2:2:end)), ...
-[s(end),s(1:end-1)];
(f(1:2:end-1)-f(2:2:end))]/sqrt(2); f = y;
f(1:2:end) = s-sqrt(3)/4*r-(sqrt(3)-
2)/4*[r(2:end),r(1)];
Haar 逆变换
f(2:2:end) = r + sqrt(3)*f(1:2:end);
function f = HaarI(h)
Daub4 K 层变换
% M = length(h)/2; function y = Daub4K(x, k)
% f = zeros(1,2*M);
% for i=1:M y = x;
% f(2*i-1:2*i) = [h(i)+h(M+i), h(i)-h(M+i)]; for i=1:k
% end
y(1:end/2^(i-1)) = Daub4(y(1:end/2^(i-1)));
end
a = h(1:end/2);
d = h(end/2+1:end); Daub4 K 层逆变换
f = reshape([a+d; a-d],1,[])/sqrt(2); function y = Daub4KI(x, k)
% Level K Inverse Daub4 Transform
K 层 Haar 变换
y = x;
function y = HaarK(f, k)
for i=k:-1:1
% k-level Haar transform
y = f; y(1:end/2^(i-1)) = Daub4I(y(1:end/2^(i-1)));
for i=1:k end
y(1:end/2^(i-1)) = Haar(y(1:end/2^(i-1)));
end 3.Daub6 变换
function y = Daub6(f)
K 层 Haar 逆变换
% level 1 Daub6 transform
function f = HaarKI(y, k)
a1 = (1+sqrt(10)+sqrt(5+2*sqrt(10)))*sqrt(2)/32;
% k-level Inverse Haar transform
a2 =
f = y;
(5+sqrt(10)+3*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
for i=k:-1:1
a3 = (10-
f(1:end/2^(i-1)) = HaarI(f(1:end/2^(i-1))); 2*sqrt(10)+2*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
end a4 = (10-2*sqrt(10)-
2*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
a5 = (5+sqrt(10)-
3*sqrt(5+2*sqrt(10)))*sqrt(2)/32;

2. a6 = (1+sqrt(10)-sqrt(5+2*sqrt(10)))*sqrt(2)/32; thetap = R0/R2;
b1 = a6; zeta = -a6/a1*R2^2;
b2 = -a5;
b3 = a4; f = y;
b4 = -a3; s2 = -R2*y(end/2+1:end);
b5 = a2; r2 = y(1:end/2)/R2;
b6 = -a1; s1 = s2 - zeta*r2;
R0 = a2 - a1*a6/a5; r1 = r2 - theta*s1 - thetap*[s1(end),s1(1:end-1)];
R1 = a4 - a3*a6/a5; f(2:2:end) = [s1(end),s1(1:end-1)]-
R2 = a3 - a1*R1/R0 - a5*R0/R1; bp*[r1(end),r1(1:end-1)]-b*r1;
a = a6/a5; f(1:2:end) = [r1(end),r1(1:end-1)]-a*f(2:2:end);
b = a5/R1;
4.Daub5/3 变换
bp = a1/R0;
function y = Daub53(f)
theta = R1/R2;
% Level 1 Daub 5/3 wavelet transform
thetap = R0/R2;
y = f;
zeta = -a6/a1*R2^2;
y(end/2+1:end) = f(2:2:end)-(f(1:2:end)+
[f(3:2:end),f(end-1)])/2;
y = f;
y(1:end/2) = f(1:2:end)+
r1 = [f(3:2:end),f(1)]+a*[f(4:2:end),f(2)];
([y(end/2+1),y(end/2+1:end-1)]
s1 = [f(4:2:end),f(2)]+bp*r1+b*[r1(2:end),r1(1)];
+y(end/2+1:end))/4;
r2 = r1+theta*s1+thetap*[s1(end),s1(1:end-1)];
s2 = s1+zeta*r2; Daub5/3 逆变换
y(1:end/2) = R2*r2;
y(end/2+1:end) = -s2/R2; function f = Daub53I(y)
% Inverse of Level 1 Daub 5/3 Wavelet
Transform
f = y;
Daub6 逆变换
f(1:2:end) = y(1:end/2)-
function f = Daub6I(y)
([y(end/2+1),y(end/2+1:end-1)]
% level 1 inverse Daub6 transform
+y(end/2+1:end))/4;
a1 = (1+sqrt(10)+sqrt(5+2*sqrt(10)))*sqrt(2)/32;
f(2:2:end) = y(end/2+1:end)+(f(1:2:end)+
a2 =
[f(3:2:end),f(end-1)])/2;
(5+sqrt(10)+3*sqrt(5+2*sqrt(10)))*sqrt(2)/32;
a3 = (10-
2*sqrt(10)+2*sqrt(5+2*sqrt(10)))*sqrt(2)/32; Daub5/3i2i 变换
a4 = (10-2*sqrt(10)- function y = Daub53i2i(f)
2*sqrt(5+2*sqrt(10)))*sqrt(2)/32; % Level 1 Daub 5/3 integer to integer wavelet
a5 = (5+sqrt(10)- transform
3*sqrt(5+2*sqrt(10)))*sqrt(2)/32; y = f;
a6 = (1+sqrt(10)-sqrt(5+2*sqrt(10)))*sqrt(2)/32; y(end/2+1:end) = f(2:2:end)-floor((f(1:2:end)+
b1 = a6; [f(3:2:end),f(end-1)])/2+1/2);
b2 = -a5; y(1:end/2) = f(1:2:end)
b3 = a4; +floor(([y(end/2+1),y(end/2+1:end-1)]
b4 = -a3; +y(end/2+1:end))/4+1/2);
b5 = a2;
Daub5/3i2i 逆变换
b6 = -a1;
function f = Daub53i2iI(y)
R0 = a2 - a1*a6/a5;
% Inverse of Level 1 Daub 5/3 integer to integer
R1 = a4 - a3*a6/a5;
Wavelet Transform
R2 = a3 - a1*R1/R0 - a5*R0/R1;
f = y;
a = a6/a5;
f(1:2:end) = y(1:end/2)-
b = a5/R1;
floor(([y(end/2+1),y(end/2+1:end-1)]
bp = a1/R0;
+y(end/2+1:end))/4+1/2);
theta = R1/R2;

3. f(2:2:end) = y(end/2+1:end)+floor((f(1:2:end)+ y = f;
[f(3:2:end),f(end-1)])/2+1/2); for i = 1:k
for j = 1:2^(i-1)
5.Daub4 2D 变换
for p = 1:2^(i-1)
function y = Daub4_2D(f)
y((j-1)*m/2^(i-1)+1:j*m/2^(i-1),(p-
[m,n] = size(f); 1)*n/2^(i-1)+1:p*n/2^(i-1)) = Daub4_2D(y((j-
y = f; 1)*m/2^(i-1)+1:j*m/2^(i-1),(p-1)*n/2^(i-
for i = 1:m 1)+1:p*n/2^(i-1)));
y(i,:) = Daub4(y(i,:)); end
end
end
for i = 1:n
end
y(:,i) = Daub4(y(:,i)')';
逆变换
end
function f = Daub4p_2D_K_I(y,k)
Daub4 2D 逆变换 [m,n] = size(y);
function f = Daub4_2D_I(y) f = y;
for i = k:-1:1
[m,n] = size(y);
for j = 1:2^(i-1)
f = y;
for i = 1:m for p = 1:2^(i-1)
f(i,:) = Daub4I(f(i,:)); f((j-1)*m/2^(i-1)+1:j*m/2^(i-1),(p-
end 1)*n/2^(i-1)+1:p*n/2^(i-1)) = Daub4_2D_I(f((j-
for i = 1:n 1)*m/2^(i-1)+1:j*m/2^(i-1),(p-1)*n/2^(i-
f(:,i) = Daub4I(f(:,i)')'; 1)+1:p*n/2^(i-1)));
end end
end
Daub4 2D K 层变换 end
function y = Daub4_2D_K(f,k)
7 图片保存信息量为原来的 99.99%
y = f; function f = Daub4p_2D_K_I(y,k)
for i = 1:k [m,n] = size(y);
f = y;
y(1:end/2^(i-1),1:end/2^(i-1)) = for i = k:-1:1
Daub4_2D(y(1:end/2^(i-1),1:end/2^(i-1)));
for j = 1:2^(i-1)
end
for p = 1:2^(i-1)
Daub4 2D K 层逆变换 f((j-1)*m/2^(i-1)+1:j*m/2^(i-1),(p-
function f = Daub4_2D_K_I(y,k) 1)*n/2^(i-1)+1:p*n/2^(i-1)) = Daub4_2D_I(f((j-
1)*m/2^(i-1)+1:j*m/2^(i-1),(p-1)*n/2^(i-
f = y; 1)+1:p*n/2^(i-1)));
for i = k:-1:1
end
f(1:end/2^(i-1),1:end/2^(i-1)) =
end
Daub4_2D_I(f(1:end/2^(i-1),1:end/2^(i-1)));
end end
6.Daub4 2D P 变换
function y = Daub4p_2D_K(f,k)
[m,n] = size(f);