Use the 'Fast-Fourier' transform to compute the convolutions of two data with zero padding.

## Arguments

- x
one-dimensional signal vector, two-dimensional image, or three-dimensional volume; numeric or complex

- filter
kernel with the same number of dimensions as

`x`

## Value

Convolution results with the same length and dimensions as `x`

.
If `x`

is complex, results will be complex, otherwise results will
be real numbers.

## Details

This implementation uses 'Fast-Fourier' transform to perform
`1D`

, `2D`

, or `3D`

convolution. Compared to implementations
using original mathematical definition of convolution, this approach is
much faster, especially for image and volume convolutions.

The input `x`

is zero-padded beyond edges. This is most common in image
or volume convolution, but less optimal for periodic one-dimensional signals.
Please use other implementations if non-zero padding is needed.

The convolution results might be different to the ground truth by a precision
error, usually at `1e-13`

level, depending on the `'FFTW3'`

library precision and implementation.

## Examples

```
# ---- 1D convolution ------------------------------------
x <- cumsum(rnorm(100))
filter <- dnorm(-2:2)
# normalize
filter <- filter / sum(filter)
smoothed <- convolve_signal(x, filter)
plot(x, pch = 20)
lines(smoothed, col = 'red')
# ---- 2D convolution ------------------------------------
x <- array(0, c(100, 100))
x[
floor(runif(10, min = 1, max = 100)),
floor(runif(10, min = 1, max = 100))
] <- 1
# smooth
kernel <- outer(dnorm(-2:2), dnorm(-2:2), FUN = "*")
kernel <- kernel / sum(kernel)
y <- convolve_image(x, kernel)
par(mfrow = c(1,2))
image(x, asp = 1, axes = FALSE, main = "Origin")
image(y, asp = 1, axes = FALSE, main = "Smoothed")
```