Settling error scaling

Settling performance is specified directly or indirectly in terms of luminance errors. This is also true for measures regarding settling time or rise and fall times, as these measures depend on the definition of according luminance error bands. In order to be able to compare the results in a meaningful way across several conditions, like the different luminance step sizes, we need to transform the physical error measures or the underlying luminance measures into something more meaningful regarding visual perception – buzz word: Weber's law.

Weber's law is often mentioned in the same breath with Fechner's law and Stevens' power-law, ever so often disregarding that Weber's law is not just an earlier variant of the other two but – in a way – fundamentally different. Weber's law is about discrimination thresholds or just noticeable differences, whereas Fechner's and Stevens' laws are about perceived stimulus magnitudes. Fechner inferred his law from Weber's law by assuming, and thereby adding a new aspect to it, that there is a direct and potentially causal link between the discrimination thresholds and the perceived stimulus magnitudes on a large scale. This assumption might actually not be true, and this might possibly explain why Stevens' power-law usually describes magnitude perception better than Fechner's law. However, this does not necessarily invalidate also Weber's law just because Fechner's law is mathematically related to Weber's law and Stevens' law is not. It might just indicate that the assumption Fechner made about the link between discrimination thresholds and magnitude perception was not quite right.

Translated to the task of measuring settling performance, this means that if we wanted to pick gray levels with a perceptually equidistant spacing, we'd rather apply Stevens' power-law when computing the according physical luminances. On the other hand, if we want to interpret luminance errors in a perceptually meaningful way, we'd rather apply Weber's law, as these errors are small and more related to discrimination thresholds rather than to magnitude percepts.

As already mentioned, it is problematic to apply these laws to real-world scenarios, especially when dynamic stimuli come into play, which is the case here – otherwise we wouldn't need to care about settling times in the first place. Therefore, it does not make too much sense to stick to some complicated formulas pretending precision that is not really there, especially if these formulas require additional parameters like the black:white contrast or absolute luminance values. We better keep it simple.

Side note: Regarding the complexity of the related formulas, the low luminance regime, including the luminance for black, is a constant source of trouble. For instance, the CIELAB defines Lightness (L*) as the perceptual equivalent for physical luminance, but tricks have been pulled to mathematically handle the low end of the according function. At least the information found on Wikipedia and also in the Information Display Measurements Standard (IDMS) leaves it unclear how to apply this formula correctly, i.e., in a way that will lead to results which are as accurate as the formula's complexity implies. This is similarly true for the formula of the gamma function used in the sRGB color space], where the definition of the low end adds quite some complexity even though the connection to some specific black-white contrast or black luminance value is not reflected in there.

But even fitting a simple Gamma function is not as trivial as it might seem. For one, it makes quite a difference whether the gamma function allows the luminance value for black to be different from zero. Besides this black luminance issue, the sample placement and the criterion for the error minimization are of importance. So whenever you see an "is equivalent to a gamma value of ...", watch out!

Gray level selection

The gray levels are distributed uniformly in pixel value space assuming the monitor was calibrated to a gamma value of 2.2. Or, equivalent, the gamma transfer function is measured directly and the pixel values are picked just so that the according luminance values follow a transfer function with a gamma value of 2.2.

This particular gamma value has been chosen because it is the gamma value used in sRGB space (best fit to the exact sRGB gamma function). The formula for the CIELAB L* actually suggests an exponent of 3, which is probably inferred from Stevens' power law. However, fitting a gamma function as it is used here (transformation of normalized pixel values ranging from 0 to 1) yields a gamma value of 2.3 (assuming Yn=100 cd/m² and a luminance range of 0.1 to 100 cd/m², corresponding to a contrast of 1:1000). Given that the power value of 3 provided by Stevens apparently refers to static stimuli measured in a dark environment, and given that power values for other stimulus conditions seem to be smaller, considerably smaller even, the choice of 2.2 seems justifiable.

Side note: A gamma transfer function is normally just used for efficient coding purposes, where efficiency is related to the overall S/N ratio in terms of perceived signal and signal errors, so Weber's law should be more relevant than Steven's law. However, presuming Weber's law we would need a gamma value of 4.4 in order to minimize perceived errors (for a luminance range of 0.1 to 100 cd/m²). Nonetheless, for the sRGB space a power-law with an effective gamma of just 2.2 is used, which only shows how arbitrary all this is.

Luminance scaling

There are two effects which play a role when looking at luminance errors. The first effect is Weber's law; a 10% luminance error relative to the maximal luminance is perceived just more easily at dark gray levels than at bright gray levels. But this is actually only true when our visual system is adapted to the current gray level. As soon as larger luminance changes occur, like the luminance steps making up the settling matrix, adaptation effects come into play. Adaption, in short, shifts the favorable operating point of the visual system regarding luminance (as an example) to the luminance of the current stimulus. It is like shifting the gaze to some interesting location on the screen in order to investigate that location in further detail, namely with the high spatial resolution only provided by the fovea. Regarding luminance, if the operating point is not set to the luminance of the stimulus, contrast detection is deteriorated, and it does not matter so much whether the stimulus is actually too bright or too dark for the currently established operating point; both will be equally bad in terms of contrast detection – driving into the dark tunnel (being bright-adapted) is as bad as coming out of it (being dark-adapted). In other words, luminance errors are less perceivable after a big luminance step, no matter whether the step went from dark to bright or from bright to dark.

There are many different adaptation processes going on in the visual system, with different spatial and temporal properties. Some of them have time constants even down to the millisecond range, which means that the interpretation of luminance errors should not only depend on the level and step size but also on the time course of the luminance signal. However, even if we were able to correctly quantify the effects of the measured luminance steps (i.e., in a perceptually meaningful way), we would still ignore that these stimuli are dissociated from any spatial or temporal context and, therefore, the results might not be very conclusive for what matters in any real world application.

The standard procedure, so far, is to normalize luminance errors to the particular luminance step size (see also Information Display Measurements Standard (IDMS)). As long as there is no empirical evidence for a perceptually more meaningful scaling, we stick to this standard, being well aware though that the perceptual relevance of the derived results is debatable. It should be mentioned here, that the number as well as the distribution of the tested gray levels also affect average results. This should be kept in mind when comparing results across different test scenarios.

Although this standard scaling might not be fully correct in reflecting perceptual effects, it goes in the right direction. Look, for example, at steps 0-1 and 7-8. An absolute luminance error of say 5 cd/m² would result in a higher %-value for the 0-1 step, i.e., at dark levels, than for the 7-8 step, because the luminance difference which the %-value refers to is, due to the non-linear choice of levels (gamma=2.2), smaller between levels 0 and 1 than it is between levels 7 and 8. This means the same absolute error is more relevant (higher %-value) at dark levels than it is at bright levels, which is well in line with Weber's law. The same absolute luminance error would result in a much smaller %-value, i.e., would become much less relevant, when it occurred for stepping from level 0 to 8 or from level 8 to 0. This is well in line with the adaptation effect mentioned above, that is, because our visual system is kind of blinded when looking at large luminance changes, we are basically also blind to errors which otherwise would have been easily detectable at the respective target luminance.