CreateL: Create Matrices necessary for Thin-Plate Spline

In Morpho: Calculations and Visualisations Related to Geometric Morphometrics

Create Matrices necessary for Thin-Plate Spline

Description

Create (Bending Engergy) Matrices necessary for Thin-Plate Spline, and sliding of Semilandmarks

Usage

CreateL(
  matrix,
  lambda = 1e-08,
  output = c("K", "L", "Linv", "Lsubk", "Lsubk3"),
  threads = 1
)

Arguments

matrix	k x 3 or k x 2 matrix containing landmark coordinates.
lambda	numeric: regularization factor
output	character vector: select which matrices to create. Can be a vector containing any combination of the strings: "K", "L","Linv","Lsubk", "Lsubk3".
threads	threads to be used for parallel execution calculating K. sliding of semilandmarks.

Value

depending on the choices in output:

L	Matrix K as specified in Bookstein (1989)
L	Matrix L as specified in Bookstein (1989)
Linv	Inverse of matrix L as specified in Bookstein (1989)
Lsubk	uper left k x k submatrix of Linv
Lsubk3	Matrix used for sliding in slider3d and relaxLM

.

Note

This function is not intended to be called directly - except for playing around to grasp the mechansims of the Thin-Plate Spline.

References

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.

Bookstein FL. 1989. Principal Warps: Thin-plate splines and the decomposition of deformations. IEEE Transactions on pattern analysis and machine intelligence 11(6).

See Also

tps3d

Examples

data(boneData)
L <- CreateL(boneLM[,,1])
## calculate Bending energy between first and second specimen:
be <- t(boneLM[,,2])%*%L$Lsubk%*%boneLM[,,2]
## calculate Frobenius norm 
sqrt(sum(be^2))
## the amount is dependant on on the squared scaling factor
# scale landmarks by factor 5 and compute bending energy matrix
be2 <- t(boneLM[,,2]*5)%*%L$Lsubk%*%(boneLM[,,2]*5)
sqrt(sum(be2^2)) # exactly 25 times the result from above
## also this value is not symmetric:
L2 <- CreateL(boneLM[,,2])
be3 <- t(boneLM[,,1])%*%L2$Lsubk%*%boneLM[,,1]
sqrt(sum(be3^2))

Morpho documentation built on Feb. 16, 2023, 10:51 p.m.