# Sub-resolution objects

Sub-resolution objects are those smaller than the Spatial Resolution of the optical device.

## My beads have a diameter of 150 nm, that is sub-resolution. Why do not their images give the PSF directly?

They might be sub-resolution in terms of the Rayleigh Criterion, but they are not small enough to probe all the necessary information to obtain a Point Spread Function (PSF). This can be better explained in terms of Fourier Transforms and frequency domain.

The PSF acts as a band-limiter. It chops off high frequencies, and let you measure only low ones: features are smoothed. It acts as a Cookie Cutter in the frequency domain.

We need to find the PSF, so we need to measure a known object with all kind of frequencies, and see which are rejected and which can pass. The ideal sub-resolution probe (a single point) is totally "space-limited", so in the Fourier domain it is not limited at all: it has all the possible frequencies. Using it as a probe, and seeing what frequencies are not detected, we obtain the PSF. That's the direct PSF imaging.

The less "space-limited" the probe is (the larger it is), the less high-frequencies it has in the frequency domain, so we can not use it to properly determine the PSF: it may lack some important high frequencies. In the practical case, the limit is established by the Nyquist Rate, and a sub-resolution point, in the terms explained here (so it has all the necessary high frequencies to probe the PSF), is smaller than 50 nm in a typical confocal case.

If you use beads larger than that, you need a Non Linear Iterative Method like the CMLE algorithm Huygens Essential uses in the Psf Distiller.