Convolution of 3D Arrays using the Fourier Transform

Convolve a three-dimensinal array with another three-dimensional arry using the Fast Fourier Transform (FFT).

convFFT(A, B, C, FFTA = NULL)

Arguments

A	is a three-dimensional array (“the template”).
B	is a three-dimensional array (“the target”).
C	is a vector of length three (the center of “the template”).
FFTA	is the three-dimensional Fourier transform of `A`, this may save time when looping over multiple “targets”.

Value

A three-dimensional array, the same dimension as the input arrays, that is the convolution of the “target” to the “template” at all spatial locations.

Details

The arrays \(A\) and \(B\) are transformed into the Fourier domain and multiplied together (equivalent to a convolution in the image domain across all spatial locations simultaneously).

References

Briggs, W.L. and Henson, V.E. (1995) The DFT: An Owner's Manual for the Discrete Fourier Transform, SIAM: Philadelphia.

Author

Brandon Whitcher bwhitcher@gmail.com

Examples


cube <- array(0, c(20,20,1))
cube[9:12,9:12,1] <- 1
tkernel <- array(0, c(20,20,1))
tkernel[,,1] <- c(.5, 1, .5, rep(0,20-3)) %o% c(.5, 1, .5, rep(0,20-3))
tcenter <- findCenter(ifelse(tkernel > 0, TRUE, FALSE))
out <- convFFT(tkernel, cube, tcenter)
out[8:13,8:13,1]  ## text output
#>      [,1] [,2] [,3] [,4] [,5] [,6]
#> [1,] 0.25 0.75    1    1 0.75 0.25
#> [2,] 0.75 2.25    3    3 2.25 0.75
#> [3,] 1.00 3.00    4    4 3.00 1.00
#> [4,] 1.00 3.00    4    4 3.00 1.00
#> [5,] 0.75 2.25    3    3 2.25 0.75
#> [6,] 0.25 0.75    1    1 0.75 0.25

par(mfrow=c(2,2))  ## graphic output
image(drop(tkernel), col=oro.nifti::tim.colors(), main="Template")
image(drop(cube), col=oro.nifti::tim.colors(), main="Target")
image(drop(out), col=oro.nifti::tim.colors(), main="Output")