So I am trying to deblur an image using the heat equation, but when I run the code, I get don't get an image now. So I did this by looking at http://www4.ncsu.edu/eos/users/w/white/www/white/ma325/MVlec3.pdf to write the code:
u0=double(imread('clock.tiff'));
[m,n,k] = size(u0); %k=conductivity
%if k==3
%u0 = rgb2gray(u0);
%end;
u0 = double(u0);
subplot(1,2,1);
imagesc(u0);
colormap gray;
title('Original');
dt = 0.2; %time step
T = 50; %stopping time
u = u0;
for t = 0:dt:T
u_xx = u(:,[2:n n]) - 2*u + u(:,[1 1:n-1]);
u_yy = u([2:m m], :) - 2*u + u([1 1:m-1],:);
u = u + k*dt*(u_xx+u_yy);
% subplot(1,2,2);
% imagesc(u);
% title(['t=',num2str(t)]);
% drawnow;
temp=u;
end;
imshow(uint8(temp));
imwrite(uint8(temp),'clockmodified.tiff','TIFF');
for t = 0:dt:T
u_xx = u(:,[2:n n]) - 2*u + u(:,[1 1:n-1]);
u_yy = u([2:m m], :) - 2*u + u([1 1:m-1],:);
L=-u_xx-u_yy;
u = u + k*dt*L;
subplot(1,2,2);
imagesc(u);
title(['t=',num2str(t)]);
drawnow;
temp=u;
end;
% imshow(uint8(temp));
imwrite(uint8(temp),'clockdeblur1.tiff','TIFF');
Related
The picture with noise is like this.
Noised picture: Image3.bmp
I was doing image processing in MatLab with some built-in and self-implemented filters.
I have already tried a combination of bilateral, median and gaussian. bilateral and gaussian code are at the end of this post.
img3 = double(imread('Image3.bmp')); % this is the noised image
lena = double(imread('lena_gray.jpg')); % this is the original one
img3_com = bilateral(img3, 3, 2, 80);
img3_com = medfilt2(img3_com, [3 3], 'symmetric');
img3_com = gaussian(img3_com, 3, 0.5);
img3_com = bilateral(double(img3_com), 6, 100, 13);
SNR3_com = snr(img3_com,img3_com - lena); % 17.1107
However, the result is not promising with SNR of only 17.11.
Filtered image: img3_com
The original picture is like this.
Clean original image: lena_gray.jpg
Could you please give me any possible ideas about how to process it? Like what noise generators generated the noised image and what filtering methods or image processing method I can use to deal with it. Appreciate!!!
My bilateral function bilateral.m
function img_new = bilateral(img_gray, window, sigmaS, sigmaI)
imgSize = size(img_gray);
img_new = zeros(imgSize);
for i = 1:imgSize(1)
for j = 1:imgSize(2)
sum = 0;
simiSum = 0;
for a = -window:window
for b = -window:window
x = i + a;
y = j + b;
p = img_gray(i,j);
q = 0;
if x < 1 || y < 1 || x > imgSize(1) || y > imgSize(2)
% q=0;
continue;
else
q = img_gray(x,y);
end
gaussianFilter = exp( - double((a)^2 + (b)^2)/ (2 * sigmaS^2 ) - (double(p-q)^2)/ (2 * sigmaI^2 ));
% gaussianFilter = gaussian((a^2 + b^2)^(1/2), sigma) * gaussian(abs(p-q), sigma);
sum = sum + gaussianFilter * q;
simiSum = simiSum + gaussianFilter;
end
end
img_new(i,j) = sum/simiSum;
end
end
% disp SNR
lena = double(imread('lena_gray.jpg'));
SNR1_4_ = snr(img_new,img_new - lena);
disp(SNR1_4_);
My gaussian implementation gaussian.m
function img_gau = gaussian(img, hsize, sigma)
h = fspecial('gaussian', hsize, sigma);
img_gau = conv2(img,h,'same');
% disp SNR
lena = double(imread('lena_gray.jpg'));
SNR1_4_ = snr(img_gau,img_gau - lena);
disp(SNR1_4_);
I am getting an error when I run this code for disc waves. The code is attached.
The Error is in line 137 and 292. Please help in resolving issue.
function waves
% WAVES Wave equation in one and two space dimensions.
% The two-dimensional domains include a pi-by-pi square, a unit disc,
% a three-quarter circular sector and the L-shaped union of three squares.
% The eigenfunctions of the square are sin(m*x)*sin(n*y). With polar
% coordinates, the eigenfunctions of the disc and the sector involve Bessel
% functions. The eigenfunctions of the L-shaped domain also involve
% Bessel functions and are computed by the MATLAB function membranetx.m.
% 2-D eigenvalues and eigenfunctions
m = 11; % Determines number of grid points
speed = 1;
bvals = [1; 0; 0; 0; 0];
t = 0;
while bvals(5) == 0
% Initialize figure
shg
clf reset
set(gcf,'doublebuffer','on','menubar','none','tag','', ...
'numbertitle','off','name','Waves','colormap',hot(64));
for k= 1:5
b(k) = uicontrol('style','toggle','value',bvals(k), ...
'units','normal','position',[.15*k .01 .14 .05]);
end
set(b(1),'style','pop','string', ...
{'1-d','square','disc','sector'})
set(b(2),'string','modes/wave')
set(b(3),'string','slower')
set(b(4),'string','faster')
set(b(5),'string','close')
if bvals(3)==1
speed = speed/sqrt(2);
set(b(3),'value',0);
end
if bvals(4)==1
speed = speed*sqrt(2);
set(b(4),'value',0);
end
bvals = cell2mat(get(b,'value'));
region = bvals(1);
modes = bvals(2)==0;
if region == 1
% 1-D
x = (0:4*m)/(4*m)*pi;
orange = [1 1/3 0];
gray = get(gcf,'color');
if modes
% 1-D modes
for k = 1:4
subplot(2,2,k)
h(k) = plot(x,zeros(size(x)));
axis([0 pi -3/2 3/2])
set(h(k),'color',orange,'linewidth',3)
set(gca,'color',gray','xtick',[],'ytick',[])
end
delta = 0.005*speed;
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
for k = 1:4
u = sin(k*t)*sin(k*x);
set(h(k),'ydata',u)
end
drawnow
bvs = cell2mat(get(b,'value'));
end
else
% 1-D wave
h = plot(x,zeros(size(x)));
axis([0 pi -9/4 9/4])
set(h,'color',orange,'linewidth',3)
set(gca,'color',gray','xtick',[],'ytick',[])
delta = 0.005*speed;
a = 1./(1:4);
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
u = zeros(size(x));
for k = 1:4
u = u + a(k)*sin(k*t)*sin(k*x);
end
set(h,'ydata',u)
drawnow
bvs = cell2mat(get(b,'value'));
end
end
elseif region <= 5
switch region
case 2
% Square
x = (0:2*m)/(2*m)*pi;
y = x';
lambda = zeros(4,1);
V = cell(4,1);
k = 0;
for i = 1:2
for j = 1:2
k = k+1;
lambda(k) = i^2 + j^2;
V{k} = sin(i*y)*sin(j*x);
end
end
ax = [0 pi 0 pi -1.75 1.75];
case 3
% Disc, mu = zeros of J_0(r) and J_1(r)
mu = [bjzeros(0,2) bjzeros(1,2)];
[r,theta] = meshgrid((0:m)/m,(-m:m)/m*pi);
x = r.*cos(theta);
y = r.*sin(theta);
V = cell(4,1);
k = 0;
for j = 0:1
for i = 1:2
k = k+1;
if j == 0
V{k} = besselj(0,mu(k)*r);
else
V{k} = besselj(j,mu(k)*r).*sin(j*theta);
end
V{k} = V{k}/max(max(abs(V{k})));
end
end
lambda = mu.^2;
ax = [-1 1 -1 1 -1.75 1.75];
case 4
% Circular sector , mu = zeros of J_(2/3)(r) and J_(4/3)(r)
mu = [bjzeros(2/3,2) bjzeros(4/3,2)];
[r,theta] = meshgrid((0:m)/m,(3/4)*(0:2*m)/m*pi);
x = r.*cos(theta+pi);
y = r.*sin(theta+pi);
V = cell(4,1);
k = 0;
for j = 1:2
for i = 1:2
k = k+1;
alpha = 2*j/3;
V{k} = besselj(alpha,mu(k)*r).*sin(alpha*theta);
V{k} = V{k}/max(max(abs(V{k})));
end
end
lambda = mu.^2;
ax = [-1 1 -1 1 -1.75 1.75];
case 5\
% L-membrane
x = (-m:m)/m;
y = x';
lambda = zeros(4,1);
V = cell(4,1);
for k = 1:4
[L lambda(k)] = membranetx(k,m,9,9);
L(m+2:2*m+1,m+2:2*m+1) = NaN;
V{k} = rot90(L,-1);
end
ax = [-1 1 -1 1 -1.75 1.75];
end
if modes
% 2-D modes
p = [.02 .52 .02 .52];
q = [.52 .52 .02 .02];
for k = 1:4
axes('position',[p(k) q(k) .46 .46]);
h(k) = surf(x,y,zeros(size(V{k})));
axis(ax)
axis off
view(225,30);
caxis([-1.5 1]);
end
delta = .08*speed;
mu = sqrt(lambda(:));
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
for k = 1:4
U = 1.5*sin(mu(k)*t)*V{k};
set(h(k),'zdata',U)
set(h(k),'cdata',U)
end
drawnow
bvs = cell2mat(get(b,'value'));
end
else
% 2-D wave
h = surf(x,y,zeros(size(V{1})));
axis(ax);
axis off
view(225,30);
caxis([-1.5 1]);
delta = .02*speed;
mu = sqrt(lambda(:));
a = 1.25./(1:4);
bvs = bvals;
while all(bvs == bvals)
t = t + delta;
U = zeros(size(V{1}));
for k = 1:4
U = U + a(k)*sin(mu(k)*t)*V{k};
end
set(h,'zdata',U)
set(h,'cdata',U)
drawnow
bvs = cell2mat(get(b,'value'));
end
end
elseif region == 6
figure
bizcard
set(b(1),'value',1)
end
% Retain uicontrol values
bvals = cell2mat(get(b,'value'));
end
close
% -------------------------------
function z = bjzeros(n,k)
% BJZEROS Zeros of the Bessel function.
% z = bjzeros(n,k) is the first k zeros of besselj(n,x)
% delta must be chosen so that the linear search can take
% steps as large as possible without skipping any zeros.
% delta is approx bjzero(0,2)-bjzero(0,1)
delta = .99*pi;
Jsubn = inline('besselj(n,x)''x','n');
a = n+1;
fa = besselj(n,a);
z = zeros(1,k);
j = 0;
while j < k
b = a + delta;
fb = besselj(n,b);
if sign(fb) ~= sign(fa)
j = j+1;
z(j) = fzerotx(Jsubn,[a b],n);
end
a = b;
fa = fb;
end
I used this link to walk through FDTD code and write it for myself to practice.
https://www.youtube.com/watch?v=y7hJAhKp2d8
This is the code that I wrote (its not verbatim, but extremely similar). When I run the program, I am told that "Field amplitude is too small to visualize properly." I don't understand why. As far as I understand Matlab (I'm very new to it), I am scaling the graph on my own. There is a function called draw1d that is in a protected file here: http://emlab.utep.edu/ee5390fdtd.htm
How should I fix this?
%FDTD1D
%Initailize
close all;
clc;
clear all;
% Units
meters = 1;
seconds = 1;
%Fundamental constants
c0 = 3e8 * meters/seconds;
e0 = 8.85e-12 * 1/meters;
u0 = 1.26e-6 * 1/meters;
%Figure Window
figure('Color', 'b');
%%%%%%%
%% FUNAMENTAL CONSTANTS
%%%%%%%%%
%Simple parameters
dz = 5 * meters;
Nz = 200;
dt = 1e-3 * seconds;
STEPS = 1000;
%%%%%%%
%% BUILD DEVICE ON GRID
%%%%%%%%%
%Grid Device - let it be air
ER = ones(1, Nz);
UR = ones(1, Nz);
%%%%%%%
%% Initialize FDTD Parameters
%%%%%%%%%
%Initialize Vectors
mEy = (c0*dt)./ER; %unique update coefficient for each place on the grid
mHx = (c0*dt)./UR; %unique update coefficent again
%%%%%%%
%% Initialize Fields
%%%%%%%%%
%Initialize Fields
Ey = zeros(1, Nz);
Hx = zeros(1, Nz);
%%%%%%%
%% IFDTD LOOP ANALYSIS
%%%%%%%%%
%Actually doing FDTD
for T = 1 : STEPS
%Update H from E
for nz = 1 : Nz - 1
Hx(nz) = Hx(nz) + mHx(nz)*(Ey(nz+1) - Ey(nz))/dz;
end
Hx(Nz) = Hx(Nz) + mHx(Nz)*(0 - Ey(Nz))/dz;
%Update E from H
Ey(1) = Ey(1) + mEy(1) * ( Hx(1) - 0 )/dz;
for nz = 2 : Nz
Ey(nz) = Ey(nz) + mEy(nz) * ( Hx(nz) - Hx(nz-1) )/dz;
end
%Show Status
if ~mod(T, 10)
draw1d(ER, Ey, Hx, dz);
xlim([dz Nz*dz]);
xlabel('z');
title(['Field at step ' num2str(T) ' of ' num2str(STEPS)]);
end
end
Actually i have two intersecting circles as specified in the figure
i want to find the area of each part separately using Monte carlo method in Matlab .
The code doesn't draw the rectangle or the circles correctly so
i guess what is wrong is my calculation for the x and y and i am not much aware about the geometry equations for solving it so i need help about the equations.
this is my code so far :
n=1000;
%supposing that a rectangle will contain both circles so :
% the mid point of the distance between 2 circles will be (0,6)
% then by adding the radius of the left and right circles the total distance
% will be 27 , 11 from the left and 16 from the right
% width of rectangle = 24
x=27.*rand(n-1)-11;
y=24.*rand(n-1)+2;
count=0;
for i=1:n
if((x(i))^2+(y(i))^2<=25 && (x(i))^2+(y(i)-12)^2<=100)
count=count+1;
figure(2);
plot(x(i),y(i),'b+')
hold on
elseif(~(x(i))^2+(y(i))^2<=25 &&(x(i))^2+(y(i)-12)^2<=100)
figure(2);
plot(x(i),y(i),'y+')
hold on
else
figure(2);
plot(x(i),y(i),'r+')
end
end
Here are the errors I found:
x = 27*rand(n,1)-5
y = 24*rand(n,1)-12
The rectangle extents were incorrect, and if you use rand(n-1) will give you a (n-1) by (n-1) matrix.
and
first If:
(x(i))^2+(y(i))^2<=25 && (x(i)-12)^2+(y(i))^2<=100
the center of the large circle is at x=12 not y=12
Second If:
~(x(i))^2+(y(i))^2<=25 &&(x(i)-12)^2+(y(i))^2<=100
This code can be improved by using logical indexing.
For example, using R, you could do (Matlab code is left as an excercise):
n = 10000
x = 27*runif(n)-5
y = 24*runif(n)-12
plot(x,y)
r = (x^2 + y^2)<=25 & ((x-12)^2 + y^2)<=100
g = (x^2 + y^2)<=25
b = ((x-12)^2 + y^2)<=100
points(x[g],y[g],col="green")
points(x[b],y[b],col="blue")
points(x[r],y[r],col="red")
which gives:
Here is my generic solution for any two circles (without any hardcoded value):
function [ P ] = circles_intersection_area( k1, k2, N )
%CIRCLES_INTERSECTION_AREA Summary...
% Adnan A.
x1 = k1(1);
y1 = k1(2);
r1 = k1(3);
x2 = k2(1);
y2 = k2(2);
r2 = k2(3);
if sqrt((x1-x2)*(x1-x2) + (y1-y2)*(y1-y2)) >= (r1 + r2)
% no intersection
P = 0;
return
end
% Wrapper rectangle config
a_min = x1 - r1 - 2*r2;
a_max = x1 + r1 + 2*r2;
b_min = y1 - r1 - 2*r2;
b_max = y1 + r1 + 2*r2;
% Monte Carlo algorithm
n = 0;
for i = 1:N
rand_x = unifrnd(a_min, a_max);
rand_y = unifrnd(b_min, b_max);
if sqrt((rand_x - x1)^2 + (rand_y - y1)^2) < r1 && sqrt((rand_x - x2)^2 + (rand_y - y2)^2) < r2
% is a point in the both of circles
n = n + 1;
plot(rand_x,rand_y, 'go-');
hold on;
else
plot(rand_x,rand_y, 'ko-');
hold on;
end
end
P = (a_max - a_min) * (b_max - b_min) * n / N;
end
Call it like: circles_intersection_area([-0.4,0,1], [0.4,0,1], 10000) where the first param is the first circle (x,y,r) and the second param is the second circle.
Without using For loop.
n = 100000;
data = rand(2,n);
data = data*2*30 - 30;
x = data(1,:);
y = data(2,:);
plot(x,y,'ro');
inside5 = find(x.^2 + y.^2 <=25);
hold on
plot (x(inside5),y(inside5),'bo');
hold on
inside12 = find(x.^2 + (y-12).^2<=144);
plot (x(inside12),y(inside12),'g');
hold on
insidefinal1 = find(x.^2 + y.^2 <=25 & x.^2 + (y-12).^2>=144);
insidefinal2 = find(x.^2 + y.^2 >=25 & x.^2 + (y-12).^2<=144);
% plot(x(insidefinal1),y(insidefinal1),'bo');
hold on
% plot(x(insidefinal2),y(insidefinal2),'ro');
insidefinal3 = find(x.^2 + y.^2 <=25 & x.^2 + (y-12).^2<=144);
% plot(x(insidefinal3),y(insidefinal3),'ro');
area1=(60^2)*(length(insidefinal1)/n);
area3=(60^2)*(length(insidefinal2)/n);
area2= (60^2)*(length(insidefinal3)/n);
I am using for my project the "LucasKanade" code in matlab. It gives me as output 2 matrices (u and v). These are(i believe so) the velocities of the image in the x and y axes respectively. Now how can i convert these velocities to object velocities(eg in meters/second)?
Thanks in advance
"LucasKanade" code:
function [u, v] = LucasKanade(im1, im2, windowSize);
%LucasKanade lucas kanade algorithm, without pyramids (only 1 level);
%REVISION: NaN vals are replaced by zeros
[fx, fy, ft] = ComputeDerivatives(im1, im2);
u = zeros(size(im1));
v = zeros(size(im2));
halfWindow = floor(windowSize/2);
for i = halfWindow+1:size(fx,1)-halfWindow
for j = halfWindow+1:size(fx,2)-halfWindow
curFx = fx(i-halfWindow:i+halfWindow, j-halfWindow:j+halfWindow);
curFy = fy(i-halfWindow:i+halfWindow, j-halfWindow:j+halfWindow);
curFt = ft(i-halfWindow:i+halfWindow, j-halfWindow:j+halfWindow);
curFx = curFx';
curFy = curFy';
curFt = curFt';
curFx = curFx(:);
curFy = curFy(:);
curFt = -curFt(:);
A = [curFx curFy];
U = pinv(A'*A)*A'*curFt;
u(i,j)=U(1);
v(i,j)=U(2);
end;
end;
u(isnan(u))=0;
v(isnan(v))=0;
%u=u(2:size(u,1), 2:size(u,2));
%v=v(2:size(v,1), 2:size(v,2));
%%
function [fx, fy, ft] = ComputeDerivatives(im1, im2);
%ComputeDerivatives Compute horizontal, vertical and time derivative
% between two gray-level images.
if (size(im1,1) ~= size(im2,1)) | (size(im1,2) ~= size(im2,2))
error('input images are not the same size');
end;
if (size(im1,3)~=1) | (size(im2,3)~=1)
error('method only works for gray-level images');
end;
fx = conv2(im1,0.25* [-1 1; -1 1]) + conv2(im2, 0.25*[-1 1; -1 1]);
fy = conv2(im1, 0.25*[-1 -1; 1 1]) + conv2(im2, 0.25*[-1 -1; 1 1]);
ft = conv2(im1, 0.25*ones(2)) + conv2(im2, -0.25*ones(2));
% make same size as input
fx=fx(1:size(fx,1)-1, 1:size(fx,2)-1);
fy=fy(1:size(fy,1)-1, 1:size(fy,2)-1);
ft=ft(1:size(ft,1)-1, 1:size(ft,2)-1);