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createGridMesh.m
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createGridMesh.m
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function output = createGridMesh(xvec, yvec, zvec)
%CREATEGRIDMESH Creates a triangular/tetrahedral finite element mesh that consists of pixels/voxels
%
%
% DESCRIPTION:
%
% This function can be used to create a mesh compatible
% for pixel or voxel based input values. The structure of the mesh
% is given in the figure below and has the same idea as the
% native MATLAB function 'meshgrid'
%
%
% o-------o--------o
% | | |
% | x | x |
% | | |
% o-------o--------o
% | | |
% | x | x |
% | | |
% o-------o--------o
%
% o = mesh cooridinates
% x = grid point (locattion given by xvec, yvec)
%
% The mesh can be either 2d (xvec, yvec given) or 3d.
% In 2d, the triangular mesh is constructed in the following fashion
%
% o--b3---o--b4---o
% | . t6 | . t8 |
% b2 . | . b5
% | t2 . |t4 . |
% o-------o-------o
% | . t5 | . t7 |
% b1 . | . b6
% | t1 . |t3 . |
% o---b8--o---b7--o
%
% t_i = triangle
% b_i = boundary segment
%
% The idea for the 3d mesh is the same. xvec, yvec and zvec
% depict the center location of each cube. Each cube contains 6
% tetrahedrons. The first tetrahedon is in the first cube, the
% second tetrahedron is in the second cube, in ascending
% y-x-z. The second tetrahedron of the first cube is the
% size(xvec)*size(yvec)*size(xvec)+1:th tetrahedron, the third
% tetrahedron of the first cube cube is the
% 2*size(xvec)*size(yvec)*size(xvec)+1:th tetrahedron and so on
%
% USAGE:
%
% vmcmesh = createGridMesh(xvec, yvec);
% vmcmesh = createGridMesh(xvec, yvec);
%
% INPUT:
%
% xvec - x coordinates for the center location of each pixel/cube
% yvec - y coordinates for the center location of each pixel/cube
%
% OPTIONAL INPUT:
%
% zvec - z coordinates for the center location of each pixel/cube
%
% This function is provided with ValoMC
if(nargin==2)
output.r = [];
output.H = [];
output.BH = [];
dx = abs(xvec(2)-xvec(1));
dy = abs(yvec(2)-yvec(1));
gridvecx = (xvec - dx/2);
gridvecy = (yvec - dy/2);
if(iscolumn(gridvecx))
gridvecx = [gridvecx; gridvecx(end)+dx];
gridvecy = [gridvecy; gridvecy(end)+dy];
else
gridvecx = [gridvecx gridvecx(end)+dx];
gridvecy = [gridvecy gridvecy(end)+dy];
end
[x,y] = meshgrid(gridvecx, gridvecy);
output.r = [x(:) y(:)];
num_x_voxels = length(xvec);
num_y_voxels = length(yvec);
k=1;
% k------k+1
% | t1 . |
% | . |
% | . t2 |
%k+xwidth-k+xwidth+1
% Build triangles
%
% 3-------6--------9
% | | |
% | x | x |
% | | |
% 2-------5--------8
% | | |
% | x | x |
% | | |
% 1-------4--------7
%
% num_y_voxels = 2;
% ysize = 3;
% TODO: for loops can be optimized away
ysize=num_y_voxels+1;
xsize=num_x_voxels+1;
output.H = zeros(num_x_voxels*num_y_voxels*2,3);
n=1;
for i = 1:num_x_voxels
for j = 1:num_y_voxels
output.H(n,:) = [k,k+1,k+ysize];
k=k+1;
n=n+1;
end
k=k+1;
end
k=1;
for i = 1:num_x_voxels
for j = 1:num_y_voxels
output.H(n,:) = [k+1,k+ysize+1,k+ysize];
n=n+1;
k=k+1;
end
k=k+1;
end
output.BH = [];
% Build boundary
%
% TL TR
% 4-------8--------12
% | | |
% | x | x |
% | | |
% 3-------7--------11
% | | |
% | x | x |
% | | |
% 2-------6--------10
% | | |
% | x | x |
% | | |
% 1-------5--------9
% BL BR
%
% xsize = 3
% ysize = 4
% BL to TL
for j = 1:num_y_voxels
output.BH = [output.BH; [j,j+1]];
end
% TL to BR
for j = 1:num_x_voxels
output.BH = [output.BH; [ysize*j,ysize*(j+1)]];
end
% TR to BR
for j = 1:num_y_voxels
output.BH = [output.BH; [ysize*xsize - (j-1), ysize*xsize - (j) ]];
end
% BR to TL
for j = 1:num_x_voxels
output.BH = [output.BH; [ysize*(xsize-1)+1-ysize*(j-1),ysize*(xsize-1)+1-ysize*(j)]];
end
else
output.r = [];
output.H = [];
output.BH = [];
dx = abs(xvec(2)-xvec(1));
dy = abs(yvec(2)-yvec(1));
dz = abs(zvec(2)-zvec(1));
gridvecx = (xvec - dx/2);
gridvecy = (yvec - dy/2);
gridvecz = (zvec - dz/2);
if(iscolumn(gridvecx))
gridvecx = [gridvecx; gridvecx(end)+dx];
gridvecy = [gridvecy; gridvecy(end)+dy];
gridvecz = [gridvecz; gridvecz(end)+dz];
else
gridvecx = [gridvecx gridvecx(end)+dx];
gridvecy = [gridvecy gridvecy(end)+dy];
gridvecz = [gridvecz gridvecz(end)+dz];
end
[x,y,z] = meshgrid(gridvecx, gridvecy,gridvecz);
output.r = [x(:) y(:) z(:)];
points = [0 0 0; 0 1 0; 1 0 0; 1 1 0; 0 0 1; 0 1 1; 1 0 1; 1 1 1];
connectivity = [5 1 2 3; 6 5 2 3; 6 7 5 3; 6 4 7 3; 6 2 4 3; 6 8 7 4];
ijk_to_index = @(i,j,k) ((k-1)*(size(x,2)*size(x,1))+(i-1)*size(x,1)+j);
num_x_voxels=size(x,2)-1;
num_y_voxels=size(x,1)-1;
num_z_voxels=size(x,3)-1;
output.H=zeros(num_x_voxels*num_y_voxels*num_z_voxels*6,4);
% Build the topology matrix
counter=1;
for t=1:6
for k=1:num_z_voxels
for i=1:num_x_voxels
for j=1:num_y_voxels
offset = connectivity(t,:);
ijkoffset = points(offset,:);
output.H(counter,:) = [ijk_to_index(i+ijkoffset(1,1),j+ijkoffset(1,2),k+ijkoffset(1,3))
ijk_to_index(i+ijkoffset(2,1),j+ijkoffset(2,2),k+ijkoffset(2,3))
ijk_to_index(i+ijkoffset(3,1),j+ijkoffset(3,2),k+ijkoffset(3,3))
ijk_to_index(i+ijkoffset(4,1),j+ijkoffset(4,2),k+ijkoffset(4,3))];
counter=counter+1;
end
end
end
end
% Build the boundary topology
counter = 1;
output.BH=zeros(num_x_voxels*num_y_voxels*2*2+num_y_voxels*num_z_voxels*2*2+num_z_voxels*num_x_voxels*2*2,3);
% XY plane
for i=1:num_x_voxels
for j=1:num_y_voxels
k=1;
output.BH(counter, :) = [ijk_to_index(i,j,k) ijk_to_index(i+1,j,k) ijk_to_index(i,j+1,k)];
counter = counter + 1;
output.BH(counter, :) = [ijk_to_index(i+1,j+1,k) ijk_to_index(i,j+1,k) ijk_to_index(i+1,j,k)];
counter = counter + 1;
k=num_z_voxels+1;
output.BH(counter, :) = [ijk_to_index(i,j,k) ijk_to_index(i+1,j,k) ijk_to_index(i,j+1,k)];
counter = counter + 1;
output.BH(counter, :) = [ijk_to_index(i+1,j+1,k) ijk_to_index(i,j+1,k) ijk_to_index(i+1,j,k)];
counter = counter + 1;
end
end
% YZ plane
for j=1:num_y_voxels
for k=1:num_z_voxels
i=1;
output.BH(counter, :) = [ijk_to_index(i,j,k) ijk_to_index(i,j+1,k) ijk_to_index(i,j,k+1)];
counter = counter + 1;
output.BH(counter, :) = [ijk_to_index(i,j+1,k+1) ijk_to_index(i,j,k+1) ijk_to_index(i,j+1,k)];
counter = counter + 1;
i=num_x_voxels+1;
output.BH(counter, :) = [ijk_to_index(i,j,k) ijk_to_index(i,j+1,k) ijk_to_index(i,j,k+1)];
counter = counter + 1;
output.BH(counter, :) = [ijk_to_index(i,j+1,k+1) ijk_to_index(i,j,k+1) ijk_to_index(i,j+1,k)];
counter = counter + 1;
end
end
% ZX plane
for k=1:num_z_voxels
for i=1:num_x_voxels
j=1;
output.BH(counter, :) = [ijk_to_index(i,j,k) ijk_to_index(i,j,k+1) ijk_to_index(i+1,j,k)];
counter = counter + 1;
output.BH(counter, :) = [ijk_to_index(i+1,j,k+1) ijk_to_index(i+1,j,k) ijk_to_index(i,j,k+1)];
counter = counter + 1;
j=num_y_voxels+1;
output.BH(counter, :) = [ijk_to_index(i,j,k) ijk_to_index(i,j,k+1) ijk_to_index(i+1,j,k)];
counter = counter + 1;
output.BH(counter, :) = [ijk_to_index(i+1,j,k+1) ijk_to_index(i+1,j,k) ijk_to_index(i,j,k+1)];
counter = counter + 1;
end
end
end