I'm writing a "Peak finder" in Matlab. I've never used Matlab or anything similar before this project, so I'm new to "vectorizing" my code. Essentially, the program needs to take a video of molecules and plot circles on the molecules present in each frame of the video. If a molecule is crowded then it gets a red circle, but if it is not crowded it gets a green circle.
My problem is that some of these videos have 2000 frames and my program takes up to ~25 seconds to process a single frame, which is not practical.
Using tic and toc I've found the trouble maker: A for-loop which calls a function that contains a for-loop.
function getPeaks( orgnl_img, processed_img, cut_off, radius )
% find large peaks peaks by thresholding, i.e. you accept a peak only
% if its more than 'threshold' higher than its neighbors
threshold = 2*std2(orgnl_img);
peaks = (orgnl_img - processed_img) > threshold;
% m and n are dimensions of the frame, named peaks
[m, n] = size(peaks);
cc_centroids = regionprops(peaks, 'centroid');
% Pre-allocate arrays
x_centroid = zeros(1, length(cc_centroids));
y_centroid = zeros(1, length(cc_centroids));
for i = 1:length(cc_centroids)
% Extract the x and y components from cc_centroids struct
x_centroid(i) = cc_centroids(i).Centroid(1);
y_centroid(i) = cc_centroids(i).Centroid(2);
row = int64(x_centroid(i));
col = int64(y_centroid(i));
% Assure that script doesnt attempt to exceed frame
if col-2 > 0 && row-2 > 0 && col+2 < m && row+2 < n
region_of_interest = orgnl_img(col-2:col+2,row-2:row+2);
local_intensity = max(region_of_interest(:));
% Do not plot circle when intensity is 'cut off' or lower
if local_intensity > cut_off
dist_bool = findDistance(cc_centroids, x_centroid(i), y_centroid(i), radius);
if dist_bool == 1
color = 'g';
else
color = 'r';
end
plotCircle(color, x_centroid(i), y_centroid(i), radius)
end
end
end
end
And here is the findDistance function which contains another for-loop and determines if the circles overlap:
function dist_bool = findDistance( all_centroids, x_crrnt, y_crrnt, radius )
x_nearby = zeros(1, length(all_centroids));
y_nearby = zeros(1, length(all_centroids));
for i = 1:length(all_centroids)
if all_centroids(i).Centroid(1) < (x_crrnt+2*radius) &&...
all_centroids(i).Centroid(1) > (x_crrnt-2*radius) &&...
all_centroids(i).Centroid(2) < (y_crrnt+2*radius) &&...
all_centroids(i).Centroid(2) > (y_crrnt-2*radius)
x_nearby(i) = all_centroids(i).Centroid(1);
y_nearby(i) = all_centroids(i).Centroid(2);
pts_of_interest = [x_nearby(i),y_nearby(i);x_crrnt,y_crrnt];
dist = pdist(pts_of_interest);
if dist == 0 || dist > 2*radius
dist_bool = 1;
else
dist_bool = 0;
break
end
end
end
end
I think that there must be much improvement to be done here. I would appreciate any advice.
Thanks! :)
UPDATE: Here are the profiler results. The first section of code is the "getPeaks" function
http://i.stack.imgur.com/VaLdH.png
The hold comes from the plot circle function:
function plotCircle( color, x_centroid, y_centroid, radius )
hold on;
th = 0:pi/50:2*pi;
xunit = radius * cos(th) + x_centroid;
yunit = radius * sin(th) + y_centroid;
plot(xunit, yunit, color);
hold off;
end
Related
While using Matlab for image processing (exactly improving img by Fuzzy Logic) I found a really strange thing. My fuzzy function is correct, I tested it on random values and they are basically simple linear functions.
function f = Udark(z)
if z < 50
f = 1;
elseif z > 125
f = 0;
elseif (z >= 50) && (z <= 125)
f = -z/75 + 125/75;
end
end
where z is a value of a pixel (in grayscale). Now there is a really strange thing going on.
f = -z/75 + 125/75;, where a is an image. However, it is giving really different results if used as an input. I.e. if I use a variable p = 99, the output of the function is 0.3467 as it should be, when if I use A(i,j) it is giving me result f=2. Since it is clearly impossible, I do not know where is the problem. I thought that maybe there is a case with the type of the variable but if I change it to uint8 it stays the same... If you know what's going on, please, let me know :)
1.Changed line:
f = (125/75) - (z/75);
After editing the third condition the resultant/transformed image has no pixel values of 2. Not sure if you intend to work with decimals. If decimals are necessary using the im2double() function to convert the image and scaling it up by a factor of 255 might suffice your needs. See heading 3 for rounding details.
2.Reading in Image and Testing:
%Reading in the image and applying the function%
Image = imread("RGB_Image.png");
Greyscale_Image = rgb2gray(Image);
[Image_Height,Image_Width] = size(Greyscale_Image);
Transformed_Image = zeros(Image_Height,Image_Width);
for Row = 1: +1: Image_Height
for Column = 1: +1: Image_Width
Pixel_Value = Greyscale_Image(Row,Column);
[Transformed_Pixel_Value] = Udark(Pixel_Value);
Transformed_Image(Row,Column) = Transformed_Pixel_Value;
end
end
subplot(1,2,1); imshow(Greyscale_Image);
subplot(1,2,2); imshow(Transformed_Image);
%Checking that no transformed pixels falls in this impossible range%
Check = (Transformed_Image > (125/75)) & (Transformed_Image ~= 1);
Check_Flag = any(Check,'all');
%Function to transform pixel values%
function f = Udark(z)
if z < 50
f = 1;
elseif z > 125
f = 0;
elseif (z >= 50) && (z <= 125)
f = (125/75) - (z/75);
end
end
3.Evaluating the Specifics of the Third Condition
Working with integers (uint8) will force the values to be rounded to the nearest integer. Any number that falls between the range (50,125] will evaluate to 1 or 0.
f = -z/75 + 125/75;
If z = 50.1,
-50.1/75 + 125/75 = 74.9/75 ≈ 0.9987 → rounds to 1
Using MATLAB version: R2019b
i am asking for help.. I want to animate the Kaczmarz method on Matlab. It's method allows to find solution of system of equations by the serial projecting solution vector on hyperplanes, which which is given by the eqations of system.
And i want make animation of this vector moving (like the point is going on the projected vectors).
%% System of equations
% 2x + 3y = 4;
% x - y = 2;
% 6x + y = 15;
%%
A = [2 3;1 -1; 6 1];
f = [4; 2; 15];
resh = pinv(A)*f
x = -10:0.1:10;
e1 = (1 - 2*x)/3;
e2 = (x - 2);
e3 = 15 - 6*x;
plot(x,e1)
grid on
%
axis([0 4 -2 2])
hold on
plot(x,e2)
hold on
plot(x,e3)
hold on
precision = 0.001; % точность
iteration = 100; % количество итераций
lambda = 0.75; % лямбда
[m,n] = size(A);
x = zeros(n,1);
%count of norms
for i = 1:m
nrm(i) = norm(A(i,:));
end
for i = 1:1:iteration
j = mod(i-1,m) + 1;
if (nrm(j) <= 0), continue, end;
predx = x;
x = x + ((f(j) - A(j,:)*x)*A(j,:)')/(nrm(j))^2;
p = plot(x);
set(p)
%pause 0.04;
hold on;
if(norm(predx - x) <= precision), break, end
end
I wrote the code for this method, by don't imagine how make the animation, how I can use the set function.
In your code there are a lot of redundant and random pieces. Do not call hold on more than once, it does nothing. Also set(p) does nothing, you want to set some ps properties to something, then you use set.
Also, you are plotting the result, but not the "change". The change is a line between the previous and current, and that is the only reason you'd want to have a variable such as predx, to plot. SO USE IT!
Anyway, this following code plots your algorithm. I added a repeated line to plot in green and then delete, so you can see what the last step does. I also changed the plots in the begging to just plot in red so its more clear what is each of the things.
Change your loop for:
for i = 1:1:iteration
j = mod(i-1,m) + 1;
if (nrm(j) <= 0), continue, end;
predx = x;
x = x + ((f(j) - A(j,:)*x)*A(j,:)')/(nrm(j))^2;
plot([predx(1) x(1)],[predx(2) x(2)],'b'); %plot line
c=plot([predx(1) x(1)],[predx(2) x(2)],'g'); %plot it in green
pause(0.1)
children = get(gca, 'children'); %delete the green line
delete(children(1));
drawnow
% hold on;
if(norm(predx - x) <= precision), break, end
end
This will show:
I'm doing this exercise by Andrew NG about using k-means to reduce the number of colors in an image. But the problem is my code only gives a black-and-white image :( . I have checked every step in the algorithm but it still won't give the correct result. Please help me, thank you very much
Here is the link of the exercise, and here is the dataset.
The correct result is given in the link of the exercise. And here is my black-and-white image:
Here is my code:
function [] = KMeans()
Image = double(imread('bird_small.tiff'));
[rows,cols, RGB] = size(Image);
Points = reshape(Image,rows * cols, RGB);
K = 16;
Centroids = zeros(K,RGB);
s = RandStream('mt19937ar','Seed',0);
% Initialization :
% Pick out K random colours and make sure they are all different
% from each other! This prevents the situation where two of the means
% are assigned to the exact same colour, therefore we don't have to
% worry about division by zero in the E-step
% However, if K = 16 for example, and there are only 15 colours in the
% image, then this while loop will never exit!!! This needs to be
% addressed in the future :(
% TODO : Vectorize this part!
done = false;
while done == false
RowIndex = randperm(s,rows);
ColIndex = randperm(s,cols);
RowIndex = RowIndex(1:K);
ColIndex = ColIndex(1:K);
for i = 1 : K
for j = 1 : RGB
Centroids(i,j) = Image(RowIndex(i),ColIndex(i),j);
end
end
Centroids = sort(Centroids,2);
Centroids = unique(Centroids,'rows');
if size(Centroids,1) == K
done = true;
end
end;
% imshow(imread('bird_small.tiff'))
%
% for i = 1 : K
% hold on;
% plot(RowIndex(i),ColIndex(i),'r+','MarkerSize',50)
% end
eps = 0.01; % Epsilon
IterNum = 0;
while 1
% E-step: Estimate membership given parameters
% Membership: The centroid that each colour is assigned to
% Parameters: Location of centroids
Dist = pdist2(Points,Centroids,'euclidean');
[~, WhichCentroid] = min(Dist,[],2);
% M-step: Estimate parameters given membership
% Membership: The centroid that each colour is assigned to
% Parameters: Location of centroids
% TODO: Vectorize this part!
OldCentroids = Centroids;
for i = 1 : K
PointsInCentroid = Points((find(WhichCentroid == i))',:);
NumOfPoints = size(PointsInCentroid,1);
% Note that NumOfPoints is never equal to 0, as a result of
% the initialization. Or .... ???????
if NumOfPoints ~= 0
Centroids(i,:) = sum(PointsInCentroid , 1) / NumOfPoints ;
end
end
% Check for convergence: Here we use the L2 distance
IterNum = IterNum + 1;
Margins = sqrt(sum((Centroids - OldCentroids).^2, 2));
if sum(Margins > eps) == 0
break;
end
end
IterNum;
Centroids ;
% Load the larger image
[LargerImage,ColorMap] = imread('bird_large.tiff');
LargerImage = double(LargerImage);
[largeRows,largeCols,~] = size(LargerImage); % RGB is always 3
% Dist = zeros(size(Centroids,1),RGB);
% TODO: Vectorize this part!
% Replace each of the pixel with the nearest centroid
for i = 1 : largeRows
for j = 1 : largeCols
Dist = pdist2(Centroids,reshape(LargerImage(i,j,:),1,RGB),'euclidean');
[~,WhichCentroid] = min(Dist);
LargerImage(i,j,:) = Centroids(WhichCentroid);
end
end
% Display new image
imshow(uint8(round(LargerImage)),ColorMap)
imwrite(uint8(round(LargerImage)), 'D:\Hoctap\bird_kmeans.tiff');
You're indexing into Centroids with a single linear index.
Centroids(WhichCentroid)
This is going to return a single value (specifically the red value for that centroid). When you assign this to LargerImage(i,j,:), it will assign all RGB channels the same value resulting in a grayscale image.
You likely want to grab all columns of the selected centroid to provide an array of red, green, and blue values that you want to assign to LargerImage(i,j,:). You can do by using a colon : to specify all columns of Centroids which belong to the row indicated by WhichCentroid.
LargerImage(i,j,:) = Centroids(WhichCentroid,:);
I'm doing this exercise by Andrew NG about using k-means to reduce the number of colors in an image. It worked correctly but I'm afraid it's a little slow because of all the for loops in the code, so I'd like to vectorize them. But there are those loops that I just can't seem to vectorize effectively. Please help me, thank you very much!
Also if possible please give some feedback on my coding style :)
Here is the link of the exercise, and here is the dataset.
The correct result is given in the link of the exercise.
And here is my code:
function [] = KMeans()
Image = double(imread('bird_small.tiff'));
[rows,cols, RGB] = size(Image);
Points = reshape(Image,rows * cols, RGB);
K = 16;
Centroids = zeros(K,RGB);
s = RandStream('mt19937ar','Seed',0);
% Initialization :
% Pick out K random colours and make sure they are all different
% from each other! This prevents the situation where two of the means
% are assigned to the exact same colour, therefore we don't have to
% worry about division by zero in the E-step
% However, if K = 16 for example, and there are only 15 colours in the
% image, then this while loop will never exit!!! This needs to be
% addressed in the future :(
% TODO : Vectorize this part!
done = false;
while done == false
RowIndex = randperm(s,rows);
ColIndex = randperm(s,cols);
RowIndex = RowIndex(1:K);
ColIndex = ColIndex(1:K);
for i = 1 : K
for j = 1 : RGB
Centroids(i,j) = Image(RowIndex(i),ColIndex(i),j);
end
end
Centroids = sort(Centroids,2);
Centroids = unique(Centroids,'rows');
if size(Centroids,1) == K
done = true;
end
end;
% imshow(imread('bird_small.tiff'))
%
% for i = 1 : K
% hold on;
% plot(RowIndex(i),ColIndex(i),'r+','MarkerSize',50)
% end
eps = 0.01; % Epsilon
IterNum = 0;
while 1
% E-step: Estimate membership given parameters
% Membership: The centroid that each colour is assigned to
% Parameters: Location of centroids
Dist = pdist2(Points,Centroids,'euclidean');
[~, WhichCentroid] = min(Dist,[],2);
% M-step: Estimate parameters given membership
% Membership: The centroid that each colour is assigned to
% Parameters: Location of centroids
% TODO: Vectorize this part!
OldCentroids = Centroids;
for i = 1 : K
PointsInCentroid = Points((find(WhichCentroid == i))',:);
NumOfPoints = size(PointsInCentroid,1);
% Note that NumOfPoints is never equal to 0, as a result of
% the initialization. Or .... ???????
if NumOfPoints ~= 0
Centroids(i,:) = sum(PointsInCentroid , 1) / NumOfPoints ;
end
end
% Check for convergence: Here we use the L2 distance
IterNum = IterNum + 1;
Margins = sqrt(sum((Centroids - OldCentroids).^2, 2));
if sum(Margins > eps) == 0
break;
end
end
IterNum;
Centroids ;
% Load the larger image
[LargerImage,ColorMap] = imread('bird_large.tiff');
LargerImage = double(LargerImage);
[largeRows,largeCols,NewRGB] = size(LargerImage); % RGB is always 3
% TODO: Vectorize this part!
largeRows
largeCols
NewRGB
% Replace each of the pixel with the nearest centroid
NewPoints = reshape(LargerImage,largeRows * largeCols, NewRGB);
Dist = pdist2(NewPoints,Centroids,'euclidean');
[~,WhichCentroid] = min(Dist,[],2);
NewPoints = Centroids(WhichCentroid,:);
LargerImage = reshape(NewPoints,largeRows,largeCols,NewRGB);
% for i = 1 : largeRows
% for j = 1 : largeCols
% Dist = pdist2(Centroids,reshape(LargerImage(i,j,:),1,RGB),'euclidean');
% [~,WhichCentroid] = min(Dist);
% LargerImage(i,j,:) = Centroids(WhichCentroid,:);
% end
% end
% Display new image
imshow(uint8(round(LargerImage)),ColorMap)
UPDATE: Replaced
for i = 1 : K
for j = 1 : RGB
Centroids(i,j) = Image(RowIndex(i),ColIndex(i),j);
end
end
with
for i = 1 : K
Centroids(i,:) = Image(RowIndex(i),ColIndex(i),:);
end
I think this may be vectorized further by using linear indexing, but for now I should just focus on the while loop since it takes most of the time.
Also when I tried #Dev-iL's suggestion and replaced
for i = 1 : K
PointsInCentroid = Points((find(WhichCentroid == i))',:);
NumOfPoints = size(PointsInCentroid,1);
% Note that NumOfPoints is never equal to 0, as a result of
% the initialization. Or .... ???????
if NumOfPoints ~= 0
Centroids(i,:) = sum(PointsInCentroid , 1) / NumOfPoints ;
end
end
with
E = sparse(1:size(WhichCentroid), WhichCentroid' , 1, Num, K, Num);
Centroids = (E * spdiags(1./sum(E,1)',0,K,K))' * Points ;
the results were always worse: With K = 16, the first takes 2,414s , the second takes 2,455s ; K = 32, the first takes 4,529s , the second takes 5,022s. Seems like vectorization does not help, but maybe there's something wrong with my code :( .
Replaced
for i = 1 : K
for j = 1 : RGB
Centroids(i,j) = Image(RowIndex(i),ColIndex(i),j);
end
end
with
for i = 1 : K
Centroids(i,:) = Image(RowIndex(i),ColIndex(i),:);
end
I think this may be vectorized further by using linear indexing, but for now I should just focus on the while loop since it takes most of the time.
Also when I tried #Dev-iL's suggestion and replaced
for i = 1 : K
PointsInCentroid = Points((find(WhichCentroid == i))',:);
NumOfPoints = size(PointsInCentroid,1);
% Note that NumOfPoints is never equal to 0, as a result of
% the initialization. Or .... ???????
if NumOfPoints ~= 0
Centroids(i,:) = sum(PointsInCentroid , 1) / NumOfPoints ;
end
end
with
E = sparse(1:size(WhichCentroid), WhichCentroid' , 1, Num, K, Num);
Centroids = (E * spdiags(1./sum(E,1)',0,K,K))' * Points ;
the results were always worse: With K = 16, the first takes 2,414s , the second takes 2,455s ; K = 32, the first took 4,529s , the second took 5,022s. Seems like vectorization did not help in this case.
However, when I replaced
Dist = pdist2(Points,Centroids,'euclidean');
[~, WhichCentroid] = min(Dist,[],2);
(in the while loop) with
Dist = bsxfun(#minus,dot(Centroids',Centroids',1)' / 2 , Centroids * Points' );
[~, WhichCentroid] = min(Dist,[],1);
WhichCentroid = WhichCentroid';
the code ran much faster, especially when K is large (K=32)
Thank you everyone!
Could someone please run this for me and tell me how long it takes for you? It took my laptop 60s. I can't tell if it's my laptop that's crappy or my code. Probably both.
I just started learning MatLab, so I'm not yet familiar with which functions are better than others for specific tasks. If you have any suggestions on how I could improve this code, it would be greatly appreciated.
function gbp
clear; clc;
zi = 0; % initial position
zf = 100; % final position
Ei = 1; % initial electric field
c = 3*10^8; % speed of light
epsilon = 8.86*10^-12; % permittivity of free space
lambda = 1064*10^-9; % wavelength
k = 2*pi/lambda; % wave number
wi = 1.78*10^-3; % initial waist width (minimum spot size)
zr = (pi*wi^2)/lambda; % Rayleigh range
Ri = zi + zr^2/zi; % initial radius of curvature
qi = 1/(1/Ri-1i*lambda/(pi*wi^2)); % initial complex beam parameter
Psii = atan(real(qi)/imag(qi)); % Gouy phase
mat = [1 zf; 0 1]; % transformation matrix
A = mat(1,1); B = mat(1,2); C = mat(2,1); D = mat(2,2);
qf = (A*qi + B)/(C*qi + D); % final complex beam parameter
wf = sqrt(-lambda/pi*(1/imag(1/qf))); % final spot size
Rf = 1/real(1/qf); % final radius of curvature
Psif = atan(real(qf)/imag(qf)); % final Gouy phase
% Hermite - Gaussian modes function
u = #(z, x, n, w, R, Psi) (2/pi)^(1/4)*sqrt(exp(1i*(2*n+1)*Psi)/(2^n*factorial(n)*w))*...
hermiteH(n,sqrt(2)*x/w).*exp(-x.^2*(1/w^2+1i*k/(2*R))-1i*k*z);
% Complex amplitude coefficients function
a = #(n) exp(1i*k*zi)*integral(#(x) Ei.*conj(u(zi, x, n, wi, Ri, Psii)),-2*wi,2*wi);
%----------------------------------------------------------------------------
xlisti = -0.1:1/10000:0.1; % initial x-axis range
xlistf = -0.1:1/10000:0.1; % final x-axis range
nlist = 0:2:20; % modes range
function Eiplot
Efieldi = zeros(size(xlisti));
for nr = nlist
Efieldi = Efieldi + a(nr).*u(zi, xlisti, nr, wi, Ri, Psii)*exp(-1i*k*zi);
end
Ii = 1/2*c*epsilon*arrayfun(#(x)x.*conj(x),Efieldi);
end
function Efplot
Efieldf = zeros(size(xlistf));
for nr = nlist
Efieldf = Efieldf + a(nr).*u(zf, xlistf, nr, wf, Rf, Psif)*exp(-1i*k*zf);
end
If = 1/2*c*epsilon*arrayfun(#(x)x.*conj(x),Efieldf);
end
Eiplot
Efplot
plot(xlisti,real(Ii),xlistf,real(If))
xlabel('x(m)') % x-axis label
ylabel('I(W/m^2)') % y-axis label
end
The cost is coming from the calls to hermiteH -- for every call, this creates a new function using symbolic variables, then evaluates the function at your input. The key to speeding this up is to pre-compute the hermite polynomial functions then evaluate those rather than create them from scratch each time (speedup from ~26 seconds to around 0.75 secs on my computer).
With the changes:
function gbp
x = sym('x');
zi = 0; % initial position
zf = 100; % final position
Ei = 1; % initial electric field
c = 3*10^8; % speed of light
epsilon = 8.86*10^-12; % permittivity of free space
lambda = 1064*10^-9; % wavelength
k = 2*pi/lambda; % wave number
wi = 1.78*10^-3; % initial waist width (minimum spot size)
zr = (pi*wi^2)/lambda; % Rayleigh range
Ri = zi + zr^2/zi; % initial radius of curvature
qi = 1/(1/Ri-1i*lambda/(pi*wi^2)); % initial complex beam parameter
Psii = atan(real(qi)/imag(qi)); % Gouy phase
mat = [1 zf; 0 1]; % transformation matrix
A = mat(1,1); B = mat(1,2); C = mat(2,1); D = mat(2,2);
qf = (A*qi + B)/(C*qi + D); % final complex beam parameter
wf = sqrt(-lambda/pi*(1/imag(1/qf))); % final spot size
Rf = 1/real(1/qf); % final radius of curvature
Psif = atan(real(qf)/imag(qf)); % final Gouy phase
% Hermite - Gaussian modes function
nlist = 0:2:20; % modes range
% precompute hermite polynomials for nlist
hermites = {};
for n = nlist
if n == 0
hermites{n + 1} = #(x)1.0;
else
hermites{n + 1} = matlabFunction(hermiteH(n, x));
end
end
u = #(z, x, n, w, R, Psi) (2/pi)^(1/4)*sqrt(exp(1i*(2*n+1)*Psi)/(2^n*factorial(n)*w))*...
hermites{n + 1}(sqrt(2)*x/w).*exp(-x.^2*(1/w^2+1i*k/(2*R))-1i*k*z);
% Complex amplitude coefficients function
a = #(n) exp(1i*k*zi)*integral(#(x) Ei.*conj(u(zi, x, n, wi, Ri, Psii)),-2*wi,2*wi);
%----------------------------------------------------------------------------
xlisti = -0.1:1/10000:0.1; % initial x-axis range
xlistf = -0.1:1/10000:0.1; % final x-axis range
function Eiplot
Efieldi = zeros(size(xlisti));
for nr = nlist
Efieldi = Efieldi + a(nr).*u(zi, xlisti, nr, wi, Ri, Psii)*exp(-1i*k*zi);
end
Ii = 1/2*c*epsilon*arrayfun(#(x)x.*conj(x),Efieldi);
end
function Efplot
Efieldf = zeros(size(xlistf));
for nr = nlist
Efieldf = Efieldf + a(nr).*u(zf, xlistf, nr, wf, Rf, Psif)*exp(-1i*k*zf);
end
If = 1/2*c*epsilon*arrayfun(#(x)x.*conj(x),Efieldf);
end
Eiplot
Efplot
plot(xlisti,real(Ii),xlistf,real(If))
xlabel('x(m)') % x-axis label
ylabel('I(W/m^2)') % y-axis label
end