Lorenz Meier
10 years ago
2 changed files with 199 additions and 6 deletions
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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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