subTriLoop

Below is a demonstration of the features of the subTriLoop function

Syntax

[Fs,Vs,Cs]=subTriLoop(F,V,n,fixBoundaryOpt);

Description

The subTriLoop function enables refinement of triangulated data using the smooth refinement method by Loop et al. 1987 https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/thesis-10.pdf

The input faces and vertices are defined by F and V respectively. Each triangle is split into 4 sub-triangles.

Examples

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

Plot Settings

```fontSize=10;
```

Example: Refining an icosahedron to illustrate smoothing effect

```[V,F]=platonic_solid(4,1);

n=0:1:3; %Number of refinement steps

cFigure;
gtitle('Loop subdivision')
for q=1:1:numel(n)
[Fs,Vs]=subTriLoop(F,V,n(q));
subplot(2,2,q); hold on;
title([num2str(n(q)),' split iterations'],'FontSize',fontSize);
hp1=gpatch(F,V,'none','k',1,2);
hp2=gpatch(Fs,Vs,'bw','k',1,2);
legend([hp1 hp2],{'Original','Refined'})
end
drawnow;
```

Example: Refinement and maintenance of face color data

```[V,F]=platonic_solid(4,1);
CF=(1:1:size(F,1))'; %Face color data

n=2; %Number of refinement steps

[Fs,Vs,Cs]=subTriLoop(F,V,n);
CFs=CF(Cs);

cFigure;
gtitle('Loop subdivision')
subplot(1,2,1); hold on;
title('Original with face colors','FontSize',fontSize);
gpatch(F,V,CF,'k',1,2);
drawnow;

subplot(1,2,2); hold on;
title('Resampled with face colors','FontSize',fontSize);
gpatch(Fs,Vs,CFs,'k',1,2);
colormap gjet; icolorbar;
drawnow;
```

Example: Refinement and resampling of vertex data

```[F,V]=stanford_bunny; V=V-mean(V,1); V=V./max(V(:));
CV=sin(2*pi*V(:,1));

n=1; %Number of refinement steps

VI=[V CV]; %Append vertex data as column(s)
[Fs,VIs]=subTriLoop(F,VI,n);
CVs=VIs(:,4);
Vs=VIs(:,1:3);

cFigure;
gtitle('Loop subdivision')
subplot(1,2,1); hold on;
title('Original with vertex colors','FontSize',fontSize);
gpatch(F,V,CV,'k',1,1);
colormap gjet;
drawnow;

subplot(1,2,2); hold on;
title('Resampled with vertex colors','FontSize',fontSize);
gpatch(Fs,Vs,CVs,'k',1,1);
colormap gjet;
drawnow;
```

Example: Forcing maintenance of boundary

```[F,V]=graphicsModels(10);

n=1; %Number of refinement steps

[Fs,Vs]=subTriLoop(F,V,n);

fixBoundaryOpt=1; %Option to constrain boundary to be linearly sampled
[Fs2,Vs2]=subTriLoop(F,V,n,fixBoundaryOpt);
```
```cFigure;
gtitle('Loop subdivision')
subplot(1,3,1); hold on;
title('Original with face colors','FontSize',fontSize);
hp1=gpatch(F,V,'w','k',1,1);
hp2=gpatch(patchBoundary(F,V),V,'none','b',1,3);
legend([hp1 hp2],{'Surface','Boundary'},'Location','SouthOutside');
view([-40 55]); zoom(1.1);

subplot(1,3,2); hold on;
title('Resampled with default smooth boundary','FontSize',fontSize);
hp1=gpatch(Fs,Vs,'w','k',1,1);
hp2=gpatch(patchBoundary(Fs,Vs),Vs,'none','b',1,3); hp2.EdgeAlpha=0.9;
legend([hp1 hp2],{'Surface','Boundary'},'Location','SouthOutside');
view([-40 55]); zoom(1.1);

subplot(1,3,3); hold on;
title('Resampled with linearly constrained boundary','FontSize',fontSize);
hp1=gpatch(Fs2,Vs2,'w','k',1,1);
hp2=gpatch(patchBoundary(Fs2,Vs2),Vs2,'none','b',1,3); hp2.EdgeAlpha=0.9;
legend([hp1 hp2],{'Surface','Boundary'},'Location','SouthOutside');
view([-40 55]); zoom(1.1);

drawnow;
```

Example: Study vertex type

An optional 4th output can provide "vertex labels", these define the vertex origins, i.e. whether they stem from the initial coordinates (iteration 0), or from iteration n.

```[F,V]=graphicsModels(10);

n=2; %Number of refinement steps

[Fs,Vs,~,CV]=subTriLoop(F,V,n);
```
```cFigure; hold on;
title('Visualizing point type','FontSize',fontSize);
gpatch(Fs,Vs,'w','k',1,1);
scatterV(Vs,50,CV,'filled')
view(2);
colormap gjet; [~,hc]=icolorbar;
hc.TickLabels={'Iter. 0';'Iter. 1';'Iter. 2'};
drawnow;
```

Example: Constraining faces to subdivide "linearly" (shape maintaining)

```% Using |triBox| to build the patch model
[F,V,faceBoundaryMarker]=triBox([2 2 2],1);
```
```n=2; %Number of refinement steps
logicConstrain=(faceBoundaryMarker==5); %Logic for faces to subdivide linearly
[Fs,Vs,Cs,CV]=subTriLoop(F,V,n,0,logicConstrain);

faceBoundaryMarker_sub=faceBoundaryMarker(Cs); %Get boundary markers for refined mesh
```
```cFigure;
subplot(1,2,1); hold on;
gpatch(F,V,faceBoundaryMarker,'k',1,1);
colormap(gca,gjet(6)); icolorbar;

subplot(1,2,2); hold on;
gpatch(Fs,Vs,faceBoundaryMarker_sub,'k',1,1);
colormap(gca,gjet(6)); icolorbar;
drawnow;
```

Example: Constraining faces but alter boundary of constrained

```% Using |triBox| to build the patch model
F=[Fq(:,[1 2 3]);Fq(:,[3 4 1]);];
faceBoundaryMarker=repmat(faceBoundaryMarker_q,2,1);
```
```n=3; %Number of refinement steps
logicConstrain=ismember(faceBoundaryMarker,[5,6]); %Logic for faces to subdivide linearly
indConstrain=find(logicConstrain);
indNotConstrain=find(~logicConstrain);

[Fs1,Vs1,Cs1]=subTriLoop(F(logicConstrain,:),V,n);
faceBoundaryMarker_sub1=faceBoundaryMarker(indConstrain(Cs1)); %Get boundary markers for refined mesh

[Fs2,Vs2,Cs2]=subTriLoop(F(~logicConstrain,:),V,n);
faceBoundaryMarker_sub2=faceBoundaryMarker(indNotConstrain(Cs2)); %Get boundary markers for refined mesh

[Fs,Vs,faceBoundaryMarker_sub]=joinElementSets({Fs1,Fs2},{Vs1,Vs2},{faceBoundaryMarker_sub1,faceBoundaryMarker_sub2});

[Fs,Vs]=patchCleanUnused(Fs,Vs);
[Fs,Vs]=mergeVertices(Fs,Vs);
```
```cFigure;
subplot(1,2,1); hold on;
gpatch(F,V,faceBoundaryMarker,'k',1,1);
colormap(gca,gjet(6)); icolorbar;

subplot(1,2,2); hold on;
gpatch(Fs,Vs,faceBoundaryMarker_sub,'k',1,1);
colormap(gca,gjet(6)); icolorbar;
drawnow;
```

GIBBON www.gibboncode.org

Kevin Mattheus Moerman, [email protected]

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