[R] solving for a 2D transformation matrix

Russell Senior seniorr at aracnet.com
Sat Jul 3 06:59:14 CEST 2004

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 ]

    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!

Russell Senior         ``I have nine fingers; you have ten.''
seniorr at aracnet.com

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