After studying the overall dark-current effect on the Photon Transfer Curve (PTC), it is now time to take a closer look to the effect of the dark-current fixed-pattern noise.  To do so, the same structure as described earlier is used.  The dark current and its shot noise component are unchanged, only the extra dark FPN is added to the model. 

Again, at each chosen exposure time, 100+ dark images are generated.  The exposure time was varied from 0 s till 65 s. The result of this exercise in dark can be seen in the following four figures :

-       Figure 1 contains the average dark signal (left axis) of the generated images and its noise component (right axis) as a function of the integration time (horizontal axis).  To find the noise, all images were averaged on pixel level and the standard deviation on the resulting image was calculated.  This method allows obtaining the FPN, because the calculation is done on the averaged pixel signals and consequently, the temporal noise component is excluded (or at least strongly reduced). 

-       Figure 2 shows the dark fixed-pattern noise versus the dark signal, based on the data shown in Figure 1.  Figure 2 is very similar to the Photon Transfer Curve, keeping in mind that this time the FPN is shown.

-       Figure 3 contains the average dark signal (left axis) of the generated images and its temporal noise component (right axis) as a function of the integration time (horizontal axis).  Because only the temporal noise is shown in this graph, it is exactly the same curve as Figure 2 in the previous blog,

-       Figure 4 shows the dark current temporal noise versus the dark signal, based on the data shown in Figure 3.  Also in this case, Figure 4 is a copy of Figure 3 of the previous blog. 

 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, indicating that the dark current is responsible for the signal in dark.  The relation between the dark signal and the exposure time (t_exp expressed in ms !) is also indicated in Figure 1. The FPN noise as a function of exposure time is shown on the right axis.  Because the FPN is composed only out of dark current non-uniformities, the relation between the FPN and the exposure time is linear as well.  The empirical relationship is shown in Figure 1 (t_exp expressed in ms).  From the two formulas shown, it can be deduced that the FPN component is 1/6.6 or 15.2 % of the dark signal.

Figure 2 : Dark FPN versus dark signal.

 The corresponding “PTC” curve is illustrated in Figure 2 : the FPN versus the dark signal is shown.  The measured signal of the pixel in dark, S_tot (expressed in DN, digital number) can be written as :

                                                S_tot = k · N_d

with k being the overall gain [DN/electron] of the image sensor and N_d [electrons] the number of electrons generated in dark.  The total noise at the output of the sensor, n_tot, is related to the noise in the pixel by (all noise sources are set to zero, except the dark current shot noise and the dark FPN) :

                                                n_tot = k ·(n_d² + n_DSNU²)^0.5

with n_d and n_DSNU being the dark current shot noise and the Dark Signal Non-Uniformity (DSNU) or dark FPN respectively.  The relation between n_DSNU and N_d is :

                                                n_DSNU = a · N_d.

with a being a constant.  In the case only FPN is considered (as shown in Figure 2 because the temporal noise is averaged out), and by combining the aforementioned formulas, the relation between the measured signal S_tot and its measured noise component n_tot can be written as :

                                                n_tot = k · a · N_d = k · a · (S_tot/k) = a · S_tot

or :      

                                                log(n_tot) = log(a) + log(S_tot).

From Figure 2, the following can be learned :

-       its slope should be equal to 1, because of the linear relationship between the (logarithmic of the) dark current signal and the (logarithmic of the) dark FPN.  The actual value of the slope is a perfect match to the theory !

-       the intersection of the curve with the horizontal axis can deliver the constant a mentioned in the formulas and indicating the amount of FPN present in the system : with an intersection at 0.821 DN, this gives a = 1/10^-0.821 = 0.155 or 15.5 %.  This result is in perfect agreement with the number obtained from Figure 1.

  Figure 3 : Dark current and its temporal noise component as a function of the exposure time.

Figure 4 : “PTC” of the sensor with only dark current shot noise.

 Figures 3 and 4 are showing the dark signal with its temporal noise component, being the dark shot noise.  These curves are basically the same as in the previous blog because they only contain the temporal noise component and that has not changed by adding dark FPN.  That means that also the same parameters can be deduced :

-       conversion gain : 1/10^0.796 = 0.160 DN/electron,

-       dark current level : 0.066·t_exp = 66 DN/pixel/second = 412.5 electrons/pixel/second at 30^oC (this is the temperature at which the images are being generated).

Knowing the conversion gain and/or the dark current, also the FPN can be calculated to be equal to : 0.010·t_exp = 10 DN/pixel/second = 62.5 electrons/pixel/second at 30^oC.

In Figure 4 the “PTC” curve after averaging all individual noise values of the pixels is shown by the thick line, but also some curves of individual pixels are shown by the thin lines.  Clearly visible is the noisy behavior of single-pixel curves, while the curve of all pixels averaged is a straight line.

Conclusion : after obtaining the 100+ images in dark two operations took place : averaging and standard deviation calculation.  Averaging first and consequently calculation of the standard deviation results in the characterization of the FPN.  Standard deviation calculation first followed by the averaging results in the characterization of the temporal noise.  

Next time the influence of the temperature on dark current, DSNU and the PTC will be described.

Albert 2009-08-31

As promised, I would like to share with you my findings w.r.t. the effect of various noise sources on the Photon Transfer Curve (PTC).  To do so, I developed a noise model for a CMOS imager with 4T pixels.  The basic architecture of the hypothetical sensor is shown in Figure 1.Figure 1 : Architecture of the hypothetical sensor used in the noise model.

The pixel is based on a pinned-photodiode in combination with a transfer gate (TX), reset transistor (RST) and addressing transistor (RS), each column contains the bias current source, the circuitry to perform the correlated-doubling sampling (with sampling pulses SR for the reset level and SS for video signal level), and on chip level, the programmable gain amplifier (PGA) and an analog-to-digital converter (ADC) are included.

In a first experiment, only the dark current and its shot noise component are included in the model, all other noise sources are set to zero.  At each chosen exposure time, 100+ dark images were generated.  The exposure time was varied from 0 s till 65 s, the latter makes sure that the pixels can be fully saturated with dark current.

The result of this exercise can be seen in the following two figures :

-          Figure 2 contains the average dark signal (left axis) of the generated images and its noise component (right axis) as a function of the integration time (horizontal axis).  To find the noise, the standard deviation on pixel level was calculated, and next the obtained noise results were averaged.  This method allows to obtain the temporal noise, because the calculation is done on pixel level and consequently, FPN is excluded.  So every point on the curves is the result of all pixels in 100+ images,

-          Figure 3 shows the dark current shot noise versus the dark signal, based on the data shown in Figure 2.  Because the noise shown is the temporal noise, Figure 3 is basically the same as the original PTC (described by Jim Janesick in his book “Photon Transfer, SPIE Press, 2007”) that can be generated by means of light input.

Figure 2 : Dark signal (red) and temporal noise (blue) versus exposure time.

As can be seen in Figure 2, the average dark signal is linear with the integration time, indicating that the dark current is responsible for the signal in dark.  The relation between dark signal and exposure time (t_exp expressed in ms !) is indicated in Figure 2 as well. The temporal noise, as a function of exposure time, is shown on the right axis.  Because the noise model only contains the dark current generation, the temporal noise component represents the dark current shot noise.

The corresponding “PTC” curve is illustrated in Figure 3 : the temporal noise versus the dark signal is shown.

Figure 3 : “PTC” of the sensor with only dark current shot noise.

The measured signal of the pixel in dark, S_tot (expressed in DN, digital number) can be written as :

                                                S_tot = k · N_d

with k being the overall gain [DN/electron] of the image sensor and N_d [electrons] the number of electrons generated in dark present in every pixel.  The total noise at the output of the sensor, n_tot, is related to the noise in the pixel by (all noise sources are set to zero, except the dark current shot noise) :

                                                n_tot = k · n_d

with n_d being the dark current shot noise.  The relation between n_d and N_d is determined by the Poisson statistics :

                                                n_d = N_d^0.5.

By combining the aforementioned formulas, the relation between the measured signal S_tot and its measured noise component n_tot can be written as :

                                                n_tot = (k · S_tot)^0.5

or :      

                                                log(n_tot) = 0.5 · log(k) + 0.5 · log(S_tot).

From the actual PTC curve, the following can be learned :

-          its slope should be equal to 0.5, because of the square-root relationship between the dark current signal and the dark current shot noise.  The actual value of the slope is 0.499, so a perfect match to the theory !

-          the intersection of the curve with the horizontal axis can deliver the gain of the overall system : a value of 0.811 is found, which can be converted into a gain of 0.155 DN/electron.

From Figure 2, it can be seen that the relationship between dark current signal and the exposure time (expressed in ms) is equal to :

0.066 · t_exp + 0.169.  

Taking into account the conversion gain, this can be translated in a dark current equal to 426 electrons/pixel/s at the data collection temperature, being 30 ^oC.

It is interesting to realize that the dark current, expressed in electrons/pixel/s, can be easily obtained from the PTC curve in dark.  One important remark : the absolute dark signal should be available, in many cameras only a relative dark signal is measurable (= dark signal in the pixel - dark signal in the black reference pixel).

Next time, the dark current fixed-pattern noise will be added to the noise model and its effect on the PTC curve will be investigated.

Albert 2009-08-19

This blog started about one month ago and the main purpose is to share information and/or to stimulate discussion in the field of solid-state imaging.  After two general topic stories, I will soon start with items having more technical content, for example, the various noise sources in image sensors.  This plan of introducing more technical items does not imply that every future article published in this blog will have technical or scientific information.  Depending on my inspiration I will publish also non-technical stuff in between.

 A while ago I started working on a new course program : a two-day or three-day course with hands-on measurements on existing, off-the-shelf solid-state cameras.  The basic idea for this new course is to use the photon-transfer curve to do the evaluation of a sensor/camera.  The photon-transfer curve (also called “mean-variance method” or “photon-shot noise method”) was published for the first time in the 80’s by Jim Janesick (those days working at JPL).  Long time ago I had a discussion with Jim about his measurement technique.  He told me that he discovered more and new benefits of the photon-transfer curve (PTC) each time he worked on it.  To my own surprise I have to admit that Jim was right.  The PTC is a great evaluation tool, because basically you do not need to know what is inside the sensor/camera under test, and absolute light measurements are not needed.  All together, the PTC method is an easy way to get a lot of information (performance parameters and characteristics) about the sensor/camera under test.

During the preparation of the new course material I developed a noise model for a 4T CMOS image sensor.  The model describes the fixed-pattern noise as well as the temporal noise present in rolling-shutter pinned-photodiode pixels (PPD).  Next to the PPD, the imager is provided with column-level CDS, a global output amplifier and an on-chip analog-to-digital converter.  The model takes into account also the noise generated by these peripheral circuits.  By using this noise model I do have the possibility to change every single noise source in the hypothetical CMOS imager and to study its effect on the PTC. 

In the near future I will share this information with you through this blog.  In the very first publication I will focus on the dark current shot noise and its effect on the PTC.  The “P” of the PTC stands for “photon”, but the basic PTC idea can be successfully applied to the dark current present in the imagers and its shot noise component as well.  In other words, in the coming weeks I will describe how to apply the photon transfer curve without a single photon coming to the imager. 

Albert 2009-08-06