Re: x = A \ B... but I know some of x!

[Marc Normandin]

Kyle Altendorf wrote:
> My math/Octave question:
> How do I solve (or best fit) Ax=B when I know the form or some values of x?
> 
> A bit more background:
> I have a set of points on a rigid plate whose positions have been
> identified with a camera.  The plate was then rotated about some point
> (not the origin) and I am trying to identify that point.  I am already
> calculating it using 'x = A \ B' (via QR decomposition produces the
> same results) where A and B are m by 3 sets of coordinates (with the
> third column filled with 1's) and then extracting the estimated angle
> of rotation and center of rotation from x.  The problem is that when I
> repeat the measurements, I can't get the repeatability I need in
> identifying the center of rotation.  I hope that constraining some
> parts of x might improve the repeatability.  I do prescribe the
> rotation, so theta is more or less known.
> 
> I suppose I should also clarify that I can identify the points about
> 20 times more consistently than the center of rotation is being
> calculated and I am using a set of 32 points.  That's why I am under
> the impression I should be able to improve the repeatability.
> 
> Even if you simply give me a term for what I'm trying to do so that I
> can go research it myself, that would be appreciated.
> 
> Thanks,
> -kyle

Hi Kyle,

Sounds like you're trying to solve for the affine transformation matrix.
 The notation typically used would be x'=Ax, where A is said matrix and
x and x' are column vectors indicating the coordinates of a point in the
native and transformed spaces, respectively.  If I understand the way
you've constructed the problem, the coordinates in A and B are given in
a row vector sense, so you'd actually be estimating the transpose of the
affine transformation matrix.

If you know some parts of x you can probably just multiply those through
by A and subtract the result from B, then solve the modified problem for
the remaining components of x.  If parts of x are known "more or less"
and not with certainty, then you have yourself a more complex
optimization problem that will require iterative fitting.

Sorry if my answer's vague or rough around the edges.  If you have any
follow-up questions or just need me to clarify I'll be glad to help, but
please note that I'm defending my thesis on Thursday so I'll be out of
commission for most of the week.

Regards,
Marc

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Marc D. Normandin              http://web.ics.purdue.edu/~mdnorman
Graduate Research Assistant                     address@hidden
Indiana University School of Medicine           317-278-9841 (tel)
Department of Radiology, Division of Research   317-274-1067 (fax)
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