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Can not understand Jacobian Computation Process in function OptimizePairwise
hello,
i read the paper Fast Global Registration, which uses Gauss-Newton method to optimize T by equation 8 at the Section3.2
I find your calculation of the Jacobian as follows：
`
int ii = corres_[c].first; int jj = corres_[c].second; Eigen::Vector3f p, q; p = pointcloud_[i][ii]; q = pcj_copy[jj]; Eigen::Vector3f rpq = p - q;
int c2 = c;
float temp = par / (rpq.dot(rpq) + par);
s[c2] = temp * temp;
J.setZero();
J(1) = -q(2);
J(2) = q(1);
J(3) = -1;
r = rpq(0);
JTJ += J * J.transpose() * s[c2];
JTr += J * r * s[c2];
r2 += r * r * s[c2];
J.setZero();
J(2) = -q(0);
J(0) = q(2);
J(4) = -1;
r = rpq(1);
JTJ += J * J.transpose() * s[c2];
JTr += J * r * s[c2];
r2 += r * r * s[c2];
J.setZero();
J(0) = -q(1);
J(1) = q(0);
J(5) = -1;
r = rpq(2);
JTJ += J * J.transpose() * s[c2];
JTr += J * r * s[c2];
r2 += r * r * s[c2];
r2 += (par * (1.0 - sqrt(s[c2])) * (1.0 - sqrt(s[c2])));`
To have a better understanding, I tried to substitute lp,q (equation 6) into objective 3 E(T;L).
Then I got:
Using chain rule:
So：
And I realize that this equation represents the following line in the code:
s[c2] = temp * temp;
As far as I am concerned, J(2) should be calculated as follows:
• I see that your calculation of Jacobian is divided into three parts which are quite different from I expected before. Could you explain more about your codes of Jacobian calculation?
Thanks,
yechuankun
See this page for detailed derivation: http://redwood-data.org/indoor/registration.html
Basically you can only linearize the transformation matrix when alpha, beta, gamma, a, b, c are all very small values. So T cannot be directly linearized in this way. Instead, you need some trick to bring T*^{-1}T to the front and linearize it. And the derivation leads to what's in the code.
