forked from Archive/PX4-Autopilot
Matlab tools: Add ellipsoid fit
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Lorenz Meier
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baca229b95
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a4a6e69521
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% Copyright (c) 2009, Yury Petrov
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% All rights reserved.
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%
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% Redistribution and use in source and binary forms, with or without
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% modification, are permitted provided that the following conditions are
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% met:
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%
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% * Redistributions of source code must retain the above copyright
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% notice, this list of conditions and the following disclaimer.
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% * Redistributions in binary form must reproduce the above copyright
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% notice, this list of conditions and the following disclaimer in
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% the documentation and/or other materials provided with the distribution
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%
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% THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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% AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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% IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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% ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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% LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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% CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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% SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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% INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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% CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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% ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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% POSSIBILITY OF SUCH DAMAGE.
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%
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function [ center, radii, evecs, v ] = ellipsoid_fit( X, flag, equals )
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%
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% Fit an ellispoid/sphere to a set of xyz data points:
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%
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% [center, radii, evecs, pars ] = ellipsoid_fit( X )
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% [center, radii, evecs, pars ] = ellipsoid_fit( [x y z] );
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% [center, radii, evecs, pars ] = ellipsoid_fit( X, 1 );
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% [center, radii, evecs, pars ] = ellipsoid_fit( X, 2, 'xz' );
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% [center, radii, evecs, pars ] = ellipsoid_fit( X, 3 );
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%
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% Parameters:
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% * X, [x y z] - Cartesian data, n x 3 matrix or three n x 1 vectors
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% * flag - 0 fits an arbitrary ellipsoid (default),
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% - 1 fits an ellipsoid with its axes along [x y z] axes
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% - 2 followed by, say, 'xy' fits as 1 but also x_rad = y_rad
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% - 3 fits a sphere
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%
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% Output:
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% * center - ellispoid center coordinates [xc; yc; zc]
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% * ax - ellipsoid radii [a; b; c]
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% * evecs - ellipsoid radii directions as columns of the 3x3 matrix
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% * v - the 9 parameters describing the ellipsoid algebraically:
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% Ax^2 + By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + 2Gx + 2Hy + 2Iz = 1
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%
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% Author:
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% Yury Petrov, Northeastern University, Boston, MA
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%
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error( nargchk( 1, 3, nargin ) ); % check input arguments
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if nargin == 1
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flag = 0; % default to a free ellipsoid
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end
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if flag == 2 && nargin == 2
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equals = 'xy';
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end
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if size( X, 2 ) ~= 3
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error( 'Input data must have three columns!' );
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else
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x = X( :, 1 );
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y = X( :, 2 );
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z = X( :, 3 );
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end
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% need nine or more data points
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if length( x ) < 9 && flag == 0
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error( 'Must have at least 9 points to fit a unique ellipsoid' );
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end
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if length( x ) < 6 && flag == 1
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error( 'Must have at least 6 points to fit a unique oriented ellipsoid' );
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end
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if length( x ) < 5 && flag == 2
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error( 'Must have at least 5 points to fit a unique oriented ellipsoid with two axes equal' );
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end
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if length( x ) < 3 && flag == 3
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error( 'Must have at least 4 points to fit a unique sphere' );
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end
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if flag == 0
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% fit ellipsoid in the form Ax^2 + By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + 2Gx + 2Hy + 2Iz = 1
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D = [ x .* x, ...
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y .* y, ...
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z .* z, ...
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2 * x .* y, ...
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2 * x .* z, ...
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2 * y .* z, ...
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2 * x, ...
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2 * y, ...
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2 * z ]; % ndatapoints x 9 ellipsoid parameters
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elseif flag == 1
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% fit ellipsoid in the form Ax^2 + By^2 + Cz^2 + 2Gx + 2Hy + 2Iz = 1
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D = [ x .* x, ...
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y .* y, ...
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z .* z, ...
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2 * x, ...
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2 * y, ...
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2 * z ]; % ndatapoints x 6 ellipsoid parameters
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elseif flag == 2
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% fit ellipsoid in the form Ax^2 + By^2 + Cz^2 + 2Gx + 2Hy + 2Iz = 1,
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% where A = B or B = C or A = C
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if strcmp( equals, 'yz' ) || strcmp( equals, 'zy' )
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D = [ y .* y + z .* z, ...
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x .* x, ...
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2 * x, ...
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2 * y, ...
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2 * z ];
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elseif strcmp( equals, 'xz' ) || strcmp( equals, 'zx' )
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D = [ x .* x + z .* z, ...
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y .* y, ...
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2 * x, ...
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2 * y, ...
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2 * z ];
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else
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D = [ x .* x + y .* y, ...
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z .* z, ...
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2 * x, ...
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2 * y, ...
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2 * z ];
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end
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else
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% fit sphere in the form A(x^2 + y^2 + z^2) + 2Gx + 2Hy + 2Iz = 1
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D = [ x .* x + y .* y + z .* z, ...
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2 * x, ...
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2 * y, ...
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2 * z ]; % ndatapoints x 4 sphere parameters
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end
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% solve the normal system of equations
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v = ( D' * D ) \ ( D' * ones( size( x, 1 ), 1 ) );
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% find the ellipsoid parameters
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if flag == 0
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% form the algebraic form of the ellipsoid
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A = [ v(1) v(4) v(5) v(7); ...
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v(4) v(2) v(6) v(8); ...
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v(5) v(6) v(3) v(9); ...
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v(7) v(8) v(9) -1 ];
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% find the center of the ellipsoid
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center = -A( 1:3, 1:3 ) \ [ v(7); v(8); v(9) ];
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% form the corresponding translation matrix
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T = eye( 4 );
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T( 4, 1:3 ) = center';
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% translate to the center
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R = T * A * T';
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% solve the eigenproblem
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[ evecs evals ] = eig( R( 1:3, 1:3 ) / -R( 4, 4 ) );
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radii = sqrt( 1 ./ diag( evals ) );
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else
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if flag == 1
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v = [ v(1) v(2) v(3) 0 0 0 v(4) v(5) v(6) ];
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elseif flag == 2
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if strcmp( equals, 'xz' ) || strcmp( equals, 'zx' )
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v = [ v(1) v(2) v(1) 0 0 0 v(3) v(4) v(5) ];
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elseif strcmp( equals, 'yz' ) || strcmp( equals, 'zy' )
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v = [ v(2) v(1) v(1) 0 0 0 v(3) v(4) v(5) ];
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else % xy
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v = [ v(1) v(1) v(2) 0 0 0 v(3) v(4) v(5) ];
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end
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else
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v = [ v(1) v(1) v(1) 0 0 0 v(2) v(3) v(4) ];
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end
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center = ( -v( 7:9 ) ./ v( 1:3 ) )';
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gam = 1 + ( v(7)^2 / v(1) + v(8)^2 / v(2) + v(9)^2 / v(3) );
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radii = ( sqrt( gam ./ v( 1:3 ) ) )';
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evecs = eye( 3 );
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end
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@ -1,6 +1,19 @@
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%
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% Tool for plotting mag data
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%
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% Reference values:
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% telem> [cal] mag #0 off: x:0.15 y:0.07 z:0.14 Ga
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% MATLAB: x:0.1581 y: 0.0701 z: 0.1439 Ga
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% telem> [cal] mag #0 scale: x:1.10 y:0.97 z:1.02
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% MATLAB: 0.5499, 0.5190, 0.4907
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%
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% telem> [cal] mag #1 off: x:-0.18 y:0.11 z:-0.09 Ga
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% MATLAB: x:-0.1827 y:0.1147 z:-0.0848 Ga
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% telem> [cal] mag #1 scale: x:1.00 y:1.00 z:1.00
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% MATLAB: 0.5122, 0.5065, 0.4915
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%
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%
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close all;
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clear all;
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@ -13,11 +26,11 @@ ymin = -ymax;
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zmax = plot_scale;
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zmin = -zmax;
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mag0_raw = load('../../mag0_raw.csv');
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mag1_raw = load('../../mag1_raw.csv');
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mag0_raw = load('../../mag0_raw2.csv');
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mag1_raw = load('../../mag1_raw2.csv');
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mag0_cal = load('../../mag0_cal.csv');
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mag1_cal = load('../../mag1_cal.csv');
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mag0_cal = load('../../mag0_cal2.csv');
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mag1_cal = load('../../mag1_cal2.csv');
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fm0r = figure();
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@ -25,15 +38,21 @@ mag0_x_scale = 1.07;
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mag0_y_scale = 0.95;
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mag0_z_scale = 1.00;
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plot3(mag0_raw(:,1) .* mag0_x_scale, mag0_raw(:,2) .* mag0_y_scale, mag0_raw(:,3) .* mag0_z_scale, '*r');
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plot3(mag0_raw(:,1), mag0_raw(:,2), mag0_raw(:,3), '*r');
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[center, radii, evecs, pars ] = ellipsoid_fit( [mag0_raw(:,1) mag0_raw(:,2) mag0_raw(:,3)] );
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center
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radii
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axis([xmin xmax ymin ymax zmin zmax])
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fm1r = figure();
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plot3(mag1_raw(:,1), mag1_raw(:,2), mag1_raw(:,3), '*r');
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[center, radii, evecs, pars ] = ellipsoid_fit( [mag1_raw(:,1) mag1_raw(:,2) mag1_raw(:,3)] );
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center
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radii
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axis([xmin xmax ymin ymax zmin zmax])
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fm0c = figure();
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plot3(mag0_cal(:,1), mag0_cal(:,2), mag0_cal(:,3), '*b');
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plot3(mag0_cal(:,1) .* mag0_x_scale, mag0_cal(:,2) .* mag0_y_scale, mag0_cal(:,3) .* mag0_z_scale, '*b');
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axis([xmin xmax ymin ymax zmin zmax])
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fm1c = figure();
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