Dear Biomch-l,

I have the following problem:

Using image processing algorithms (sub-voxel accuracy) on medical
imaging datasets, four marker centroids are determined, combined
and normalized to define a local reference frame Rloc within a
global reference frame Rglob.

The next step is to find the transformation matrix between both Rloc
and Rglob. To do this I am using, the unit vectors of both Rloc and
Rglob into a SVD algorithm (Challis, J. of Biomechanics, 28, 733-
737, 1995). Root Mean Square Error (RMS) the position was found
as well (Soderkvist, J. of Biomechanics, 26, 1473-1477). Mean
Absolute Relative Difference (MARD) as well (see Challis).

It seems to work because further use of the results are satisfactory.

My problem is the following:
the RMS errors I found are very small (between 0.000016 mm and
0.00004 mm) and are not accepted by one of the reviewers of a
paper we are trying to publish.

I agree very much that such error, I quote the reviewer, "is
unrealistically small" ... but we checked, re-checked our code, let
our data processed by an external dept. and cannot locate any
problems in our protocol: the above error seems really real, except
of course if we are missing something ...

I would greatly appreciate if somebody could process the following
data and compared it to our following results. Could you, please,
take a look at it ? Many thanks !!

THE DATA

1 0 134.820068 110.476936 -383.543427
1 1 134.725296 111.466965 -383.647430
1 2 135.702469 110.512123 -384.012543
1 3 134.359253 110.340698 -384.420410
2 0 0.000000 0.000000 0.000000
2 1 1.000000 0.000000 0.000000
2 2 0.000000 1.000000 0.000000
2 3 0.000000 0.000000 1.000000

First digit indicates the frame (1: Rloc, 2: Rglob).
Second digit indicates the landmark.
Then, the X Y Z coordinates of each of the 4 landmarks.

THE RESULTS

Transformation matrix
-0.094769 0.990052 -0.104002 -136.490646
0.882429 0.035186 -0.469128 -302.787415
-0.460802 -0.136233 -0.876985 -259.185822
0.000000 0.000000 0.000000 1.000000

RMS error
0.000016 (small, isn't !?, is it normal or not ?).

MARD error
inf (to small to be handled in single floating point precision ...)

Rotation amplitude in radians (in degrees).
0.252528 (14.468779)

Translation (mm)
-251.301172

To me, such a small error can be explained by the fact we are using
the unit vectors of both reference systems into the SVD routines to
process the transformation matrix, meaning the cluster is very rigid
and therefore the matrix well defined.

Please, can you comment this ?

Again, thank you very much in advance !!

Serge

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Serge VAN SINT JAN, Ph.D. | Marie Curie Fellow
VAKHUM project Co-ordinator | phone: + 32 2 555 6325
Dept. of Anatomy (CP 619) | + 32 2 555 6376
University of Brussels | fax: + 32 2 555 6378
808 Lennik Street | email: sintjans@ulb.ac.be
1070 Brussels - Belgium | WWW: http://homepages.ulb.ac.be/~sintjans
VAKHUM project: http://www.ulb.ac.be/project/vakhum
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