Frequency response and sampling theorem for triangular function

Question

The triangular function is defined as follows:

$h_l(x) = \begin{cases}1-|x|,&|x|<1;\\0&\text{otherwise}.\end{cases}$

According to ccrma.stanford.edu:

"If the output of the interpolator is also sampled, this can be modeled by sampling the continuous-time interpolation result, thereby aliasing the $\text{sinc}^2$ frequency response. [...] The Fourier transform of $h_l(nT/L)$ is the same function aliased on a block of size $ f_s=L/T$ Hz."

What I am thinking: If the Fourier transform $\hat{f}$ has bounded support (bandlimited), then $f$ has unbounded support (infinite length). This is a consequence of the uncertainty principle. But the triangular function (i.e. $f$) is $0$ for $|x| > 1$, so $f$ has bounded support (and $\hat{f}$ cannot be bandlimited). The sampling theorem requires $\hat{f}$ to be supported on some interval $[-B, B]$ (bandlimited). Then we cannot apply the sampling theorem, because $\hat{f}$ is not bandlimited. So aliasing will occur because we cannot find a good sampling rate.

Is my interpretation correct? Aliasing will always occur for $h_l$ no matter whether we look at $\text{DTFT}(\text{sample}(h_l))$ or at $\hat{h_l}$ (Fourier transform)? Then why is linear interpolation so popular in image processing (bilinear interpolation) if it has such a bad frequency response?

Answer 1

You're right that a triangular function is not band-limited and that it can't be sampled without introducing aliasing. However, this doesn't mean that linear interpolation can't be useful. The usefulness of linear interpolation depends on the data to be interpolated and on the desired interpolation factor. If the data have a low pass character, i.e., if they are already sufficiently oversampled, then linear interpolation can be good enough if the interpolation factor is not too large.

The obvious advantage of (bi-)linear interpolation is its simplicity and computational efficiency. In image processing it performs better than the common nearest-neighbor interpolation. However, if there are no real-time requirements and if computational efficiency is no big issue, then the standard interpolation method in image processing is the more complex bicubic interpolation.