# Why are real-world digital images not bandlimited?

In the materials about image resampling, it always mentions that real-world digital images not bandlimited. However no explanation is provided.
For example,

Sinc resampling in theory provides the best possible reconstruction for a perfectly bandlimited signal. In practice, the assumptions behind sinc resampling are not completely met by real-world digital images. Lanczos resampling, an approximation to the sinc method, yields better results. Bicubic interpolation can be regarded as a computationally efficient approximation to Lanczos resampling.

from the link https://en.wikipedia.org/wiki/Image_scaling.

Why are real-world digital images not bandlimited? Is there any phenomenon that can prove the statement is correct?

• dsp.stackexchange.com/q/58026/11256
– MBaz
Jul 15, 2022 at 14:14
• Real world images are actually bandlimited. What could possibly be not bandlimited are those computer generated (synthetic) images or vector fonts etc... Jul 16, 2022 at 11:32
• Sampling theory assumes an infinite signal. It is the finite domain that causes the assumptions to not be met. You his has nothing to do with the signal not being bandlimited. Jul 16, 2022 at 13:44
• @Fat32 If the images are bandlimited, sinc function can reconstruct the original image perfectly. Artifacts will no occur if images are resampling. However it is very easy to observe artifacts if images ar scaled. Jul 18, 2022 at 1:34

The notion of "digital image" is not so precise, when talking about some 2D multidimensional signal. One instance I know of is a seismic gather: a line of surface-based sensors, located every x meters, each acquiring a 1D vibration/acceleration signal. The seismic signal can be band-pass (say 5 Hz to 120 Hz) and well-acquired, but the ground spacing (50-100 meters) may yield aliasing in the other dimension.

Hence, several issues:

• limited support data always has infinite bandwidth. Therefore, short image dimensions may have not bandlimited content.
• natural images often have content with sharp features like edges and textures, that are likely to possess high frequency content. As said by Jogging Song in comments, 2D features of interest can be sharp in one direction (non bandlimited) and smooth in the other. The 1D features are rarely both.
• a classical model is the illumination-reflectance factorization, where image $$D=IR$$, that accounts for instance for shadow effects. It is akin to the Beer-Lambert law, details are gived in the presentation Illumination and Reflectance (Michael Langer, 2018). It can be processed in a $$\log$$ (homomorphic) domain.
• image formation, including occlusion (front objects hiding background), is quite non-linear. Hence, this does not satisfies the classical linear settings of the Nyquist theory.

Another hint is that people tend to use nonlinear tools, long median or range filters, homomorphic processing, mathematical morphology on images.

• From the link inst.eecs.berkeley.edu/~cs283/sp13/lectures/283-lecture2.pdf, Real world: lines have infinitely high frequencies, can't sample at high enough resolution. Maybe this is the reason that image scaling will produce artifacts if the original image is looks  pleasant. Jul 18, 2022 at 6:27
• How is the frequency in digital images are defined? It seems that digital frequence and spatial frequency are used in different papers. Is there any resources about how to calculate the two terms? Jul 18, 2022 at 6:33
• Indeed. A 2D line strucrure can be meaningful, while the 1D equivalent, a Dirac impilse, is rarely so. I'll work on your comment to update my answer Jul 18, 2022 at 15:29

The statement is either misleading or wrong, depending on interpretation. I defer to this answer for details, and for short version I cite my comments from under this answer

What I'm saying is, it's the right idea used in the wrong way. Yes, strictly speaking, bandlimited implies infinite knowledge: infinite sampling rate for all time. At the same time, even if we had a physical process that is truly periodic for as long as it exists, it still doesn't qualify, because it hasn't existed for infinite time in the past, and is yet to exist for infinite time in the future. Yet, this process qualifies perfectly for the motivation of the Fourier transform as a periodicity-mapping tool, and this motivation is lost when we try to be "strictly correct"

I insist on this point because of the misleading implication that, essentially, a periodic physical process can't exist. I'm no quantum mechanic but I think EM waves in vacuum defy this. The full model of "frequency" then consists of at least two relations, one being the laws that govern the photon, another the Fourier transform, and using latter without former is simply wrong. A "system of equations" if you will. If we know something is physically periodic, we should be able to call it bandlimited.

Moreover, to comment on the quote in question, the only thing that Fourier theory says about "not bandlimited" is, that sinc interpolation won't recover it perfectly. That doesn't mean nothing else will. For example, if we make a synthetic, perfectly bandlimited image, we can do full recovery perfectly. But if we simply crop the same image, it's aliasing, and we can no longer do so. I doubt anyone would otherwise call cropping "aliasing".

This is an old chestnut. One of the earlier (but relatively recent) musings about this topic for general signals is Slepian's On Bandwidth.. A screenshot of the abstract is below.