Re: [Algorithms] solving for multiple matrices
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Re: [Algorithms] solving for multiple matrices
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From: Nguyen B. <ng...@gm...> - 2009-09-18 15:36:45
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You gotta be more precise about your definition of "transformation matrix" here. Normally, we refer to them as "rotation matrix" which is a member of SO(3) group. This type of group has been extensively studied and there are many mathematical tools for you to use. If you refer to them as homogeneous transformation which also contains translation then they belong to the set SO(3)xR^3. I don't think you can do a lot here. Note that when talking about symmetry, people usually mean "rotation matrix" as homogeneous matrix is not symmetry by definition. -------------------------------------------------- Binh Nguyen Computer Science Department Rensselaer Polytechnic Institute Troy, NY, 12180 -------------------------------------------------- On Fri, Sep 18, 2009 at 10:51 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 > |
