February 1, 2001

from: Dennis Couzin

to: Ian Tutt

J Rennilson

Dave King

Yoshi Ohno

Nick Hale

Richard Austin

re: **Allard\'s method**

Last April I proposed to CIE TC 2-49 a \"new method\" for the I~e~
measure on flashing lights:

**1. Perform a mathematical convolution of the pulse function with a
visual impulse function (having time integral = 1 cd-sec).**

**2. Define Ie as the maximum value of the convolution.**

I now see that my method is but a generalization of the old method of
Allard. This short report is a reappraisal of Allard\'s 1876 method

Allard\'s method is based on a differential equation:

$\frac{\text{di}}{\text{dt}} = \frac{I(t) - i(t)}{A}$ (1)

I(t) is the physical flash intensity as a function of time.

i(t) is the physiologically delayed flash intensity as a function of
time.

A is a constant, like 0.2 seconds.

Differential equation (1) is familiar to electrical engineers. If I(t)
is the voltage applied to a simple RC circuit, I(t) is the delayed
output voltage, and constant A=RC.

Figure 1

![](media/image1.wmf)

Equation (1) governs time delays generally. Consider the following
heuristic derivation.

$\frac{\text{di}}{\text{dt}} = \underset{\Delta \rightarrow 0}{\text{lim}}\frac{i(t + \Delta) - i(t)}{\Delta}$

$$\frac{\text{di}}{\text{dt}} \approx \frac{i(t + A) - i(t)}{A}$$

$$\frac{\text{di}}{\text{dt}} \approx \frac{I(t) - i(t)}{A}$$

After determining function i(t) Allard defined I~e~=max(i(t)).

It turns out that Allard\'s function i(t) is equal to a convolution of
function I(t) with a q(t) defined as follows:

q(t) $= e^{- \frac{t}{A}}A$ when t ≥ 0

q(t) = 0 when t \< 0. (2)

![](media/image2.wmf){width="5.170138888888889in"
height="3.3006944444444444in"}

Figure 2

The convolution of the Allard \"impulse function\" q(t) with a 0.5
second rectangular pulse is shown in figure 2. Constant A has been set
to 0.2 seconds. When the Allard method is described as a convolutional
method, the only difference between my proposal and Allard\'s is that he
specified q(t) while I left the choice of the \"visual impulse
function\" wide open.

Blondel-Rey, Blondel-Rey-Douglas, and Form Factor, all of Allard\'s
replacements, fiddle directly with the physical pulse function I(t) to
produce I~e~, rather than producing an intermediate delayed pulse
function i(t). Were the later workers modeling the visual mechanism at
all? Allard\'s modeling can be questioned in two places. He used a
simplistic visual impulse function that can\'t be quite right. Also
making I~e~ the maximum of the i(t) function is a fast move (shared by
method).

One theoretical advantage of the Allard method over the later methods is
that it could give different values for I~e~ when the

![](media/image3.wmf){width="3.1798611111111112in"
height="2.245138888888889in"}![](media/image4.wmf){width="3.1798611111111112in"
height="2.245138888888889in"}

Figure 3

time direction of the pulse was reversed. See Figure 3 illustrating
forward and backward triangle pulses.

Another theoretical advantage of the Allard method over the later
methods is that it handles trains of pulses in a not grossly implausible
manner.

The 1977 Technical Note from the I.A.L.A. (\"Recommendations on the
determination of the luminous intensity of a marine aid-to-navigation
light\") put Allard on an equal footing with the later methods in terms
of accuracy. That paper calculated I~e~ for several simple pulse forms
for each of Allard, Blondel-Rey-Douglas, and Form Factor, finding
differences up to about 30%. Apparently the available psycho-physical
data was insufficient to decide between the methods.

![](media/image5.wmf){width="5.504166666666666in" height="1.8125in"}

Form Factor fails horribly for the non-simple pulse I exhibited last
April. See figure 4. None of the other methods suffer this
counterexample.

Figure 4

If any of the later researchers have data to show their methods more
accurate than Allard for simple pulses, that data can probably be
accommodated by my generalization to Allard. Mathematically, the visual
impulse function can be any function with maximum value equal to 5× its
time integral (t in seconds). Causality requires that the visual impulse
function should be 0 when t \< 0. Figure 5 shows a convolution result of
a \"beautified\" version of Allard\'s q(t), given by formula (3):

q(t) $= e^{- \text{ln}(7\text{.}1t)}0\text{.}2t$ when t \> 0

q(t) = 0 when t ≤ 0. (3)

![](media/image6.wmf){width="5.5in" height="3.1368055555555556in"}

Figure 5

Allard\'s method gives I~e~ for the 0.5 second rectangular pulse which
is considerably higher (1.29×) than the value given by
Blondel-Rey-Douglas or Form Factor. Leaving aside the question of which
method is correct, the small change to Allard\'s q(t) given by equation
3 results in I~e~ for the 0.5 second rectangular pulse which is slightly
lower (0.90×) than the value given by Blondel-Rey-Douglas or Form
Factor. Compare Figure 2 with Figure 5.