Generating a 3D binary mask of geometric shapes in Matlab - matlab

Question

I would like to generate a 3D binary mask which represents an ellipsoid with centers xc,yc,zc and radiuces xr,yr,zr.
I noticed that the function ellipsoid generates a mesh of points given these parameters. However, I want the data to be represented by a binary matrix (in my case, of size [100,100,100]), and not a mesh.
My Parameters are:
mask = zeros(100,100,100);
xc = 50; yc = 50; zc = 50;
xr = 15; yr = 15; zr = 15;
Thanks in advance!

Answer 1

To generate a binary mask of shapes which can use an equation you can follow the steps:
Generate a mesh (with ndgrid). Make sure the domain limits includes the volume/surface mask, and choose the mesh resolution according to your needs.
Use the volume/surface equation to generate a binary mask, by doing a simple logical comparison of the coordinates with the equation.
Simple 2D example:
Let's define a simple ellipse (in 2D).
%% // Simple 2D example
xc = 5 ; yc = 6 ; %// ellipse center = (5,6)
xr = 3 ; yr = 2 ; %// ellipse radiuses
xbase = linspace(0,10,11) ; %// temporary variable used to send to "ndgrid"
[xm,ym] = ndgrid( xbase , xbase ) ; %// generate base mesh
mask = ( ((xm-xc).^2/(xr.^2)) + ((ym-yc).^2/(yr.^2)) <= 1 ) %// get binary mask
Gives you the binary mask:
mask =
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 1 1 1 0 0 0
0 0 0 0 0 1 1 1 0 0 0
0 0 0 0 1 1 1 1 1 0 0
0 0 0 0 0 1 1 1 0 0 0
0 0 0 0 0 1 1 1 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
Granted you can hardly recognise an ellipse in the mask but I kept the resolution low to be able to display it as text. You can easily increase the resolution by defining a finer mesh.
3D Ellipsoid:
Well, it's exactly the same method, except we are going to add one dimension to the mesh, and use the 3D equation of the ellipsoid. So for your precise case:
%% // 3D ellipsoid
xc = 50; yc = 50; zc = 50;
xr = 15; yr = 15; zr = 15;
xbase = linspace(1,100,100) ;
[xm,ym,zm] = ndgrid( xbase , xbase , xbase) ;
mask = ( ((xm-xc).^2/(xr.^2)) + ((ym-yc).^2/(yr.^2)) + ((zm-zc).^2/(zr.^2)) <= 1 ) ;
I cannot show you a text output with these kind of 3D arrays, but your mask is now a 3D logical array containing true inside the ellipsoid and false elsewhere.

Answer 2

If I understood your question correctly, this should work:
mask = zeros(100, 100, 100);
%//your ellipsoid properties
xc = 50; yc = 50; zc = 50;
xr = 15; yr = 15; zr = 15;
for x=1:100
for y=1:100
for z=1:100
if ( ((x-xc)/xr)*((x-xc)/xr) + ((y-yc)/yr)*((y-yc)/yr) + ((z-zc)/zr)*((z-zc)/zr) < 1 )
mask(x,y,z) = 1; %//set elements within ellipsoid to 1
end
end
end
end

Answer 3

You can do this with Heaviside functions, this probably needs a bit more thought to get exactly what you want but as a start,
close all
clear all
%Setup Domain
maxdomain = 100;
mindomain = 0.;
step = 1.0;
X = mindomain:step:maxdomain;
Y = mindomain:step:maxdomain;
Z = mindomain:step:maxdomain;
[x,y,z] = meshgrid(X,Y,Z);
xc = 50; yc = 50; zc = 50;
xr = 15; yr = 15; zr = 15;
r2 = xr/2;
r = sqrt((x-xc).^2/xr + (y-yc).^2/yr + (z-zc).^2/zr);
u = heaviside(r-r2);
%Plot Surface of sphere
p = patch(isosurface(x,y,z,u));
isonormals(x,y,z,u,p)
set(p,'FaceColor','red','EdgeColor','none');
camlight ; alpha(0.6);
xlabel('x'); ylabel('y'); zlabel('z');
daspect([1,1,1]); view(3);
axis tight; camlight; camlight(-80,-10);
lighting gouraud;
which for your values above looks like,
and forxr = 15; yr = 45; zr = 15;,
The Heaviside function can be defined using,
function [out]=heaviside(x)
out=0.5.*(sign(x)+1.0);
end
if the Symbolic Math Toolbox is not available.