Doug Kerr
Well-known member
Introduction
An important imperfection in digital imaging systems is "noise", by which we essentially mean a random variation in the reported luminance of points in the image having an actual constant luminance.
Our human perception of this is such that the greater the actual luminance, the greater must be the amount of random variation to produce a certain "degradation". Thus, borrowing a term from electrical engineering, we speak of the quantity "signal to noise ratio" (SNR) as the metric of noise impact.
In our case, "signal" means the digital output that corresponds to the actual luminance (essentially, the average of the digital outputs for many instances), and "noise" mans a characterization of the degree of (random) variation of the digital output over those many instances.
Often the SNR is described in the "unit" decibel (dB). Here we will see what that means.
In the electrical domain
Since the concept of SNR is adapted from the work of electrical engineering, it is useful to see how it works there.
An electrical "signal" is a variation of instantaneous voltage. The instantaneous power represented by the signal at any instant is proportional to the square of the instantaneous voltage.
The average power of the signal (what a "signal power" meter would indicate) is thus proportional the the average over time of the square of the instantaneous voltage. It is the (average) power of an audio signal (converted to an acoustic signal), for example, which the human ear essentially recognizes as the "potency" of the signal.
In electrical engineering, we often use "decibel (dB) measure" to compare the potency of two signals (for example, the potency of the signal delivered by an amplifier vs. the potency of the signal sent into the amplifier).
The decibel measure of the comparison of two powers, P2 and P1, is:
where log is the common, or base 10, logarithm function. The values P2 and P1 are the average powers (over perhaps a short instant of time).
It turns out that if we take the instantaneous voltage of the signal, square it, average that over time, and take the square root of that, we will have a metric that, when squared, will indicate the relative (average) power of the signal.
This is called the root-mean-square (RMS) metric of the signal voltage. It is what a "true RMS" AC voltmeter is intended to indicate.
Now, of we wish to compare the relative powers of two signals, using decibel measure, when we know their RMS voltages, V1 and V2, the value is this
Suppose that our signal comprised an intended signal (which was in fact a constant voltage) plus an unwanted random variation: noise.
We can determine the (relative) power of the noise by subtracting out the constant voltage (which we know to be the signal), taking the RMS value of what remains, and squaring that. But mathematically, determining the RMS value is identical to determining the standard deviation of the overall voltage (not from zero, but in the usual way: relative to the mean voltage, which is in fact the "intended signal").
Back to photography
In digital imaging, the original "signal" is the profile of luminance across the image.
Luminance is actually directly related to the electrical concept of power. Nevertheless, in dealing with such concepts as the "noise" in our signal, we treat luminance as if it were instantaneous voltage.
The reason is that we have learned that the response of the human eye to random variations in luminance is roughly proportional to the square of the standard deviation of the luminance. This corresponds in the electrical case of the ear being responsive to the square of the standard deviation of the signal voltage.
Thus, in describing the signal-to-noise ratio, we compare:
• The standard deviation of the digital indication of luminance
to
• The mean (average) of the digital indication of luminance
We take that ratio, determine its logarithm, take 20 times that, and report the result in decibels (dB). Why 20 times? Because we treat luminance as instantaneous voltage in this matter.
Best regards,
Doug
An important imperfection in digital imaging systems is "noise", by which we essentially mean a random variation in the reported luminance of points in the image having an actual constant luminance.
Our human perception of this is such that the greater the actual luminance, the greater must be the amount of random variation to produce a certain "degradation". Thus, borrowing a term from electrical engineering, we speak of the quantity "signal to noise ratio" (SNR) as the metric of noise impact.
In our case, "signal" means the digital output that corresponds to the actual luminance (essentially, the average of the digital outputs for many instances), and "noise" mans a characterization of the degree of (random) variation of the digital output over those many instances.
Often the SNR is described in the "unit" decibel (dB). Here we will see what that means.
In the electrical domain
Since the concept of SNR is adapted from the work of electrical engineering, it is useful to see how it works there.
An electrical "signal" is a variation of instantaneous voltage. The instantaneous power represented by the signal at any instant is proportional to the square of the instantaneous voltage.
The average power of the signal (what a "signal power" meter would indicate) is thus proportional the the average over time of the square of the instantaneous voltage. It is the (average) power of an audio signal (converted to an acoustic signal), for example, which the human ear essentially recognizes as the "potency" of the signal.
In electrical engineering, we often use "decibel (dB) measure" to compare the potency of two signals (for example, the potency of the signal delivered by an amplifier vs. the potency of the signal sent into the amplifier).
The decibel measure of the comparison of two powers, P2 and P1, is:
10 log P2/P1 (decibels, abbreviated dB)
where log is the common, or base 10, logarithm function. The values P2 and P1 are the average powers (over perhaps a short instant of time).
The factor 10 comes from the fact that the basic logarithmic calculation gives a result in the unit bel, which is never actually used; rather we use a unit 1/10 that size (the decibel), and thus the number of them is 10 times as great.
In practice, we normally observe the instantaneous voltage of the signals, but need to infer from that their powers. To do that, we need to have some metric of the signal voltage that, when squared, will indicate the (average) signal power (at least in the relative sense that the powers of two signals can be compared).It turns out that if we take the instantaneous voltage of the signal, square it, average that over time, and take the square root of that, we will have a metric that, when squared, will indicate the relative (average) power of the signal.
This is called the root-mean-square (RMS) metric of the signal voltage. It is what a "true RMS" AC voltmeter is intended to indicate.
Now, of we wish to compare the relative powers of two signals, using decibel measure, when we know their RMS voltages, V1 and V2, the value is this
20 log (V1/V1) (dB)
The additional factor of 2 compared to the earlier expression reflects the fact that the relative power is proportional to the square of the relative voltage) and thus its logarithm would be twice as great.
Now it turns out that definition of the RMS value of the instantaneous voltage of a signal is exactly the same as the mathematical function standard deviation, relative to zero (not from the mean voltage, as would be the case in most mathematical uses of the standard deviation function).Suppose that our signal comprised an intended signal (which was in fact a constant voltage) plus an unwanted random variation: noise.
We can determine the (relative) power of the noise by subtracting out the constant voltage (which we know to be the signal), taking the RMS value of what remains, and squaring that. But mathematically, determining the RMS value is identical to determining the standard deviation of the overall voltage (not from zero, but in the usual way: relative to the mean voltage, which is in fact the "intended signal").
Back to photography
In digital imaging, the original "signal" is the profile of luminance across the image.
Luminance is actually directly related to the electrical concept of power. Nevertheless, in dealing with such concepts as the "noise" in our signal, we treat luminance as if it were instantaneous voltage.
The reason is that we have learned that the response of the human eye to random variations in luminance is roughly proportional to the square of the standard deviation of the luminance. This corresponds in the electrical case of the ear being responsive to the square of the standard deviation of the signal voltage.
Thus, in describing the signal-to-noise ratio, we compare:
• The standard deviation of the digital indication of luminance
to
• The mean (average) of the digital indication of luminance
We take that ratio, determine its logarithm, take 20 times that, and report the result in decibels (dB). Why 20 times? Because we treat luminance as instantaneous voltage in this matter.
Best regards,
Doug
