Below is a demonstration of the features of the subQuad function

## Description

The subQuad function enables refinement of quadrangulated data. The quadrilateral faces defined by the patch format data F (faces) and V (vertices). Each face is split n times using the specified split method (splitMethod). The user may request the following outputs: The new faces: Fs The new coordinates: Vs Face color labels: C Nodal labels: CV

Four split methods are defined: 1: General linear resampling 2: Linear resampling only in the first direction 3: Linear resampling only in the second direction

## Examples

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

Plot Settings

```fontSize=15;
faceAlpha=1;
edgeColor=0.2*ones(1,3);
edgeWidth=1.5;
markerSize=35;
markerSize2=20;
```

## Refining a 2D quadrilateral with boundary edges

```V=[0 0; 1 0; 1 1; 0 1];
F=[1 2 3 4];

n=0:1:3; %Number subdivision iterations
pColors=gjet(numel(n));

cFigure;
for q=1:1:numel(n)
subplot(2,2,q); hold on;
title([num2str(n(q)),' split iterations'],'FontSize',fontSize);
gpatch(Fs,Vs,pColors(q,:),'k');
plotV(Vs,'k.','markerSize',markerSize2);
plotV(V,'k.','markerSize',markerSize);

axis equal; axis tight; view(2);
end
drawnow;
```

## Refining a cube (3D closed mesh)

```[V,F]=platonic_solid(2,1);

n=0:1:3; %Number subdivision iterations
pColors=gjet(numel(n));

cFigure;
for q=1:1:numel(n)
subplot(2,2,q); hold on;
title([num2str(n(q)),' split iterations'],'FontSize',fontSize);
gpatch(Fs,Vs,pColors(q,:),'k');
%     patchNormPlot(Fs,Vs);
plotV(Vs,'k.','markerSize',markerSize2);
plotV(V,'k.','markerSize',markerSize);
axisGeom(gca,fontSize);
end
drawnow;
```

## Refining 3D quadrilateral surfaces with boundary edges

```[X,Y,Z]=peaks(10);
Z=Z/5;
[F,V]=surf2patch(X,Y,Z);

n=[0 1 2 3]; %Number of refinement steps

pColors=gjet(numel(n));

cFigure;
for q=1:1:numel(n)
subplot(2,2,q); hold on;
title([num2str(n(q)),' split iterations'],'FontSize',fontSize);
gpatch(Fs,Vs,pColors(q,:),'k');
%     patchNormPlot(Fs,Vs);
axisGeom(gca,fontSize);
end
drawnow;
```

## Example: Subdividing in 1 direction by specifying splitMethod

```V=[0 0 0; 1 0 0; 1 1 0; 0 1 0];
F=[1 2 3 4];

pColors=gjet(4);

n=2;

cFigure;
subplot(2,2,1); hold on;
title('Original','FontSize',fontSize);
gpatch(F,V,pColors(1,:),'k',1,3);
patchNormPlot(F,V);
axisGeom(gca,fontSize);

for q=1:1:3
splitMethod=q;
subplot(2,2,q+1); hold on;
title(['Refinement method: ',num2str(q)],'FontSize',fontSize);
gpatch(Fs,Vs,pColors(q+1,:),'k',1,3);
patchNormPlot(Fs,Vs);
axisGeom(gca,fontSize);
%     view(2);
end
drawnow;
```

## Example: Maintaining/resampling face data (e.g. face color)

```[X,Y,Z]=peaks(15); Z=Z/5;
[F,V,CV]=surf2patch(X,Y,Z,Z);
C=vertexToFaceMeasure(F,CV);
```

Requesting additional output to allow for "book keeping" of face data

```[Fs,Vs,Css]=subQuad(F,V,2,1);
```

The additional output Css contains face indices, i.e. each refined face in Fs belongs to an initial face in F and this mapping is defined in Css. Hence the following operation "picks" out the appropriate color data for each face from the original array.

```Cs=C(Css); %Get colors for refined faces from color data of original
```
```cFigure;
subplot(1,2,1); hold on;
title('Original','FontSize',fontSize);
gpatch(F,V,C,'k');
axisGeom(gca,fontSize);
colormap gjet;

subplot(1,2,2); hold on;
title('Refined','FontSize',fontSize);
gpatch(Fs,Vs,Cs,'k');
axisGeom(gca,fontSize);
colormap gjet;

drawnow;
```

## Example: Maintaining/resampling vertex data (e.g. vertex color)

Requesting additional output to allow for "book keeping" of face data

```VI=[V CV]; %Append color as artificial additional coordinate
Vs=VsI(:,1:3); %Pick out coordinates from first 3 columns
CVs=VsI(:,4); %Pick out resampled vertex data from 4th column
```
```cFigure;
subplot(1,2,1); hold on;
title('Original','FontSize',fontSize);
gpatch(F,V,C,'k');
axisGeom(gca,fontSize);
colormap gjet;

subplot(1,2,2); hold on;
title('Refined','FontSize',fontSize);
gpatch(Fs,Vs,CVs,'k');
axisGeom(gca,fontSize);
colormap gjet;

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.

```n=2; %Number of refinement steps
```
```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={'Origin';'Mid-edge';'Mid-face'};
drawnow;
```

GIBBON www.gibboncode.org

Kevin Mattheus Moerman, [email protected]

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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.

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