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Given the equation of perspective projection of a set of 3D points in an image:

lambda_ij * x_ij = P_i * X_j;

how can one estimate the optimal values of lambda matrix using linear least squares given that:

 x_ij is 3 by n matrix

 P_i is 3 by 4 matrix

and 

 X_j is 4 by n matrix.

optional: for each element lambda_i in lambda:

 -1 <= lambda_i <= 1

EDIT:

What I mean by optimal is any values of lambda_ij beteween -1 and 1. What I have in entry is the image points x_ij of size 3 x n, the 3D points X_j of size 4 x n and the projection matrices P_i of size 3 x 4 x m and. The objective is to compute lambda_ij of size 1 x n x m that best minmize the above least-squares

Gamba Osaca
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    Define what you mean by "optimal". Because this is an optimization framework, there is surely some criteria you're using to determine whether or not the parameters estimated are optimal? What's the cost function you're using? Do you have sample data? Can you show us any code you have written so far? – rayryeng Dec 22 '14 at 02:52
  • I have tried `P_i * X_j \ x_ij` by I got a matrix of size `n x n` that I didn't understand how to interpret – Gamba Osaca Dec 22 '14 at 17:56
  • Ah OK. You are searching for weighted linear least squares. Should have figured that out from your post. I'll provide an answer soon. I'm not in a place where I can write a detailed answer for you. – rayryeng Dec 22 '14 at 20:19
  • I forgot to mention that `lambda_ij .* x_ij` is element by element multiplication. (if I put `lambda_ij` in the form of a `3 x n` matrix, then all elements at the same column are equal) – Gamba Osaca Dec 22 '14 at 21:31

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