I have been struggling with this problem for a while and I would appreciate if anyone can help me out. I am able to generate a 10 by 10 matrix and have it randomly assign "1"s in the matrix. My goal is to plot a "star" at the location of each element in the vector that has a value of "1", but I can't seem to figure out how to map the vector to a x-y coordinate system. The code I wrote below generates a plot of 100 stars at each cell and also generates a vector "v", but I don't know how I can link the plot to the vector that instead of having 100 "star"s in my plot, I have however many that there is a value of "1" at the corresponding location of the element.
Thanks!!
David
davidtongg#gmail.com
close all
clear all
clc
a=10;b=10;
v = zeros(a,b);
xy = int32(randi(a, 100, 2));
z = randi(1, 100, 1); % 100 values.
indexes = sub2ind([a, b], xy(:,1), xy(:,2))
v(indexes) = z
m=length(v);
ctr=0;
for i=1:m^2
x_cor(i)=(i-(floor(i/m)*m))*200-100;
y_cor(i)=(floor(i/m)+1)*200-100;
for j=1:m
if i==j*m
x_cor(i)=((i-(floor(i/m)*m))*200-100)+(2*m*100);
y_cor(i)=(floor(i/m))*200-100;
end
end
end
figure(1)
plot(x_cor,y_cor,'*');
grid on
I may of course have misinterpreted this because that code is confusingly complicated, but this is what I think you're after.
For an axb matrix with a random number of ones:
v = randi([0 1], a, b);
Or for a specific number n of ones, in random locations:
v = zeros(a, b);
idx = randi([1 numel(v)], n, 1);
v(idx) = 1; % linear indexing into a matrix
Then to plot them in arbitrarily scaled coordinates:
[y x] = find(v);
x = x * xscale + xoffset;
y = y * yscale + yoffset;
plot(x, y, '*');
Or the really cheaty way:
spy(v);
You can do it easily taking into account that plot(A) , where A is a matrix, plots the columns of the matrix vs their index, and that NaNs are not plotted:
v =[ 1 0 0 0
1 1 0 0
0 0 0 1
1 1 1 1
0 1 1 0 ]; %// example data
v2 = double(v); %// create copy; will be overwritten
v2(~v2) = NaN; %// change zeros to NaNs
plot(bsxfun(#plus, fliplr(v2.'), 0:size(v,1)-1) ,'b*')
%'// transpose and flip from left to right.
%// Add 1 incrementally to each column to have all of them "stacked" in the plot
axis([0 size(v,2)+1 0 size(v,1)+1]) %// set axis limits
set(gca,'xtick',1:size(v,2),'ytick',1:size(v,1)) %// set ticks
grid
Related
So essentially I have this code here that I can use to generate a 2D Random Walk discretely along N number of steps with M number of walkers. I can plot them all on the same graph here.
clc;
clearvars;
N = 500; % Length of the x-axis, also known as the length of the random walks.
M = 3; % The amount of random walks.
x_t(1) = 0;
y_t(1) = 0;
for m=1:M
for n = 1:N % Looping all values of N into x_t(n).
A = sign(randn); % Generates either +1/-1 depending on the SIGN of RAND.
x_t(n+1) = x_t(n) + A;
A = sign(randn); % Generates either +1/-1 depending on the SIGN of RAND.
y_t(n+1) = y_t(n) + A;
end
plot(x_t, y_t);
hold on
end
grid on;
% Enlarge figure to full screen.
set(gcf, 'Units', 'Normalized', 'Outerposition', [0, 0.05, 1, 0.95]);
axis square;
Now, I want to be able to Create graphs that show the distribution in space of the positions of a large number
(e.g. n = 1000) random walkers, at three different time points (e.g. t = 100, 200 and 300 or any three time points really).
I'm not sure how to go about this, I need to turn this into a function and iterate it through itself three different times and store the coordinates? I have a rough idea but iffy on actually implementing. I'd assume the safest and least messy way would be to use subplot() to create all three plots together in the same figure.
Appreciate any assistance!
You can use cumsum to linearize the process. Basically you only want to cumsum a random matrix composed of [-1 and 1].
clc;
close all;
M = 50; % The amount of random walks.
steps = [10,200,1000]; % here we analyse the step 10,200 and 1000
cc = hsv(length(steps)); % manage the color of the plot
%generation of each random walk
x = sign(randn(max(steps),M));
y = sign(randn(max(steps),M));
xs = cumsum(x);
xval = xs(steps,:);
ys = cumsum(y);
yval = ys(steps,:);
hold on
for n=1:length(steps)
plot(xval(n,:),yval(n,:),'o','markersize',1,'color',cc(n,:),'MarkerFaceColor',cc(n,:));
end
legend('10','200','1000')
axis square
grid on;
Results:
EDIT:
Thanks to #LuisMendo that answered my question here, you can use a binomial distribution to get the same result:
steps = [10,200,10000];
cc = hsv(length(steps)); % manage the color of the plot
M = 50;
DV = [-1 1];
p = .5; % probability of DV(2)
% Using the #LuisMendo binomial solution:
for ii = 1:length(steps)
SDUDx(ii,:) = (DV(2)-DV(1))*binornd(steps(ii), p, M, 1)+DV(1)*steps(ii);
SDUDy(ii,:) = (DV(2)-DV(1))*binornd(steps(ii), p, M, 1)+DV(1)*steps(ii);
end
hold on
for n=1:length(steps)
plot(SDUDx(n,:),SDUDy(n,:),'o','markersize',1,'color',cc(n,:),'MarkerFaceColor',cc(n,:));
end
legend('10','200','1000')
axis square
grid on;
What is the advantage ? Even if you have a big number of steps, let's say 1000000, matlab can handle it. Because in the first solution you have a bruteforce solution, and in the second case a statistical solution.
If you want to show the distribution of a large number, say 1000, of these points, I would say the most suitable way of plotting is as a 'point cloud' using scatter. Then you create an array of N points for both the x and the y coordinate, and let it compute the coordinate in a loop for i = 1:Nt, where Nt will be 100, 200, or 300 as you describe. Something along the lines of the following:
N = 500;
x_t = zeros(N,1);
y_t = zeros(N,1);
Nt = 100;
for tidx = 1:Nt
x_t = x_t + sign(randn(N,1));
y_t = y_t + sign(randn(N,1));
end
scatter(x_t,y_t,'k*');
This will give you N x and y coordinates generated in the same way as in the sample you provided.
One thing to keep in mind is that sign(0)=0, so I suppose there is a chance (admittedly a small one) of not altering the coordinate. I am not sure if you intended this behaviour to be possible (a walker standing still)?
I will demonstrate the 1-dimensional case for clarity; you only need to implement this for each dimension you add.
Model N steps for M walkers using an NxM matrix.
>> N = 5;
>> M = 4;
>> steps = sign(randn(N,M));
steps =
1 1 1 1
-1 1 -1 1
1 -1 -1 -1
1 1 -1 1
1 -1 -1 -1
For plotting, it is useful to make a second NxM matrix s containing the updated positions after each step, where s(N,M) gives the position of walker M after N steps.
Use cumsum to vectorize instead of looping.
>> s = cumsum(steps)
s =
1 1 1 1
0 2 0 2
1 1 -1 1
2 2 -2 2
3 1 -3 1
To prevent plot redraw after each new line, use hold on.
>> figure; hold on
>> plot(1:N, s(1:N, 1:M), 'marker', '.', 'markersize', 20, 'linewidth', 3)
>> xlabel('Number of steps'); ylabel('Position')
The output plot looks like this: picture
This method scales very well to 2- and 3-dimensional random walks.
I am wanting to do conditional plotting of vertical lines, that change color based on the value of an integer vector. Those values are integers that range from 0-4.
Currently, I am using a loop to go through the tables to plot the lines. This works, but for LARGE amounts of data it takes time, and I'm wondering if it can be vectorized.
Attached is a stripped down version of the script to loop through a data vector(sample) that simply Loops through the vector, and plots a vertical line based on the value of the integer.
I will also attach the simple variable I created called 'SAMPLE' below to paste into your workspace.
for i=1:size(sample,1)
if sample(i)==1
line( [i i] ,[0 10], 'Marker','.','LineStyle','-','Color','r');
elseif sample(i)==2
line( [i i] ,[0 10], 'Marker','.','LineStyle','-','Color','b');
elseif sample(i)==3
line( [i i] ,[0 10], 'Marker','.','LineStyle','-','Color',[1 .5 0]);
elseif sample(i)==4
line( [i i] ,[0 10], 'Marker','.','LineStyle','-','Color','g');
end
end
Variable:
sample=[[3;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;4;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;2;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;3;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;4;0;0;0;0]];
But is is possible to 'vectorize' plotting in this way w/o having to do it iteratively in a loop as I have done?
Take advantage of the fact that when plotting a line, MATLAB will skip points whose value is NaN.
% Your vector
sample=[3;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;4;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;2;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;3;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;4;0;0;0;0];
% Your colors
colors = [
1 0 0
0 0 1
1 .5 0
0 1 0];
for idx = 1:4
% Find the index of each of your integers
X = find(sample == (idx));
% Force X to be a row vector
X = X(:)';
% Stack two X's on top of one another with a third row filled
% with NaNs. Fill in your Y values in the same way while
% you're at it.
Y = [zeros(size(X)); 10 + zeros(size(X)); nan(size(X))];
X = [X; X; nan(size(X))]; %#ok<AGROW>
% Matlab is column major. By using the colon here, you
% produce a vector that is [X1 X1 nan X2 X2 nan ... etc.]
X = X(:);
Y = Y(:);
% Draw the line
line(X, Y, 'Marker', '.', 'LineStyle', '-', 'Color', colors(idx, :))
end
There's still a loop, but now you're just looping over the possible values instead of looping over the each value in the vector. I think you will find that this will scale much better.
Changing input to:
sample = zeros(1, 1e6);
for idx = 1:4
sample(randi(1e6, 1, 1000)) = idx;
end
and benchmarking with timeit gives a time of 0.0065706 seconds on my machine, while the OP code benchmarks at 1.4861 seconds.
I'd change to something like:
colors=[1 0 0,
0 1 0,
1 0.5 0,
0 0 1];
nnsamples=samples(samples~=0);
for ii=1:size(nnsamples,1)
line( [ii ii] ,[0 10], 'Marker','.','LineStyle','-','Color',colors(nnsamples(ii),:));
end
I have a data set that looks like this
140400 70.7850 1
140401 70.7923 2
140402 70.7993 3
140403 70.8067 4
140404 70.8139 5
140405 70.8212 3
Where the first column corresponds to time (one second intervals between data points) and will be on the x axis, the second column corresponds with distance and will be on the y axis. The third column is a number (one through five) that is a qualification of the movement.
I want to make a plot that changes the color of the line between two points depending on what the number of the previous data point was. For example, I want the line to be red between the first and second data points because the qualification value was 1.
I've seen a lot of posts about making a sliding scale of colors depending on an intensity value, but I just want 5 colors: (red, orange, yellow, green, and blue) respectively.
I tried doing something like this:
plot(x,y,{'r','o','y','g','b'})
But with no luck.
Any ideas of how to approach this? Without looping if possible.
You can also do it with a trick which works with Matlab version anterior to 2014b (as far back as 2009a at least).
However, is will never be as simple as you expected (unless you write a wrapper for one of the solution here you can forget about plot(x,y,{'r','o','y','g','b'})).
The trick is to use a surface instead of a line object. Surfaces benefit from their CData properties and a lot of useful features to exploit color maps and texture.
Matlab surf does not handle 1D data, it needs a matrix as input so we are going to give it by just duplicating each coordinate set (for example xx=[x,x]).
Don't worry though, the surface will stay as thin as a line, so the end result is not ugly.
%% // your data
M=[140400 70.7850 1
140401 70.7923 2
140402 70.7993 3
140403 70.8067 4
140404 70.8139 5
140405 70.8212 3];
x = M(:,1) ; %// extract "X" column
y = M(:,2) ; %// same for "Y"
c = M(:,3) ; %// extract color index for the custom colormap
%% // define your custom colormap
custom_colormap = [
1 0 0 ; ... %// red
1 .5 0 ; ... %// orange
1 1 0 ; ... %// yellow
0 1 0 ; ... %// green
0 0 1 ; ... %// blue
] ;
%% // Prepare matrix data
xx=[x x]; %// create a 2D matrix based on "X" column
yy=[y y]; %// same for Y
zz=zeros(size(xx)); %// everything in the Z=0 plane
cc =[c c] ; %// matrix for "CData"
%// draw the surface (actually a line)
hs=surf(xx,yy,zz,cc,'EdgeColor','interp','FaceColor','none','Marker','o') ;
colormap(custom_colormap) ; %// assign the colormap
shading flat %// so each line segment has a plain color
view(2) %// view(0,90) %// set view in X-Y plane
colorbar
will get you:
As an example of a more general case:
x=linspace(0,2*pi);
y=sin(x) ;
xx=[x;x];
yy=[y;y];
zz=zeros(size(xx));
hs=surf(xx,yy,zz,yy,'EdgeColor','interp') %// color binded to "y" values
colormap('hsv')
view(2) %// view(0,90)
will give you a sine wave with the color associated to the y value:
Do you have Matlab R2014b or higher?
Then you could use some undocumented features introduced by Yair Altman:
n = 100;
x = linspace(-10,10,n); y = x.^2;
p = plot(x,y,'r', 'LineWidth',5);
%// modified jet-colormap
cd = [uint8(jet(n)*255) uint8(ones(n,1))].' %'
drawnow
set(p.Edge, 'ColorBinding','interpolated', 'ColorData',cd)
My desired effect was achieved below (simplified):
indices(1).index = find( data( 1 : end - 1, 3) == 1);
indices(1).color = [1 0 0];
indices(2).index = find( data( 1 : end - 1, 3) == 2 | ...
data( 1 : end - 1, 3) == 3);
indices(2).color = [1 1 0];
indices(3).index = find( data( 1 : end - 1, 3) == 4 | ...
data( 1 : end - 1, 3) == 5);
indices(3).color = [0 1 0];
indices(4).index = find( data( 1 : end - 1, 3) == 10);
indices(4).color = [0 0 0];
indices(5).index = find( data( 1 : end - 1, 3) == 15);
indices(5).color = [0 0 1];
% Loop through the locations of the values and plot their data points
% together (This will save time vs. plotting each line segment
% individually.)
for iii = 1 : size(indices,2)
% Store locations of the value we are looking to plot
curindex = indices(iii).index;
% Get color that corresponds to that value
color = indices(iii).color;
% Create X and Y that will go into plot, This will make the line
% segment from P1 to P2 have the color that corresponds with P1
x = [data(curindex, 1), data(curindex + 1, 1)]';
y = [data(curindex, 2), data(curindex + 1, 2)]';
% Plot the line segments
hold on
plot(x,y,'Color',color,'LineWidth',lineWidth1)
end
When the result figure of two variables plotted is a circle, will be necessary to add the time in z axes.
For example the figure of induction machine rotor velocity vs electric torque in one laboratory test is: 2d plot figure
In the last figure the direction of the time point plotting could be clockwise or counter clockwise. For the last reason will be added time in z axis.
% Wr vs Te
x = logsout.getElement( 'Wr' ).Values.Data;
y = logsout.getElement( '<Te>' ).Values.Data;
z = logsout.getElement( '<Te>' ).Values.Time;
% % adapt variables for use surf function
xx = zeros( length( x ) ,2 );
yy = zeros( length( y ) ,2 );
zz = zeros( length( z ) ,2 );
xx (:,1) = x; xx (:,2) = x;
yy (:,1) = y; yy (:,2) = y;
zz (:,1) = z; zz (:,2) = z;
% % figure(1) 2D plot
figure (1)
hs = surf(xx,yy,zz,yy,'EdgeColor','interp') %// color binded to "y" values
colormap('hsv')
view(2)
% %
figure(2)
hs = surf(xx,yy,zz,yy,'EdgeColor','interp') %// color binded to "y" values
colormap('hsv')
view(3)
Finally we can view the 3d form and detect that counterwise is the real direction of the time plotting is: 3d plot
Scatter can plot the color according to the value and shows the colormap of the range of values. It's hard to interpolate the color though if you want continuous curves.
Try:
figure
i = 1:20;
t = 1:20;
c = rand(1, 20) * 10;
scatter(i, t, [], c, 's', 'filled')
colormap(jet)
The figure looks like
I'm new to Matlab but I know a bit about programming.
For class, we have been asked to generate a matrix that gives the vertices of a two dimensional n-sided shape where n>=4. Then, generate the vectors to connect the vertices. We were also given a hint: a vector for each segment can be found by adding the vectors drawn from the origin to each of two adjacent vertices.
I know how to create a matrix using A = [1 1; 1 2; 2 2; 2 1] but I'm not sure how to draw the vectors given this or any other matrix.
The plot() function looks promising, but I'm unsure how to use it with the matrix.
Thank you for any suggestions.
Btw, I'm using Matlab 2011a
I'm not exactly sure how your matrix represents your shape but you might for example let the x-coordinates of the shape be the first column of your array, then let the y-coordinates be the 2nd column, like:
A = [1 1; 1 2; 2 2; 2 1];
x = A(:,1);
y = A(:,2);
fill(x,y,'g');
axis([0 3 0 3]);
axis square;
Which in your case plots a square from the matrix A:
Or construct something a little more complicated like a pentagon:
theta = [0:pi/2.5:2*pi];
x = sin(theta);
y = cos(theta);
% your matrix is then:
B(:,1) = x;
B(:,2) = y;
B
figure;fill(x,y,'g');
axis square;
Which gives:
If you just want to plot the outline with plot (not fill the interior with fill), just remember you have to repeat the initial point at the end so that the polygonal line is closed:
A = [1 1; 1 2; 2 2; 2 1];
B = [A; A(1,:) ]; %// repeat first row at the end
plot(B(:,1),B(:,2))
axis equal %// same scale on both axes
axis([min(x)-.5 max(x)+.5 min(y)-.5 max(y)+.5]) %// larger axes for better display
I have a 3-D grayscale volume corresponding to ultrasound data. In Matlab this 3-D volume is simply a 3-D matrix of MxNxP. The structure I'm interested in is not oriented along the z axis, but along a local coordinate system already known (x'y'z'). What I have up to this point is something like the figure shown below, depicting the original (xyz) and the local coordinate systems (x'y'z'):
I want to obtain the 2-D projection of this volume (i.e. an image) through a specific plane on the local coordinate system, say at z' = z0. How can I do this?
If the volume was oriented along the z axis this projection could be readily achieved. i.e. if the volume, in Matlab, is V, then:
projection = sum(V,3);
thus, the projection can be computed just as the sum along the 3rd dimension of the array. However with a change of orientation the problem becomes more complicated.
I've been looking at radon transform (2D, that applies only to 2-D images and not volumes) and also been considering ortographic projections, but at this point I'm clueless as to what to do!
Thanks for any advice!
New attempt at solution:
Following the tutorial http://blogs.mathworks.com/steve/2006/08/17/spatial-transformations-three-dimensional-rotation/ and making some small changes, I might have something which could help you. Bear in mind, I have little or no experience with volumetric data in MATLAB, so the implementation is quite hacky.
In the below code I use tformarray() to rotate the structure in space. First, the data is centered, then rotated using rotationmat3D to produce the spacial transformation, before the data is moved back to its original position.
As I have never used tformarray before, I handeled datapoints falling outside the defined region after rotation by simply padding the data matrix (NxMxP) with zeros all around. If anyone know a better way, please let us know :)
The code:
%Synthetic dataset, 25x50x25
blob = flow();
%Pad to allow for rotations in space. Bad solution,
%something better might be possible to better understanding
%of tformarray()
blob = padarray(blob,size(blob));
f1 = figure(1);clf;
s1=subplot(1,2,1);
p = patch(isosurface(blob,1));
set(p, 'FaceColor', 'red', 'EdgeColor', 'none');
daspect([1 1 1]);
view([1 1 1])
camlight
lighting gouraud
%Calculate center
blob_center = (size(blob) + 1) / 2;
%Translate to origin transformation
T1 = [1 0 0 0
0 1 0 0
0 0 1 0
-blob_center 1];
%Rotation around [0 0 1]
rot = -pi/3;
Rot = rotationmat3D(rot,[0 1 1]);
T2 = [ 1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1];
T2(1:3,1:3) = Rot;
%Translation back
T3 = [1 0 0 0
0 1 0 0
0 0 1 0
blob_center 1];
%Total transform
T = T1 * T2 * T3;
%See http://blogs.mathworks.com/steve/2006/08/17/spatial-transformations-three-dimensional-rotation/
tform = maketform('affine', T);
R = makeresampler('linear', 'fill');
TDIMS_A = [1 2 3];
TDIMS_B = [1 2 3];
TSIZE_B = size(blob);
TMAP_B = [];
F = 0;
blob2 = ...
tformarray(blob, tform, R, TDIMS_A, TDIMS_B, TSIZE_B, TMAP_B, F);
s2=subplot(1,2,2);
p2 = patch(isosurface(blob2,1));
set(p2, 'FaceColor', 'red', 'EdgeColor', 'none');
daspect([1 1 1]);
view([1 1 1])
camlight
lighting gouraud
The arbitrary visualization below is just to confirm that the data is rotated as expected, plotting a closed surface when the data passed the value '1'. With blob2, you should know be able to project by using simple sums.
figure(2)
subplot(1,2,1);imagesc(sum(blob,3));
subplot(1,2,2);imagesc(sum(blob2,3));
Assuming you have access to the coordinate basis R=[x' y' z'], and that those vectors are orthonormal, you can simply extract the representation in this basis by multiplying your data with the the 3x3 matrix R, where x',y',z' are column vectors.
With the data stored in D (Nx3), you can get the representation with R, by multiplying by it:
Dmarked = D*R;
and now D = Dmarked*inv(R), so going back and forth is stragihtforward.
The following code might provide help to see the transformation. Here I create a synthetic dataset, rotate it, and then rotate it back. Doing sum(DR(:,3)) would then be your sum along z'
%#Create synthetic dataset
N1 = 250;
r1 = 1;
dr1 = 0.1;
dz1 = 0;
mu1 = [0;0];
Sigma1 = eye(2);
theta1 = 0 + (2*pi).*rand(N1,1);
rRand1 = normrnd(r1,dr1,1,N1);
rZ1 = rand(N1,1)*dz1+1;
D = [([rZ1*0 rZ1*0] + repmat(rRand1',1,2)).*[sin(theta1) cos(theta1)] rZ1];
%Create roation matrix
rot = pi/8;
R = rotationmat3D(rot,[0 1 0]);
% R = 0.9239 0 0.3827
% 0 1.0000 0
% -0.3827 0 0.9239
Rinv = inv(R);
%Rotate data
DR = D*R;
%#Visaulize data
f1 = figure(1);clf
subplot(1,3,1);
plot3(DR(:,1),DR(:,2),DR(:,3),'.');title('Your data')
subplot(1,3,2);
plot3(DR*Rinv(:,1),DR*Rinv(:,2),DR*Rinv(:,3),'.r');
view([0.5 0.5 0.2]);title('Representation using your [xmarked ymarked zmarked]');
subplot(1,3,3);
plot3(D(:,1),D(:,2),D(:,3),'.');
view([0.5 0.5 0.2]);title('Original data before rotation');
If you have two normalized 3x1 vectors x2 and y2 corresponding to your local coordinate system (x' and y').
Then, for a position P, its local coordinate will be xP=P'x2 and yP=P'*y2.
So you can try to project your volume using accumarray:
[x y z]=ndgrid(1:M,1:N,1:P);
posP=[x(:) y(:) z(:)];
xP=round(posP*x2);
yP=round(posP*y2);
xP=xP+min(xP(:))+1;
yP=yP+min(yP(:))+1;
V2=accumarray([xP(:),yP(:)],V(:));
If you provide your data, I will test it.