approx_tensors: Tensor Interpolation
In astamm/fdatractography: Functional Data Analysis for Tractography in R

Description Usage Arguments Details Value Examples

The function approx_tensors is the analog to the approx function for interpolating tensor-valued data.

1 2	approx_tensors(x, tensors, xout, method = "linear", n = 50, yleft, yright, rule = 1, ties = mean)

`x`	Numeric vector specifying the locations at which tensors are defined.
`tensors`	List of `tensor` objects to be interpolated.
`xout`	An optional set of numeric values specifying where interpolation is to take place.
`method`	Specifies the interpolation method to be used. Choices are "linear" [default] or "log" for interpolating in the log-Euclidean space.
`n`	If `xout` is not specified, interpolation takes place at `n` equally spaced points spanning the interval `[min(x), max(x)]`. Default is `n = 50`.
`yleft`	The value to be returned when input `x` values are less than `min(x)`. The default is defined by the value of `rule` given below.
`yright`	The value to be returned when input `x` values are greater than `max(x)`. The default is defined by the value of `rule` given below.
`rule`	An integer (of length 1 or 2) describing how interpolation is to take place outside the interval `[min(x), max(x)]`. If `rule` is 1 then `NA`s are returned for such points and if it is 2, the value at the closest data extreme is used. Use, e.g., `rule = 2:1`, if the left and right side extrapolation should differ.
`ties`	Handling of tied `x` values. Either a function with a single vector argument returning a single number result or the string `"ordered"`.

If method == "linear", it performs linear tensor interpolation according to the paper by Gahm, J. K., Wisniewski, N., Kindlmann, G., Kung, G. L., Klug, W. S., Garfinkel, A., & Ennis, D. B. (2012, October). Linear invariant tensor interpolation applied to cardiac diffusion tensor MRI. In International Conference on Medical Image Computing and Computer-Assisted Intervention (pp. 494-501). Springer, Berlin, Heidelberg. If method == "log", it performs log-Euclidean tensor interpolation according to the paper by Arsigny, V., Fillard, P., Pennec, X. and Ayache, N. (2006) ‘Log-Euclidean metrics for fast and simple calculus on diffusion tensors.’, Magnetic resonance in medicine : official journal of the Society of Magnetic Resonance in Medicine / Society of Magnetic Resonance in Medicine, 56(2), pp. 411–21. doi: 10.1002/mrm.20965.

A list with components x and y, containing n coordinates which interpolate the given data points according to the method (and rule) desired.

theta <- c(0, pi/6, pi/4, pi/3, pi/2)
R <- NULL
for (i in seq_along(theta)) {
  th <- theta[i]
  R[[i]] <- cbind(
    c(cos(th), sin(th), 0),
    c(-sin(th), cos(th), 0),
    c(0, 0, 1)
  )
}
L <- diag(c(1.7, 0.3, 0.1))
D <- purrr::map(R, ~ . %*% L %*% t(.))
s <- seq(0, 1, length.out = length(theta))
approx_tensors(s, D)
approx_tensors(s, D, method = "log")