# Three-Dimension Resolution Enhanced Microscopy Based on Parallel Detection
## 2 Methods  
### Theoretical Analysis  
Here will show the theoretical analysis of 3D-PRFED. At first, CLSM with a single detector and pinhole can be theoretically described by [16]  
\[\text{I}_{\text{c}}=\text{S}\otimes(\text{PSF}_{\text{ex}}\times(\text{PSF}_{ \text{em}}\otimes\text{Pinhole})) \tag{1}\]  
where \(\text{I}_{\text{c}}\) is the intensity distribution of the confocal image, S is the intensity distribution of a sample, \(\text{PSF}_{\text{ex}}\) is the point spread function of excitation light at the sample plane, \(\text{PSF}_{\text{em}}\) is the point spread function of emission light for the system and Pinhole is the pinhole function, \(\otimes\) is the convolution operator. Because of the existence of a pinhole, CLSM can achieve much more excellent lateral and axial resolution performance than conventional wide-field microscopy. However, with the pinhole size decreasing, a more effective signal would be blocked by the pinhole, as in Figure 1A. PRISM can further improve the resolution of CLSM without reducing signal, which can be achieved by replacing a single pinhole with a detector array. The imaging process for different detectors in the array can be mathematically described by [8; 17; 18]  
\[\mathrm{I_{n}(x,y,z)=S\otimes(PSF_{ex}(x,y,z)\times(PSF_{em}(x,y,z)\otimes D(x+d_{n} )))} \tag{2}\]  
\[\mathrm{I_{PRISM}=\sum_{i=1}^{n}I_{n}(r-\frac{d_{n}}{2}\,,\varphi)} \tag{3}\]  
where \(I_{n}(x,y,z)\) is the volume image recorded at the nth pinhole, \(S\) is the sample distribution function, \(PSF_{ex}(x,y,z)\) is the volume point spread function of excitation light, \(PSF_{em}(x,y,z)\) is the volume point spread function of emission light, \(D(x+d_{n})\) is the pinhole function of the nth pinhole in the array, \(d_{n}\) is the relative distance between the nth detector and the center of the array, \(\mathrm{I_{PRISM}}\) is the reconstruction image based on PRISM algorithm, and \(I_{n}(r-d_{n}/2,\varphi)\) is the image shifted by a distance \(d_{n}/2\). According to Equations (1)-(3) and without considering Stokes shift, the equivalent PSF (point spread function) analysis of confocal and PRISM was shown in Figure 1. The detector array used for simulation was depicted in Figure 1D. Figure 1A shows us the relative intensity and resolution change for different pinhole sizes, which indicates the smaller the pinhole size is, the worse the signal-to-noise ratio will be gotten. Figure 1B demonstrates the image translation for different detectors at the detector array because of the existence of pinhole offset. By measuring the full width at half maximum (FWHM) of the normalization profile crossing the PSF  
Figure 1: (**A**) The relative intensity and resolution change for different pinhole size in CLSM (confocal laser scanning microscopy); (**B**) Cross-section through the images of a point object detected by the pinholes with different offset distance dn; (**C**) The normalization cross-section profile comparison between the CLSM image detected by a pinhole with one AU and the PRISM (Pixel reassignment image scanning microscopy) image detected by a detector array which is shown in (**D**); (**D**) The detector array pattern with 19 pinholes used in this simulation and the diameter of the array is one AU. The wavelength of excitation light and emission light: 640 nm. AU: Airy Unit. Gaussian center of CLSM and PRISM, the resolution of PRISM is 1.25 times enhanced than conventional CLSM with one AU (Airy Unit) pinhole size, as in Figure 1C. To further enhance three-dimension resolution of PRISM, here three-dimension fluorescence emission difference microscopy is introduced. Then a novel method called three-dimension pixel reassignment fluorescence emission difference microscopy (3D-PRFED) is proposed.