From: Prof Brian Ripley <ripley_at_stats.ox.ac.uk>

Date: Sat 03 Jul 2004 - 16:07:40 EST

Date: Sat 03 Jul 2004 - 16:07:40 EST

The best least-squares fit maps centroid to centroid. So find the mean of
the observed and target, and shift the observed to match the target. That
just leaves a rotation, and I would directly maximize the sum of squared
errors over that using optimize().

On 2 Jul 2004, Russell Senior wrote:

*>
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> We have recently digitized a set of points from some scanned

*> engineering drawings (in the form of PDFs). The digitization resulted
**> in x,y page coordinates for each point. The scans were not aligned
**> perfectly so there is a small rotation, and furthermore each
**> projection (e.g. the yz-plane) on the drawing has a different offset
**> from the page origin to the projection origin. From the dimensions
**> indicated on the drawing, I know the intended "world" coordinates of a
**> subset of the points. I want to use this subset of points to compute
**> a best-fit transformation matrix so that the remaining points can be
**> converted to world coordinates.
**>
**> The transformation matrix is (I think) of the form:
**>
**> [ x' ] [ a11 a12 a13 ] [ x ]
**> | y' | = | a21 a22 a23 | | y |
**> [ w' ] [ a31 a32 a33 ] [ w ]
**>
**> where:
**>
**> x,y = page coordinates
**> x',y' = world coordinates
**>
**> a13 = translation of x
**> a23 = translation of y
**>
**> a11 = scale * cos(theta)
**> a12 = sin(theta)
**> a21 = -sin(theta)
**> a22 = scale * cos(theta)
**>
**> a31 = 0
**> a31 = 0
**> a33 = 1
**> w' = 1
**> w = 1
**>
**> Can anyone give me a pointer on how to go about solving for the
**> transformation matrix given a set of points, where x,y and x',y' are
**> available? I sense the presence a solution lingering in the murky
**> mists, (some kind of least squares?) but I am not sure what it is or
**> how to go about it exactly.
**>
**> Thanks for your help!
**>
**>
*

-- Brian D. Ripley, ripley@stats.ox.ac.uk Professor of Applied Statistics, http://www.stats.ox.ac.uk/~ripley/ University of Oxford, Tel: +44 1865 272861 (self) 1 South Parks Road, +44 1865 272866 (PA) Oxford OX1 3TG, UK Fax: +44 1865 272595 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.htmlReceived on Sat Jul 03 16:11:10 2004

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