## 2 Literature Review
### Combining the Geometries
#### 2.3.6 Image Scanning Microscopy  
We have already discussed the use of a detector array in a confocal microscope. The name image scanning microscopy (ISM) was introduced by [PERSON] and [PERSON] to describe their microscope system using a detector array [110]. It is not clear exactly to which feature they meant for this term to refer to. The main novel feature is obviously not the addition of a detector array to the confocal microscope, as this was not a new idea. There are two concepts that they describe. First, the recognition that an off-axis element of the detector does not represent a signal from the point of the sample that is being illuminated. This means that processing of the data must take account of this effect. Second, the most simple way to process the data is to shift the measured signal to the coordinates of the object to which it refers. This operation is called pixel reassignment. Thus, our understanding of the terms is that pixel reassignment is a particular strategy of processing the data, whereas ISM can refer to methods of processing other than simple pixel reassignment. In fact, both the principle of image scanning microscopy and pixel reassignment were introduced much earlier [111]. It was mentioned that such a system is especially useful for confocal fluorescence, as the resulting signal is much stronger than in a confocal microscope with a small pinhole. The case of a confocal reflection system was also discussed. This paper also gave the PSF and OTF for a system with an infinitely large detector array after summing the reassigned pixel signals. The overall PSF is the convolution of the two rescaled(compressed) PSFs, \(h(v)=h_{1}(2v)\otimes h_{2}(2v)\), while the OTF is the product of two rescaled (enlarged) OTFs, \(C(m)=C_{1}(m/2)C_{2}(m/2)\). Note that for confocal microscopy with a small pinhole, the overall PSF is given by a product and the OTF by a convolution. A plot of the OTF was given, showing that the high spatial frequency response is increased even compared with confocal imaging with a small pinhole. The reason for this is that, as was already known, a displaced point detector, surprisingly, results in a sharper PSF [112; 113; 114]. It is interesting to note that these papers assumed that the image had been reassigned, but as only a single displaced detector was used, no reassignment was necessary. Thus, ISM gives a much stronger signal while, at the same time, giving better resolution than true confocal microscopy, with an infinitely small pinhole. It was also explained that the image formation model justifies why resolution is improved in a confocal microscope with a small pinhole, whereas you might think that throwing away light would not do this: increasing the size of the pinhole introduces light coming from neighbouring parts of the object, thus reducing the resolution. Thus, the confocal pinhole actually adds information about the position in the detector plane. A detector array can therefore be seen to add even more information. [PERSON] and [PERSON] were the first to perform ISM experimentally [110]. They used an EMCCD detector, applied pixel reassignment and summation, and then also used post-processing by linear deconvolution to obtain a doubling in resolution. There are many possible alternative ways to process the ISM data. The measured signal from a 2D object is a function of four variables: \(S(\mathbf{x}_{1},\mathbf{x}_{2})\), where \(\mathbf{x}_{1},\mathbf{x}_{2}\) are coordinates for the scan position and the detector location. The 4D signal in a reflection-mode spinning disk microscope was presented many years ago and applies equally to ISM [115]. Other papers that consider imaging in spinning disk microscopes include Refs. [116; 117]. The 4D nature of the signal was also apparent in Refs. [91; 96].