Evolution Meetings poster

1
Extension of 2D Thin-Plate Splines to 3D for
Morphometric Studies Richard Strauss1, Momchil
Atanassov2, and Eric Dyreson3 1Department of
Biological Sciences and 2Museum, Texas Tech
University, Lubbock TX 3Department of
Mathematics, University of Montana Western,
Dillon MT
ABSTRACT
ONE-DIMENSIONAL SPLINES
Example Shape changes in Caiman crania
THREE-DIMENSIONAL SPLINES

• Thin-plate splines are commonly used in
  morphometric studies of biological forms to
  describe a shape difference as a deformation of
  one form into another in terms of a set of
  putatively homologous anatomical landmarks on the
  two forms. An infinite number of interpolated
  deformations between two forms are possible. An
  advantage of the thin-plate spline over other
  kinds of interpolation functions is that it can
  be easily decomposed into globally uniform
  (affine) and non-uniform (non-affine) components.
  (The non-uniform component can be further
  decomposed into principal or relative warps,
  although that is not our concern here.) A
  disadvantage of current implementations of
  thin-plate splines is that they utilize only
  two-dimensional images (sections or projections)
  of typically three-dimensional organisms. We (1)
  describe the extrapolation of thin-plate splines
  from 2D to 3D and (2) illustrate an application
  to form- and growth-differences in crocodile
  crania. We leave open the possibility of using
  4D splines (in which time is the 4th dimension)
  to model differences in growth patterns among
  forms.

• Bookstein (1989) noted that the 2D thin-plate
  spline is the natural generalization of the
  familiar one-dimensional cubic spline that is
  often used to fit smoothed curves to bivariate
  data. In the 1D case the cubic spline is a
  thin-wire spline that minimizes the bending
  energy of a thin wire passing through a set of
  landmark points. The 1D spline uses a weighting
  function to establish a shift map, indicating the
  direction and magnitude by which an arbitrary
  point must be shifted in the deformation (Fig.
  1). The shifts in landmark positions are given
  by the observed landmark configurations of the
  two forms. The shift in any arbitrary point is
  given by a weighted sum of landmark shifts,
  wherein each landmark shift is weighted according
  to its distance from the point. For the 1D case,
  the weights U are
• for distance d from point to landmark the larger
  the absolute weight, the less influence a
  specific landmark shift has on a point.
• The total deformation portrayed in Fig. 1a can be
  decomposed into a linear affine (uniform)
  component and a non-linear non-affine
  (non-uniform) component. The total deformation
  is the sum of these two.

• The 3D thin-hyperplate spline corresponds to
  the distortion of a malleable cube. The weights
  for interpolated shifts in arbitrary points in
  three dimensions are
  (Bookstein 1989). The stacked 2D planar grids
  for the reference form are a sample of slices
  through what is actually a solid
  three-dimensional grid, but are useful for
  visualization. The affine component keeps
  parallel lines parallel in three dimensions. The
  corresponding layers in the final deformed cube
  are, of course, not necessarily planar even for
  subsets of landmarks that lie in a plane in both
  the reference and target forms (e.g., the bottom
  slice of the cube).

• Selected anatomical landmarks on crania of Caiman
  crocodilus fuscus from Colombia were digitized
  using a Polhemus? Fastrack II 3D digitizer.
  Three representative individuals were selected
  for this example a juvenile, adult male, and
  adult female. 3D deformation using the
  thin-hyperplate interpolation function was used
  to illustrate shape differences between (1) the
  juvenile and male (simulating a growth series),
  and (2) the female and male (representing sexual
  dimorphism). Procrustes rotation and scaling in
  three dimensions was used to standardize the
  forms prior to deformation. These figures show
  the digitized crania of the male in two different
  projections.

INTRODUCTION

• In morphometric studies of biological forms, an
  important model of shape change is the
  deformation of one form into another. The
  concept of form-change as a deformation dates
  back to DArcy Thompsons (1917) well-known
  transformation grids, which mapped a set of
  morphological landmarks on one form onto the
  corresponding (presumably homologous) landmarks
  on another form and carried along
  (interpolated) the deformed regions of the form
  among the landmarks. Thompsons transformation
  grids were subjective artistic renderings rather
  than computational structures, and numerous
  attempts have made since Thompsons work to
  objectively derive deformation grids (Bookstein
  1978). Regardless of how it is computed, a
  deformation is a smooth interpolation function
  that predicts the positions of points on the form
  that lie among the known landmark positions.
• More recently, Bookstein (1989) has shown how the
  thin-plate spline, a conventional mathematical
  tool for interpolating surfaces among scattered
  points in a plane (Meinguet 1978), can be used to
  model shape changes among biological forms. An
  important advantage of the thin-plate spline over
  other kinds of interpolation functions is that it
  can easily be decomposed into global and local
  components of the deformation. The global affine
  (uniform) component is a single linear function
  that best describes the overall deformation of
  one form into another by best matching the
  corresponding landmarks in the two forms. The
  local non-affine (non-uniform) component
  describes the localized, nonlinear residual
  adjustments that must be made to force the
  landmark positions in the two forms to match
  exactly. The non-affine component can be
  expressed as a sum of principal warps of
  successively larger physical scales, and a sample
  of shape changes can be decomposed into a series
  of independent relative warps. These are used
  (often incorrectly) to interpret the nature of
  shape differences among forms, but we are
  concerned here only with the affine and
  non-affine components of the deformation.
• A second, conceptual advantage of the thin-plate
  spline interpolation is that the affine and
  non-affine components have natural, geometric
  interpretations in terms of the rotation and
  bending in three dimensions of a uniformly thin
  metal plate. The total deformation can be viewed
  as the projection into two dimensions of a plate
  that has been rotated and deformed in three
  dimensions. The amount of local deformation is
  then measured as the bending energy required to
  deform such a plate, and the thin-plate spline is
  that deformation that minimizes the total bending
  energy.
• A disadvantage of current implementations of
  thin-plate splines is that they utilize only
  two-dimensional images (sections or projections)
  of typically three-dimensional organisms. Here
  we (1) describe thin-plate splines in 1D, 2D and
  3D and (2) illustrate a 3D application to form-
  and growth-differences in caiman skulls.

• The following figures illustrate the total
  deformation from juvenile to male. Because the
  forms were standardized, the resulting
  deformation is a function of allometric change
  plus a small amount of individual variation. The
  primary differences are a flattening of the roof
  of the cranium and elongation of the rostrum.

Figure 1. Deformation of a reference form to a
target form based on the 1D thin-plate spline
interpolation. (A) Interpolated correspondence
between reference and target forms, linear
organisms each having four homologous landmarks.
An arbitrary linear grid has been superimposed on
the reference form and interpolated onto the
target form. Shown are the affine and non-affine
components of the landmark shifts. (B)
Interpolated shifts for points between landmarks,
decomposed into affine and non-affine components.

• The following figures show the total deformation
  from female to male. The individuals were
  approximately the same size, so the slight
  difference observed is due to sexual dimorphism.
  The pattern is subtle but similar to that of the
  ontogenetic comparison a somewhat more flattened
  cranial roof and elevated supraoccipital region
  in the male in relation to the female.

TWO-DIMENSIONAL SPLINES
Figure 3. Deformation of a reference form (a
cube) to a target form based on the 3D
thin-hyperplate spline interpolation.

• Figure 2 illustrates a typical 2D thin-plate
  spline. Interpolated shifts in arbitrary points
  are now in two dimensions rather than three but,
  as in the 1D case, the shifts are weighted sums
  of landmark shifts. For 2D interpolation the
  weights are
  for distance d from point to landmark (Meinguet
  1979, Bookstein 1989). The affine component in
  this case is a tilted plane viewed in
  perspective. The non-affine component
  characterizes the regional deformations (warping
  of the thin plate) that are added to the affine
  component to give the total deformation.

DIFFERENCES BETWEEN 2D AND 3D INTERPOLATIONS

• A question immediately arises Can 3D
  differences in form can be correctly inferred
  from examination of separate 2D projections or
  sections? Our examples suggest that the
  deformation of a 2D slice using a 2D spline
  provides a different picture (and, of course,
  numerical solution) than the corresponding 2D
  slice of a 3D spline, and the difference can be
  substantial. This is because, in the 3D spline,
  the landmarks outside of a particular 2D slice
  contribute to the deformation within that slice.
  These results suggest that, for three-dimensional
  organisms, 3D splines should always be preferred
  over 2D splines of projections, despite the
  increased complexity of interpretation of 3D
  solutions.

SUGGESTIONS FOR FURTHER WORK

• Although, in parallel to the 2D case, the
  non-affine component of the spline can be
  decomposed into eigenfunctions in the 3D case,
  there are a number of computational details that
  need to be resolved. Bookstein (1989) listed
  several aspects of the 3D extension that will
  require considerable imagination. In
  particular, since 2D splines are visualized in
  3D, it is not clear how best to draw a
  thin-hyperplate spline that is a 3D projection of
  a 4D geometric object.
• Despite these (and other) difficulties,
  thin-hyperplate splines can in principle be
  extrapolated to higher dimensions if they can be
  adequately manipulated and visualized. We
  suggest that 4D splines, in which time or a proxy
  for time comprised the 4th dimension, might allow
  the direct comparison of differences in growth
  patterns among taxa.

REFERENCES Bookstein, F.L. 1978. The
Measurement of Biological Shape and Shape Change.
Lecture Notes in Biomathematics, vol. 24. New
York Springer-Verlag. Bookstein, F.L. 1984. A
statistical method for biological shape
comparisons. Journal of Theoretical Biology
107475-520. Bookstein, F.L. 1989. Principal
warps thin-plate splines and the decomposition
of deformations. IEEE Transactions on Pattern
Analysis and Machine Intelligence
11567-585. Meinguet, J. 1979. An intrinsic
approach to multivariate spline interpolation at
arbitrary points. In B. Sahney (ed.),
Polynomial and Spline Approximation, pp. 163-190.
Dordrecht Reidel. Thompson, D.W. 1917. On
Growth and Form. Cambridge Cambridge University
Press.
Figure 2. Deformation of a reference form to a
target form based on the 2D thin-plate spline
interpolation. Also shown are the affine and
non-affine components of the landmark shifts.