Last time we discussed the effects of column noise on the PTC, this time we highlight the row noise. Row noise can have a temporal character as well as an FPN part, but moreover, this FPN can be repetitive. For instance, a certain FPN pattern will be repeated every x lines. The strategy in this study is always the same : set all other noise sources to zero, except the generation of the dark current and the dark-current non-uniformities (DSNU), and check how the row noise components influence the PTC.
By means of the mathematical model 100+ dark images were generated at various exposure times (between 0 s and 65 s). The result of this exercise in dark can be seen in the following six figures :
- Figure 1 contains the average dark signal (left axis), and its fixed-pattern noise component (right axis) as a function of the integration time (horizontal axis). (See previous blogs to learn how the calculation of the fixed-pattern noise is done.)
- Figure 2 shows the dark fixed-pattern noise versus the dark signal, based on the data shown in Figure 1.
- Figure 3 shows the row fixed-pattern noise (measured in dark) as a function of the integration time.
- Figure 4 shows the Fourier transform of the row fixed-pattern noise.
- Figure 5 contains the average dark signal (left axis), and its temporal noise component (right axis) as a function of the integration time (horizontal axis). (See previous blogs to learn how the calculation of the fixed-pattern noise is done.)
- Figure 6 shows the dark current temporal noise versus the dark signal, based on the data shown in Figure 5.
Figure 1 : Dark current and its FPN component as a function of the exposure time.
As can be seen in Figure 1, the average dark signal is linear with the integration time, at least for the exposure times that do not saturate the pixel. This indicates that the dark current is responsible for the signal in dark. The linear relation between the dark signal and the exposure time (texp expressed in ms !), shown in Figure 1, holds for the linear part of the curve. Notice that the expression, as well as the curve, shows the presence of a DC offset. The curve of the fixed-pattern noise, shown on the right axis, is not influenced by this DC offset. The slope of the curve representing the FPN as a function of the integration or exposure time is not influenced by the row FPN either. This is not surprising because any row FPN will be totally independent of the exposure time.
From the two formulas shown, it can be deduced that the FPN component is 1/6.6 or 15.2 % of the dark signal in the linear region and becomes 4.9 % of the full-well level when the pixels are saturated. The latter is representing the pixel non-uniformity in saturation. Notice also the change in offset contained in the noise formula : the offset was (see previous blogs) 0.168 DN, and is now changed in 0.260 DN. The offset is referring to the FPN at an exposure time equal to 0 s, and the increase in FPN can be attributed for 100 % to the increase in row FPN. But the curve mentioned in Figure 1 is the FPN on pixel level !
Figure 2 : Dark FPN versus dark signal.
The corresponding “PTC” curve is illustrated in Figure 2 : the FPN versus the (dark) signal is shown. From this PTC curve several interesting parameters can be deduced :
- The DSNU can be found to be equal to : 1/100.816 = 0.153 or 15.3 % at 30oC,
- The saturation non-uniformity : 102.138 = 137 DN.
Notice that also Figure 2 shows the data on pixel level. Next, to find more information with respect to the row behavior, the FPN is calculated on row level. This can be done by calculating an average value for every row, and then, calculating the standard deviation on these row-average values. The result of this exercise is shown in Figure 3.
Figure 3 : Row-level FPN.
The graph of row FPN versus the exposure time can be explained as follows :
- For very low values of the exposure time, the FPN is dominated by the FPN introduced by the row circuitry, and is independent of the exposure time. It can be found that the row FPN is equal to 10-0.268 DN = 0.54 DN,
- For larger values of the exposure time, the FPN proportionally increases with the exposure time. In this region, the DSNU on pixel level is the dominant source of FPN, even if all pixels within a row are averaged. The DSNU (on row level !) found is equal to 1/Sd at the moment the noise is equal to 1 DN. This situation happens at an exposure time equal to 103.090 ms = 1.230 s. From Figure 1, it can be deduced that for an integration time of 1.230 s a signal value equal to 80 DN can be found. Or the row level FPN is equal to 1/Sd = 1/80 = 0.0125 = 1.25 %, which is a factor of 16/1.25 = 12.8 lower than the same parameter on pixel level presented in Figure 2.
- The FPN curve reaches a maximum and then the FPN moves to a steady-state value of 101.030 DN = 10.72 DN due to the anti-blooming non-uniformities on row level. This value is a factor 137/10.72 = 12.8 lower than the same parameter on pixel level.
Notice that the reduction of FPN (in the non-saturated region as well as in the saturated region) from pixel level to row level equals to a factor of 12.8. This value should not be a surprise, because it is the reduction in noise after averaging the noise of 160 pixel values into one row value : sqrt(160) = 12.6 !
Figure 4 : Fourier transform of the row fixed-pattern noise.
Something not yet discussed in the study of the PTC in relation to the various noise sources is the calculation illustrated in Figure 4 : the row-FPN is shown in the frequency domain. To get this result, the fixed-pattern noise on row level, as shown in Figure 3, is Fourier transformed. The outcome is shown on the vertical axis of Figure 4, as a function of the sample frequency. (The sample frequency is defined by the number of lines.) As can be seen, a very regular frequency pattern is popping up at multiples of 0.0625 times the sample frequency. In combination with the 512 lines of the imager, this corresponds to a repetitive row fixed-pattern noise with a repetition period of 0.0313 x 512 = 16 lines. Finding a certain repetitive pattern in the FPN can give very important information to find the root cause of the FPN during the optimization process of a sensor and/or camera design. [Remark : all data shown in this blog is based on an imager size of 160 (H) x 120 (V) pixels, except Fig.4 which is based on an imager with 160 (H) x 512 (V) pixels, all other parameters remained the same for all calculations.]
Figure 5 : Dark current and its temporal noise component as a function of the exposure time.
Figure 5 shows the dark signal and the temporal noise as function of the exposure time. Not really that much new information can be extracted from these graphs, except the minimum temporal pixel noise being equal to 0.98 DN.
Figure 6 : “PTC” of the sensor.
Figure 6 shows the real Photon Transfer Curve, in which the temporal noise is shown as a function of the signal. The curve shows an indication of the part that is (nearly) independent of the dark signal (with a slope of 0) as well as the part that is directly depending on the dark signal (with a slope of 0.5). The region in the graph showing a collapsing curve indicates the saturation of the pixels.
From the PTC curve the following parameters can be deduced :
- The conversion gain, being equal to 1/100.724 DN/e- = 0.188 DN/e-,
- The total temporal pixel noise (without any influence of the dark current) = 100.010 DN = 1.02 DN, this minimum noise level is representing the temporal row noise because all other temporal noise sources are set to zero (even the dark current shot noise is zero at an exposure time equal to zero),
- The onset of anti-blooming = 103.32 DN = 2090 DN,
- The saturation level of the pixels = 103.45 DN = 2818 DN,
Conclusion : by generating images in dark at different exposure times, very important information can be extracted w.r.t. to further optimization of a new sensor/camera or to evaluate an existing sensor/camera. The methods described are strong tools when it comes down to benchmark existing products available on the market.
Next blog will focus on the noise contribution of the output amplifier (CDS, AGC, …).
Albert 2010-01-21