Shifting

Using the Fourier shift property one can implement shifting of arrays not only over pixel but also sub-pixel amount.

Examples

For full interactivity, have a look at this Pluto notebook.

begin
    f(x) = cos(4π * x / 30)
    x1 = 1:30
    x2 = x1 .+ 3
end

begin
    y1 = f.(x1)
    y2 = f.(x2)
    offset = 2.01
    y3 = shift(y2, tuple(offset))
end

begin
    plot(y1, label="Original signal")
    plot!(y2, label="Shifted signal")
    plot!(y3, label="Fourier shifted with $offset")
end

Function references

FourierTools.shift — Function

shift(arr, shifts)

Returning a shifted array. See shift! for more details

source

FourierTools.shift! — Function

shift!(arr, shifts)

Shifts an array in-place. For real arrays it is based on rfft. For complex arrays based on fft. shifts can be non-integer, for integer shifts one should prefer circshift or ShiftedArrays.circshift because a FFT-based methods introduces numerical errors.

Memory Usage

Note that for complex arrays we can avoid any large memory allocations because of fft!. For rfft there does not exist a usable implementation yet, so for real arrays there might be a temporary larger memory usage.

Examples

julia> x = [1.0 2.0 3.0; 4.0 5.0 6.0]
2×3 Matrix{Float64}:
 1.0  2.0  3.0
 4.0  5.0  6.0

julia> shift!(x, (1, 2))
2×3 Matrix{Float64}:
 5.0  6.0  4.0
 2.0  3.0  1.0

julia> x = [0, 1.0, 0.0, 1.0]
4-element Vector{Float64}:
 0.0
 1.0
 0.0
 1.0

julia> shift!(x, 0.5)
4-element Vector{Float64}:
 0.49999999999999994
 0.5
 0.49999999999999994
 0.5

source