# Effect of the Transition Points Mismatch on Quanta Image Sensors
## 3 Imaging Models for the Effect of the Transition Points Mismatch
### Multi-bit QIS  
The imaging model of the multi-bit QIS is established based on that of the single-bit QIS. The illumination factor \(h_{0}\) here is \((2^{n}{-}1)/255\). In the quantization process, the signal of a jet is compared to a group of transition points (\(U_{th}\) (1) to \(U_{th}\) (\(2^{n}{-}1\))) to obtain an \(n\) bits number. All columns of jet signals could be compared to the same group of transition points for ideal ADCs or a different group of transition points for non-ideal ADCs. Note that \(2^{n}{-}m^{2}z\) is 4096; however, it is not the equivalent FWC. In fact, the equivalent FWC is \((2^{n}{-}1){\cdot}m^{2}z\), which is a little smaller than 4096. In the image reconstruction process, the sum of a jet data cubic is multiplied by \(255/(2^{n}{-}m^{2}z)\). In the discussion about the mismatch of ADC transition points, a pre-amplifier is expected to be used before an ADC due to the extremely small signals (e.g., maximally about 10 millivolts for a bit depth \(n=4\)), so the \(2^{n}{-}1\) transition points of an ADC have the same deviation from their ideal values. The reference image is reconstructed with ideal ADC transition points \(U_{th}(k)=k{-}0.5\) for \(m=4\), \(z=16\), and an ADC bit depth \(n=4\) as shown in Figure 10(a). In Figure 10(b), the mismatch of transition points with standard deviation \(U_{thn}=0.21\), which is suggested in Section 2, is introduced into the quantization process, and its GSI is 1.13 LSB. Compared to Figure 8(c), there are no obvious stripes even though the \(m\) value is smaller. The reason is that there are only two jet states in the single-bit QIS and a bit error means a totally opposite result; however, in the multi-bit QIS, for example, quantizing eight photoelectrons in a jet as nine photoelectrons has a very small impact on a four-bit QIS with 16 states in total. That could also explain why the BER could not be used to compare reconstructed images with different bit depths. Figure 10(c) is a reconstructed image for a lower bit depth, and some slight stripes can be observed. The GSI of this image is 1.93 LSB. It proves that, among the multi-bit QISs with different bit depths, the one with the highest bit depth has the highest tolerance for the mismatch of ADC transition points. Just like the single-bit QIS, the spatial oversampling factor in the multi-bit QIS influences the image quality. Figure 10(d) shows an image reconstructed from the same condition as Figure 10(b), except that \(m=1\) and \(z=256\). Its GSI is 2.06 LSB, and slight stripes can be observed. The GSI values for different bit depths and spatial oversampling factors are summarized in Table 2.  
The design value of the transition point \(U_{th}\) is 0.5 in the single-bit QIS and \(U_{th}(k)\) is \(k{-}0.5\) in the multi-bit QIS here; however, it could be adjusted to the global exposure level (controlled by the illumination factor \(h_{0}\) in the models) to achieve a lower BER [12]. So, in a practical application, the QIS may obtain better images with the same transition points mismatch conditions that were simulated in the former discussion through adjusting the design values of the ADC transition points. Similarly, the relationship of the FPN of the reconstructed images and the standard deviations of the transition points of the multi-bit QIS is plotted in Figure 12 with \(m=4\), \(z=16\), and \(n=4\). It also proves that the FPN increases with a stronger transition point mismatch. Usually, the FPN could be calibrated through image-processing techniques; however, this is not suggested in a QIS. A QIS is actually a photon counting system and needs to record the temporal and spatial arrival information of photons as much as possible. Image-processing techniques could smooth the image but lead to arrival information loss and distortion; so, realizing a small transition points mismatch through circuit design is a better choice.