Added proper curvature analysis based on polynomials

docs/HELP_patchCurvaturePolynomial.m

@@ -0,0 +1,94 @@
+%% patchCurvature
+% Below is a demonstration of the features of the |patchCurvature| function
+
+%% Syntax
+% |[Vd,Fd,Fds]=patchCurvature(V,F);|
+
+%% Description
+% Computes curvature metrics for the patch data defined by the faces F and
+% the vertices V. 
+
+%% Examples
+
+%%
+clear; close all; clc;
+
+%%
+% Plot settings
+cMap=warmcold(250);
+
+%%
+
+% [F,V]=geoSphere(4,5.25);
+[F,V]=graphicsModels(9);
+% [F,V]=stanford_bunny;
+% [F,V]=tri2quad(F,V);
+% [F,V]=patchcylinder(60,100,60,60,'tri');
+
+
+%% Compute curvature
+[U1,U2,K1,K2,H,G] = patchCurvaturePolynomial(F,V);
+
+%% Visualize curvature on mesh
+
+% Compute plot variables
+vecPlotSize=mean(patchEdgeLengths(F,V)); %Vector plotting size
+
+% Visualize
+cFigure; 
+subplot(1,2,1); hold on;
+title('K1');
+hp=gpatch(F,V,K1,'none',0.9);
+hp.FaceColor='interp';
+colormap(gca,cMap); colorbar;
+quiverVec(V,U1,vecPlotSize,'k');
+axisGeom; 
+c=max(abs(K1(:)));
+caxis(0.1*[-c c]);
+camlight headlight;
+
+subplot(1,2,2); hold on;
+title('K2');
+hp=gpatch(F,V,K2,'none',0.9);
+hp.FaceColor='interp';
+quiverVec(V,U2,vecPlotSize,'k');
+colormap(gca,cMap); colorbar;
+axisGeom;
+c=max(abs(K2(:)));
+caxis(0.1*[-c c]);
+camlight headlight;
+
+drawnow;
+
+%%
+%
+% <<gibbVerySmall.gif>>
+%
+% _*GIBBON*_
+% <www.gibboncode.org>
+%
+% _Kevin Mattheus Moerman_, <gibbon.toolbox@gmail.com>
+
+%% 
+% _*GIBBON footer text*_ 
+% 
+% License: <https://github.com/gibbonCode/GIBBON/blob/master/LICENSE>
+% 
+% GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for
+% image segmentation, image-based modeling, meshing, and finite element
+% analysis.
+% 
+% Copyright (C) 2006-2023 Kevin Mattheus Moerman and the GIBBON contributors
+% 
+% This program is free software: you can redistribute it and/or modify
+% it under the terms of the GNU General Public License as published by
+% the Free Software Foundation, either version 3 of the License, or
+% (at your option) any later version.
+% 
+% This program is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+% GNU General Public License for more details.
+% 
+% You should have received a copy of the GNU General Public License
+% along with this program.  If not, see <http://www.gnu.org/licenses/>.

lib/patchCurvature.m

@@ -7,6 +7,10 @@
 % 
 % ------------------------------------------------------------------------
 
+%%
+
+warning('This function is obsolete, and will be removed, use patchCurvaturePolynomial instead');
+
 %%
 % Get connectivity matrices
 connecticityStruct=patchConnectivity(F,V);

lib/patchCurvaturePolynomial.m

@@ -0,0 +1,93 @@
+function [U1,U2,K1,K2,H,G] = patchCurvaturePolynomial(F,V)
+
+% function [U1,U2,K1,K2,H,G] = patchCurvaturePolynomial(F,V)
+% ------------------------------------------------------------------------
+%
+% This function computes the mesh curvature at each vertex for the input mesh 
+% defined by the face `F` and the vertices `V`. A local polynomial is fitted to 
+% each point's "Laplacian umbrella" (point neighbourhood), and the curvature of 
+% this fitted form is derived. 
+% 
+% The reference below [1] provides more detail on the algorithm. In addition, this 
+% implementation was created with the help of this helpful document:
+% https://github.com/alecjacobson/geometry-processing-curvature/blob/master/README.md, 
+% which features a nice overview of the theory/steps involved in this algorithm. 
+% 
+% References 
+% 1. [F. Cazals and M. Pouget, _Estimating differential quantities using polynomial fitting of osculating jets_, Computer Aided Geometric Design, vol. 22, no. 2, pp. 121-146, Feb. 2005, doi: 10.1016/j.cagd.2004.09.004](https://doi.org/10.1016/j.cagd.2004.09.004)
+% 
+% ------------------------------------------------------------------------
+
+%%
+m = size(V,1);
+CC = patchConnectivity(F,V,{'vv'});
+con_V2V = CC.vertex.vertex;
+[~,~,NV]=patchNormal(F,V);
+nz = [0.0 0.0 1.0]; % z-vector
+
+K1 = zeros(m,1);
+K2 = zeros(m,1);
+U1 = zeros(m,3);
+U2 = zeros(m,3);
+for q = 1:1:m
+    n = NV(q,:);
+    [a,d]=vectorOrthogonalPair(n);
+    Q=[a; d; n];
+    ind = con_V2V(q,:); ind = ind(ind>0);
+    vr = (V(ind,:)-V(q,:))*Q';
+
+    % Set up polynomial fit
+    T = zeros(length(ind),5);
+    w = zeros(length(ind),1);
+    for i = 1:length(ind)
+        T(i,:) = [vr(i,1),vr(i,2),vr(i,1)^2,vr(i,1)*vr(i,2),vr(i,2)^2];
+        w(i) = vr(i,3);
+    end
+    a = T\w;
+
+    E = 1.0 + a(1)^2;
+    F = a(1)*a(2);
+    G = 1.0 + a(2)^2;
+    d = sqrt(a(1)^2+1.0+a(2)^2);
+    e = (2.0*a(3)) / d;
+    f =       a(4) / d;
+    g = (2.0*a(5)) / d;
+
+    S = -[e f; f g] * inv([E F; F G]);
+    [u,k] = eig(S); % Eigen decomposition to get first/second eigenvalue and vectors    
+
+    % Store derived quantities
+    K1(q) = k(2,2);
+    K2(q) = k(1,1);
+    U1(q,:) = [u(1,2) u(2,2) 0.0]*Q;
+    U2(q,:) = [u(1,1) u(2,1) 0.0]*Q;   
+
+end
+
+H = 0.5 * (K1+K2); % Mean curvature
+G = K1.*K2; % Gaussian curvature
+
+end
+%% 
+% _*GIBBON footer text*_ 
+% 
+% License: <https://github.com/gibbonCode/GIBBON/blob/master/LICENSE>
+% 
+% GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for
+% image segmentation, image-based modeling, meshing, and finite element
+% analysis.
+% 
+% Copyright (C) 2006-2023 Kevin Mattheus Moerman and the GIBBON contributors
+% 
+% This program is free software: you can redistribute it and/or modify
+% it under the terms of the GNU General Public License as published by
+% the Free Software Foundation, either version 3 of the License, or
+% (at your option) any later version.
+% 
+% This program is distributed in the hope that it will be useful,
+% but WITHOUT ANY WARRANTY; without even the implied warranty of
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+% GNU General Public License for more details.
+% 
+% You should have received a copy of the GNU General Public License
+% along with this program.  If not, see <http://www.gnu.org/licenses/>.