ICvectorfields (version 0.1.2)

Xcov2D: Cross-covariance in two spatial dimensions

Description

This function efficiently computes two dimensional cross-covariance of two equal dimensioned matrices of real numbers using efficient discrete fast Fourier trasforms.

Usage

Xcov2D(mat1, mat2)

Arguments

mat1

a real valued matrix

mat2

a real valued matrix of equal dimension to mat1

Value

a real valued matrix showing cross-covariance in each direction

Details

The algorithm first pads each matrix with zeros so that the outer edges of the matrices do not interact with one another due to the circular nature of the discrete fast Fourier transform. Cross-covariance calculations require computation of the complex conjugate of one of the two imput matrices. Assuming all of it's elements are real, computing the complex conjugate is equivalent to flipping the matrix in the horizontal and vertical directions. Then to compute cross-covariance, the first matrix is convolved with the flipped second matrix as described in the convolution theorem.

This function is called by the main functions that compute displacement fields and vector fields and is included here primarily for demonstration purposes. Specifically, the method for computing the magnitude and direction of shifts is demonstrated in the examples.

The shift that produces the maximum cross-covariance between the two input matrices can be obtained by finding the row and column indices associated with the maximum cross-covariance. The shift in each direction is obtained by subracting one plus the half the dimension of the output matrix (the same for rows and columns) from the row and column values that are associated with the maximum cross-covariance as demonstrated in the examples below. Note that shifts to the right and up are denoted with positive numbers and shifts to the left and down are denoted by negative numbers. This is contrary to some conventions but efficient for producing vector fields. For more details on cross-covariance see cross-correlation.

Examples

# NOT RUN {
matrix(c(1:6, rep(0, 3)), nrow = 3); matrix(c(rep(0, 3), 1:6), nrow = 3)
dim(Xcov2D(matrix(c(1:6, rep(0, 3)), nrow = 3),
    matrix(c(rep(0, 3), 1:6), nrow = 3)))
ICvectorfields::GetRowCol(
    which.max(Xcov2D(matrix(c(1:6, rep(0, 3)), nrow = 3),
        matrix(c(rep(0, 3), 1:6), nrow = 3))),
    dim1 = dim(Xcov2D(matrix(c(1:6, rep(0, 3)), nrow = 3),
        matrix(c(rep(0, 3), 1:6), nrow = 3)))[1],
    dim2 = dim(Xcov2D(matrix(c(1:6, rep(0, 3)), nrow = 3),
        matrix(c(rep(0, 3), 1:6), nrow = 3)))[2]
        )
# This implies that the shift is 6 - (10/2 + 1) in the vertical
# direction and 7 - (10/2 + 1) in the horizonatal direction.
# }