Tom,
Triangles only have shape in two dimensions. In fact, 2D triangles
represent objects with the fewest number of landmarks that have
shapes: objects with 2 landmarks in 2-dimensions only have a
length. For 3D data, a tetrahedron contains the fewest number of
landmarks to have a shape. A triangle represented by 3D landmarks
lies in a sub-space of the full 3p space, and thus reduces to 2D
landmarks for the purposes of shape analysis.
Because of this I am a bit confused by your comment about not wanting
to reduce the 3D data down to 2 dimensions "That reduction in
dimensionality loses important information that is contained within
the 3D location of the landmarks". Strictly speaking, this statement
is not correct, because triangles only have shape in 2D. I can hazard
a guess that your statement reflects the fact that when you view the
raw data, the position of the triangle in 3D relative to your vantage
point is important to you. However, what this means is that you have
mentally incorporated an additional landmarks (or a reference axis)
to provide you that mental perspective.
If that is indeed what you are doing, and what you wish to capture
with the data, then you need to explicitly digitize this point (or
points) during the preliminary data collection step. For me this is
best done prior to Procrustes standardizations.
Dean
Dean C. Adams
Associate Professor
Department of Ecology, Evolution, and Organismal Biology, and
Department of Statistics
253 Bessey Hall
Iowa State University
Ames, IA 50011
tel: (515) 294-3834
fax: (515) 294-1337
web: http://www.public.iastate.edu/~dcadams
At 03:24 PM 1/18/2007, you wrote:
>Hello All,
>
>I've been trying to work on a 3D data set that consists of only three
>landmarks - 3D triangles. The data are collected in a way that they are
>already aligned by translation and rotation. At this point, the
>rotational and translational differences that remain among the subjects
>are aspects of the essential differences of "shape" that I want to be
>able to analyze. Only scaling is left to do to place the subjects into
>shape space.
>
>Here's the problem. With 3D triangles the problem would get reduced to 2
>(3 x 3 - 7) dimensions. That reduction in dimensionality loses important
>information that is contained within the 3D location of the landmarks,
>and so I want to avoid the reduction to two dimensions. The question is,
>how?
>
>I could add the invariant 0,0,0 landmark to each subject. This would
>create a 3D tetrahedron, and reduce the analysis to 5 dimensions. But I
>am not sure that this would be statistically correct. After all, does
>this point really add degrees of freedom to the data set when its value
>is invariant across all subjects?
>
>Another question would be when to add the 0,0,0 point? If it is added
>before scaling is applied, then this point will have an impact in
>determining the centroid size of each subject. If it is added after the
>scaling that wouldn't be a problem, but then I still wonder if this is
>methodologically correct.
>
>So, what are the opinions from the morphometric experts on how to handle
>this data?
>
>Thanks for your opinions,
>
>Thomas M. Greiner, Ph.D.
>Assistant Professor of Anatomy
>University of Wisconsin - La Crosse
>Department of Health Professions
>4054 Health Science Center
>1725 State Street
>La Crosse, WI 54601
>
>Phone: (608) 785-8476
>Email: [EMAIL PROTECTED]
>
>
>
>
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