Lanczos resampling

It can be used as a low-pass filter or used to smoothly interpolate the value of a digital signal between its samples.

Given a one-dimensional signal with samples si, for integer values of i, the value S(x) interpolated at an arbitrary real argument x is obtained by the discrete convolution of those samples with the Lanczos kernel:[3] where a is the filter size parameter, and

Therefore, the reconstructed signal exactly interpolates the given samples: we will have S(x) = si for every integer argument x = i. Lanczos resampling is one form of a general method developed by Lanczos to counteract the Gibbs phenomenon by multiplying coefficients of a truncated Fourier series by

[4] The same reasoning applies in the case of truncated functions if we wish to remove Gibbs oscillations in their spectrum.

Thus, by varying the 2a parameter one may trade computation speed for improved frequency response.

The parameter also allows one to choose between a smoother interpolation or a preservation of sharp transients in the data.

For image processing, the trade-off is between the reduction of aliasing artefacts and the preservation of sharp edges.

Turkowski and Gabriel claimed that the Lanczos filter (with a = 2) is the "best compromise in terms of reduction of aliasing, sharpness, and minimal ringing", compared with truncated sinc and the Bartlett, cosine-, and Hann-windowed sinc, for decimation and interpolation of 2-dimensional image data.

"[5] Lanczos interpolation is a popular filter for "upscaling" videos in various media utilities, such as AviSynth[6] and FFmpeg.

When using the Lanczos filter for image resampling, the ringing effect will create light and dark halos along any strong edges.

While these bands may be visually annoying, they help increase the perceived sharpness, and therefore provide a form of edge enhancement.

This may improve the subjective quality of the image, given the special role of edge sharpness in vision.

[8] In some applications, the low-end clipping artifacts can be ameliorated by transforming the data to a logarithmic domain prior to filtering.

This is rather academic, since using a single-lobe kernel (a = 1) loses all the benefits of the Lanczos approach and provides a poor filter.

Lanczos windows for a = 1, 2, 3.
Lanczos kernels for the cases a = 1, 2, and 3, with their frequency spectra. A sinc filter would have a cutoff at frequency 0.5.
The incipit of a black-and-white image. Original, low-quality expansion with JPEG artifacts.
The same image resampled to five times as many samples in each direction, using Lanczos resampling. Pixelation artifacts were removed changing the image's transfer function.
A discrete Lanczos window and its frequency response ; see Window function for comparison with other windows.