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%
% Usage:
%
% [R,t,S] = factor(W)
%
% Function to factor a matrix of input data (W) into the camera
% rotation matrix (R), translation (t), and the shape matrix (S).
% Three-dimensional version. Failure of normalization results in
% empty R and S.
function [R,t,S] = factor(W)
pts = size(W,2);
t = W*ones(pts,1)/pts;
W = W - t*ones(1,pts);
% Use SVD to factor W.
[a,b,c] = svd(W,0);
smallb = b(1:3,1:3); % Since W is rank 3, b has only three meaningful values
sqrtb = sqrt(smallb);
Rhat = a(:,1:3) * sqrtb;
Shat = sqrtb * c(:,1:3)';
G = findG(Rhat);
if size(G,1) == 0,
R = [];
S = [];
else
R = Rhat*G;
S = inv(G)*Shat;
% rotation matrix that aligns the reference frame with the first camera
F = size(R,1)/2;
R1 = R(1,:);
R1 = R1/norm(R1);
R2 = R(F+1,:);
R2 = R2/norm(R2);
R3 = cross(R1,R2);
R3 = R3/norm(R3);
P = [R1; R2; R3];
P = P';
R = R*P;
S = inv(P)*S;
end
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