# The Quanta Image Sensor: Every Photon Counts
## 4 Read Noise and Counting Error Rates
### Quantization and Bin Counts  
In a single-bit QIS, the threshold level \(U_{TI}\) for setting the output to a logic \"1\" is \(U_{TI}\) = 0.5, so that the digital output is \"1\" for \(U>0.5\) and otherwise \"0\". For multi-bit QIS, additional thresholds are set at integer increments above \(U_{TI}\) (e.g., \(U_{TI}\) = 1.5, U\({}_{T3}\) = 2.5, etc.). All read out signals lying between two adjacent thresholds (a bin) result in the count for that bin \(C_{N}\) being incremented by one. False positive counts are generated when a read out signal is misquantized into the wrong bin due to noise  
Figure 7: Readout signal probability density as a function readout signal for a quanta exposure \(H\) = 2 and for various read noise levels. Examples with CG variation are shown in [21; 22]. and conversion gain variation. These false positives (and their corresponding false negatives) give rise to an error in the total count. An ensemble of \(M\) jots results in a total of \(M\) counts spread across the \(2^{n}\) bins of a multi-bit QIS. The total count in each bin can be used to determine the expected total number of photoelectrons collected by the ensemble \(N_{TOT}\) such that [21]:  
\[N_{TOT}=M\sum_{N=\ 0}^{\infty}N\cdot C_{N} \tag{11}\]  
For \(\sigma_{k}\lesssim 0.50\) e\(-\) r.m.s., the expected count in each bin for a single jot is given by:  
\[C_{N}\cong\sum_{k=\ N-\ 1}^{N+\ 1}\frac{1}{2}\mathbb{P}[k]\left[\text{erf} \left(\frac{N+\frac{1}{2}-k}{\sigma_{k}\sqrt{2}}\right)-\text{erf}\left(\frac {N-\frac{1}{2}-k}{\sigma_{k}\sqrt{2}}\right)\right] \tag{12}\]  
except for bin 0 and last bin \(2^{n}-1\) where in the former the bin extends to \(U=-\infty\), and in the latter to \(U=+\infty\), so that:  
\[C_{0}=\frac{1}{2}\sum_{k=\ 0}^{\infty}\mathbb{P}[k]\left[\text{erf}\left(\frac{ \frac{1}{2}-k}{\sigma_{k}\sqrt{2}}\right)+1\right] \tag{13}\]  
and:  
\[C_{2^{n}-\ 1}=\frac{1}{2}\sum_{k=\ 0}^{\infty}\mathbb{P}[k]\left[1-\text{erf} \left(\frac{(2^{n}-1)-\frac{1}{2}-k}{\sigma_{k}\sqrt{2}}\right)\right] \tag{14}\]  
In [21] it was found that for higher quanta exposures (\(H>0.2\)), the count was not significantly affected by (deep sub-electron) read noise nor conversion gain variation in the ensemble, assuming \(M\) was sufficiently large, and the non-linear Hurter-Driffield response dominated counting error in a predictable way. For lower quanta exposure, systematic count error was introduced by read noise. Essentially under sparse illumination conditions (\(H<0.1\)), even a small amount of read noise can cause excess counting by the occasional misquantization of the dominant \"0\" signal as \"1\". This systematic count error can result in counting rate error of 34% for \(H=0.1\) and read noise of 0.30 e\(-\) r.m.s. yet nearly no error at a read noise of 0.20 e\(-\) r.m.s. The systematic counting rate error increases dramatically for lower exposures, strongly indicating that for applications requiring accurate photon counting in this realm, read noise should be 0.15 e\(-\) r.m.s. or smaller. Count vs. quanta exposure is shown in Figure 8. Ideally the count should be equal to the quanta exposure leading to a linear relationship shown by the diagonal gray line (mostly obscured). For a 4b QIS with read noise of 0.15 e\(-\) r.m.s., (purple solid line), the count is nearly indistinguishable from the linear relationship. However, for higher read noise levels, significant systematic departure from the ideal behavior under sparse illumination conditions can be observed, independent of bit depth. At higher exposures, the Hurter-Driffield response dominates the non-linear behavior independently of read noise. In all cases, the impact of conversion gain variation in the ensemble is negligible if \(M\) is sufficiently large, and if not, then photoresponse non-uniformity can be an issue as in conventional CIS devices. The expected count in Equations (11)-(14) can be used to estimate the count for an ensemble of \(M\) jots.