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In matlab, although my matrix is ​​full rank, the solution is still incorrect.​Ax=b,A(100​00*10000),​x、b(10000*​1)

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道夫
道夫 on 9 Jul 2024 at 7:38
Commented: 道夫 on 13 Jul 2024 at 6:55
This is equation solving
this is D matrics
this is J matrics
I want to test
but the result is not good
My ultimate goal is to use the constructed inverse matrix to restore the image, because the deviation in the solution causes the restoration effect not to be very good：
The above picture is restored by me directly using the inverse matrix. The picture below is restored by using the constructed inverse matrix. The inverse matrix is ​​J and the constructed inverse matrix is ​​Dx1.
Thank you guys
  2 Comments
Torsten
Torsten on 9 Jul 2024 at 8:25
Edited: Torsten on 9 Jul 2024 at 9:08
We can't use your D and J. Include them in a .mat-file instead of an image.
道夫
道夫 on 10 Jul 2024 at 2:00
Edited: Torsten on 10 Jul 2024 at 15:11
Ran in:
Sorry, I didn't take that into consideration
tenpixel is the code that generates the D matrix，ITM is the J matrix.D is saved through cell,As long as it is arranged in rows, a D matrix of 10000×10000 is formed.
J = load("ITM.mat")
J = struct with fields:
W2_s: [100x100 double]
J = J.W2_s;
J = J(:);
um=1e-2;
lambda=1.153*um;
zsize=10*um;%Axial propagation distance
zsize2=12*um;
rect_size=0.875*um;
FL=50*um;
Ni=10;
Nk=100;
Nis=Ni^2;
Nks=Nk^2;
TMC=zeros(Nis,Nis);%Transmission matrix size
dx1=FL/Ni;%Input and output surface element spacing
dx2=FL/Nk;%Mask unit spacing
lefts1=(FL-dx1)/2;
lefts2=(FL-dx2)/2;
xj=linspace(-lefts1,lefts1,Ni);
yj=xj;
xi=xj;
yi=yj;
xk=linspace(-lefts2,lefts2,Nk);
yk=xk;
k0 = 2*pi/lambda;
[XI,YI] = meshgrid(xi,yi);
[XK,YK] = meshgrid(xk,yk);
Wkj=zeros(Nks,Nis);%The size of the right propagation matrix w
Wki=zeros(Nis,Nks);%left propagation matrix size
%
% input matrix
for jj=1:1:Nis
rho_p1 = sqrt((XK-XI(jj)).^2+...
(YK-YI(jj)).^2+zsize^2);
f_xy = k0.*zsize./(2i*pi).*exp(k0*rho_p1*1i)./rho_p1.^2.*(1-1./(k0*rho_p1*1i));
f_xy=reshape(f_xy,[Nks,1]);
Wkj(:,jj)=f_xy;
end
%Output matrix
for jj=1:1:Nis
rho_p1 = sqrt((XK-XI(jj)-4*um).^2+...
(YK-YI(jj)+um).^2+zsize2^2);
f_xy = k0.*zsize2./(2i*pi).*exp(k0*rho_p1*1i)./rho_p1.^2.*(1-1./(k0*rho_p1*1i));
f_xy=reshape(f_xy,[Nks,1]);
Wki(jj,:)=f_xy.';
end
clear TMC rect_size f_xy rho_p1 zsize YI YK XI XK yk xk lefts1 lefts2
C=cell(Ni^2,1);
% ab=single(Wki);
%
% How to calculate the final matrix may still be a problem. The C matrix is ​​the D matrix of the next code, D=C=RVL, R is the output propagation matrix (Relso Murphy propagation), L is the input matrix,
% V is the modulation matrix, which is equivalent to phase spatial light modulation. The size of C is 10000*10000
for jj=1:1:Nis
BIJK=zeros(Nis,Nks);
for i=1:1:Nis
BIJK(i,:)=Wki(i,:).*Wkj(:,jj).';
% BIJK(i,:)=Wki(i,:).*Wki(jj,:);
end
C{jj,1}=BIJK;
end
D = [];
for i = 1:100
D = [D;C{i}];
end
cond(D)
ans = 1.0839e+11
dD = decomposition(D);
X1 = dD\J
X1 =
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0.0000i 0.0000 + 0.0000i -0.0000 + 0.0000i 0.0000 - 0.0000i 0.0000 + 0.0000i -0.0001 - 0.0000i 0.0001 - 0.0000i -0.0000 + 0.0001i -0.0001 - 0.0002i 0.0003 + 0.0000i -0.0002 + 0.0004i -0.0004 - 0.0004i 0.0008 - 0.0003i 0.0002 + 0.0011i -0.0014 - 0.0000i 0.0002 - 0.0016i 0.0016 + 0.0004i -0.0005 + 0.0015i -0.0015 - 0.0007i 0.0007 - 0.0016i 0.0016 + 0.0002i 0.0004 + 0.0011i -0.0004 + 0.0005i -0.0002 + 0.0000i
norm(D*X1-J)
ans = 189.9414

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Answers (1)

Torsten
Torsten on 10 Jul 2024 at 15:12
Moved: Torsten on 10 Jul 2024 at 15:12
The condition number of your matrix D is around 1e11 (see above). This is near to singular. I think you cannot expect a better result.
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Torsten
Torsten on 10 Jul 2024 at 22:31
Edited: Torsten on 10 Jul 2024 at 22:49
Ran in:
You can reduce the residual norm by post-iteration. I don't know if this leads to a "better" solution from the physical point of view.
J = load("ITM.mat");
J = J.W2_s;
J = J(:);
um=1e-2;
lambda=1.153*um;
zsize=10*um;%Axial propagation distance
zsize2=12*um;
rect_size=0.875*um;
FL=50*um;
Ni=10;
Nk=100;
Nis=Ni^2;
Nks=Nk^2;
TMC=zeros(Nis,Nis);%Transmission matrix size
dx1=FL/Ni;%Input and output surface element spacing
dx2=FL/Nk;%Mask unit spacing
lefts1=(FL-dx1)/2;
lefts2=(FL-dx2)/2;
xj=linspace(-lefts1,lefts1,Ni);
yj=xj;
xi=xj;
yi=yj;
xk=linspace(-lefts2,lefts2,Nk);
yk=xk;
k0 = 2*pi/lambda;
[XI,YI] = meshgrid(xi,yi);
[XK,YK] = meshgrid(xk,yk);
Wkj=zeros(Nks,Nis);%The size of the right propagation matrix w
Wki=zeros(Nis,Nks);%left propagation matrix size
%
% input matrix
for jj=1:1:Nis
rho_p1 = sqrt((XK-XI(jj)).^2+...
(YK-YI(jj)).^2+zsize^2);
f_xy = k0.*zsize./(2i*pi).*exp(k0*rho_p1*1i)./rho_p1.^2.*(1-1./(k0*rho_p1*1i));
f_xy=reshape(f_xy,[Nks,1]);
Wkj(:,jj)=f_xy;
end
%Output matrix
for jj=1:1:Nis
rho_p1 = sqrt((XK-XI(jj)-4*um).^2+...
(YK-YI(jj)+um).^2+zsize2^2);
f_xy = k0.*zsize2./(2i*pi).*exp(k0*rho_p1*1i)./rho_p1.^2.*(1-1./(k0*rho_p1*1i));
f_xy=reshape(f_xy,[Nks,1]);
Wki(jj,:)=f_xy.';
end
clear TMC rect_size f_xy rho_p1 zsize YI YK XI XK yk xk lefts1 lefts2
C=cell(Ni^2,1);
% ab=single(Wki);
%
% How to calculate the final matrix may still be a problem. The C matrix is the D matrix of the next code, D=C=RVL, R is the output propagation matrix (Relso Murphy propagation), L is the input matrix,
% V is the modulation matrix, which is equivalent to phase spatial light modulation. The size of C is 10000*10000
for jj=1:1:Nis
BIJK=zeros(Nis,Nks);
for i=1:1:Nis
BIJK(i,:)=Wki(i,:).*Wkj(:,jj).';
% BIJK(i,:)=Wki(i,:).*Wki(jj,:);
end
C{jj,1}=BIJK;
end
D = [];
for i = 1:100
D = [D;C{i}];
end
%cond(D)
dD = decomposition(D);
X1 = dD\J;
norm(X1)
ans = 8.8946e+08
res = D*X1-J;
norm(res)
ans = 189.9414
deltax = dD\res;
norm(deltax)
ans = 9.0709e+03
X2 = X1-deltax;
res = D*X2-J;
norm(res)
ans = 0.8860
道夫
道夫 on 13 Jul 2024 at 6:55
Thank you for providing me with ideas to solve the problem and allowing me to discover the main reasons,That my condition number is too large.I have almost solve this problem now.

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