Dye densities of multi-layered films are determined by applying a regression analysis to the spectral response of the composite transparency. The amount of dye in each layer is determined by fitting the sum of the individual dye layer densities to the measured dye densities. From this, dye content constants are calculated. Methods of calculating equivalent exposures are discussed. Equivalent exposures are a constant amount of energy over a limited band-width that will give the same dye content constants as the real incident energy. Methods of using these equivalent exposures for analysis of photographic data are presented

Scarpace, F. L.

Voss, A. W.

English

NASA Technical Reports Server

ANALYSIS OF MULTI-LAYERED FILMS
Dr. Frank L. Scarpace and Alan W. Voss
Environmental Monitoring and Data Acquisition Group
Institute for Environmental Studies
University of Wisconsin
Madison, Wisconsin 53706
ABSTRACT
Dye densities of multi-layered films are determined by applying a
regression analysis to the spectral response of the composite transparency.
The amount of dye in each layer is determined by fitting the sum of the
individual dye layer densities to the measured dye densities. From this,
dye content constants are calculated.
Methods of calculating "equivalent exposures" are discussed.
"Equivalent exposures" are a constant amount of energy over a limited
band-width that will give the same dye content constants as the real
incident energy. Methods of using these equivalent exposures for analysis
of photographic data are presented.
INTRODUCTION
All the information recorded at each point on multi-layered films
can be characterized by one constant for each layer of the film. The
constant for each layer is the number a normalized dye density must be
multiplied by in order to equal the dye in that layer for a particular
scene response.
All multi-layered films have characteristic colors associated with
each layer of the film. The spectral properties of the dyes that make up
these colors are well knownI , though not necessarily stable with time.
An example of the spectral characteristics for Kodak Ektachrome MS
Aerographic film type 2448 is given in Figure 1. In this case the film
(NASA-CR-137284) ANALYSIS OF N74-20017
HULTI-LAYERED FILMS (Wisconsin Univ.)
-23* p HC $4.25 CSCL 14E
'7 Unclas
G3/14 33540
https://ntrs.nasa.gov/search.jsp?R=19740011904 2020-03-23T11:16:42+00:00Z
SPECTRAL SENSITIVITY CURVES
2.0
yellow magenta cyan
er orming orming
>-ayer .layer
c0
-
1.0
400 450 500 '550 600 650 700
WAVELENGTH (nm)
SPECTRAL DYE DENSITY CURVES
1.5
neutral
1.0
SWcyan
. yellow dye magents dye dye
- 0.5
00 ,
400 450 500 550 600 650 700
WAVELENGTH (nm)
Figure 1. Spectral Characteristics of Kodak color film type 2448,
1°
has three layers as do most multi-layered films. The colors formed in
the three layers are yellow, magenta and cyan. The properties of this
film are well known.2 The method of analysis is developed using type
2448 as an example though the methods can be easily generalized to any
multi-layered film.
The spectral response or color at each point on a transparency is
due to the dyes in the three layers at that point. The spectral properties
of the dyes in each of the layers at a point can be represented by a
constant (one for each dye) times a normalized dye density at each wave-
length. Figure 1 shows the three dye density curves for film type 
2448.
Thus the spectral properties of the transmitted light through the developed
film transparency are determined by a linear superposition of constants
times these dye density curves. Mathematically this can be written:
D(x,y,X) = Kl(x,y)Y(X)+ K2 (x,y)M() + K3 (x,y)C(O) + B(X) (1)
and I(x,y,X) = I (x,y,X)10-D(x,y,X) (2)
where D(x,y,X) is the density at any point as a function of x, y and
of wavelength (X)
Y(X) is the normalized dye density for the yellow layer as a
function of wavelength
M(X) is the normalized dye density for the magenta layer as a
function of wavelength
C(A) is the normalized dye density for the cyan layer as a
function of wavelength
B(X) is the density of the film base as a function of wavelength
Kl (x,y), K2 (x,y), K 3 (x,y) are constants related to the amount of
dye on the three respective film layers and only a function of
position on the film format (dye content constants)
2<
I(x,y,X) is the transmitted light through the transparency at
each point on the film and is a function of wavelength
Io(x,y,X) is the incident light on the transparency
At any given point, the quantities experimentally determined are
usually the total transmitted light and the incident light. From this
the density can be calculated for that point. Because these quantities
are a function of the wavelength of the light (color), they must be
measured at a single wavelength. Actually, for this treatment, measurements
must be acquired for at least three different wavelengths at the point of
interest on the transparency. Three equations can be written and solved
for the three dye content constants. It is then possible to determine an
equivalent integral spectral density or analytical spectral density
4 at
any wavelength.
The transmittance of a typical film base is represented in Figure 2.
Predicted and observed values of transmittance for colored obj.ects are
shown in Figures 3, 4 and 5.
Equivalent Energies
An equivalent energy is defined as a constant amount of energy through-
out the spectrum that will give the same dye constant in a film layer as the
real incoming radiation. There will be a separate equivalent energy for
each layer of the film. These equivalent energies can be calculated by
using the known sensitivities of the film layers.
Let: S.(X) be the sensitivity of the ith layer of the film as
a function of wavelength
c(X) be the radiation falling at a point on the film (ergs/cm2/)
th
E. be the equivalent energy for the i film layer
i = 1 for the yellow layer
3<~
4<
TRANSMITTANCE OF FILM BASE
1.00
.80
.60
0
S.40 -
.20 -
.00 I
400 500 600 700
WAVELENGTH (NM)
17- r4 ..
FRAME 3 BLUE OBJECI
Figure 3. Transmittance vs. wavelength of a blue object.
.50
40--- predicted values
measured values used in
prediction
& measured values not used in
prediction
U .30
z Dye Content .Constants
K1  1.14
K2 = 1.40
K3:= 2.29
C)
z .20
I-
.10,
400 500 600 700
WAVELENGTH (NM)5<
FRAME 9 -YELLOW OBJECT
Figure 4. Transmittance vs. wavelength of a yellow object.
.50
- Predicted values using 7 measured values
--- Predicted values using 3 measured values
* Measured values
O Measured values not used in solid line fit
.40
0
iI. .30C)
-
S.20
< Dye Content Constants
K1 = 1.939
K2.= .426
K3 = .289
0t0
SI
400 500 600 700
WAVELENGTH (NM)
FRAME 9 RED OBJECT
Figure 5. Transmittance vs. wavelength of a red object.
.50
Predicted values using 7 measured values
--- Predicted values using 3 measured values
* Measured values
O Measured values not used in solid line fit
.40
W .30
z
Dye Content Constants
-K 1 = 2.13
K2 = 2.14
C) K = .64
.20 3
F-
.10
400 500 600 700
WAVELENGTH (NM) ,
"7<( ..
i = 2 for the magenta layer
i = 3 for the cyan layer
Mathematically we can write:
IE.S.i()dX = le(1)S.i()dX - (3)
where the integral is over all wavelengths.
The units of Si(1) are in 1/(ergs/cm 2) to give a density of 1 with
standard processing. Thus the integral of radiation times the sensitivity
will give the fraction of exposure required to give a final density of 1
with standard processing for that radiation.
Since E...on the left of (3) is a constant we can write (3) as:
1
E. = fE()Si(X)dX (4)i /Si(X)dX
In essence, this E. is a weighted average of incoming radiation over the1
sensitivity of the layer.
In themselves these E's are not very useful. They represent three
different constant energy spectrums. More useful would be one equivalent
energy spectrum that can replace the incoming radiation spectrum and give
the same dye densities. Using the same type of treatment, this equivalent
energy spectrum can be found in terms of the E's and thus in terms of
incident energy spectrum e(X) and the sensitivity of the film layer, Si(X).
For color and color IR films these three equivalent energies are
designated as EB, EG and ER and EG, ER and El respectively. An example
of these equivalent energies is graphically represented in Figure 3. EB*
is a constant amount of energy between .4 and .5 microns, EG is a constant
amount between .5 and .6 microns, ER is a constant amount between .6 and
.7 microns, and EI is a constant amount between .7 and .9 microns.
The actual values of the three equivalent energies for each of the
film types can be easily found by substituting the equivalent energy spectrum
*Kodak filter 2B must be used for it if EB is to correspond to the range
.4 to .5p.
84:<
into equation (4) in the place of c(X), being careful to integrate over
the proper limits. In the case of color film this reduces to:
f.5EB S.()dX + f6EG S.(X)dX + f7 ER S (X)dX
E. 1 .5 i .6 1 (5)Ei f.7 s-- -- i- ~- - -f4 Si(X)dX
Equation (5) is really 3 equations that can be written explicitly as:
E = A' EB + A' EG + A' ER (6)1 11 12 13
E = A' EB + A' EG + A' ER (7)2 21 22 23
E = A' EB + A' EG + A' ER (8)3 31 32 33
where 1 5S1()d A2 7 56S1(3)d 3 6 7S1(A)d
f S1 (X)dX f, S(X)dX f7 S1 (X)dX
' 5 A6 7S2 ( ) d h 156S2(X)dX A S 2 (X)dX
21 7S 2(X)d 22 7S2 ()dX 23 4 7S2 ()dX
S3 ( )dX *56S3(X)dX f67S3(X)dX
A'1 - A' Al .6
31 7S3 (X)d 32 , 7S3 ( ) d  33 4 7S3 ()d
and S1 () is the sensitivity of the yellow layer
S2(A) is the sensitivity of the magenta layer
S3(A) is the sensitivity of the cyan layer
-The nine "A primes" (above) are just constants dependent only on the
film sensitivity. The equations (6) through (8) can be solved for EB, EG and
ER in terms of El, E2 and E3 by several methods.
5 The solutions to equations
(6), (7) and (8) for EB, EG and ER in terms of E1 , E2 and E3 can be written as:
EB = A11E1 + A12E2 + A13E3 (9)
EG = A21E1 + A22E2 + A23E3 (10)
ER = A31 E + A32E2 + A 33E3 (11)
These A's are again only a function of the film sensitivity. The
numerical values of these A's are given in Table 2 for two different types
of film. An example of an equivalent energy spectrum is given in Figure 6
for a blue object.
9<
REAL AND EQUIVALENT REFLECTED ENERGY
Figure 6. Typical values of real and equivalent reflected energy.
10
EB =7.8
BLUE OBJECT
5
SY EG = I.,5
-o. ER = . 17
400 500 600 7009-
Cn,
E
. 10
a YELLOW OBJECT ER 7.8
EG =5.8
5
C(
Li -O
LLI -0
400 500 600 700
0
F-
Ll
LL- 10
: RED OBJECT
5
ER = 2.6
EB =.35 EG,= .64
0 
-
400 500 600 700
WA/VFl rN MTH (Ni M41 10<
FILM TYPE All A12 A13 A21 A22 A23 A31 A32 A33
Kodak 2443 1.03 -.0011 -.02 -.16 1.197 -.032 -.18 -.55 1.7
Kodak 2448 1.05 -.0256 -.0108 -.134 1.176 -.0155 -.0008 -.184 1.2
Table 2
In the case of color IR film, these equations can be written as:
EG = A11E1 + A12E2 + A13E3 (12)
ER = A21 E + A22E2 + A23E 3  (13)
EI = A31E1 + A32E2 + A33E3 (14)
Thus, when E1 , E2 and E3 have been found, EB, EG and ER can be calculated
for color film or EG, ER and EI can be calculated for color IR film. The
physical significance of these equivalent energies, EB, EG, ER and EI, is
that they represent numbers proportional to average energies over a limited
band width very similar to the data acquired by multi-spectral scanners.
EG and ER .correspond exactly to the ERTS satellite MSS bands 4 and 5.
EI corresponds to a larger band pass than ERTS MSS band 6.
The calculation for E1, E2 and E3 may be difficult if proper procedures
are not followed. The critical step in the procedure is the exposure of a
calibrated step wedge on the film before development. The important thing
here is to be able to calculate E1 , E2 and E3 for each step of the wedge
from equation (4). In this case the E(X) is the radiation falling on the
nth step of the film wedge, the film wedge being the exposed step wedge
on the film of interest. These values are designated as E (i = 1,3)
for the nth step on the film wedge.
Film Calibration
The dyes used in color and color IR film are not stable with time.
On different rolls of film processing differences will cause different
amounts of dye to be formed in each of the layers for the same exposures.
12<
For these reasons the familiar D log E or characteristic curve must be
used for calibration.
The two ingredients involved in successful calibration of a multi-
layered film are (1) the step wedge densities as a function of wavelength
must be known for each step, and (2) the spectral distribution of the
light incident on the step wedge must be known.
Let Dn(X) = the density of the nth step of the step wedge as a
S
function of X
Eo (A) = the incident energy on the step wedge
En = the exposure value for the it h layer from the nt h step
i
of the step wedge
j Iso(X) 10D s S (X)dX (15)
S fS. (X)dX
1
If Co(), D (X), and Si (X) are known, the exposures due to the light
transmitted through each of the steps of the step wedge for each layer can
be calculated.
One of the usual practices in sensitometric analysis is to plot the
density vs (-) the step wedge densities as a characteristic curve. Let
us look carefully at the assumption one must make for this to be valid.
If so(X) is a constant, Eo and Dn(X) do not depend on X, then equation
(15) reduces to
Ei = Col0 s  (16)
Taking logs of both sides
log El = -Dn + log Eo (17)
1 S
If the assumptions of D s(A) and Co(X) do not depend on wavelengths are true,n
the log of the exposure due to the step wedge can be represented by -D
The term log so is only necessary if absolute calibration of the system
is necessary. For most purposes, a relative characteristic curve is
12<~
sufficient, and only the values of the relative densities of the step wedge
are necessary. It should be noted that in this approximation, the value
of log o0 is the same for each layer.
In many cases the density of the step wedge is a constant over the
relevant wavelengths, but the incident energy on the step wedge has a
distribution like a black body at 58000 K. This is a reasonable assumption
since most sensitometers are color corrected to this temperature. In this
approximation, equation (15) reduces to
n -Dn [G0Si(X) (18)E = 10 s X5(e hc/XKT-l)
fS.(X)dX
1
n
=  
-Dn
or E = 10 s G. (19)
1 i
where G. represents the three different values of ratio of the integrals
1
in equation (18). The log of the exposure for each layer then reduces to
log E. = -Dn + log G. (20)1 S 1
The important thing to note here is that the term log G. is different for1
each of the layers. If the exposure for each layer is determined for a
scene response and these exposures are used 
independently, only the -Dn
term need be used. But if the sums of these exposures are used, to
calculate an equivalent energy spectrum for example, or ratios of the
exposures are needed, the log G. term must be included.
In the cases that neither D n() or eo(X) are constant over the relevantS
wavelengths, equation (15) can be used throughout the film analysis.
Measurements can be made on each step of the film wedge and dye contents
constants can be calculated. Three characteristic curves can then be
plotted; K.'s are plotted against the log of the E 's as calculated in
equation (15). Three different curves (one for each layer) will have
been plotted. From these curves, for any selected point on the film, values
of El, E2 and E3 can be found for calculated values of KI, K2 and K3. EB,
EG and ER or EG, ER and El can be calculated from El, E2 and E3 for color
or color IR film, respectively from equations (9) to (14).
A necessary ingredient for the calculation of the dye constants K1 ,
K2 and K3 is the knowledge of the spectral densities of the dyes in the
film. If the shapes of these curves are known and do not change with time,
the procedures described so far are all that is necessary to calibrate
multi-layered films. If the height of these curves change with time, the
step wedge calibration will adequately deal with this problem. But if
the relative shape of this curve changes with time, or if the spectral
densities are not known, additional calibrations must be made.
The easiest way to calibrate the spectral density curves for a film
is to expose three different step wedges. The best way to expose these
step wedges is through narrow band interference filters. For color film
the filters would be centered at 400 nm, 550 nm, and 6.50 nm; for color IR
film the filters would be centered at 550 nm, 650 nm, and 750 nm.
In the case of color film, the film wedge exposed at 650 nm will
give the spectral density of the cyan layer. The film wedge exposed
at 550 nm will contain information on the spectral density of both the
magenta and cyan layers. If the cyan spectral density has been determined
then the magenta spectral density can be determined from combination
density. The film wedge exposed at 400 nm will give the spectral density
of the yellow dye. A similar argument can be made for color IR film.
If this technique is employed, the exposures used to plot the
characteristic curves must be calculated by equation (15). For color
film this reduces to:
SI(40 Q)-D '( 4 0 0 )
En IE (400)10 s F1S1(400) d X (21)1 = 1Sl(x)dX
14<r
n
-Dn(400)
En (400)10 s 1(400) S( 0) (22)
-1
fS1(x)dX
n 0-Dn(550)
E2 = o( 5 5 0 )10 s S2(550) (23)
fS 2 ()dX
n
n -D (650)
E = e (650)10 s S (650) .(24)3 o 3
fS 3 (A)dX
The characteristic curve for K1 is determined from the film wedge exposed
at 400 nm; the curve for K2 is determined from wedge exposed at 550 nm;
and the curve for K3 is determined from the wedge exposed at 650 nm. The
curve for K3 could be determined from the wedge exposed at 550 nm also.
In that case, En would be:
-D n (550)
En = co (550)10 S3 (550) (25)
fS3 (X)dX
An alternate method of arriving at the same result is to expose one
step wedge with a sensitometer and also one step of the wedge through the
three different narrow band filters. Using the step exposed through the
filters the spectral densities of the dyes can be determined. The charac-
teristic curves are then plotted from the complete step wedge exposed by
the sensitometer.
Summary
We have presented a method of quantitatively analyzing multi-layered
films. This type of analysis uses the physical characteristics of the film
to relate the film's spectral response to the spectral distribution of the
incoming radiation. A method has been presented to determine three dye
content constants at each point on the film that can be easily related
to the incoming radiation. The equivalent energies are simply a linear
transformation of these dye content constants which are more closely
related physically to the incoming radiation. These equivalent energies
can be thought of as the output from a three channel multi-spectral
scanner. In this regard this method is well suited for comparison to
ERTS satellite data.
REFERENCES
1. Kodak Data for Aerial Photography, M-29, Eastman Kodak Company,
1971, 62p.
2. Ibid.
3. Ibid.
4. Manual of Color Aerial Photography, John T. Smith, Jr., (Ed.),
American Society of Photogrammetry, 1968, 308p.
5. Brauer, Nohel, Schneider, Linear Mathematics, W. A. Benjamin, Inc.
New York, 1970.


Analysis of multi-layered films

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