Uneducated people think it should be "double the lumens, double the brightness", which is clearly wrong. I thought that the general consensus in CPF was that to double the perceivable brightness, one needs to quadruble the intensity. That is the number I have referred multiple times outside CPF. I have also seen a reference to natural logarithm. And now you say that the intensity should be tripled.
I guess you should know the best, so, could you give a scientifically solid source which explains this phenomena. Preferably one which can be found online (generally accepted articles or good reviews in scientific journals are fine, too, I can access them through University). Next time I need to explain the phenomena to someone else, I want neither be uncertain about it myself, nor start the explanation with: "I don't know exactly, but...". Thanks.
The definitive work on detecting differences in stimuli was done by the perceptual psychologist S.S. Stevens. I have several of his classic research papers in PDF format, but none of the free image hosting services support multipage PDF's, so I can't post any of them.
Discussions of Stevens' work can be found in most any textbook on sensory perception or visual perception. One of the best, although it is a bit out of date now, has not been updated or revised, is
Visual Perception by Stephen Palmer, who was a professor of mine when I was an undergraduate at Cal Berkeley. Stevens proposed an exponential equation that describes the relationship between objective and perceived intensity, that largely holds for just about all common stimuli.
It's true that perceptual increases in many cases are logarithmic in nature, but in Stevens' equation, this is represented simply by using a fractional exponent. Each stimulus and its mode of presentation has its own exponent. To the best of my memory, Stevens found an average exponent of 3 across a wide variety of light sources and sizes, but I'll have to look it up to provide an exact value. So, as I mentioned, it is a tripling of the objective intensity of a light that produces a perceived doubling in brightness.
Mathematically, the total output in lumens = the intensity in lux multiplied by the surface area of illumination.
One of the problems with light is that the word "brightness" can mean both the total lumen output, or the intensity of light.
Thus, a greater total output in lumens is said to be brighter.
A greater intensity in lux, is also said to be brighter too.
So indeed, if we double the lumens, we double the brightness.
However, if we use the specific term lux, then doubling the lumens doesn't necessarily double the brightness in lux; it may be the same lux, illuminating double the surface area.
The eye's perception of the brightness is apparently logarithmic; I don't know the exact power of n.
However, with sound, we must multiply the power to the power of three, ie eight times, to double the sound volume...
The problem with this is that this is the objective measure only; it does not take into account the particulars of how the visual system works as a neurological information processing mechanism. This is why I consider lumens to be an inadequate measure of brightness, because it doesn't take actual the actual characteristics of visual perception into account.
And, things get a whole lot worse when you start varying the power spectra involved as well. For example, take two objectively bright lights, one red, and one blue (both of equivalent lumens, lux, etc.) Have an observer compare them side by side, and they will perceive the red one as brighter than the blue one, despite them both being of equal objective intensity. Why is this?
The answer lies in the proportion of the three types of photoreceptors in the retina. On average, 60% of all photoreceptors are long-wavelength (red) cones, while only about 10% are short-wavelength (blue) cones. The brain encodes brightness by the amount of incoming signal from the different types of receptors. Because we have many more red cones than blue cones, the signal the red cones sends to the brain is much stronger than the signal sent by the blue cones, and we perceive the red as brighter as a result. I won't even get into the Purkinje shift in color brightness that occurs in mesopic vision right now... I don't want to type it all out.
It's an interesting discussion, but when it comes to buying a new or upgraded light, a 'just noticeable difference' is meaningless to me. I'm much more interested in what the eye would consider a "significant difference," where you're confident one light is definitely brighter than the other, without needing to do careful side-by-sides or same-light-but-increase-brightness comparisons.
What I'm saying is, I'm more interested in the percentage increase where you'd say, "I can see more with the new model, so maybe it's time to buy a new light," rather than "I think maybe I saw a little difference on a side by side comparison, possibly." The lower boundary of perception just isn't compelling to me.
I've gotten the impression that the "significantly brighter" standard was around +100% brighter, and that tends to be how I space out my purchases, if possible. If my current light does 150, I probably won't rebuy until it gets close to 300, etc. Although I bet the "rebuy" percentage is different for everyone. There may be people who are willing to rebuy when the light is +50% brighter, or maybe even +25% brighter, just because they like having the latest and greatest. But that's a different sort of motivation. That's not a functional motivation anymore.
I would not even consider replacing for a +10-20% increase, because that (to me) isn't a functional difference for the use I put my lights to.
By definition, a JND is the smallest difference in objective intensity between two stimuli (in this case lights) for an observer to recognize they're different brightnesses. It is the intensity difference that produces a correct distinction 50% of the time an observer tries to tell if the two lights are of different brightnesses. So, the use of the term "significant" here really is only in terms of probability, not in terms of what would be considered a "useful" or "usable" difference.
The probability computations in all of this can get a little daunting, and require a fair amount of mathematics and statistics to accurately describe and compute. Simply put, in perceptual psychology we borrow the methods used in communications theory for signal detection. When an observer is trying to determine if two stimuli are different brightnesses, there are four possible outcomes; a "hit" is when the two are actually different, and the observer says they see that one is brighter than the other; a "miss" is when they are actually different, but the observer doesn't perceive the difference; a "false alarm" is when the two are not different, but the observer reports they are, and a "correct rejection" is when the lights aren't different, and the observer says they're not.
The advantage of this is this method provides a way to determine an observer's response bias; whether they tend to say "no" more often, or whether they tend to say "yes" more often. This can also be changed by motivations, or possible penalties for the detection task. If you reward observers with, say, 25 cents for every Hit, and penalize them 5 cents for every wrong answer, they're going to say Yes more often to maximize their reward. They'll get more Hits, but also more False Alarms. If you reverse these amounts, they'll say No more often to minimize their penalty. The important measure that's computed from this is called D-prime; the ratio of an observer's Hits to False Alarms. A D-prime value of 1 indicate's their just guessing; performing at random chance, and their individual ability to discriminate differences between the stimuli is poor, or they don't care about their performance and literally are just guessing. As D-prime increases, so does the observer's sensitivity to detect the difference; they get more and more Hits, but their False Alarm rate stays low. So, one person may produce more Hits than another, but if they also have a high False Alarm rate, they're actually not as sensitive as someone who produced less Hits, but many fewer False Alarms.
So, as you can see there is a lot of individual difference between observers in this. As I mention about Stevens' work above, the exponents he produced originate from averaging the results across a large sample of observers. What might be a usable difference for one person might not even be noticed by another person, such as someone with nyctalopia (night blindness). Stevens used a method called magnitude estimation to quantify his perceptual results. While there has been some justified criticism of the method, in general it works quite well, and Stevens' results have been very robust over time and are still quite valid and useful. Better methods do exist, such as signal detection methods I mention above, but they are much more labor and data intensive, even though they do allow the quantification of individual differences and more precise results.
In general though, across a large sample of observers, Stevens' Law and his results (unique exponents in his equation for different stimuli) are still the best heuristic we have for evaluating perceptual changes in an observer.
