R/curve_to_q.R

In fdasrvf: Elastic Functional Data Analysis

#' Curve to SRVF Space
#'
#' This function computes the SRVF of a given curve. The function also outputs
#' the length of the original curve and the \eqn{L^2} norm of the SRVF.
#'
#' @param beta A numeric matrix of shape \eqn{L \times M} specifying a curve on
#'   an \eqn{L}-dimensional space observed on \eqn{M} points.
#' @param scale A boolean value specifying whether the output SRVF function
#'   should be scaled to the hypersphere of \eqn{L^2}. When `scale` is `TRUE`,
#'   the length of the original curve becomes irrelevant. Defaults to `TRUE`.
#'
#' @return A list with the following components:
#' - `q`: A numeric matrix of the same shape as the input matrix `beta` storing
#' the (possibly scaled) SRVF of the original curve `beta`;
#' - `len`: A numeric value specifying the length of the original curve;
#' - `lenq`: A numeric value specifying the \eqn{L^2} norm of the SRVF of the
#' original curve.
#'
#' @keywords srvf alignment
#'
#' @references Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape
#'   analysis of elastic curves in Euclidean spaces. IEEE Transactions on
#'   Pattern Analysis and Machine Intelligence, **33**(7), 1415-1428.
#'
#' @export
#'
#' @examples
#' q <- curve_to_q(beta[, , 1, 1])$q
curve_to_q <- function(beta, scale = TRUE) {
  n <- nrow(beta)
  T1 <- ncol(beta)
  v <- t(apply(beta, 1, gradient, binsize = 1.0 / T1))

  q <- matrix(0, n, T1)
  for (i in 1:T1) {
    L <- max(sqrt(pvecnorm(v[, i], p = 2)), 1e-4)
    q[, i] <- v[, i] / L
  }

  len <- innerprod_q2(q, q)
  len_q <- sqrt(len)
  if (scale) q <- q / len_q

  list(q = q, len = len, lenq = len_q)
}