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The following is my implementation of a paper for cauchy noise removal.
The psnr value of the noisy image is 19 as the paper but when I compute the psnr value for the restored image un, it returns 17 which is even smaller than the psnr value of the noisy image. I guess there is something wrong in psnr computation with my code.
%%
clear memory;
clear all
close all
clc;
%% Initialization
refimg = im2double(imread('cameraman256.png')); % original image
img_height = size(refimg,1);
img_width = size(refimg,2);
refimg = refimg(1:img_height,1:img_width);
padNum = 5;
refimg = padarray(refimg,[padNum,padNum],'symmetric');
[mm,nn]=size(refimg);
img_height = size(refimg,1);
img_width = size(refimg,2);
%% Producing the degraded image
A = 1; % A =1 for image denoising
sz = size(refimg);
rng(0);
r1 = randn(sz); % (using randn because I don't have the statistics toolbox)
r2 = randn(sz);
n = 0.02; % the noise level
u0 = refimg + n.*(r1./r2);
u0 = min(u0,1); % clamp large values to 1
u0 = max(u0,0); % clamp small values to 0
figure(1); imshow(u0(padNum+1:mm-padNum,padNum+1:nn-padNum),'border','tight');
%% Initial values for the primal-dual algorithm
tol = 1e-3;
nIter = 1e3;
options.order = 1; options.bound = 'sym';
un = u0;
wn = u0;
bun = un;
bwn = wn;
pxn = zeros(ny,nx);
pyn = zeros(ny,nx);
q = zeros(ny,nx);
[gxn,gyn] = grad(u0,options);
bgxn = gxn;
bgyn = gyn;
gamma = sqrt(2)/10;
lambda = 0.7;
mu = 6.25;
tau = 0.3;
sigma = 0.3;
%% Primal-dual Algorithm
for j = 1:nIter
%%%%%%%%%%solve the subproblem p
[ux,uy]=grad(bun,options);
pxn = pxn+sigma*(bgxn-ux);
pyn = pyn+sigma*(bgyn-uy);
%%%%%%%%%%%solve the subproblem q
AUk = Au(bun);
q = q+sigma*(bwn-AUk);
%%%%%%%%%%solve the subproblem g
goldxn = gxn;
goldyn = gyn;
txn = gxn-tau*pxn;
tyn = gyn-tau*pyn;
sn = max(1e-6,sqrt(txn.^2+tyn.^2));
gxn = txn./sn.*max(0,sn-tau);
gyn = tyn./sn.*max(0,sn-tau);
%%%%%%%%%%%solve the subproblem wn
u_medfilter = medfilt2(u0);
wold = wn;
a = mu.*lambda.*tau+1;
b = -(mu.*lambda.*tau.*(2.*u0+u_medfilter)-tau.*q+2.*u0+wold);
c = tau.*lambda+mu.*lambda.*tau.*(gamma.^2+u0.^2+2.*u_medfilter.*u0)-2.*tau.*q.*u0+...
gamma.^2+u0.^2+2.*wold.*u0;
d = -tau.*lambda.*u0-mu.*lambda.*tau.*u_medfilter.*(gamma.^2+u0.^2)+tau.*q.*(gamma.^2+u0.^2)...
-wold.*(gamma.^2+u0.^2);
qval = (3.*a.*c-(b.^2))./(9.*(a.^2));
rval = (9.*a.*b.*c-27.*(a.^2).*d-2.*(b.^3))./(54.*(a.^3));
deltaval = qval.^3+rval.^2;
wn = nthroot(rval+real(sqrt(deltaval)),3)+nthroot(rval-real(sqrt(deltaval)),3)-((b)./(3.*a));
%%%%%%%%%%%solve the subproblem un
uold = un;
Asqk = Atu(q);
un = un+tau*(Asqk-div(pxn,pyn,options)); % the restored image
% un = min(1,max(0.01,un));
if (norm(un-uold, 'fro')/norm(uold,'fro')<tol)
break;
end
bun = 2*un-uold;
bwn = 2*wn-wold;
bgxn = 2*gxn-goldxn;
bgyn = 2*gyn-goldyn;
% PSNR_restoredimage = psnr(refimg(padNum+1:mm-padNum,padNum+1:nn-b
padNum),un(padNum+1:mm-padNum,padNum+1:nn-padNum))
end
%% Dispaly results
New_un = un;
refimg = refimg(padNum+1:mm-padNum,padNum+1:nn-padNum);
u0 = u0(padNum+1:mm-padNum,padNum+1:nn-padNum);
u_medfilter = u_medfilter(padNum+1:mm-padNum,padNum+1:nn-padNum);
New_un = New_un(padNum+1:mm-padNum,padNum+1:nn-padNum);
PSNR_noisy = psnr(refimg,u0)
PSNR_med = psnr(refimg,u_medfilter)
PSNR_restoredimage = psnr(New_un,refimg)
figure(2); imshow([refimg,u_medfilter,New_un],'border','tight');
%%
The problem was not in psnr computation. The problem with my code in this question was that I was considering the denoising case but I had forgotten to omit the blur kernel in primal-dual algorithm. Also, the initial parameters corresponding to the deblurring case must be zero. The following is the corrected code. Now the psnr value is 28 similar to the paper.
%%
clear;
%% Initialization
refimg = im2double(imread('cameraman256.png')); % original image
img_height = size(refimg,1);
img_width = size(refimg,2);
refimg = refimg(1:img_height,1:img_width);
%% Producing the noisy image
sz = size(refimg);
rng(0);
r1 = randn(sz); % (using randn because I don't have the statistics toolbox)
r2 = randn(sz);
n = 0.02; % the noise level
u0 = refimg + n.*(r1./r2);
u0 = min(u0,1); % clamp large values to 1
u0 = max(u0,0); % clamp small values to 0
figure(1); imshow(u0,'border','tight');
%% Initial values for the primal-dual algorithm
init = u0;
[ny,nx] = size(init);
tol = 1e-3;
nIter = 1e3;
options.order = 1; options.bound = 'sym';
un = u0;
wn = zeros(ny,nx);
bun = un;
bwn = zeros(ny,nx);
pxn = zeros(ny,nx);
pyn = zeros(ny,nx);
q = zeros(ny,nx);
[gxn,gyn] = grad(u0,options);
bgxn = gxn;
bgyn = gyn;
gamma = sqrt(2)/10;
lambda = 0.7;
mu = 6.25;
tau = 0.3;
sigma = 0.3;
%% Primal-dual Algorithm
for j = 1:nIter
%%%%%%%%%%solve the subproblem p
[ux,uy]=grad(bun,options);
pxn = pxn+sigma*(bgxn-ux);
pyn = pyn+sigma*(bgyn-uy);
%%%%%%%%%%%solve the subproblem q
AUk = bun;
q = q+sigma*(bwn-AUk);
%%%%%%%%%%solve the subproblem g
goldxn = gxn;
goldyn = gyn;
txn = gxn-tau*pxn;
tyn = gyn-tau*pyn;
sn = max(1e-6,sqrt(txn.^2+tyn.^2));
gxn = txn./sn.*max(0,sn-tau);
gyn = tyn./sn.*max(0,sn-tau);
%%%%%%%%%%%solve the subproblem wn
u_medfilter = medfilt2(u0);
wold = wn;
a = mu.*lambda.*tau+1;
b = -(mu.*lambda.*tau.*(2.*u0+u_medfilter)-tau.*q+2.*u0+wold);
c = tau.*lambda+mu.*lambda.*tau.*(gamma.^2+u0.^2+2.*u_medfilter.*u0)-2.*tau.*q.*u0+...
gamma.^2+u0.^2+2.*wold.*u0;
d = -tau.*lambda.*u0-mu.*lambda.*tau.*u_medfilter.*(gamma.^2+u0.^2)+tau.*q.*(gamma.^2+u0.^2)...
-wold.*(gamma.^2+u0.^2);
qval = (3.*a.*c-(b.^2))./(9.*(a.^2));
rval = (9.*a.*b.*c-27.*(a.^2).*d-2.*(b.^3))./(54.*(a.^3));
deltaval = qval.^3+rval.^2;
wn = nthroot(rval+real(sqrt(deltaval)),3)+nthroot(rval-real(sqrt(deltaval)),3)-((b)./(3.*a));
%%%%%%%%%%%solve the subproblem un
uold = un;
Asqk = q;
un = un+tau*(Asqk-div(pxn,pyn,options)); % the restored image
if (norm(un-uold, 'fro')/norm(uold,'fro')<tol)
break;
end
bun = 2*un-uold;
bwn = 2*wn-wold;
bgxn = 2*gxn-goldxn;
bgyn = 2*gyn-goldyn;
end
%% Dispaly results
PSNR_noisy = psnr(u0,refimg)
PSNR_med = psnr(u_medfilter,refimg)
PSNR_restoredimage = psnr(un,refimg)
figure(2); imshow([refimg,u_medfilter,un],'border','tight');
%%
I'm trying to implement Divide and Conquer SVD of an upper bidiagonal matrix B, but my code is not working. The error is:
"Unable to perform assignment because the size of the left side is
3-by-3 and the size of the right side is 2-by-2.
V_bar(1:k,1:k) = V1;"
Can somebody help me fix it? Thanks.
function [U,S,V] = DivideConquer_SVD(B)
[m,n] = size(B);
k = floor(m/2);
if k == 0
U = 1;
V = 1;
S = B;
return;
else
% Divide the input matrix
alpha = B(k,k);
beta = B(k,k+1);
e1 = zeros(m,1);
e2 = zeros(m,1);
e1(k) = 1;
e2(k+1) = 1;
B1 = B(1:k-1,1:k);
B2 = B(k+1:m,k+1:m);
%recursive computations
[U1,S1,V1] = DivideConquer_SVD(B1);
[U2,S2,V2] = DivideConquer_SVD(B2);
U_bar = zeros(m);
U_bar(1:k-1,1:k-1) = U1;
U_bar(k,k) = 1;
U_bar((k+1):m,(k+1):m) = U2;
D = zeros(m);
D(1:k-1,1:k) = S1;
D((k+1):m,(k+1):m) = S2;
V_bar = zeros(m);
V_bar(1:k,1:k) = V1;
V_bar((k+1):m,(k+1):m) = V2;
u = alpha*e1'*V_bar + beta*e2'*V_bar;
u = u';
D_tilde = D*D + u*u';
% compute eigenvalues and eigenvectors of D^2+uu'
[L1,Q1] = eig(D_tilde);
eigs = diag(L1);
S = zeros(m,n)
S(1:(m+1):end) = eigs
U_tilde = Q1;
V_tilde = Q1;
%Compute eigenvectors of the original input matrix T
U = U_bar*U_tilde;
V = V_bar*V_tilde;
return;
end
With limited mathematical knowledge, you need to help me a bit more -- as I cannot judge if the approach is correct in a mathematical way (with no theory given;) ). Anyway, I couldn't even reproduce the error e.g with this matrix, which The MathWorks use to illustrate their LU matrix factorization
A = [10 -7 0
-3 2 6
5 -1 5];
So I tried to structure your code a bit and gave some hints. Extend this to make your code clearer for those people (like me) who are not too familiar with matrix decomposition.
function [U,S,V] = DivideConquer_SVD(B)
% m x n matrix
[m,n] = size(B);
k = floor(m/2);
if k == 0
disp('if') % for debugging
U = 1;
V = 1;
S = B;
% return; % net necessary as you don't do anything afterwards anyway
else
disp('else') % for debugging
% Divide the input matrix
alpha = B(k,k); % element on diagonal
beta = B(k,k+1); % element on off-diagonal
e1 = zeros(m,1);
e2 = zeros(m,1);
e1(k) = 1;
e2(k+1) = 1;
% divide matrix
B1 = B(1:k-1,1:k); % upper left quadrant
B2 = B(k+1:m,k+1:m); % lower right quadrant
% recusrsive function call
[U1,S1,V1] = DivideConquer_SVD(B1);
[U2,S2,V2] = DivideConquer_SVD(B2);
U_bar = zeros(m);
U_bar(1:k-1,1:k-1) = U1;
U_bar(k,k) = 1;
U_bar((k+1):m,(k+1):m) = U2;
D = zeros(m);
D(1:k-1,1:k) = S1;
D((k+1):m,(k+1):m) = S2;
V_bar = zeros(m);
V_bar(1:k,1:k) = V1;
V_bar((k+1):m,(k+1):m) = V2;
u = (alpha*e1.'*V_bar + beta*e2.'*V_bar).'; % (little show-off tip: '
% is the complex transpose operator; .' is the "normal" transpose
% operator. It's good practice to distinguish between them but there
% is no difference for real matrices anyway)
D_tilde = D*D + u*u.';
% compute eigenvalues and eigenvectors of D^2+uu'
[L1,Q1] = eig(D_tilde);
eigs = diag(L1);
S = zeros(m,n);
S(1:(m+1):end) = eigs;
U_tilde = Q1;
V_tilde = Q1;
% Compute eigenvectors of the original input matrix T
U = U_bar*U_tilde;
V = V_bar*V_tilde;
% return; % net necessary as you don't do anything afterwards anyway
end % for
end % function
Trying to find the optimal hyperparameters for my svm model using a grid search, but it simply returns 1 for the hyperparameters.
function evaluations = inner_kfold_trainer(C,q,k,features_xy,labels)
features_xy_flds = kdivide(features_xy, k);
labels_flds = kdivide(labels, k);
evaluations = zeros(k,3);
for i = 1:k
fprintf('Fold %i of %i\n',i,k);
train_data = cell2mat(features_xy_flds(1:end ~= i));
train_labels = cell2mat(labels_flds(1:end ~= i));
test_data = cell2mat(features_xy_flds(i));
test_labels = cell2mat(labels_flds(i));
%AU1
train_labels = train_labels(:,1);
test_labels = test_labels(:,1);
[k,~] = size(test_labels);
%train
sv = fitcsvm(train_data,train_labels, 'KernelFunction','polynomial', 'PolynomialOrder',q,'BoxConstraint',C);
sv.predict(test_data);
%Calculate evaluative measures
%svm_outputs = zeros(k,1);
sv_predictions = sv.predict(test_data);
[precision,recall,F1] = evaluation(sv_predictions,test_labels);
evaluations(i,1) = precision;
evaluations(i,2) = recall;
evaluations(i,3) = F1;
end
save('eval.mat', 'evaluations');
end
an inner-fold cross validation function
and below the grid function where something seems to be going wrong
function [q,C] = grid_search(features_xy,labels,k)
% n x n grid
n = 3;
q_grid = linspace(1,19,n);
C_grid = linspace(1,59,n);
tic
evals = zeros(n,n,3);
for i = 1:n
for j = 1:n
fprintf('## i=%i, j=%i ##\n', i, j);
svm_results = inner_kfold_trainer(C_grid(i), q_grid(j),k,features_xy,labels);
evals(i,j,:) = mean(svm_results(:,:));
% precision only
%evals(i,j,:) = max(svm_results(:,1));
toc
end
end
f = evals;
% retrieving the best value of the hyper parameters, to use in the outer
% fold
[M1,I1] = max(f);
[~,I2] = max(M1(1,1,:));
index = I1(:,:,I2);
C = C_grid(index(1))
q = q_grid(index(2))
end
When I run grid_search(features_xy,labels,8) for example, I get C=1 and q=1, for any k(the no. of folds) value. Also features_xy is a 500*98 matrix.
I'm trying to covert this Matlab code to Scilab, but I have some problems.
N = 101;
L = 4*pi;
x = linspace(0,L,N);
% It has three data set; 1: past, 2: current, 3: future.
u = zeros(N,3);
s = 0.5;
% Gaussian Pulse
y = 2*exp(-(x-L/2).^2);
u(:,1) = y;
u(:,2) = y;
% Plot the initial condition.
handle_line = plot(x,u(:,2),'LineWidth',2);
axis([0,L,-2,2]);
xlabel('x'); ylabel('u');
title('Wave equation');
% Dirichet Boundary conditions
u(1,:) = 0;
u(end,:) = 0;
filename = 'wave.gif';
for ii=1:100
disp(['at ii= ', num2str(ii)]);
u(2:end-1,3) = s*(u(3:end,2)+u(1:end-2,2)) ...
+ 2*(1-s)*u(2:end-1,2) ...
- u(2:end-1,1);
u(:,1) = u(:,2);
u(:,2) = u(:,3);
handle_line.YData = u(:,2);
drawnow;
frame = getframe(gcf);
im = frame2im(frame);
[A,map] = rgb2ind(im,256);
if ii==1
imwrite(A,map,filename,'gif','LoopCount',Inf,'DelayTime',0.05);
else
imwrite(A,map,filename,'gif','WriteMode','append','DelayTime',0.05);
end
end
I get an error for this line:
handle_line = plot(x,u(:,2),'LineWidth',2);
Error states: Wrong number of output arguments
What should i change to fix it?
The line
axis([0,L,-2,2]);
has to be translated in Scilab to
set(gca(),"data_bounds",[0,L,-2,2]);
Try this out:
N = 101;
L = 4*pi;
x = linspace(0,L,N);
% It has three data set; 1: past, 2: current, 3: future.
u = zeros(N,3);
s = 0.5;
% Gaussian Pulse
y = 2*exp(-(x-L/2).^2);
u(:,1) = y;
u(:,2) = y;
% Define a standard plot range for x and y
x_range=[min(x) max(x)];
y_range=[-max(y) max(y)];
% Plot the initial condition.
plot(x,u(:,2),'LineWidth',2);
axis([0,L,-2,2]);
xlabel('x'); ylabel('u');
title('Wave equation');
% Dirichet Boundary conditions
u(1,:) = 0;
u(end,:) = 0;
filename = 'wave.gif';
for ii=1:100
disp(['at ii= ', num2str(ii)]);
u(2:end-1,3) = s*(u(3:end,2)+u(1:end-2,2)) ...
+ 2*(1-s)*u(2:end-1,2) ...
- u(2:end-1,1);
u(:,1) = u(:,2);
u(:,2) = u(:,3);
plot(x,u(:,2),'LineWidth',2);
axis([x_range y_range]);
frame = getframe(gcf);
im = frame2im(frame);
[A,map] = rgb2ind(im,256);
if ii==1
imwrite(A,map,filename,'gif','LoopCount',Inf,'DelayTime',0.05);
else
imwrite(A,map,filename,'gif','WriteMode','append','DelayTime',0.05);
end
end
I removed the output and added axis limit independently.
I have a linear system Ay = b, which is created by matrix looks like this:
Here attempt to find the curves based on the matrix in the image description:
n = 10;
x0 = 0;
xn = 1;
h = 1/n;
y0 = 0;
y1 = 0;
x = zeros(1:n-1);
for i = 1:n-1;
x(i) = i*h
end
A =zeros(n-1);
for j = 1:n-2;
A(j,j+1) = (1+h/2);
A(j,j) = (h*exp(x(j))-2);
A(j+1,j) = (1-h/2);
end
A(n-1,n-1) = (h*exp(x(n-1))-2);
b = zeros(1,n-1); %Right-hand side vector
for i = 1:n-1
b(i)=h^2*((exp(x(i))-pi^2)*sin(pi*x(i))+pi*cos(pi*x(i)));
end
b=b';
y = zeros(1,n-1);
y = inv(A)*b % Solving for y
figure
plot(x,y,x,sin(x))
This is code that I create but the curves disappear, anyone can help me to check my code?