Determining the dynamic range of a digital imaging system

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You can already read off the dynamic range from the x-axis of the previously determined characteristic curves, and it becomes clear that the dynamic range decreases with increasing sensitivity. Why this is the case becomes clear when we determine the corresponding values decisively and enter them in a table. To achieve this, we employ the following procedure to determine the missing values from the existing ones. It is sufficient to do this in each ISO level for the green channel. In the previously created table, all quantities are available in data values. To determine the dynamic range, we need them in electrons, and the conversion between the two is done with the conversion factor (gain). Additionally, we require the readout noise for our calculations. We determined the standard deviation at the beginning, which represents the combination of readout noise and recording noise. We determine both the gain and readout noise in the following way:

Convert the average data values and standard deviations
Depending on your camera, you may need to adjust the bit width of the A/D converter to get the correct values in electrons from the RAW data. If your model quantizes at 12-bit, you need to divide the values by 16. If it is 14-bit, you must divide by 4. The most practical approach is to add two more columns to the table you created above.

Determine the conversion factor (gain) and readout noise
For each ISO value and exposure level, have the spreadsheet plot the square of the standard deviation against the average data value on a graph with logarithmic axis division. Print the plot and connect the points with a straight line that fits as exactly as possible. The slope (gamma value) of this straight line is the inverse of the gain. You can read off the square of the readout noise where the straight line intersects the y-axis.

Determine the maximum signal and readout noise in electrons
Multiply the maximum data value and the readout noise in each ISO level by the gain to determine both values in electrons.

Determining the dynamic range
Divide the maximum signal by the readout noise to determine the linear dynamic range and then calculate the Log₂ of this value. This is the dynamic range in exposure stops.

The following tables show the values for two cameras with different pixel sizes. From these tables, it is evident that the dynamic range decreases as the maximum signal and readout noise, defined by their ratio in electrons, decrease with each increase in ISO. The context also shows that highlights lose dynamic range: Because there are fewer electrons per data value at higher sensitivities than at lower ones, the 4,096 data values provided by, say, 12-bit quantization are enough to accommodate 53,300 electrons at ISO 100 (4,096*13.02=53,300). At ISO 200, however, only 26,665 electrons fit into this range (4,096*6.51=26,665). Greater signal strength means greater brightness, but at ISO 200 all brightnesses above the calculated maximum signal are cut off (clipped).

Canon EOS 1D Mark II, 8,2 µm pixel pitch
ISO	Conversion factor (Gain, elektrons per data value)	Maximum signal in electrons	Readout noise in electrons	Dynamik range linear	Dynamik range in exposure stops	Maximum SNR
100	13,02	53 000	16,61	3190	11,6	230
200	6,51	26 500	8,95	2960	11,5	163
400	3,25	13 200	5,56	2380	11,2	115
800	1,63	6 620	4,04	1640	10,7	81
1600	0,84	3 310	3,90	850	9,7	58
3200	0,41	1 650	3,93	420	8,7	41

Canon S 70, 2,3 µm pixel pitch
ISO	Conversion factor (Gain, electrons per data value)	Maximum signal in electrons	Readout noise in electrons	Dynamik range linear	Dynamik range in exposure stops	Maximum SNR
50	2,06	8200	4,1	2000	11,0	91
100	1,03	4300	3,4	1260	10,3	66
200	0,51	2150	3,2	670	9,4	46
400	0,26	1080	4,3	250	8,0	3

And since we already have the quantities of recording noise and sensitivity together, let’s immediately clarify the relationship between the two, which tends to cause confusion.

We can model the total noise R according to the following formula:

Formula 15

Here, R0 is the noise component before the ISO amplifier, and R1 is the noise component after the ISO amplifier, whose gain factor is the value of G. According to this model, the noise component measured in digital data values increases with each increase in sensitivity because its partial value R1 is multiplied by an ever greater factor. So far, until it represents the full readout noise almost all by itself in the highest ISO level.

If, on the other hand, we represent the noise in electrons (RE2), which corresponds to its input quantity in photoelectrons, the picture is different. We perform the conversion by multiplying with the conversion factor g=U/G, where U represents a constant known as the universal gain and G represents the ISO setting.

Formula 16

Now the noise component R1 is divided by the ISO gain and becomes smaller with each increase in sensitivity. The R0 component, on the other hand, remains constant. In sum, the total readout noise measured in electrons thus decreases when the ISO value is increased. This behavior and the reciprocal relationship between exposure strength and ISO value described in the section „Sensitivity adjustment„, which leads to decreasing signal strength with increasing sensitivity, explain why the quantities of maximum signal and noise carpet and thus the dynamic range decrease with increasing sensitivity.

This leads to the conclusion, which at first glance seems to contradict intuition, that it is best for the signal-to-noise ratio to select the highest possible ISO setting for a given time/aperture combination because it is associated with the lowest readout noise. I will illuminate this thought process comprehensively in the section „Exposure determination for digital imaging systems“ .

Ah yes. Now, you’re absolutely correct if you believe that determining dynamic range is as simple as shooting a grayscale wedge and counting the exposure levels that show detail. Would you have learned as much about your digital camera’s inner workings if I had just explained and shown two examples?

Provided that you test all available ISO levels, i.e., also the thirds levels 125-160-250-320-500-640-1000-1250, you will notice with some cameras (preferably Canon models) that these show a stronger readout noise than the full levels 50-100-200-400-800-1600. This is because the generation of these intermediate values differs from that of the full levels.

On the xxD and xxxD models (30D, 40D, etc) Canon implements the intermediate values through software multiplication. For instance, when the camera is set to ISO 125, the electronics are actually set to ISO 100. However, the exposure is adjusted to a 1/3 higher value, resulting in a 1/3 stop underexposure. The RAW values are then multiplied by 1.25. This increases the readout noise without any other benefit. At ISO 160, the situation reverses. In this case, we set the electronics to ISO 200, adjust the exposure to a 1/3 lower value (1/3 stop underexposure), and then divide the .raw values by 1.25. While this lowers the readout noise, it also causes a loss of 1/3 stop latitude in the highlights. The histograms of the .raw data show that this is the case. They show holes in the x1.25 steps because the multiplication leaves 25% of the RAW values free. In the /1.25 steps, on the other hand, peaks can be seen because the division merges adjacent RAW values. 

On xD models (5D, 1D, etc), Canon achieves the intermediate values by using a second voltage amplifier before the A/D converter, which amplifies the values by factors of 1.25 or 1.6. This step adds additional noise to the signal. Quantitatively, the x1.25 position is somewhat as bad as the next full ISO step. The x1.6 step even has more readout noise than that. 

The lesson from this story is that only the full ISO levels, which have the least readout noise in each case, are suitable for achieving the best signal-to-noise ratio for camera models that exhibit this behavior.

Next The bit width of the A/D conversion

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