Resampling

To sinc interpolate a signal, it is possible to zero pad a signal in Fourier space and to do an inverse Fourier transform effectively evaluating the Fourier series at more samples. If the signal was initially band-limited, sinc interpolation leads to smoother, perfectly interpolated signals

Sinc interpolation

Below you can find a simple example for up sampling using resample. Furthermore, there is an image interpolation Pluto.jl notebook in the examples folder. We can see that the interpolated signal matches the higher sampled signal well.

 begin
	N_low = 128
	x_min = 0.0
	x_max = 16π
	
	xs_low = range(x_min, x_max, length=N_low+1)[1:N_low]
	xs_high = range(x_min, x_max, length=5000)[1:end-1]
	f(x) = sin(0.5*x) + cos(x) + cos(2 * x) + sin(0.25*x)
	arr_low = f.(xs_low)
	arr_high = f.(xs_high)
end

begin
	N = 1000
	xs_interp = range(x_min, x_max, length=N+1)[1:N]
	arr_interp = resample(arr_low, N)
end

begin
	scatter(xs_low, arr_low, legend=:bottomleft, markersize=2, label="Low sampling")
	plot!(xs_interp, arr_interp, label="FFT based sinc interpolation", linestyle=:dash)
	plot!(xs_high, arr_high, linestyle=:dashdotdot, label="High sampling")
end

Downsampling

32 samples in the downsampled signal should be sufficient for Nyquist sampling. And as we can see, the downsampled signal still matches the original one.

begin
	N_ds = 32
	xs_ds = range(x_min, x_max, length=N_ds+1)[1:N_ds]
	arr_ds = resample(arr_high, N_ds)
end

begin
	scatter(xs_low, arr_low, legend=:bottomleft, markersize=2, label="Low sampling")
	plot!(xs_interp, arr_interp, label="FFT based sinc interpolation", linestyle=:dash)
	plot!(xs_ds, arr_ds, label="resampled array", linestyle=:dot)	
end

Image Upsampling

Having a Nyquist sampled image, it is possible to perform a sinc interpolation and creating visually much nicer images. However, the information content does not change between both images. The full Pluto notebook is here. The right image is the upsampled version of the left one.

There are fast versions (upsample2) for upsampling by a factor of 2.

Function References

Missing docstring.

Missing docstring for FourierTools.resample. Check Documenter's build log for details.

Missing docstring.

Missing docstring for FourierTools.resample_by_FFT. Check Documenter's build log for details.

Missing docstring.

Missing docstring for FourierTools.resample_by_RFFT. Check Documenter's build log for details.

Missing docstring.

Missing docstring for FourierTools.resample_nfft. Check Documenter's build log for details.

Missing docstring.

Missing docstring for FourierTools.resample_czt. Check Documenter's build log for details.

Missing docstring.

Missing docstring for FourierTools.upsample2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for FourierTools.upsample2_abs2. Check Documenter's build log for details.

Missing docstring.

Missing docstring for FourierTools.barrel_pin. Check Documenter's build log for details.