Doug Kerr
Well-known member
The emergence of Canon dSLR cameras with 14-bit digitization of the sensor signals systems naturally increases interest in what properties of the imaging system are changed by the increase in "bit depth" from 12 to 14 bits, and how.
On area of inquiry is the effect of this change on the noise in the digitized sensor "signal". I have recently done an investigation of this, and would like to briefly report my findings here. A compete description of my investigation and discussion of the results are planned for a tutorial article that will hopefully issue in a short time.
What this is not about
Quantization can be described as taking a continuous variable (one that can take on any imaginable value, at least over a certain range) and "rounding" it to one of a finite set of discrete values. Doing so of course introduces error into the "delivered" data. If rounding is , for example, to integral multiples of 0.1 unit, then a value of the actual data of 0.10639285 units will be "reported" (and generally digitized) as 0.1 unit - an error of -0.00639285 units.
In the field of the quantizing of an audio or video waveform (in connection with converting it to digital form), it is common to speak of the ongoing discrepancy between the original and delivered waveforms, as a result of this quantizing error in the individual samples of the waveform, as "quantizing noise". This conceit is very useful in that area of work, as it allows the degree of quantizing distortion, as perceived by the listener or viewer, to be "scored" with the same doctrines used to assess the degree of noise of the more conventional sort.
It is my opinion that this outlook has no clear direct application to the matter of the quantization of sensor levels in a digital camera. The arguments for my position are not complex, but I will forgo them here in the interest of moving ahead. They will be presented in the forthcoming article.
In any case, we are not speaking here of "noise created by quantizing".
Rather, we will speak of the effect of quantizing on noise already in the analog signal delivered by the sensor.
Predicates
In what follows, I have assumed:
--That the noise on the sensel voltage is random and follows a classical normal statistical distribution.
--That we characterize the extent of this source noise in terms of its RMS (root-mean-square) value. Note that this is synonymous with the standard deviation of the "signal" with the noise on it.
--That we will assess the extent of the noise in the quantized signal on that same basis.
The anaysis
We start with a certain assumed level of noise on the sensel signals, and model the resulting composite signal as quantized with a certain quantizing interval. We then look at the RMS value of the random component of the quantized signal (the measure of the noise seen in the quantized signal).
Then we maintain the same level of noise on the sensel signals, but increase the quantizing interval (for example, if we were to reduce the bit depth of digitization from 14 bits to 12, the quantizing interval is increased by a factor of 4). Again, we look at the RMS value of the random (noise) component of the quantized signal.
The result
We find that with the larger (coarser) quantizing interval, the RMS value of the noise in the quantized signal is indeed larger than with the smaller quantizing interval.
But the change is very small.
For example, the noise on the sensel signals is 8 units RMS, and the quantizing interval is 1 unit, then the noise in the quantized signal will be about 7.92 units RMS.
Now, if we enlarge the quantizing interval to 4 units (as if by reducing the bit depth by two bits), the noise in the quantized signal increases to about 8.03 units RMS.
For the purists here, this is based on the assumption that the base sensel signal (without the noise riding on it) falls in the center of a quantizing band. For the more general case in which the value of the underlying sensel voltage is random over the range of the quantizer, the result is not much different.
The bottom line
Does an increase in bit depth result in a decrease in the RMS value of the sensel noise as seen in the quantized (and of course digitized) signal? Yes.
By a significant amount? No.
Effect of original level of the noise
By the way, the greater the level of the sensor noise, the less is it increased in the quantized signal by enlargement of the quantizing interval (reduction of the bit depth).
On area of inquiry is the effect of this change on the noise in the digitized sensor "signal". I have recently done an investigation of this, and would like to briefly report my findings here. A compete description of my investigation and discussion of the results are planned for a tutorial article that will hopefully issue in a short time.
What this is not about
Quantization can be described as taking a continuous variable (one that can take on any imaginable value, at least over a certain range) and "rounding" it to one of a finite set of discrete values. Doing so of course introduces error into the "delivered" data. If rounding is , for example, to integral multiples of 0.1 unit, then a value of the actual data of 0.10639285 units will be "reported" (and generally digitized) as 0.1 unit - an error of -0.00639285 units.
In the field of the quantizing of an audio or video waveform (in connection with converting it to digital form), it is common to speak of the ongoing discrepancy between the original and delivered waveforms, as a result of this quantizing error in the individual samples of the waveform, as "quantizing noise". This conceit is very useful in that area of work, as it allows the degree of quantizing distortion, as perceived by the listener or viewer, to be "scored" with the same doctrines used to assess the degree of noise of the more conventional sort.
It is my opinion that this outlook has no clear direct application to the matter of the quantization of sensor levels in a digital camera. The arguments for my position are not complex, but I will forgo them here in the interest of moving ahead. They will be presented in the forthcoming article.
In any case, we are not speaking here of "noise created by quantizing".
Rather, we will speak of the effect of quantizing on noise already in the analog signal delivered by the sensor.
Predicates
In what follows, I have assumed:
--That the noise on the sensel voltage is random and follows a classical normal statistical distribution.
--That we characterize the extent of this source noise in terms of its RMS (root-mean-square) value. Note that this is synonymous with the standard deviation of the "signal" with the noise on it.
--That we will assess the extent of the noise in the quantized signal on that same basis.
The anaysis
We start with a certain assumed level of noise on the sensel signals, and model the resulting composite signal as quantized with a certain quantizing interval. We then look at the RMS value of the random component of the quantized signal (the measure of the noise seen in the quantized signal).
Then we maintain the same level of noise on the sensel signals, but increase the quantizing interval (for example, if we were to reduce the bit depth of digitization from 14 bits to 12, the quantizing interval is increased by a factor of 4). Again, we look at the RMS value of the random (noise) component of the quantized signal.
The result
We find that with the larger (coarser) quantizing interval, the RMS value of the noise in the quantized signal is indeed larger than with the smaller quantizing interval.
But the change is very small.
For example, the noise on the sensel signals is 8 units RMS, and the quantizing interval is 1 unit, then the noise in the quantized signal will be about 7.92 units RMS.
Now, if we enlarge the quantizing interval to 4 units (as if by reducing the bit depth by two bits), the noise in the quantized signal increases to about 8.03 units RMS.
For the purists here, this is based on the assumption that the base sensel signal (without the noise riding on it) falls in the center of a quantizing band. For the more general case in which the value of the underlying sensel voltage is random over the range of the quantizer, the result is not much different.
The bottom line
Does an increase in bit depth result in a decrease in the RMS value of the sensel noise as seen in the quantized (and of course digitized) signal? Yes.
By a significant amount? No.
Effect of original level of the noise
By the way, the greater the level of the sensor noise, the less is it increased in the quantized signal by enlargement of the quantizing interval (reduction of the bit depth).