Fourier Upsampling methods

A small collection of upsampling utilities that lean on Fourier analysis. I wrote some of them for other projects and decided that it might be useful to share them.

The repository deals with localised upsampling: evaluating a dense slice of a coarsely sampled 1D signal without the need for zero-padding or reconstructing the entire high-resolution grid. This is similar to FFT with zero padding (Dirichlet kernel interpolation), but implemented more efficiently using the CZT.

Several boundary conditions are supported, including periodic, reflective (DCT), and “free” boundaries. For general upscaling tasks without specific boundary assumptions, dct_upscale_with_boundaries is usually the most convenient choice.

Requirements

• Python
• NumPy
• SciPy

Install the dependencies with pip:

python -m pip install numpy scipy

Quick start

import numpy as np
from fourier_upsampling import dirichlet_upscale_zoomfft

# Low-rate signal
x = np.sin(np.linspace(0, 6 * np.pi, 128, endpoint=False))
X = np.fft.fft(x)

# Evaluate indices [400, 600) on the grid that is 8× finer
start, end, q = 400, 600, 8
y = dirichlet_upscale_zoomfft(X, start, end, q).real

The FFT-based function assumes the signal is periodic. If your data is more naturally reflective, use the DCT variant:

from fourier_upsampling import upscale_region_via_dct

y = upscale_region_via_dct(x, start=400, end=600, q=8)

For signals with unknown boundary behavior, first blend the edges toward a constant before upsampling:

from fourier_upsampling import dct_upscale_with_boundaries

safe = dct_upscale_with_boundaries(x, start=400, end=600, q=8)

Another option is to fit an overcomplete cosine/sine dictionary and then evaluate the resulting continuous curve. This approach is considerably more computationally demanding than the others and seldom offers an advantage over dct_upscale_with_boundaries. It is the only upscaler in this repository that doesn't fully adhere to the data and tends to dampen higher frequency content.

from fourier_upsampling import free_boundary_upscale

free = free_boundary_upscale(x, start=400, end=600, q=8)

Each helper except for free_boundary_upscale only computes the requested slice, which keeps things quick even for large upsampling factors.

Boundary handling overview

• Periodic – dirichlet_upscale_zoomfft identical to zero padding in the Fourier domain, transforming back and slicing, but more efficient.

• Reflective – upscale_region_via_dct works in the DCT-II domain, applying the transform before upsampling. Slightly faster than the above method.

• Free / blended – spectral_upscale uses periodic boundaries but subtracts a weighted line fit, periodicises the residual by blending the edges, then restores the trend.
  dct_upscale_with_boundaries also uses reflective boundaries but applies smoothing to the result, which in practice often gives the best outcome.
  free_boundary_upscale solves a ridge-regularised least squares problem in an overcomplete dictionary of DCT and DST basis functions to smoothly extrapolate the edges. This is relatively slow.

Examples and comparison plots

examples.py visualises the different methods on some demo signals. It also compares them with linear and quadratic interpolation for reference. Matplotlib is an additional requirement:

python examples.py