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Can Kabsch's algorithm also provide the covariance of its solution?

0

$begingroup$

Say you have a collection of data points $p_i in Bbb R^3$, and a collection of corresponding reference points $r_i in Bbb R^3$. Kabsch's algorithm, which relies on SVD decomposition, provides an efficient way to determine the rigid transformation that best (in a least squares of errors sense) transforms the $r_i$ to $p_i$.

This is equivalent to saying that if each $p_i$ is known to have been chosen as a 3-D Gaussian variate with unit variance and mean at the transformed value of $r_i$, then the solution found is the transformation with the highest likelihood.

Having determined that transformation, which is described by a translation vector $vec t$ and a rotation $R$ characterized as the product of three axial rotations
$$
R = R_z(psi)R_y(theta)R_x(phi)
$$
I would like to calculate the covariance matrix among $t_x, t_y, t_z, psi, theta, phi$.

Is there a "standard" or particularly elegant/efficient way to find that covariance, given that we already have the intermediate information used by Kabsch's algorithm?

linear-algebra least-squares data-analysis

share|cite|improve this question

asked Mar 25 at 19:30

Mark FischlerMark Fischler

34.2k12552

$endgroup$

add a comment | 

0

$begingroup$

Say you have a collection of data points $p_i in Bbb R^3$, and a collection of corresponding reference points $r_i in Bbb R^3$. Kabsch's algorithm, which relies on SVD decomposition, provides an efficient way to determine the rigid transformation that best (in a least squares of errors sense) transforms the $r_i$ to $p_i$.

This is equivalent to saying that if each $p_i$ is known to have been chosen as a 3-D Gaussian variate with unit variance and mean at the transformed value of $r_i$, then the solution found is the transformation with the highest likelihood.

Having determined that transformation, which is described by a translation vector $vec t$ and a rotation $R$ characterized as the product of three axial rotations
$$
R = R_z(psi)R_y(theta)R_x(phi)
$$
I would like to calculate the covariance matrix among $t_x, t_y, t_z, psi, theta, phi$.

Is there a "standard" or particularly elegant/efficient way to find that covariance, given that we already have the intermediate information used by Kabsch's algorithm?

linear-algebra least-squares data-analysis

share|cite|improve this question

asked Mar 25 at 19:30

Mark FischlerMark Fischler

34.2k12552

$endgroup$

add a comment | 

$begingroup$

Say you have a collection of data points $p_i in Bbb R^3$, and a collection of corresponding reference points $r_i in Bbb R^3$. Kabsch's algorithm, which relies on SVD decomposition, provides an efficient way to determine the rigid transformation that best (in a least squares of errors sense) transforms the $r_i$ to $p_i$.

This is equivalent to saying that if each $p_i$ is known to have been chosen as a 3-D Gaussian variate with unit variance and mean at the transformed value of $r_i$, then the solution found is the transformation with the highest likelihood.

Having determined that transformation, which is described by a translation vector $vec t$ and a rotation $R$ characterized as the product of three axial rotations
$$
R = R_z(psi)R_y(theta)R_x(phi)
$$
I would like to calculate the covariance matrix among $t_x, t_y, t_z, psi, theta, phi$.

Is there a "standard" or particularly elegant/efficient way to find that covariance, given that we already have the intermediate information used by Kabsch's algorithm?

linear-algebra least-squares data-analysis

share|cite|improve this question

asked Mar 25 at 19:30

Mark FischlerMark Fischler

34.2k12552

$endgroup$

share|cite|improve this question

asked Mar 25 at 19:30

Mark FischlerMark Fischler

34.2k12552

share|cite|improve this question

asked Mar 25 at 19:30

Mark FischlerMark Fischler

34.2k12552

asked Mar 25 at 19:30

Mark FischlerMark Fischler

34.2k12552

asked Mar 25 at 19:30

Mark FischlerMark Fischler

34.2k12552