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Subject: The reference configuration and the BE matrix in TPS
Date: Tue, 21 Oct 2008 04:48:27 -0700 (PDT)
From: Dennis E. Slice <[EMAIL PROTECTED]>
To: morphmet <[email protected]>
I have received two inquiries from different parts of the Northern
hemisphere in the past few days that suggest the answers might be of
general utility.
Q 1) Why does one get different bending energies (BE) when one computes
the TPS of configuration from A to B than when one computes the TPS from
B to A.
Q 2) How does one compute the BE matrix for a single specimen. What is
it bending to?
The answer to both can be found in the observation that the BE matrix
(specifically, its eigenvectors) exhaustively describe via geometrically
orthogonal combinations of landmark coordinates how a reference can be
nonlinearly (*) deformed. The deformation of any reference specimen (**)
to any other target specimen of the same landmark number and
dimensionality can then be described as some weighted combination of
these eigenvectors.
A 1) The eigenvectors of a BE matrix describe how a specific reference
might be deformed. Deforming a specific reference A to target B will
require different deformations that when deforming B to A and, hence,
possibly different BEs.
A 2) The BE matrix is derived from the L matrix in the usual notation.
The elements of this matrix make no reference to any configuration other
than the reference. The end result is a set of descriptors for potential
deformation, not any specific deformation.
Best, ds
(*) For the computation of the uniform components of deformation see the
elegant results of: Rohlf, FJ, and FL Bookstein. 2003. Computing the
uniform component of shape variation. SYSTEMATIC BIOLOGY 52, no. 1
(February): 66-69.
(**) There is an unusual exception of the case of the reference having
two or more coincident points. The TPS cannot map two points from the
same location to two different locations. To do so would have grave
consequences for the very fabric of space and time.
--
Dennis E. Slice
Associate Professor
Dept. of Scientific Computing
Florida State University
Dirac Science Library
Tallahassee, FL 32306-4120
-
Guest Professor
Department of Anthropology
University of Vienna
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