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Surface Fitting using gridfit

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Surface Fitting using gridfit

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11 Nov 2005 (Updated )

Model 2-d surfaces from scattered data

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test_main.m
% Gridfit test script

% This script file is designed to be used in cell mode
% from the matlab editor, or best of all, use the publish
% to HTML feature from the matlab editor. Older versions
% of matlab can copy and paste entire blocks of code into
% the Matlab command window.

%% Topographic data
load bluff_data;
x=bluff_data(:,1);
y=bluff_data(:,2);
z=bluff_data(:,3);
% Two ravines on a hillside. Scanned from a
% topographic map of an area in upstate New York.
plot3(x,y,z,'.')

%%

% Turn the scanned point data into a surface
gx=0:4:264;
gy=0:4:400;
g=gridfit(x,y,z,gx,gy);
figure
colormap(hot(256));
surf(gx,gy,g);
camlight right;
lighting phong;
shading interp
line(x,y,z,'marker','.','markersize',4,'linestyle','none');
title 'Use topographic contours to recreate a surface'

%% Fitting a trigonometric surface
clear

n1 = 15;
n2 = 15;
theta = rand(n1,1)*pi/2;
r = rand(1,n2);

x = cos(theta)*r;
y = sin(theta)*r;
x=x(:);
y=y(:);

x = [[0 0 1 1]';x;x;1-x;1-x];
y = [[0 1 0 1]';y;1-y;y;1-y];
figure
plot(x,y,'.')
title 'Data locations in the x-y plane'

%%
z = sin(4*x+5*y).*cos(7*(x-y))+exp(x+y);

xi = linspace(0,1,51);
[xg,yg]=meshgrid(xi,xi);
zgd = griddata(x,y,z,xg,yg);

figure
surf(xi,xi,zgd)
colormap(hot(256))
camlight right
lighting phong
title 'Griddata on trig surface'
% Note the wing-like artifacts along the edges, due
% to the use of a Delaunay triangulation in griddata.

%%

zgrid = gridfit(x,y,z,xi,xi);

figure
surf(xi,xi,zgrid)
colormap(hot(256))
camlight right
lighting phong
title('Gridfit to trig surface')

%% The trig surface with highly different scalings on the x and y axes
xs = x/100;
xis = xi/100;

ys = y*100;
yis = xi*100;

% griddata has problems with badly scaled data
[xg,yg]=meshgrid(xis,yis);
zgd = griddata(xs,ys,z,xg,yg);

figure
surf(xg,yg,zgd)
colormap(hot(256))
camlight right
lighting phong
title 'Serious problems for griddata on badly scaled trig surface'

%%

% autoscaling off
zgrid = gridfit(xs,ys,z,xis,yis,'autoscale','off');

% autoscaling on (the default)
zgrids = gridfit(xs,ys,z,xis,yis,'autoscale','on');

% plot the unscaled result
figure
surf(xis,yis,zgrid)
colormap(hot(256))
camlight right
lighting phong
title 'Gridfit (unscaled) on trig surface'

% plot the unscaled result
figure
surf(xis,yis,zgrids)
colormap(hot(256))
camlight right
lighting phong
title 'Default gridfit (automatically scaled) on trig surface'

%% Comparison of extrapolation behaviors
% Note: griddata will not extrapolate using most most of
% its methods, and is very slow/memory intensive for the
% 'v4' method

clear

n = 50000;
x = min(2,max(-2,randn(n,1)));
y = min(2,max(-2,randn(n,1)));
z = exp(x+y);
nodes = -3:.1:3;
zg = gridfit(x,y,z,nodes,nodes,'reg','gradient','sm',.01);
zl = gridfit(x,y,z,nodes,nodes,'reg','laplacian','sm',.01);
zs = gridfit(x,y,z,nodes,nodes,'reg','springs','sm',.01);

% plot the results
figure
surf(nodes,nodes,zg)
colormap(bone(256))
camlight right
lighting phong
hold on
plot3(x,y,z,'r.')
hold off
title 'Extrapolation with gridfit (gradient regularization)'

figure
surf(nodes,nodes,zl)
colormap(bone(256))
camlight right
lighting phong
hold on
plot3(x,y,z,'r.')
hold off
title 'Extrapolation with gridfit (Laplacian regularization)'

figure
surf(nodes,nodes,zs)
colormap(bone(256))
camlight right
lighting phong
hold on
plot3(x,y,z,'r.')
hold off
title 'Extrapolation with gridfit (gradient regularization)'

%% Fitting the "peaks" surface with fairly sparse data

clear

n = 100;
x = (rand(n,1)-.5)*6;
y = (rand(n,1)-.5)*6;

z = peaks(x,y);

xi = linspace(-3,3,101);
zpgf = gridfit(x,y,z,xi,xi);

[xg,yg]=meshgrid(xi,xi);
zpgd = griddata(x,y,z,xg,yg,'cubic');

figure
surf(xi,xi,zpgd)
colormap(jet(256))
camlight right
lighting phong
title 'Griddata (method == cubic) on peaks surface'

figure
surf(xi,xi,zpgf)
colormap(hsv(256))
camlight right
lighting phong
title('Gridfit on peaks surface')

%% Fitting the seamount surface

clear

load seamount

xg = linspace(min(x),max(x),50);
yg = linspace(min(y),max(y),50);

zsgf = gridfit(x,y,z,xg,yg);

figure
surf(xg,yg,zsgf)
colormap(hot(256))
camlight right
lighting phong
title('Gridfit applied to seamount data')

% Note that gridfit will extrapolate smoothly to the
% corners of the grid, whereas griddata cannot, unless
% the 'v4' option is chosen. But 'v4' is slow on even
% moderately large problems. 

[xg,yg]=meshgrid(xg,yg);
zsgd = griddata(x,y,z,xg,yg);

figure
surf(xg,yg,zsgd)
colormap(pink(256))
camlight right
lighting phong
title 'Griddata'

%% Using tiles in gridfit

% Users of gridfit who have really huge problems now have
% an option. I'll generate a large amount of data,
% and hope to model a fairly large grid - 800 x 800. This
% would normally require gridfit to solve a system of
% equations with 640,000 unknowns. It might be a solveable
% (though very slowly so) problem for my computer, were I
% to use gridfit on the full problem. Gridfit allows you to
% break the problem into smaller tiles if you choose. In
% this case each tile is 120x120, with a 25% (30 element)
% overlap between tiles. 

% Relax, this demo may take a couple of minutes to run!!!!

n = 100000;
x = rand(n,1);
y = rand(n,1);
z = x+y+sin((x.^2+y.^2)*10);

xnodes = 0:.002:1;
ynodes = xnodes;

[zg,xg,yg] = gridfit(x,y,z,xnodes,ynodes,'tilesize',120,'overlap',0.25);

surf(xg,yg,zg)
shading interp
colormap(jet(256))
camlight right
lighting phong
title 'Tiled gridfit'

%%

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