basis: Bivariate Spline Basis Function

In FIRST-Data-Lab/BPST: Bivariate Spline over Triangulation

Bivariate Spline Basis Function

Description

This function generates the basis for bivariate spline over triangulation.

Usage

basis(V, Tr, d = 5, r = 1, Z, Hmtx = TRUE, Kmtx = TRUE, QR = TRUE, TA = TRUE)

Arguments

V	The N by two matrix of vertices of a triangulation, where N is the number of vertices. Each row is the coordinates for a vertex.
Tr	The triangulation matrix of dimention nT by three, where nT is the number of triangles in the triangulation. Each row is the indices of vertices in V.
d	The degree of piecewise polynomials – default is 5, and usually d is greater than one. -1 represents piecewise constant.
r	The smoothness parameter – default is 1, and 0 \le r < d.
Z	The cooridinates of dimension n by two. Each row is the coordinates of a point.
Hmtx	The indicator of whether the smoothness matrix H need to be generated – default is TRUE.
Kmtx	The indicator of whether the energy matrix K need to be generated – default is TRUE.
QR	The indicator of whether a QR decomposition need to be performed on the smoothness matrix – default is TRUE.
TA	The indicator of whether the area of the triangles need to be calculated – default is TRUE.

Details

This R program is modified based on the Matlab program written by Ming-Jun Lai from the University of Georgia and Li Wang from the Iowa State University.

Value

A list of vectors and matrice, including:

B	The spline basis function of dimension n by nT*{(d+1)(d+2)/2}, where n is the number of observationed points, nT is the number of triangles in the given triangulation, and d is the degree of the spline. If some points do not fall in the triangulation, the generation of the spline basis will not take those points into consideration.
Ind.inside	A vector contains the indexes of all the points which are inside the triangulation.
H	The smoothness matrix.
Q2	The Q2 matrix after QR decomposition of the smoothness matrix H.
K	The thin-plate energy function.
tria.all	The area of each triangle within the given triangulation.

Examples

# example 1
xx = c(-0.25, 0.75, 0.25, 1.25)
yy = c(-0.25, 0.25, 0.75,1 .25)
Z = cbind(xx, yy)
d = 4; r = 1;
V0 = rbind(c(0, 0), c(1, 0), c(1, 1), c(0, 1))
Tr0 = rbind(c(1, 2, 3), c(1, 3, 4))
basis(V0, Tr0, d, r, Z)

FIRST-Data-Lab/BPST documentation built on Sept. 18, 2023, 7:31 a.m.