# The Quanta Image Sensor: Every Photon Counts
## 3 Imaging Characteristics
### Signal-to-Noise Ratio (SNR) and Dynamic Range (DR)  
The use of exposure-referred \(SNR\) is useful for non-linear devices, especially when intrinsic signal noise drops near saturation. The \(SNR_{H}\) for normal readout of a single-bit QIS is given by [9]:  
Figure 4: Log signal and noise as a function of log exposure for multi-bit QIS jots with varying bit depth. The signal is the expected sum over 4096 jots (e.g., 16 \(\times\) 16 \(\times\) 16). Saturation signal is 4096 (\(2^{n}-1\)) (From [14]).  
\[SNR_{H}=\sqrt{M}\frac{H}{\sqrt{e^{H}-1}} \tag{8}\]  
where \(M\) is the number of jets in the ensemble or cubic used for the read signal sum. This assumes the read noise is low enough that the readout bit error rate (BER) [9; 14] does not significantly affect the sum--a reasonable assumption for the QIS target read noise of less than 0.15 e-- r.m.s. [9; 21]. For normal readout, the \(SNR_{H}\) reaches a maximum value at \(H\simeq\) 1.6 and the maximum value of \(SNR_{H}\) is approximately 0.8\(\sqrt{M}\). For multi-bit QIS, maximum \(SNR_{H}\) increases as \(\sim\)\(\sqrt{FWC}\).  
Dynamic range (\(DR\)) for the QIS is defined as the range between low signal \(H_{min}\) where \(SNR_{H}=1\) (essentially where the ensemble of read out jets has a total sum of one photoelectron) and high signal \(H_{max}\) where \(SNR_{H}\) drops back down to unity and lower due to saturation [9]. The \(DR\) depends on the size of the ensemble--more jets, higher dynamic range, and the \(DR\) scales approximately as \(M\), where \(M\) is the number of jets in the ensemble. The \(DR\) and the maximum value of \(SNR_{H}\) are shown in Figure 5 as a function of ensemble size, calculated using the expressions derived in [9]. At \(M=\) 4096, for example, the \(DR\) is 95 dB and the maximum value of \(SNR_{H}\) is 34 dB. This can be compared to a conventional CIS with FWC of 4096 e-- and read noise of 1.5 e-- r.m.s. The CIS would have a \(DR\) of 68 dB and maximum SNR of 36 dB assuming linear response. Increasing the bit depth does not substantially increase the \(DR\) due to the reduction in non-linearity (overexposure latitude) as the bit depth is increased. For example, going from single-bit to 2b QIS (3\(\times\) increase in FWC) only increases the \(DR\) by approximately 3 dB.  
The response of the single-bit QIS is only linear for \(H\lesssim\) 0.1 with increasing non-linearity above this exposure level. The non-linearity is a desirable feature for photography and cinematography, but not as much for some other applications, and undesirable for most scientific photon-counting applications. In the latter case, the linear response can be retrieved from the non-linear signal or the exposure must be kept so \(H\lesssim\) 0.1. Multi-bit QIS has a larger range of linearity as can be readily seen in Figure 4. As was noted in [14] multi-bit signals can be easily transformed into lower bit depth signals, or single-bit signals, by post-readout digital signal processing. This permits some flexibility in the response curve by trading FWC or flux capacity for linearity.  
Figure 5: Single-bit QIS normal readout dynamic range (blue) and maximum \(SNR_{H}\) as a function of ensemble or cubic size \(M\). For convenience, \(M=\) 4096 is highlighted by the purple dashed line. Note that \(M\) is confined to integer values despite the continuous nature of the curves in this figure.