27 September 2001

from: Dennis Couzin

to: ASTM Task Group on Flashing Light Measurement

re: Allard 1876

[Introduction]{.underline}

Emile Allard in 1876 proposed a remarkably simple formula \-- simple to
state, not to compute \-- for the visibility of flashing light.

> Visibility = the maximum value of i(t), where i(t) is the solution of
> the differential equation
>
> $\frac{\text{di}}{\text{dt}} = m \cdot \left( I(t) - i(t) \right)$
>
> where I(t) is the intensity of the light at time t, and m is a time
> constant for the visual system.

This is not intended as a measure of degree of visibility, but for
determining whether a detection threshold is met.

Françoise Viénot kindly supplied the twelve pages pertaining to flashing
light from Allard\'s 1876 publication. The differential equation is
reached by the third page. I\'ve studied these first pages, but not the
rest, which seem to be mathematical exercises upon the formula. The
pages leading to the differential equation are attached below as
Appendix A.

Allard arrives at his differential equation *a* *priori*. He appears to
make just one empirical assumption, about exponential decay, from
Newton! However his concept of \"impression élémentaire\" embodies
deeper assumptions, and when I reconstruct his mathematically soft
argument leading to the equation I find that his assumption about
exponential decay is superfluous \-- it follows from the deeper
assumptions.

The reconstruction puts Allard\'s argument on firmer mathematical
concepts, making his assumptions more obvious. It is given in the
exegesis below.

We will not learn about the visibility of a flash of light directly from
Allard\'s result. The value of his result is that it suggests a more
general model: the \"impression\" i(t), equals the mathematical
convolution of the \"lumiére\" I(t), with what I have elsewhere called
the visual impulse function q(t). By Allard\'s reasoning, q(t) must
equal m\*exp(-m\*t) for non-negative t. (Causality requires q(t) = 0 for
negative t.) I believe the visual impulse function is different from
Allard\'s q(t); it does not reach its maximum value at t = 0.

Possibly, with a well-chosen q(t), the visibility of a flash of light
can be calculated from the maximum value of a new i(t), the mathematical
convolution of I(t) with q(t). Possibly not. When we have good
psycho-physical data for a variety of flash forms, or better, a good
visual model, will we know.

Allard\'s original formula allowed a simple instrumental measurement of
flash visibility (although not in his time, and this was not his
reasoning). If a voltage proportional to light I(t) is applied to an
simple R-C circuit then a voltage proportional to impression i(t) is

![](media/image1.wmf){width="2.6479166666666667in"
height="0.9013888888888889in"}

measured after the capacitor. Indeed Allard\'s m = 1/RC. If we are very
very lucky, there is a new circuit producing the new i(t) from I(t).

[Exegesis of Allard 1876]{.underline}

p\. 62 Linearity. What Allard calls \"impressions\" are, for constant
lights, proportional to intensities.

p\. 63 Exponential decay of impressions. When a constant light ends
abruptly at time t=0, the impression, originally equal to I, obeys the
function:

i(t) = Ie^-mt^ (1)

Allard cites analogy with Isaac Newton\'s considerations on the cooling
of small bodies (and no more).

p\. 63 Perceptibility based on instantaneous impression. If j gives the
least perceptible impression, then setting making i(t) = j in equation
(1) gives q = log~e~(I/j)/m. q measures the duration of the perception.

p\. 63 Idea of the elementary impression. An abruptly ending light
decays exponentially.

di/dt = -mIe^-mt^ = -mi(t) (2)

So in the first dt of decay

> di = -mIdt (3)

If having the light off during dt results in an mIdt loss in impression,
then for any dt while the light is on, there must be mIdt gained in
impression. mIdt is the \"impression éleméntaire\" produced by the light
I during the time dt.

At the abrupt end of a steady light I the falloff follows function f(t).
So the falloff in the first dt equals

> If\'(0)dt (4)

\(A\) So during a steady light I, the light\'s \"contribution\" to
impression i from t to t+dt must equal - If\'(0)dt.

Allard next reasons on the onset of impression for a suddenly started
light. The light goes to I. The impression starts at 0, \[how do we know
this?\] and rises as function i(t). Consider the time interval t~1~ to
t~1~+dt.

\(B\) For the impression to just remain steady at level i(t) should
require a contribution of -i(t~1~)f\'(0)dt.

\(C\) However, the light I makes a contribution of -If\'(0)dt to
impression during the interval

\(D\) Therefore the net rise in impression during the time interval t~1~
to t~1~+dt is given by

di = -If\'(0)dt + i(t~1~)f\'(0)dt

di/dt = -f\'(0)(I - i) (5)

Aside from the strange use of contributions, the particular weakness in
the argument is at step (B). Step (B) appropriates from step (A)
although in (A), i(t) was steady while in (B) i(t) is rising. There is a
hidden assumption that what an impression does from t~1~ to t~1~+dt can
be determined from i(t~1~) together with I(t~1~) to I(t~1~+dt), i.e.,
that the past history of i need not be given.

Allard is presuming for decay

> f(t) = e^-mt^ (6)

so that f\'(0) = -m, and

di/dt = m(I - i) (7)

(Allard\'s argument would seem to work as well for any other falloff
function f, but we will see shortly why this isn\'t so.)

\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*\*

I now have an improved reconstruction of Allard\'s argument for equation
(7) which does not require his assumption of (6) for decay. Indeed it
implies (6).

Allard\'s notion of \"impression éleméntaire\" amounts to a contribution
by the moment\'s light to the moment\'s impression. Since the whole
impression is greater than that contribution we must wonder where it
comes from. It must come from the past. Suppose a simple conservative
model[^1]. Light I turns on at t=0, and follows function I(t).

From t=0 to arbitrary time t, the light gives

> $$\int_{u = 0}^{u = t}{I(u)\text{du}}$$

toward impression. If the impression follows function i(t) then during
the same time, it uses

$\int_{u = 0}^{u = t}{i(u)\text{du}}$

from what I gave. The remainder

> $\int_{u = 0}^{u = t}{\left( I(u) - i(u) \right)\text{du}}$

is stored for the future.

Now suppose the light is abruptly turned off at time t~0~. Allard\'s
prime assumption is that the decay curve for the impression follows some
function f(t), regardless of the light\'s and impression\'s history.
That is, if at t~0~ the impression has value i(t~0~), then after t~0~ it
must follow

i(t~0~)f(t-t~0~)

Conservation requires that the stored impression equals the integrated
impression through the decay.

> $$\int_{u = 0}^{u = t_{0}}{\left( I(u) - i(u) \right)\text{du} = \int_{u = t_{0}}^{\infty}{i(t_{0})f(u - t_{0})\text{du}}}$$
>
> $= i(t_{0})\int_{u = t_{0}}^{\infty}{f(u - t_{0})\text{du}}$
>
> $= i(t_{0})\int_{u = 0}^{\infty}{f(u)\text{du}}$

Whatever f may be its integral from 0 to ∞, equals some number K. So

$\int_{u = 0}^{u = t_{0}}{\left( I(u) - i(u) \right)\text{du} = K\ i(t_{0})}$

t~0~ could be any time t, so we have arrived at the integral version of
Allard\'s differential equation (7).

$\int_{u = 0}^{u = t}{\left( I(u) - i(u) \right)\text{du} = K\ i(t)}$
(8)

$I(t) - i(t) = K\ \text{di}/\text{dt}$ (9)

The number K = 1/m from equation (7). Moreover, if equation (9) is now
applied to a constant flash of impression = 1 which ends suddently at
t~f~, we get

$- i(t)\ \text{dt} = K\text{di}/\text{dt}$ for t \> t~f~

So i(t) = i(t~f~)e^-t/K^ for t \> t~f~

This is Allard\'s equation (6) with K=1/m.

The reconstruction gives Allard\'s \"impression éleméntaire\" a
conservative mechanistic interpretation and then using no more than his
prime assumption from step (B) shows how he does not assume exponential
decay of impressions, but deduces this.

Allard\'s equation (7) or (9) imply that impressions must begin to
diminish as soon as a light is turned off. This precludes visual delay.
Indeed there are plausible visual impulse functions which exhibit such
increase.

Allard\'s prime assumption (B) must be denied.

Appendix A

from \"Mémoire sur l\'Intensité et la Portée des Phares\",

M.E. Allard, Paris, 1876

\[p. 62\]

Étude Théorique de la vision des Feux Scintillants.

L\'effet que les feux scintillants produisent sur l\'organe de la vue
peut donner lieu à une étude intéressante, dans laquelle intervient le
phénomène connu sous le nom de persistance des impressions sur la
rétine. On pourrait croire, au premier abord, qu\'en faisant tourner
assez rapidement un appareil de feu scintillant, l\'impression produite
par un des éclats, persistant pendant
![](media/image2.png){width="0.24930555555555556in"
height="0.20277777777777778in"} de seconde, viendrait se confondre avec
le commencement de l\'impression due à l\'éclat suivant, de sorte qu\'on
éprouverait la sensation d\'un feu fixe beaucoup plus intense que celui
qu\'on pourrait obtenir de la même lampe avec l\'appareil ordinaire. Une
proposition dans ce sens avait été faite à l\'Administration et a dû
être expérimentée au Dépot des phares. Le résultat annoncé ne fut pas
obtenu, et on reconnut que l\'intensité du feu fixe dont la rotation
rapide de éclats donnait la sensation avait à peine l\'intensité de
celui qu\'on produit avec un appareil immobile.

Nous allons chercher à expliquer ce qui se passe dans un cas semblable,
en étudiant théoriquement la question.

Lorsque l\'œil est soumis à l\'action d\'une source lumineuse
d\'intensité constant I, la partie de la rétine qu\'atteignent les
rayons émanés de cette source éprouve une impression dont l\'intensité
est proportionnelle à celle de la lumière et dont la mesure peut être
prise égale à I. Si la source lumineuse vient à s\'anéantir subitement,
l\'impression sur la rétine ne cesse pas de suite, mais elle diminue
suivant une certain loi et devient bientôt assez faible pour ne plus
être perceptible. La loi de cette décroissance n\'est pas connue, mais
il est naturel de supposer qu\'elle est analogue à celle que Newton a
indiquée pour le refroidissement d\'un corps \[p. 63\] de petites
dimensions. On peut donc admettre que la vitesse avec laquelle
l\'impression diminue est à chaque instant proportionnelle à la grandeur
de cette impression, ce qui conduit à l\'équation

$\frac{\text{di}}{\text{dt}} = \text{-mi}$, d\'où
$i = {Ie}^{\text{-mt}}$,

I étant l\'impression au moment où la source lumineuse s\'anéantit,
c\'est-à-dire pour *t* = *0*, *i* l\'impression après le temps *t*, *e*
la base des logarithmes népériens et *m* une constante.

Si on appelle *j* las plus petite impression perceptible, la valeur de
*θ* tirée de l\'équation

*j*=I*e^-mθ^*

sera la durée de la sensation lumineuse que suit la disparition de la
lumière. Cette durée variera avec I, mais la variation sera d\'autant
moindre que *m* sera plus grand. Au delà de la durée *θ*, les valeurs de
l\'impression ne deviennent jamais théoriquement nulles, mais elles sont
extrêmement petites et ne produisent aucune sensation.

L\'équation précédente donne, pour *t*=0,

*di*=-*m*I*dt*;

c\'est la perte que l\'impression éprouve dans le premier instant *dt*
qui suit l\'extinction de la lumière, et comme l\'impression que fait
éprouver une lumière constante est elle-même uniforme, cette quantité
*m*I*dt* doit aussi représenter l\'impression élémentaire que la lumière
I produit pendant chaque instant *dt*. Cette remarque va nous servir
dans la question suivante.

Supposons maintenant qu\'une lumière I commence subitement à frapper la
rétine. L\'impression n\'atteindra pas tout de suite sa valeur
définitive I, mais elle y parviendra en augmentant à partir de zéro
suivant une certain loi qu\'il est facile de déterminer. Soit en effet
*i* la valeur de l\'impression au bout du temps *t*. Supposons qu\'à ce
moment la lumière s\'anéantisse: l\'impression diminuera suivant la loi
précédemment admise, et, dans l\'instant *dt* qui suivra l\'extinction
de la lumière, la perte sera \-- *midt*. Mais si la source lumineuse, au
lieu de disparaître, continue à agir, elle ajoutera pendant ce même
instant *dt* une impression élémentaire \[p. 64\] qui, comme nous
l\'avons remarqué ci-dessus, aura pour valeur *m*I*dt*; de sorte que
l\'accroissement effectif de l\'impression sera

*di* = *m*(I-*i*)*dt*, d\'où *i*=I(1-*e*^-*mt*^).

[^1]: A simple model has photons flowing into the eye and electrons
    flowing out the nerves. This can be conserving or not. Simplest if
    it is conserving, with delays (= storage) by the eye. Since (7)
    leads to a convolution model it must be conserving.