Image 2x downsampling with Lanczos filter

Question

I'm trying to implement image downsampling with Lanczos2.

However, the kernel seems to have zeros everywhere (since sin(pi*x)=0 if x is integer) except at the center pixel.

Thus, if the downsampling factor is an integer number (e.g. the output size is 1/2 of the original size at each dimension), then the Lanczos downsampling yields the exact same result as nearest neighbor interpolation (just taking every other pixel in 2X downsampling).

I believe that this is not intended to be the case, so my question is: What am I missing?

How to use lanczos2 filter for 2x downsampling and is the result expected to be different than simply take every other pixel?

Answer 1

The kernel for 2x downsampling is given in section "Decimation by factor of 2 with the Lanczos2 sinc function" on page 10 of the reference you linked, with the coefficients:

0, -0.032, 0, 0.284, 0.496, 0.284, 0, -0.032, 0

This kernel is obtained by evaluating the given lanczos2(x) function at values of x=0.5n where n is the sample number (an integer). This reflect the fact that the output rate is half the original sampling rate (thus requires a half-band filter before pixel decimation to avoid aliasing).

P.S.: the kernel with zeros everywhere except at the center pixel that you have obtained, would typically be used (although implementations would usually optimize this kernel out as a simple pixel copy) in conjunction with a phase 1/2 kernel for interpolation by a factor 2.