After discussing the FPN and temporal noise generated in the pixels, the attention will now be shifted to the noise generated in the column circuitry. This can be FPN as well as temporal noise. To study these column-wise noise sources, all other noise sources will be set to zero, except the generation of the dark current and the dark-current non-uniformities (DSNU).
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 five 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 column fixed-pattern noise (measured in dark) as a function of the integration time.
- Figure 4 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 5 shows the dark current temporal noise versus the dark signal, based on the data shown in Figure 4.
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 show the presence of a DC offset. No effect of the column FPN can be seen. This is not surprising because the curve shown refers to the FPN on pixel level as a function of the integration time, and any column 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-uniformities in saturation.
Also notice the change in offset contained in the noise formula : the offset was (see previous blogs) 0.168 DN, and is now changed in 1.212 DN. The offset is referring to the FPN at an exposure time equal to 0 s, and the increase can be attributed for 100 % to the increase in column FPN. But the parameter 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 : 1/100.782 = 0.165 or 16.5 % at 30oC,
- The pixel FPN (without DSNU) : 100.278 DN = 1.897 DN (or after finding the conversion gain, this corresponds to 11.9 e-),
- The saturation non-uniformity : 102.138 = 137 DN (or after finding the conversion gain, this corresponds to 859 e-).
Notice that also Figure 2 shows the data on pixel level. To find more information with respect to the column behavior, the FPN is calculated on column level as well. This can be done by calculating an average value for all pixels in every column, and next, calculating the standard deviation on these column-average values. The result of this exercise is shown in Figure 3.
Figure 3 : Column-level FPN.
The graph of 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 column circuitry. It can be found that the column FPN is equal to 100.283 DN = 1.92 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 column are averaged. The DSNU (on column 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 103.003 ms = 1.007 s. From Figure 1, it can be deduced that for an integration time of 1.007 s a signal value equal to 66 DN can be found. Or the column level FPN is equal to 1/Sd = 1/66 = 0.015 = 1.5 %, which is a factor of 16/1.5 = 10.7 lower than the same parameter on pixel level,
- The FPN curve reaches a maximum and the FPN moves to a steady-state value of 101.112 DN = 12.94 DN due to the anti-blooming non-uniformities on column level. This value is a factor 137/12.94 = 10.6 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 column level equal is to a factor of 10.7. This value should not be a surprise, because it is the reduction in noise after averaging the noise of 120 pixel values into one column value : sqrt(120) = 11 !
Figure 4 : Dark current and its temporal noise component as a function of the exposure time.
Figure 4 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 0.97 DN.
Figure 5 : “PTC” of the sensor.
Figure 5 shows the real Photon Transfer Curve, in which the temporal noise is shown as a function of the signal. The curve shows a part that is independent of the dark signal (with a slope of 0) and a 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 1/100.753 DN/e- = 0.176 DN/e-, (this is a relative large value, due to the fact that the slope of the shot-noise limited part is not really equal to 0.5),
- The total temporal pixel noise (without any influence of the dark current) = 100.012 DN = 1.02 DN = 6.4 e-, this minimum noise level is representing the temporal column 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 = 13480 e-,
- The saturation level of the pixels = 103.45 DN = 2818 DN = 18180 e-,
Conclusion : it is amazing how many parameters can be deduced from the various curves shown in this part of the study. But it should be remarked that in this discussion the specific column-related parameters can be calculated in a simple way because all other noise sources are set to zero. It will be shown later that when more noise sources are influencing the output signal, it will be much more difficult to extract the column parameters …
Next time the effect of row noise will be investigated. As can be expected, it will be very similar to the discussion of the column noise, but with an very interesting extra characteristic : repetitive row noise. Come and see next time !
Albert 2010-01-06