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Re: [Algorithms] solving for multiple matrices
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From: Andras B. <and...@gm...> - 2009-09-18 22:11:45
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Well, I'm not sure how a different factorization will help, but I'll look into that. At the same time, I'm also starting to consider an iterative approach. What do you guys think? It seems to me that this is basically an inverse kinematics problem, right? Any pointers/advice, in case I'll decide to take that route? Thanks, Andras On Fri, 18 Sep 2009 10:44:12 -0600, Bill Baxter <wb...@gm...> wrote: > That was my first thought too, but you don't need to a *Cholesky* > decomp on C and D. Another kind of factorization will do, like a > diagonalization -- > http://en.wikipedia.org/wiki/Square_root_of_a_matrix > > --bb > > On Fri, Sep 18, 2009 at 7:51 AM, Andras Balogh <and...@gm...> > wrote: >> Hmm, I just looked up Cholesky decomposition, and it says you can only >> do >> that if the matrix is symmetric and positive definite. Since the >> original >> definition of C = A1^-1 * A2, where A1 and A2 both contain arbitrary >> translation and rotation, I don't think C will be symmetric. Or am I >> missing something? >> >> Andras >> >> >> On Fri, 18 Sep 2009 01:58:41 -0600, Gino van den Bergen >> <gin...@gm...> wrote: >> >>> If Y can be considered a rotation then Y is orthogonal and thus Y^-1 = >>> Y^T, in which case this equation can be solved through Cholesky >>> decomposition: >>> >>> For C = Y^T * D * Y, let's decompose >>> >>> C = U^T * U, and >>> D = V^T * V >>> >>> then >>> >>> U^T U = Y^T * V^T * V * Y >>> = (V * Y) ^T * (V * Y) >>> >>> this gives U = V * Y >>> >>> and thus Y = V^-1 * U >>> >>> Basically you are taking the square root of a matrix. >>> >>> Hope this helps, >>> >>> Gino >>> >>> >>> >>> Andras Balogh wrote: >>>> Both X and Y matrices represent a simple translation and rotation. For >>>> the >>>> case of the Y matrix, the translation part will likely to be very >>>> small, >>>> so I could probably pretend it's only rotation. >>>> >>>> What I would really like though, is to find a solution, where I could >>>> use >>>> more than two equations, eg: >>>> A1 = X * B1 * Y >>>> A2 = X * B2 * Y >>>> A3 = X * B3 * Y >>>> ... >>>> An = X * Bn * Y >>>> >>>> And then compute a least squares solution from this over-constrained >>>> system. >>>> BTW, when I said that I'm looking for an analytical solution, I just >>>> meant >>>> something that is not based on an iterative approach. As long as I can >>>> get >>>> to a part where I have to solve a large system of over-constrained >>>> linear >>>> equations, I'm home. Unfortunately, I don't know how to make this >>>> linear.. >>>> >>>> >>>> Andras >>>> >>>> >>>> >>>> >>>> On Thu, 17 Sep 2009 16:32:25 -0600, Jon Watte <jw...@gm...> >>>> wrote: >>>> >>>> >>>>> Andras Balogh wrote: >>>>> >>>>>> Then it becomes: >>>>>> C = Y^-1 * D * Y >>>>>> >>>>>> Now, how do I solve this for Y? This form lookes strangely familiar, >>>>>> but I >>>>>> cannot figure out what to do from here (wish I knew how to Google >>>>>> this >>>>>> ;). >>>>>> Hopefully there's an analytic solution to this. Any ideas? >>>>>> >>>>>> >>>>>> >>>>> That's the formula for applying a rotation in the reference frame of >>>>> another rotation. >>>>> >>>>> Do you know anything more about these matrices than that they are >>>>> matrices? Are they supposed to contain no scale? No translation? If >>>>> you >>>>> can formulate them as quaternions, writing out the analytical answer >>>>> is >>>>> a lot simpler :-) >>>>> >>>>> Sincerely, >>>>> >>>>> jw >>>>> >>>>> >>>>> >>>>> >>>> >>>> >>>> >>>> ------------------------------------------------------------------------------ >>>> Come build with us! The BlackBerry® Developer Conference in SF, CA >>>> is the only developer event you need to attend this year. Jumpstart >>>> your >>>> developing skills, take BlackBerry mobile applications to market and >>>> stay >>>> ahead of the curve. Join us from November 9-12, 2009. Register >>>> now! >>>> http://p.sf.net/sfu/devconf >>>> _______________________________________________ >>>> GDAlgorithms-list mailing list >>>> GDA...@li... >>>> https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list >>>> Archives: >>>> http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list >>>> >>> >>> >>> ------------------------------------------------------------------------------ >>> Come build with us! The BlackBerry® Developer Conference in SF, CA >>> is the only developer event you need to attend this year. Jumpstart >>> your >>> developing skills, take BlackBerry mobile applications to market and >>> stay >>> ahead of the curve. Join us from November 9-12, 2009. Register >>> now! >>> http://p.sf.net/sfu/devconf >>> _______________________________________________ >>> GDAlgorithms-list mailing list >>> GDA...@li... >>> https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list >>> Archives: >>> http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list >> >> >> >> ------------------------------------------------------------------------------ >> Come build with us! The BlackBerry® Developer Conference in SF, CA >> is the only developer event you need to attend this year. Jumpstart your >> developing skills, take BlackBerry mobile applications to market and >> stay >> ahead of the curve. Join us from November 9-12, 2009. Register >> now! >> http://p.sf.net/sfu/devconf >> _______________________________________________ >> GDAlgorithms-list mailing list >> GDA...@li... >> https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list >> Archives: >> http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list >> > > ------------------------------------------------------------------------------ > Come build with us! The BlackBerry® Developer Conference in SF, CA > is the only developer event you need to attend this year. Jumpstart your > developing skills, take BlackBerry mobile applications to market and stay > ahead of the curve. Join us from November 9-12, 2009. Register > now! > http://p.sf.net/sfu/devconf > _______________________________________________ > GDAlgorithms-list mailing list > GDA...@li... > https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list > Archives: > http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list |
