## 2 Literature Review
### Combining the Geometries
#### 2.3.6 Image Scanning Microscopy
For fluorescence ISM, the 4D signal is [118]  
\[I(\mathbf{x}_{1},\mathbf{x}_{2})=\iint H_{1}(\mathbf{x}_{1}-\mathbf{x}_{0})H _{2}(\mathbf{x}_{2}-\mathbf{x}_{0})t(\mathbf{x}_{0})d^{2}\mathbf{x}_{0}, \tag{3}\]  
where \(\mathbf{x}_{0}\) is the coordinate vector in the object plane, which is basically a cross-correlation function. We can see that \(\mathbf{x}_{1}=\mathbf{x}_{0}\) gives a conventional image, \(\mathbf{x}_{2}=\mathbf{x}_{0}\) gives a scanning image, and \(\mathbf{x}_{1}=\mathbf{x}_{2}\) gives a confocal image. If \(H_{1}=H_{2}\), the signal for \(\mathbf{x}_{1},\mathbf{x}_{2}\) gives an image of the point \((\mathbf{x}_{1}+\mathbf{x}_{2})/2\), i.e., midway between the illumination point and the detected point. Importantly, it does not give an image of the illuminated point \(\mathbf{x}_{1}\). Of course, the PSFs \(H_{1},H_{2}\) are compact, so the complete 4D signal contains redundant information. Soon after my paper was published, [PERSON] et al. published a paper where they applied, almost as an afterthought, deconvolution by Tikhonov regularisation to 1D pixel reassignment[119]. Their conclusion was  
'It is seen that [PERSON]'s method provides an improvement in resolution with respect to conventional CSLM which is smaller than the improvement provided by the method discussed in this paper.' However, in my opinion, this was not a fair comparison, as better results can be obtained by multi-image deconvolution of ISM data. Further, pixel reassignment is much simpler than their algorithm. Later, however, they further investigated a more general algorithm, for which the [PERSON] et al. algorithm and the pixel reassignment algorithm were special cases [120]. A disadvantage of SIM is that thick samples distort the fringes and hence limit the sample thickness. [PERSON] et al. described a system that could image samples eight-times thicker than SIM while retaining a comparable resolution and high speed (one 2D image/s) [121]. Their name'multifocal SIM' can be explained in terms of their view of their system as SIM with multi-point illumination, but we prefer to consider it as a multifocal ISM. In fact, they used exactly the pixel reassignment and summation algorithm of ISM. Multi-focal ISM is possible because only a small region of pixels gives an appreciable signal for each illumination spot. In a later paper, they increased the speed to 100 \(\times\) 2D image/s by performing the reconstruction optically, using a matched microlens array to locally contract each pinhole emission, with a galvanometric mirror to translate the excitation pattern and the summation of the fluorescence emissions with a camera [122]. [PERSON] et al. used a galvanometric mirror to perform the pixel reassignment and scaling [123]. The reassignment factor can be altered as required. They called their method Rescan Confocal Microscopy. The system is similar to an architecture for confocal microscopy with a pinhole and without pixel reassignment proposed by [PERSON] and [PERSON] [124]. [PERSON] et al. constructed a confocal spinning disk ISM [125]. A CCD detector images a region of pixels around the centre of each illumination spot. A similar system was described by [PERSON] and [PERSON] [126]. [PERSON] et al. described an ISM with optical pixel reassignment [127]. They called their system OPRA (Optical Photon Reassignment Microscopy). Later, they combined OPRA with patterned illumination to improve axial imaging [128]. [PERSON] et al. presented a theoretical analysis of ISM [129]. This paper introduced the term pixel reassignment. ISM with an infinitely large detector array gives no optical sectioning. In fact, the size of the detector array plays a similar role to the size of the pinhole in confocal microscopy. It was shown how the resolution and signal collection efficiency of ISM varied with the size of the detector array. The FWHM of the PSF for ISM was found to be about 10% narrower than for confocal microscopy with a small pinhole.