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Topic: Non-linear optimization
Replies: 32   Last Post: Mar 8, 2013 2:22 AM

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Matt J

Posts: 4,997
Registered: 11/28/09
Re: Non-linear optimization
Posted: Mar 6, 2013 9:59 AM
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"Toan Cao" <> wrote in message <kh5at7$iv2$>...
> I will explain more detail about my cost function and hope to receive your suggestion.
> Given two 3D point clouds (source point cloud (SPC) and target point cloud (TPC)). I would like to move each point of SPC to be coincide with each corresponding point of TPC.
> Each movement of each point of SPC is described by a Rotation matrix Ri and a translation vector Ti.
> Rotation matrix Ri is constrained:
> Rot(Ri)= (C1.C2)^2 + (C1.C3)^2 + (C2.C3)^2 +(C1.C1 -1)^2 +(C2.C2 -1)^2 + (C3.C3 -1)^2, where C1, C2, C3 are 3x1 column vectors of Ri.
> Given m points in SPC, the first term of cost function is: Sum(Rot(Ri)) where i =1:m
> If we call a point in SPC is Vi, its corresponding point in TPC is Ui, its transformed point is V'i. So, the second term of cost function is: Sum((V'i - Ui)^2), i=1:m
> Finally, my cost function is F = Sum(Rot(Ri)) +Sum((V'i - Ui)^2), i=1:m
> Now, i want to find all Ci of Rotation matrices Ri as well all elements of translation vectors Ti. What should i do to obtain local minimum value of this function ?


The problem has a closed form solution, so iterative minimization is unnecessary. Here is one implementation

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