Determining Transformation

Creator
Creator
Seonglae Cho
Created
Created
2023 Apr 4 1:37
Editor
Edited
Edited
2023 Apr 4 1:51
Refs
Refs
L2 Norm

xx' is measured location and MM is unknown matrix

x=Mxx' = Mx

Least Square Error Function f(xi;p)f(x_i; p) is predicted location

 
ELS=Σif(xi;p)xi2E_{LS} = \Sigma_i||f(x_i;p) - x'_i||^2

pp is parameter

p^=argminpELS\hat{p} = argmin_p E_{LS}
 
 

General form of linear least squares in matrix form

ELS=Axb2E_{LS} = ||Ax - b||^2=xT(ATA)x2xT(ATb)+b2= x^T(A^TA)x - 2x^T(A^Tb) + ||b||^2

When you solve derivative

(ATA)x=ATb(A^TA)x = A^Tbx=(ATA)1ATbx = (A^TA)^{-1}A^Tb
but inverse of Matrix sucks
 
 
 
 
 

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