Morpho (version 2.1)

procSym: Procrustes registration

Description

procSym performs Procrustes superimposition including sliding of semi-landmarks on curves/outlines in 2D and 3D.

Usage

procSym(dataarray, scale = TRUE, reflect = TRUE, CSinit = TRUE,
  orp = TRUE, tol = 1e-05, pairedLM = NULL, sizeshape = FALSE,
  use.lm = NULL, center.part = FALSE, distfun = c("angle", "riemann"),
  SMvector = NULL, outlines = NULL, deselect = FALSE, recursive = TRUE,
  iterations = 0, initproc = FALSE)

Arguments

dataarray

Input k x m x n real array, where k is the number of points, m is the number of dimensions, and n is the sample size.

scale

logical: indicating if scaling is requested

reflect

logical: allow reflections.

CSinit

logical: if TRUE, all configurations are initially scaled to Unit Centroid Size.

orp

logical: if TRUE, an orthogonal projection at the meanshape into tangent space is performed.

tol

numeric: Threshold for convergence in the sliding process

pairedLM

A X x 2 matrix containing the indices (rownumbers) of the paired LM. E.g. the left column contains the lefthand landmarks, while the right side contains the corresponding right hand landmarks.

sizeshape

Logical: if TRUE, a log transformed variable of Centroid Size will be added to the shapedata as first variable before performing the PCA.

use.lm

vector of integers to define a subset of landmarks to be used for Procrustes registration.

center.part

Logical: if TRUE, the data superimposed by the subset defined by use.lm will be centered according to the centroid of the complete configuration. Otherwise orp will be set to FALSE to avoid erroneous projection into tangent space.

distfun

character: "riemann" requests a Riemannian distance for calculating distances to mean, while "angle" uses an approximation by calculating the angle between rotated shapes on the unit sphere.

SMvector

A vector containing the landmarks on the curve(s) that are allowed to slide

outlines

A vector (or if threre are several curves) a list of vectors (containing the rowindices) of the (Semi-)landmarks forming the curve(s) in the successive position on the curve - including the beginning and end points, that are not allowed to slide.

deselect

Logical: if TRUE, the SMvector is interpreted as those landmarks, that are not allowed to slide.

recursive

Logical: if TRUE, during the iterations of the sliding process, the outcome of the previous iteration will be used. Otherwise the original configuration will be used in all iterations.

iterations

integer: select manually how many iterations will be performed during the sliding process (usefull, when there is very slow convergence). 0 means iteration until convergence.

initproc

Logical: indicating if the first Relaxation step is performed against the mean of an initial Procrustes superimposition. Symmetric configurations will be relaxed against a perfectly symmetrical mean.

Value

• size
  a vector containing the Centroid Size of the configurations
• rotated
  k x m x n array of the rotated configurations
• Sym
  k x m x n array of the Symmetrical component - only available for the "Symmetry"-Option (when pairedLM is defined)
• Asym
  k x m x n array of the Asymmetrical component - only available for the "Symmetry"-Option (when pairedLM is defined)
• asymmean
  k x m matrix of mean asymmetric deviation from symmetric mean
• mshape
  sample meanshape
• symmean
  meanshape of symmetrized configurations
• tan
  if orp=TRUE: Residuals in tangentspace else, Procrustes residuals - only available without the "Symmetrie"-Option
• PCs
  Principal Components - if sizeshape=TRUE, the first variable of the PCs is size information (as log transformed Centroid Size)
• PCsym
  Principal Components of the Symmetrical Component
• PCasym
  Principal Components of the Asymmetrical Component
• PCscores
  PC scores
• PCscore_sym
  PC scores of the Symmetrical Component
• PCscore_asym
  PC scores of the Asymmetrical Component
• eigenvalues
  eigenvalues of the Covariance matrix
• eigensym
  eigenvalues of the "Symmetrical" Covariance matrix
• eigenasym
  eigenvalues of the "Asymmetrical" Covariance matrix
• Variance
  Table of the explained Variance by the PCs
• SymVar
  Table of the explained "Symmetrical" Variance by the PCs
• AsymVar
  Table of the explained "Asymmetrical" Variance by the PCs
• orpdata
  k x m x n array of the rotated configurations projected into tangent space
• rho
  vector of Riemannian distance from the mean
• dataslide
  array containing slidden Landmarks in the original space - not yet processed by a Procrustes analysis. Only available if a sliding process was requested

Details

This function performs Procrustes registration, allowing a variety of options, including scaling, orthogonal projection into tangentspace and relaxation of semi-landmarks on curves (without reprojection onto the surface/actual outline). It also allows the superimpositioning to be performed using only a subset of the available landmark. For taking into account object symmetry, pairedLM needs to be set. This generates an object of class "symproc". Otherwise an object of class "nosymproc".

References

Dryden IL, and Mardia KV. 1998. Statistical shape analysis. Chichester.

Klingenberg CP, Barluenga M, and Meyer A. 2002. Shape analysis of symmetric structures: quantifying variation among individuals and asymmetry. Evolution 56(10):1909-1920.

Gunz, P., P. Mitteroecker, and F. L. Bookstein. 2005. Semilandmarks in Three Dimensions, in Modern Morphometrics in Physical Anthropology. Edited by D. E. Slice, pp. 73-98. New York: Kluwer Academic/Plenum Publishers.

See Also

slider3d

Examples

require(rgl)
data(boneData)

### do an analysis of symmetric landmarks
## visualize landmarks on surface
spheres3d(boneLM[,,1])
wire3d(skull_0144_ch_fe.mesh,col=3)
## get landmark numbers
text3d(boneLM[,,1],text=paste(1:10),adj = 1, cex=3)
## determine paired Landmarks left side:
left <- c(4,6,8)
## determine corresponding Landmarks on the right side:
# important: keep same order
right <- c(3,5,7)
pairedLM <- cbind(left,right)
symproc <- procSym(boneLM, pairedLM=pairedLM)
## visualize first 3 PCs of symmetric shape
pcaplot3d(symproc, sym=TRUE)
## visualize first 3 PCs of asymmetric shape
pcaplot3d(symproc, sym=FALSE)

## visualze distribution of symmetric PCscores population
pop <- name2factor(boneLM, which=3)
require(car)
spm(~symproc$PCscore_sym[,1:5], groups=pop)
## visualze distribution of asymmetric PCscores population
spm(~symproc$PCscore_asym[,1:5], groups=pop)