Microscopy images normally look blurry because light originating from one point in the sample is not all detected at a single pixel: usually it is detected over several pixels and z-slices. This is not simply because we cannot use perfect lenses; rather, it is caused by a fundamental limit imposed by the nature of light. The end result is as if the light that we detect is redistributed slightly throughout our data (Figure 1).

This is important for three reasons:

Therefore, almost every measurement we might want to make can be affected by blurring to some degree.

That is the bad news about blur. The good news is that it is rather well understood, and we can take some comfort that it is not random. In fact, the main ideas have already been described in Filters, because blurring in fluorescence microscopy is mathematically described by a convolution involving the microscope’s Point Spread Function (PSF). In other words, the PSF acts like a linear filter applied to the perfect, sharp data we would like but can never directly acquire.

Previously, we saw how smoothing (e.g. mean or Gaussian) filters could helpfully reduce noise, but as the filter size increased we would lose more and more detail. At that time, we could choose the size and shapes of filters ourselves, changing them arbitrarily by modifying coefficients to get the noise-reduction vs. lost-detail balance we liked best. But the microscope’s blurring differs in at least two important ways. Firstly, it is effectively applied to our data before noise gets involved, so offers no noise-reduction benefits. Secondly, because it occurs before we ever set our eyes on our images, the size and shape of the filter (i.e. the PSF) used are only indirectly (and in a very limited way) under our control. It would therefore be much nicer just to dispense with the blurring completely since it offers no real help, but unfortunately light conspires to make this not an option and we just need to cope with it.

The purpose of this chapter is to offer a practical introduction to why the blurring occurs, what a widefield microscope’s PSF looks like, and why all this matters. Detailed optics and threatening integrals are not included, although several equations make appearances. Fortunately for the mathematically unenthusiastic, these are both short and useful.

As stated above, the fundamental cause of blur is that light originating from an infinitesimally small point cannot then be detected at a similar point, no matter how great our lenses are. Rather, it ends up being focused to some larger volume known as the PSF, which has a minimum size dependent upon both the light’s wavelength and the lens being used.

This becomes more practically relevant if we consider that any fluorescing sample can be viewed as composed of many similar, exceedingly small light-emitting points – you may think of the fluorophores. Our image would ideally then include individual points too, digitized into pixels with values proportional to the emitted light. But what we get instead is an image in which every point has been replaced by its PSF, scaled according to the point’s brightness. Where these PSFs overlap, the detected light intensities are simply added together. Exactly how bad this looks depends upon the size of the PSF (Figure 2).

The chapter on Filters gave one description of convolution as replacing each pixel in an image with a scaled filter – which is just the same process. Therefore it is no coincidence that applying a Gaussian filter to an image makes it look similarly blurry. Because every point is blurred in the same way (at least in the ideal case; extra aberrations can cause some variations), if we know the PSF then we can characterize the blur throughout the entire image – and thereby make inferences about how blurring will impact upon anything we measure.

We can gain an initial impression of a microscope’s PSF by recording a z-stack of a small, fluorescent bead, which represents an ideal light-emitting point. Figure 3 shows that, for a widefield microscope, the bead appears like a bright blob when it is in focus. More curiously, when viewed from the side (xz or yz), it has a somewhat hourglass-like appearance – albeit with some extra patterns. This exact shape is well enough understood that PSFs can also be generated theoretically based upon the type of microscope and objective lenses used (C).

So much for appearances. To judge how the blurring will affect what we can see and measure, we need to know the size of the PSF – where smaller would be preferable.

The size requires some defining: the PSF actually continues indefinitely, but has extremely low intensity values when far from its center. One approach for characterizing the Airy disk size is to consider its radius $r_{airy}$ as the distance from the center to the first minimum: the lowest point before the first of the outer ripples begins. This is is given by:

$$r_{airy} = \frac{0.61 \lambda}{\textrm{NA}}$$

where $\lambda$ is the light wavelength and NA is the numerical aperture of the objective lens^[1].

A comparable measurement to $r_{airy}$ between the center and first minimum along the z axis is:

$$z_{min} = \frac{2 \lambda \times \eta}{\textrm{NA}^2}$$

where $\eta$ is the refractive index of the objective lens immersion medium (which is a value related to the speed of light through that medium).