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

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Toan Cao

Posts: 52
Registered: 10/15/10
Re: Non-linear optimization
Posted: Mar 7, 2013 3:35 PM
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"Matt J" wrote in message <kharnu$599$1@newscl01ah.mathworks.com>...
> "Toan Cao" <toancv3010@gmail.com> wrote in message <khaats$7h6$1@newscl01ah.mathworks.com>...
> > "Matt J" wrote in message <kh8dvs$imu$1@newscl01ah.mathworks.com>...
> > > "Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <kh8bsc$bov$1@newscl01ah.mathworks.com>...
> > > > "Matt J" wrote in message <kh89he$4al$1@newscl01ah.mathworks.com>...
> > > >

> > > > > f(x)=(sqrt(F(x)-f_low))^2
> > > >
> > > > What you write is
> > > >
> > > > f(x) = abs(F(x)-f_low) = F(x)-f_low
> > > >
> > > > Minimizing f(x) is the same as minimizing F(x). How far do we get?

> > > ===============
> > >
> > > That should really have been
> > >
> > > f(x)=sqrt(F(x)-f_low)
> > >
> > > min (f(x))^2
> > >
> > > But yes, the above is equivalent to minimizing F(x). That's what we want. Now, however, you can feed f(x) to LSQNONLIN and run its Levenberg-Marquardt routine.

> >
> > Hi Matt J,
> >
> > I will summarize my function here again and wish to receive feedback!
> >

> > >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
> > --------------------------
> > I assume that (for simplicity) the third term for my cost function is Sum(norm(Ri-Rj)^2), i,j=1:m, i ~=j
> > Now, my cost functiion : F = Sum(Rot(Ri)) +Sum((V'i - Ui)^2) +Sum(norm(Ri-Rj)^2), i=1:m
> >
> > I read document of optimization toolbox of matlab, It suggests that i can use LSQNONLIN for this function with Levenberg-Marquardt algorihm. With this routine, it requires we provide a vector-value function f(x)=[f{1}(x),f{2}(x),...,f{n}(x)]' for LSQNONLIN.

>
>

> > Now, to use this routine, i will do:
> > 1) f{1}(x)= (C1{i}.C2{i}), f{2}(x)= (C1{i}.C3{i}),..., (a set of functions for first term).
> > f{k}(x) =(V'{i} - U{j}), f{k+1}(x) =(V'{i+1} - U{j+1}),...., (a set of functions for second term).
> > f{h}(x)= norm(R{i}-R{j}) {i=h},... ,f{m}(x)=norm(R{i}-R{j}) {i=m}. (a set of functions for third term).
> > => So, f(x) = [f{1}(x), f{2}(x),..., f{k}(x), f{k+1}(x),..., f{h}(x),..., f{m}(x)]'
> >
> > OR, i just give:
> > 2) f{1}(x) = sqrt(Sum(Rot(Ri))), f{2}(x) = sqrt(Sum((V'i - Ui)^2)), f{3}(x) =sqrt(Sum(norm(Ri-Rj)^2)).
> > => So, f(x) = [f{1}(x), f{2}(x), f{3}(x)]'
> > With your experience, Which option (1 or 2) should i follow ?

> ============
>
> Toan,
>
> I think neither option is a good one because, as I have already told you, it looks like the Ri parameters are unnecessary. Also, I can give you a solution right now that you can verify, by direct subsititution, will globally minimize the function you have written with F(Ri,Ti)=0. The solution is
>
> Ri=eye(3)
> Ti=Ui-Vi
>
> for all i=1..m


Hi Matt J,

Thanks for your consideration!
Yes, i know what you mean in order to implement easily for my problem. However, in limited space of the text, i can not explain why i need rotation matrices to describe movements of points. It is reason i just explain main idea of the cost function.
If you would like to know more, i can show the paper i read and want to implement, its title: "Embedded deformation for shape manipulation". Its cost function is expressed in section 4.
So, when applying LSQNONLIN, i am confused which option (1 or 2) i use for f(x)=[f{1}(x),f{2}(x),...,f{n}(x)]' ?
I am willing to receive any your suggestion for my case.

Best regards,
Toan



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