# pointLocationTR

Below is a demonstration of the features of the pointLocationTR function

## Syntax

[TI,BC]=pointLocationTR(TR,QP,distCropOpt,chullCropOpt,waitbarOpt,toleranceMagnitude); [TI,BC]=pointLocationTR(TR,QP,optionStruct);

## Description

This function finds the triangles or tetrahedrons in which the query points QP are contained. In addition it outputs the barycentric coordinates.

Default options: optionStructDef.distCropOpt=1; optionStructDef.chullCropOpt=1; optionStructDef.waitbarOpt=1; optionStructDef.toleranceMagnitude=0;

## Examples

```clear; close all; clc;
```

Plot settings

```fontSize=15;
faceColor1='g';
faceColor2='r';
faceAlpha1=0.3;
faceAlpha2=1;
edgeColor=0.4*ones(1,3);
edgeWidth=2;
markerSize=2;
cMap=gjet(250);
```

## Example: Finding points inside a triangulated mesh

```%Boundary 1
ns=150;
t=linspace(0,2*pi,ns);
t=t(1:end-1);
r=6+2.*sin(5*t);
[x,y] = pol2cart(t,r);
V1=[x(:) y(:)];

%Boundary 2
[x,y] = pol2cart(t,ones(size(t)));
V2=[x(:) y(:)+4];

%Boundary 3
[x,y] = pol2cart(t,2*ones(size(t)));
V3=[x(:) y(:)-0.5];

regionCell={V1,V2,V3}; %A region between V1 and V2 (V2 forms a hole inside V1)
pointSpacing=1;
[F,V]=regionTriMesh2D(regionCell,pointSpacing,1,0);

V(:,3)=sin(V(:,1)); %Z coordinate

[~,QP]=subtri(F,V,2);
```

Convert tesselation to triangulation class

```TR = triangulation(F,V);
```

Use pointLocationTR to test points

```optionStruct.toleranceMagnitude=1e-3;
[ti,bc]=pointLocationTR(TR,QP,optionStruct);
L=any(isnan(bc),2);
```
```logicPlot=~isnan(ti); %Point selection logic

cFigure;
hold on;
gpatch(F,V,'kw','k',0.5);
plotV(QP,'k.','markerSize',25);
plotV(QP(L,:),'r.','markerSize',50);
scatterV(QP(logicPlot,:),50,ti(logicPlot),'fill','markerEdgeColor','k');
axisGeom;
colormap(cMap(randperm(size(cMap,1))',:));
drawnow;
```

## Example: Finding points inside a tetrahedral mesh

Create a test tesselation

```r=1; %Radius of tetrahedron circumsphere
[V,~]=platonic_solid(1,r);

E=[1 2 3 4];

n=3;
E=E; V=V;
for q=1:1:n
[E,V]=subTet(E,V,1);
end
C=(1:size(E,1))';
```

Define query points

```n=50;
[X,Y,Z]=meshgrid(linspace(-r,r,n));

QP=[X(:) Y(:) Z(:)];
```

Plotting the test case

```[F,CF]=element2patch(E,C);

cFigure;
title(['A ',num2str(size(E,1)),' element tesselation with ',num2str(size(QP,1)),' query points'],'FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

patch('Faces',F,'Vertices',V,'FaceColor','flat','CData',CF,'EdgeColor',edgeColor,'FaceAlpha',faceAlpha1,'lineWidth',edgeWidth);
plotV(QP,'b.','MarkerSize',markerSize);

colormap(cMap); colorbar;
view(3); grid on; axis equal; axis tight;
view([-50,12])
set(gca,'FontSize',fontSize);
drawnow;
```

Convert tesselation to triangulation class

```TR = triangulation(E,V);
```

Use pointLocationTR to test points

```[ti,bc]=pointLocationTR(TR,QP,1,1,1);
```

Plot results and color points to the element number they are contained in

```cFigure;
title('Interior query points colored to element index','FontSize',fontSize);
hold on;

patch('Faces',F,'Vertices',V,'FaceColor','flat','CData',CF,'EdgeColor',edgeColor,'FaceAlpha',faceAlpha1,'lineWidth',edgeWidth);

logicPlot=~isnan(ti); %Point selection logic
% plotV(QP(logicPlot,:),'b.','MarkerSize',markerSize);
scatter3(QP(logicPlot,1),QP(logicPlot,2),QP(logicPlot,3),75,ti(logicPlot),'fill','markerEdgeColor','k');

colormap(cMap); colorbar;
view(3); axis equal; axis tight; axis off;
view([-50,12])
set(gca,'FontSize',fontSize);
drawnow;
```

## Example: Using barycentric coordinates to interpolate data (map points in deformed model)

Changing shape

```V=TR.Points;
V(:,1)=V(:,1)+0.5.*sin(pi*V(:,3));
V(:,2)=V(:,2)+0.5.*cos(pi*V(:,3));

% Fixing triangulation
TR = triangulation(E,V);
QPm = barycentricToCartesian(TR,ti(logicPlot),bc(logicPlot,:));
```

Plotting deformed shape

```cFigure;
title('Mapping of points in deformed object','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

patch('Faces',F,'Vertices',V,'FaceColor','flat','CData',CF,'EdgeColor',edgeColor,'FaceAlpha',faceAlpha1,'lineWidth',edgeWidth);

scatter3(QPm(:,1),QPm(:,2),QPm(:,3),75,ti(logicPlot),'fill','markerEdgeColor','k');

colormap(cMap); colorbar;
view(3); axis equal; axis tight; axis off;
view([-50,12])
set(gca,'FontSize',fontSize);
drawnow;
```

## Example: Finding voxels inside a tetrahedral mesh

Simulate image data

```load mri;
M=double(squeeze(D)); %example image data set
clear D;
M=M(1:2:end,1:2:end,:);

voxelSize=2./[1,1,.4]; %Voxel size
voxelSize(1)=voxelSize(1)*2;
voxelSize(2)=voxelSize(2)*2;
siz=size(M); %Image size
FOV=siz.*voxelSize; %Field of view size
```

Get patch data

```T_low=min(M(:))+((max(M(:))-min(M(:)))/10); %Threshold example
logicVoxels=(M>T_low);
[Fv,Vv,Cv]=ind2patch(logicVoxels,M,'vb');
Cv=Cv./max(Cv(:));
[Vv(:,1),Vv(:,2),Vv(:,3)]=im2cart(Vv(:,2),Vv(:,1),Vv(:,3),voxelSize);
```

Simulate a tesselated model

```Vv_mean=mean(Vv,1);
[V,~]=platonic_solid(1,max(FOV)/2);
V=V+Vv_mean(ones(1,size(V,1)),:);
E=[1 2 3 4];

n=4;
for q=1:1:n
[E,V]=subTet(E,V,1);
end
C=(1:size(E,1))';
```
```[F,CF]=element2patch(E,C);

cFigure;
title('Image data and a tetrahedral tesselation','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

patch('Faces',F,'Vertices',V,'FaceColor','flat','CData',CF,'EdgeColor',edgeColor,'FaceAlpha',faceAlpha1,'lineWidth',edgeWidth);
hp1= patch('Faces',Fv,'Vertices',Vv,'FaceColor','flat','FaceVertexCData',Cv(:,ones(1,3)),'EdgeColor','k','FaceAlpha',faceAlpha1);

colormap(cMap); colorbar;
view(3); grid on; axis equal; axis tight;
drawnow;
```

Setting up query points, in this case voxel coordinates

```[Jv,Iv,Kv]=meshgrid(1:1:size(M,2),1:1:size(M,1),1:1:size(M,3)); %Voxel image coordinates
[Xv,Yv,Zv]=im2cart(Iv,Jv,Kv,voxelSize); %Voxel cartesian coordinates
Vq=[Xv(:) Yv(:) Zv(:)];
```

Find what elements the voxels are contained in

```TR = triangulation(E,V);
[TI,BC]=pointLocationTR(TR,Vq,1,1,1);

% Reshaping the output TI leads in effect to a labeled image whereby each
% voxels containes an index (or group/label number) for an element
ML=reshape(TI',size(M));
```

Plotting the result

```cFigure;
title('Image data found inside the tetrahedral tesselation','FontSize',fontSize);
xlabel('X','FontSize',fontSize);ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

patch('Faces',F,'Vertices',V,'FaceColor','g','EdgeColor','none','FaceAlpha',0.1);

logicVoxels=~isnan(ML);
[Fm,Vm,Cm]=ind2patch(logicVoxels,M,'vb');
[Vm(:,1),Vm(:,2),Vm(:,3)]=im2cart(Vm(:,2),Vm(:,1),Vm(:,3),voxelSize);
patch('Faces',Fm,'Vertices',Vm,'FaceColor','flat','CData',Cm,'EdgeColor','k','FaceAlpha',faceAlpha2);

colormap gray; colorbar;
axis equal; view(3); axis tight;  grid on;  set(gca,'FontSize',fontSize);
drawnow;
```

## Example: Interpolation of image data onto volume elements

Hence deriving image based intensity values becomes a matter of averaging across each element index label. If the voxels are sufficiently small with respect to the elements then this method actually partially takes into account partial voluming effects. i.e. the intensity of an element becomes the mean of the voxels contained within it which in some cases is more accurate then an interpolation which does not take into account the volumetric nature of the mesh (e.g. methods based on sampling/interpolating the intensities at the centre point or nodal points of the mesh only instead).

```[E_color]=imlabelMean(M,ML); %The image based element colors
```

Plotting results

```cFigure;
title('The image data and the tesselation colored towards the image data','FontSize',fontSize);
xlabel('X','FontSize',fontSize);ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

[F,C]=element2patch(E,E_color);

L=~isnan(C);
patch('Faces',F(L,:),'Vertices',V,'FaceColor','flat','CData',C(L),'EdgeColor','k','FaceAlpha',faceAlpha2);

L_plot=false(siz);
L_plot(:,:,round(size(M,3)/2))=1;
L_plot=L_plot;
[Fm,Vm,Cm]=ind2patch(L_plot,double(M),'sk');
[Vm(:,1),Vm(:,2),Vm(:,3)]=im2cart(Vm(:,2),Vm(:,1),Vm(:,3),voxelSize);
patch('Faces',Fm,'Vertices',Vm,'FaceColor','flat','CData',Cm,'EdgeColor','k','FaceAlpha',faceAlpha2);

L_plot=false(siz);L_plot(round(size(M,1)/2),:,:)=1;
L_plot=L_plot;
[Fm,Vm,Cm]=ind2patch(L_plot,M,'si');
[Vm(:,1),Vm(:,2),Vm(:,3)]=im2cart(Vm(:,2),Vm(:,1),Vm(:,3),voxelSize);
patch('Faces',Fm,'Vertices',Vm,'FaceColor','flat','CData',Cm,'EdgeColor','k','FaceAlpha',faceAlpha2);

L_plot=false(siz);L_plot(:,round(size(M,2)/2),:)=1;
L_plot=L_plot;
[Fm,Vm,Cm]=ind2patch(L_plot,M,'sj');
[Vm(:,1),Vm(:,2),Vm(:,3)]=im2cart(Vm(:,2),Vm(:,1),Vm(:,3),voxelSize);
patch('Faces',Fm,'Vertices',Vm,'FaceColor','flat','CData',Cm,'EdgeColor','k','FaceAlpha',faceAlpha2);

colormap gray; colorbar;
axis equal; view(3); axis tight;  grid on;  set(gca,'FontSize',fontSize);
drawnow;
```

GIBBON www.gibboncode.org

Kevin Mattheus Moerman, gibbon.toolbox@gmail.com

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GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for image segmentation, image-based modeling, meshing, and finite element analysis.

Copyright (C) 2019 Kevin Mattheus Moerman

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