HELP_ellipsoidFit_centered

Below is a demonstration of the features of the ellipsoidFit_centered function

Contents

Syntax

[M,ellipStretch,R,MU]=ellipsoidFit_centered(X,MU);

Description

The ellipsoidFit_centered function fits an ellipsoid to data when the ellipsoid centre is known. If the centre is not provided the mean of the input point set will be assumed to be the centre.

Examples

clear; close all; clc;

Plot settings

figColor='w';
figColorDef='white';
fontSize=11;

Example: Using ellipsoidFit_centered to fit an ellipsoid to a point cloud with known centre

Simulating an ellipsoid with known directions

% Ellipsoid axis stretch factors
ellipStretchTrue=[pi 2 0.5];
MU_true=[1 6 pi];

% Create ellipsoid patch data
[F,X,~]=geoSphere(3,1);
x=X(:,1);
FX=mean(x(F),2);
logicKeep=FX>0;
F=F(logicKeep,:);
indKeep=unique(F(:));
indFix=nan(size(X,1),1);
indFix(indKeep)=1:numel(indKeep);
X=X(indKeep,:);
F=indFix(F);
X=X.*ellipStretchTrue(ones(size(X,1),1),:);

%Create Euler angles to set directions
E=[0.25*pi 0.25*pi -0.25*pi];
[R_true,~]=euler2DCM(E); %The true directions for X, Y and Z axis
X=(R_true*X')'; %Rotate polyhedron

X=X+MU_true(ones(size(X,1),1),:); %Centre points around mean

%Add noise
n_std=0.2;  %Standard deviation
Xn=X+n_std.*randn(size(X));

This is the true axis system

R_true
R_true =

    0.5000    0.5000    0.7071
   -0.1464    0.8536   -0.5000
   -0.8536    0.1464    0.5000

These are the true stretch factors

ellipStretchTrue
ellipStretchTrue =

    3.1416    2.0000    0.5000

[M,ellipStretchFit,R_fit,MU]=ellipsoidFit_centered(Xn,MU_true);

This is the fitted axis system. The system axes should be colinear with the true axes but can be oposite in direction.

R_fit=R_fit(1:3,1:3)
R_fit =

    0.5071   -0.5000   -0.7020
   -0.1723   -0.8569    0.4859
   -0.8445   -0.1255   -0.5207

These are the fitted stretch factors

ellipStretchFit
ellipStretchFit =

    2.9737    1.9852    0.6399

Building a fitted (clean) ellipsoid for visualization

%Create sphere
[F_fit,V_fit,~]=geoSphere(4,1);

%Transforming sphere to ellipsoid
V_fit_t=V_fit;
V_fit_t(:,end+1)=1;
V_fit_t=(M*V_fit_t')'; %Rotate polyhedron
V_fit=V_fit_t(:,1:end-1);

Visualizing results

MU=mean(X,1); %Origin for vectors
a=[7 7]; %Vector size

[Fq,Vq,Cq]=quiver3Dpatch(MU(1)*ones(1,3),MU(2)*ones(1,3),MU(3)*ones(1,3),R_fit(1,:),R_fit(2,:),R_fit(3,:),eye(3,3),a); %Fitted vectors
[Fq2,Vq2,Cq2]=quiver3Dpatch(MU(1)*ones(1,3),MU(2)*ones(1,3),MU(3)*ones(1,3),R_true(1,:),R_true(2,:),R_true(3,:),eye(3,3),a); %True vectors

figuremax(figColor,figColorDef);
title('The true (green) and fitted ellipsoid (red) and axis directions (solid, transparant respectively)','FontSize',fontSize);
xlabel('X','FontSize',fontSize); ylabel('Y','FontSize',fontSize); zlabel('Z','FontSize',fontSize);
hold on;

plotV(Xn,'k.','MarkerSize',15);

hp=patch('Faces',F,'Vertices',X);
set(hp,'FaceColor','g','FaceAlpha',1,'EdgeColor','k');

hp=patch('Faces',F_fit,'Vertices',V_fit);
set(hp,'FaceColor','r','FaceAlpha',0.2,'EdgeColor','none');

patch('Faces',Fq,'Vertices',Vq,'FaceColor','flat','FaceVertexCData',Cq,'FaceAlpha',1);
patch('Faces',Fq2,'Vertices',Vq2,'FaceColor','flat','FaceVertexCData',Cq2,'FaceAlpha',0.2,'EdgeColor','none');
camlight('headlight');
axis equal; view(3); axis vis3d; axis tight;  grid on;
set(gca,'FontSize',fontSize);
drawnow;

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Kevin Mattheus Moerman, [email protected]