ELLIPSOID_TOOL_optimisation_function.m
792da49477d7command_line

ELLIPSOID_TOOL_optimisation_function.m
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			function [f, df] = ELLIPSOID_TOOL_optimisation_function(x, Points)
			% Function used by fmincon to compute the cost function for the ellipsoid.
			%--------------------------------------------------------------------------
			% Syntax :
			% [f, df] = ELLIPSOID_TOOL_optimisation_function(x, Points)
			%--------------------------------------------------------------------------
			%
			% EPFL - LA - LBO SHOULDER MODEL
			% Created by : David Ingram (LA - EPFL)
			%
			% File Description :
			% This function computes the cost function for the ribcage ellipsoid
			% optimisaiton. The cost function is the sum of the norms between points on
			% the ribcage meshing and their projection onto the surface of the
			% ellipsoid.
			%
			%--------------------------------------------------------------------------

			% Get Ellipsoid Variables
			xo = x(1); a = x(4);
			yo = x(2); b = x(5);
			zo = x(3); c = x(6);

			% Ellipsoid Matrix
			E = diag([1/a^2; 1/b^2; 1/c^2]);

			% Ellipsoid Centre
			O = [xo; yo; zo];

			% Initialise the output
			f = 0;

			% The gradient is computed using MATLAB (finite differencing)
			df = zeros(6,1);

			% Compute the cost function
			for i = 1:size(Points,1)
			% Get the coordinates of point to be projected
			u = Points(i,1);
			v = Points(i,2);
			w = Points(i,3);
			Q = [u; v; w];

			% Compute the projection scalar
			Lam = Lambda_Compute(xo, yo, zo, a, b, c, u, v, w);

			% Compute the projected point
			P = inv(eye(3) + 2LamE)(Q + 2LamEO);

			% Augment the cost function
			f = f + 0.5(P-Q)'(P-Q);
			end
			return;

			% Small Sub-function for computing Lambda the projection scalar
			function Lam = Lambda_Compute(xo, yo, zo, a, b, c, u, v, w)
			% The Projected Point Coordinates satisfy the equations
			% u = x + 2 * lam * (x - xo)/a^2 -> x = (u * a^2 + 2 * lam * xo)/(a^2 + 2 * lam)
			% v = y + 2 * lam * (y - yo)/b^2 -> y = (v * b^2 + 2 * lam * yo)/(b^2 + 2 * lam)
			% w = z + 2 * lam * (z - zo)/c^2 -> z = (w * c^2 + 2 * lam * zo)/(c^2 + 2 * lam)
			%
			% Injecting the relations into the ellipsoid equation
			%
			% a^2(u - xo)^2/(a^2 + 2 lam) + b^2(v - yo)^2/(b^2 + 2 lam) + c^2(w - zo)^2/(c^2 + 2 lam) - 1 = 0
			%
			% List of Polynomial Coefficients for Computing lambda, the projection
			% parameter. These coefficients where computed using MATLAB's symbolic
			% toolbox.
			K6 = - 64;

			K5 = - 64*(a^2 + b^2 + c^2);

			K4 = - 16*(a^4 + b^4 + c^4)...
			- 64(a^2c^2 + b^2c^2 + a^2b^2)...
			+ 16(a^2u^2 + b^2v^2 + c^2w^2)...
			- 32(a^2uxo + b^2vyo + c^2w*zo)...
			+ 16(a^2xo^2 + b^2yo^2 + c^2zo^2);

			K3 = - 64(a^2b^2*c^2)...
			- 16(a^4b^2 + a^4c^2 + b^4a^2 + b^4c^2 + c^4a^2 + c^4*b^2)...
			+ 16(a^2u^2b^2 + a^2u^2c^2 + b^2v^2a^2 + b^2v^2c^2 + c^2w^2a^2 + c^2w^2*b^2)...
			+ 16(c^2zo^2b^2 + c^2zo^2a^2 + a^2xo^2c^2 + a^2xo^2b^2 + b^2yo^2c^2 + b^2a^2*yo^2)...
			- 32(a^2uxoc^2 + a^2uxob^2 + b^2vyoc^2 + b^2vyoa^2 + c^2wzob^2 + c^2wzo*a^2);

			K2 = + 16(a^2b^2c^2u^2 + a^2b^2c^2v^2 + a^2b^2c^2w^2)...
			+ 16(a^2b^2c^2xo^2 + a^2b^2c^2yo^2 + a^2b^2c^2zo^2)...
			- 32(a^2b^2c^2uxo + a^2b^2c^2vyo + a^2b^2c^2w*zo)...
			- 4(a^4b^4 + a^4c^4 + b^4c^4) - 16(a^4b^2c^2 + a^2b^4c^2 + a^2b^2*c^4)...
			+ 4(a^2b^4u^2 + a^2c^4u^2 + b^2a^4v^2 + b^2c^4v^2 + c^2a^4w^2 + c^2b^4*w^2)...
			- 8(a^2c^4uxo + a^2b^4uxo + b^2c^4vyo + b^2a^4vyo + c^2a^4wzo + c^2b^4w*zo)...
			+ 4(a^2b^4xo^2 + a^2c^4xo^2 + b^2a^4yo^2 + b^2c^4yo^2 + c^2a^4zo^2 + c^2b^4*zo^2);

			K1 = - 4(a^2b^4c^4 + b^2a^4c^4 + c^2b^4*a^4)...
			+ 4(a^2b^2c^4u^2 + a^2b^4c^2u^2 + a^4b^2c^2v^2 + a^2b^2c^4v^2 + a^4b^2c^2w^2 + a^2b^4c^2*w^2)...
			- 8(a^2b^2c^4uxo + a^2b^4c^2uxo + a^2b^2c^4vyo + a^4b^2c^2vyo + a^4b^2c^2wzo + a^2b^4c^2w*zo)...
			+ 4(a^2b^2c^4xo^2 + a^2b^4c^2xo^2 + a^4b^2c^2yo^2 + a^2b^2c^4yo^2 + a^4b^2c^2zo^2 + a^2b^4c^2*zo^2);

			K0 = - (a^4b^4c^4)...
			+ (a^2b^4c^4u^2 + a^4b^2c^4v^2 + a^4b^4c^2*w^2)...
			+ (a^2b^4c^4xo^2 + a^4b^2c^4yo^2 + a^4b^4c^2*zo^2)...
			- 2(a^2b^4c^4uxo + a^4b^2c^4vyo + a^4b^4c^2w*zo);

			% Select the smallest real root
			% K6lam^6 + K5lam^5 + K4lam^4 + K3lam^3 + K2lam^2 + K1lam + K0 = 0
			LamRoots = roots([K6, K5, K4, K3, K2, K1, K0]); % Compute roots
			idx = find(abs(imag(LamRoots)) < 1.e-6); % Identify real roots
			[Lam, idy] = min([abs(LamRoots(idx(1),1)), abs(LamRoots(idx(2),1))]); % Select smallest value
			Lam = LamRoots(idx(idy),1);
			return;

msm/TOOLBOX_Ribcage_Ellipsoid/ELLIPSOID_TOOL_optimisation_function.m792da49477d7command_line

ELLIPSOID_TOOL_optimisation_function.mView Options

msm/TOOLBOX_Ribcage_Ellipsoid/ELLIPSOID_TOOL_optimisation_function.m
792da49477d7command_line

ELLIPSOID_TOOL_optimisation_function.m
View Options