CALIBRATION OF NEUTRAL DENSITY FILTERS

• Document Type: CREST
• Collection: NGA Records (Formerly NIMA)
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• Original Classification: K
• Document Page Count: 16
• Document Creation Date: December 28, 2016
• Document Release Date: March 30, 2005
• Sequence Number: 34
• Publication Date: August 31, 1964
• Content Type: MEMO

STAT
STAT
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To:
From:
Subject.
CC:
Calibration of Neutral Density Filters
31 August 1964
WT:bb:jg -405
997-112
Declass Review by NGA.
Introduction
This memo presents the results of the calibration of glass neutral
density filters manufactured by
STAT
The densities of the filters tested, as determined by
were .41, .69, 1.06, 1.51, and 1.83.
Experimental Procedure
The glass filters were tested using the Spectrometer,
employing a 931A phototube. The illuminating source was a tungsten
filament. Each of the filters tested was placed in front of the entrance slit
to the spectrometer. Care had to be taken to prevent the phototube from
saturating. The spectrometer then automatically scanned from approxi-
mately 3790 to 7860 R and recorded the intensity as a function ci wave-
length. A tungsten response curve with no filter present was also recorded.
It was not possible to test densities greater than 1.83 due to the large amount
of noise present in the output when high gain was used on the spectrometer
recorder. It is believed that most of the noise was due to erratic line
voltage fluctuations caused by the machines in the shops. The tests were
repeated several times and the results averaged to reduce the errors caused
by noise.
Results
Three independent functions of the wavelength, A , are present
In this test; (1) the intensity of the tungsten source, I(A) , (2) the trans-
mission of the filters, -r (x) , and (3) the response of the phototube, Ps(??)
The recorder output, W (A), can be written as:
\es/ (x) r- I (A) T R( A
If no filter is present, the tungsten response curve will be:
(),) R (A)
Thus,
T( >i) = wow
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31 August 1964
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can be determined by dividing point for point the filter response curve
by the tungsten response curve. The results of TCA) vs. for each
filter are plotted in Figures 1 and 2.
It is seen that T(i) is not a constant. It is therefore necessary
to obtain an "average" density as indicating the "neutral" density. The
simplest "average" to define is 5, , which is found by calculating the
mean transmission, T, , and converting this value to density
-r ,17k
T-,
v-i a
and tcli.(1-4)
Simple numerical approximations to these integrals can be defined as:
T
ens I
=
fr
where the ave rage is taken over t' 1.Z transmission readings for A running
from 3790 A tr) 7860 R. The results of these calculations are tabulated in
Table 1 and the values of T , are indicated on Figures 1 and 2.
TABLE I
Labeled Density
Labeled Transmission
Mean _
Transmission T.
Density
D,
.21
.62
.58
.24
.69
.20
.18
.74
1.06
.087
.074
1.13
1.51
.031
.031
1.51
1.83
.015
.015
1.83
The above averaging process is not, in general, characteristic
of the way that a densitometer averages the transmission of the light in
making a density measurement. A densitometer will measure the trans-
mittance of a filter as a, weighted average, where the weighting factor for
each wavelength is the system response.
Since both the Ansco Model 4 Microden.sitometer and the MacBeth
Densitometer use tungsten sources and 931A phototubes, it is reasonable
to take the response to be ido(A . The data recorded by the spectrometer
for the filter transmissions, w (A) , is the product of the filter response
and the system response, -r(-A) vio(A)
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31 August 1964
rage i WT:bb:jg
Thus, T'z can be defined as a weighted average of "T(X) by:
ci
f WOC A) )
Simple numerical approximations of these integrals can be defined
(
= I
as:
Liz
where the summations are over the M=I2 recorded values of the functions.
Increasing M to 24 did not significantly change the results of calculations.
The results for the calculation of 7.2 for >% running from
3790 to 7860 A are shown in Table 2 and on Figures 1 and 2. Dz. is
defined as logio() .
TABLE 2
-72
.21
.69
1.06
1.51
1.83
Conclusions
.62
.20
.087
.031
.015
.65
.20
.091
.034
.017
.19
.69
1.04
1.47
1.77
The results for Tz are too high, thus making the Dx values less
than the la,beled filter va4es. In Figures 1 and all experimental curves show
a peak near A = 5540 .X, the same wavelength at which the W0(A) response
Curve peaks. It is not known whether or not these peaks are characteristic
of the filters or of the apparatus. However, nothing could be done to reduce
the:m.The sample curves of transmittance as a function of wavelength provided
by Schott &Gen. do not show these peaks. The sample curves also show
the transmittance remaining approximately constant for A up to 10000 A.
The experimental results in Figures 1 and Z show a definite falling off of
transmittance above 6000 A.
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Page 4
31 August 1964
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Recommendations
In the event that future tests require measurements of the type
described above, it is recommended that they be conducted when none
of the shops are operating. It might also be worthwhile to check the
voltage output versus light intensity relation for
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.F7Le ix) notawerI TAT
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TWO-DIMENSIONAL SPATIAL FILTERING AND COMPUTERS
ABSTRACT
Processing of two-dimensional signals has important applications, for example,
in photographic image analysis, but when the weighting function of a two-dimensional
linear filter extends over a large area, e.g. , smoothing filters, digital realization
via a two-dimensional convolution is prohibitively time consuming, and analog reali-
zation is extremely difficult. The principal purpose of this paper is to show how a
broad and useful class of two-dimensional filtering operations can have notably
shortened execution time in the digital case, and be put into a particularly convenient
form for electrical filtering. The procedure includes a reduction of dimension from
two to one; transformation of two-sided (weighting function extends into both past and
future) operations into one-sided (physically realizable) operations; and finally, for
the digital case, the transformation of a direct many-term convolution expression into
a compact recursive form ideally suited for digital computation. An important class
of smoothing filters, with weighting functions approximately Gaussian, is derived and
used for illustration. The result is a several-order-of-magnitude reduction in time
for digital two-dimensional filters, and some interesting results applicable to one-
dimensional zero-phase-shift filters.
I. INTRODUCTION
Analog or digital processing of two-dimen-
sional signals has many important applications,
notably in photographic image analysis for mili-
tary or commercial purposes. Much of this type
of ptocessing is special purpose, tailored to the
physically significant details within an image.
But there are certain basic operations (e. g. high
and low-pass filtering) that are extensions of
or responding one-dimensional operations, and
which have a similar range of usefulness.
Because in two-dimensional processing by
digital means, or by means of electrical filters,
storage and processing time requirements are
much greater than for one dimension, there is
a genuine need for nontrivial methods to allevi-
ate time and storage problems. This need is
greatest when each computed point in the output
image is affected by values from a relatively
large area of the original or input image. Such
lAn exception is the optical filter, not
considered in this paper.
filters, called area filters, are exemplified by
many smoothing operations; they form the
principal topic of this paper. In contrast to the
area filter is the local filter, in which the val,Ir
of an output image point depends only upon a
small neighborhood of the corresponding input
image point.
II. THE DIRECT CONVOLUTION APPROACH
The direct approach to realization of an
arbitrary filter operating on a two-dimensional
input is explicit convolution of the impulse re-
sponse of the filter with the input. In the digital
case this takes the form:
my' (1)
where subscripts on y (output) and (input)
give the digitized coordinates of the image point,
and hi, is the discrete two-dimensional impulse
response or weighting function.
In the case of a local filter the4144tinber of
terms in this expression is small, 1&;-4- the method
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is practical and often useful. For example,
Kovasznay and Joseph (Ref. 1) describe an inter-
esting analog equivalent to (1) used for outline
enhancement. David (Ref. 2) reports a digital
application of linear and nonlinear local opera-
tions for noise reduction.
In the case of an area filter, the large
number of terms (121, for example, if / =1=5 )
required for each output point makes this approach
impractical.
III, THE SIMPLIFICATION STEPS
Equation (1) of the direct method is the
starting point for a number of drastic simplifica-
tions. The goal is a set of one-dimensional
filters, of simple recursive form for digital com-
putation, and of physically realizable form for
analog use. The simplification steps are:
(a) Restriction of the weighting function
/id1 or h (t, , ) to a product form
9, or l'(t,) ) ; this special
form allows the two-dimensional prob-
lem to be decomposed into two one-
dimensional problems.
(b) Transformation of the one-dimension
filtering operation, which has an impulse
response extending into both positive and
negative time values, with two filtering
operations of the "physically realizable"
form, in which output depends upon the
"past" (impulse response vanishes for
negative argument).
(c) In the digital case, conversion of the
one-dimensional filter into a recursive
form so that only very few terms appear
in the computation of an output value,
even though the effective memory
extends far into the past.
These items are discussed in the following
four sections.
IV. REDUCTION OF DIMENSIONALITY
The first major simplification of (1) is the
reduction in dimensionality from two to one, by
restricting (with some loss of generality) the
weighting function of two variables to be of
product form
= 69., (discrete) or
h(i,, ez)., r(e,)g(i,)
(continuous)
Then the complete double summation of (1) may be
replaced by two sets of calculations, each only
having a single sum; omitting details, and again
writing only the digital equations, we have:
:nyint? ,
9,71, n - r
(2)
(3)
Here, is an intermediate result, com-
puted from the x's, then regarded as input
variable for final computation of the desired y?,,?
values, The first computation (2) is (for any
fixed r ) an ordinary one-dimensional filter,
operating on one horizontala line of the image;
the parameter r identifies which line. Similarly,
the second computation (3) is, for any fixed rn ,
an ordinary one-dimensional filter, operating on
a vertical line of the y image; the parameter
OF identifies which vertical line.
To illustrate the general effect in terms of
saving computing time, suppose that a 100 x 100
grid of picture intensity values is to be filtered,
and assume that the weighting function of (1)
extends 10 terms in each of the possible directions
( I = J = 10 ). With edge effects ignored, the
application of (1) directly would require
21 x 21 = 441 operations (an operation consisting
of a multiplication and addition) for each proces-
sed point, for a total number of 4,410, 000 oper-
ations. The corresponding double application of
one-dimensional filters, according to (2) and (3),
would require 21 + 21 = 42 operations per proces-
sed point, for a total number of 420, 000 operations
-- less than 1/10 as many.
zWe arbitrarily identify horizontal lines with
fixed second subscript, varying first subscript,
and vertical lines with contrary conditions, in
analogy with the continuous form x (t,, e, ,
where the first variable normally gives the
abscissa value and the second gives the ordinate
value.
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V. SIMPLIFICATION OF THE
ONE-DIMENSIONAL FILTERS
The following approach towards simplifying
the one-dimensional filters of (2) and (3) gives
results definitely useful in originally one-dimen-
sional filtering problems, as well as in the pres-
ent context as intermediate aids for two-dimen-
sional filtering.
It is convenient to drop the redundant double
subscript, and begin with the generic form (open-
loop, double-sided) form that is implied by (2),
(3) and the preceding discussion:
"Li Xn-A
?lc
YOl'ih(r) 21(1-r)dr
(discrete)
(4)
(continuous),
and it is convenient to regard a subscript as a
time value (e. g., bey is the value of y at the
quantized time value n )
Equation (4) could be used in its existing
form for digital computation. But for the case
where the number of terms is large, there is a
more efficient method moat conveniently derived
by orienting our terminology towards the continu-
ous-time situation, so that we can simultaneously
develop a method suited to electrical filtering and
of a form that can be converted to an efficient re-
cursive digital filter.
Write h (r) ? /7(r )7, h_ (r) , where h,
vanishes for negative time, and h_ vanishes for
positive time. The h? part could be the im-
pulse response of a realizable filter, assumed to
have rational transfer function (s) with
Its poles in the left half plane. The h_ part may
be achieved with a realizable filter by reversing
the time variable, for example, by storing the
input on magnetic tape, then playing the tape
backwards. A reversed-time signal passed
through a realizable filter with transfer function
6(s) , gives a result that is formally equi-
valent (in the sense of a bilateral Laplace trans-
form) to passing the original forward-time signal
through a filter with transfer function 6(- .) .
Thus we may speak of filters with poles in the
right half plane, with the understanding that they
refer to a physically realizable filter driven by a
time-reversed version of an input signal.
3
Thus, if the original picture is processed
with filter having transfer function f7 (s) ,
then the original picture is filtered independently
with filter having transfer function F, (- s )
(actually accomplished by running the input signal
backwards through a filter with transfer function
F(c) ), and the results are added, the result-
ant impulse response will be desired k. ,
and the total effective transfer function will be
G(') ?'/(J)
The rational functions ) and f; s )
may be combined into a single multiplicative
expression, of the form
where contains all of the poles in the left half
plane and contains all of the poles in the
right half plane. Thus, we have a cascade form:
rather than filter the original picture twice and
add results, as in the previous paragraph, one
could filter the original picture with 6, , then
filter that result with q, . In the digital case,
the latter has the advantage of eliminating the
need for duplicate storage space, by means
which, if not obvious, are simple. For analog
filtering the' cascade form is more convenient and
avoids various practical synchronization diffi-
culties.
VI. DIGITAL ONE-SIDED RECURSION FILTERS
An open-loop one-sided digital filter has the
form
yn = hox, A, 'z hz ? ? ? ,
the expression possibly being infinite in extent.
"Open-loop" refers to the fact that y is expres-
sed only in terms of z s (inputs); "one-sided"
refers to the fact that only present (time in ) and
past values of input are used to determine output.
For example, the one-sided open-loop digital
filter corresponding to an exponential impulse
response (simple RC lag filter) is
Y." " ? 1 In 7? n- azin-z *ctiX4-11?
-
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If, as in this example, the number of terms is
infinite, then digital computation can be perform-
ed only by approximating with a finite number of
terms. But if re in the expression above is
fairly close to unity, (e.g., O19 ), it is possible
that several hundreds of terms would be required
for satisfactory approximation.
A one-sided feedback digital filter equivalent
to (5) has the form
1n a6Yn-/ Xn
(6)
where oe is the same one as in (5). This form
is of course particularly suited to digital compu-
tation, since it contains only two terms. "Feed-
back filter" here refers to the fact that output
depends explicitly on prior output, as well as the
1, input; this type of filter is also commonly
called a recursive (or recursion) filter.
VII. RECURSION FORMULAS FROM
TRANSFER FUNCTIONS
The general problem of converting a transfer
function (such as our F; (5) , if additive form
is used, or Os (9 ? if cascade form is used)
Into a digital recursion relationship is easily
solved by means of a method presented at the
1961 NEC (Ref. 3, 4), which is described here
only in necessarily very sketchy form. In the
expression for a transfer function 0(3 ) , make
the substitution s-I... 2 , clear of extra-
-r
neous fractional forms and normalize numerator
and denominator into the following form:
a,Z ?rc ce5Z1# ? ? ? # a?,Zm
/ 6,Z bz Z2 o- ? ? ? hrzr
(7)
( Z can be interpreted as the delay operator,
with Laplace transform e-'r ).
Then the recursion formula becomes
ao14,, t a/ 3117-, # ? # "s-m
"'Br?Yo-r. 6zYn-z ?. ? ?bp yn-r ?
The quantity r is a time scaling parameter; in
this application, it relates the time variable of
(8)
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the Laplace transform to the separation between
adjacent sample values.
To illustrate, we use a transfer function that
will later be used to illustrate another aspect of
the two-dimensional filtering problem;
21(7
60) -
s' #.3.s
The impulse response has time constants of the
order of one second in real time. Suppose we
desire that over one of these time constants there
would be 10 picture elements (in a rough sense,
the memory covers ten picture elements, if
memory is taken to be a nominal time constant).
Then 7- = 0.1. Use of the substitution described
above gives for the final recursion formula:
a008.99 (Y,2 )77-1) h
VIII, ILLUSTRATIVE EXAMPLE
In many two-dimensional filtering applica-
tions, it is desired to have an impulse response
that is circularly symmetrical. The purpose of
this section is to show how one class of such
impulse responses can be approximately realized
with the multiple application of one-dimensional
filters as described above.
Circular symmetry requires that the filter
Impulse response 17 be a function of t,1 7? i; ?
Thus the one-dimensional filter y(e) which is to
be applied in each direction must be chosen such
that
h(e?t5)- 5,(t3 ),9(f22 )
with 9(tf) 9, (11) - # ) .
A solution of this functional equation is
(9)
(10)
In accordance with the previously prescribed
procedure, we now attempt to find a one-sided,
one-dimensional filter whose impulse response is
= 0 t .< 0 (11)
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There is no filter whose Laplace transform
is a rational function of 5 which has the impulse
response (11). However, there is a class of
filters with transfer functions rational in 5 ,
and with impulse responses approximately (11)
in the vicinity of -E = 0. These can be found by
expanding (11) in a Taylor series about the origin,
transforming the result, and equating the coeffi-
cient to the large 5 expansion of a rational
function. Thus, for
fre) = - t I 4- - 4- ? ? ? d5 0 (11)
2 6
we want
2 12 120
F(5) ????? ? ? --? 7.- CO s-s-oa (13)
5 Si 52' 7
The first few of a family of approximations to
(13) are:
S4. a
-(5) Z as 4- 2
(5) =
5 asz 65 2a
fi(5)
5 as3 ,/2.s2 0. 6as # /2
a5.27, /05
Here Fn(s) is an approximation to (13) good
through terms in s'n . All of the transfer
functions are stable for any positive value of the
parameter a . A numerical example of the two-
dimensional filter related to f; (i) with a = 3
has been worked out. The results are presented
as contours of constant impulse response in
Fig. 1. Note that in the vicinity of the origin the
contours are approximately circular, as predicted.
Two cross sections of the "impulse mountain" are
shown in Fig. 2.
The example above can be continued to display
explicitly the transfer function to be used for each
separate filtering operation. The one-sided one-
dimensional transfer function is
5 3
F(5 ) - ? (14)
5
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The two-sided one-dimensional transfer function
is then (in summation form)
,5iJ- 4?.3
51 7.- 35 2 - #
We can combine the terms to obtain
/2 0T7
?
(s'#35 7" 2 )('52-35 #2.) sz#3,3 5 -35 * 2
It can thus be seen that the required transfer
function to produce the two-dimensional impulse
response of Fig. 1 is 2 . This trans-
fer function is used for four cascaded filtering
operations: positive t, direction, negative e,
direction, positive ez direction, negative t,
direction.
Examination of the two-sided transfer function
shows that this filter has a high frequency attenu-
ation of 24 db per octave and no phase shift at any
frequency.
IX. CONCLUSION
When a two-dimensional image is to be pro-
cessed with a filter for which the response at an
output point depends upon a large region of the
input image, the methods of this paper find their
principal utility. When the methods are applica-
ble, there is a potential for decreasing execution
time by several orders of magnitude in the digital
case, or for putting the processing steps into a
form suited for physically realizable filters in the
analog case.
These simplifications are achieved by
(a) restricting the two-dimensional impulse re-
sponse to the product of two one-dimensional
impulse responses -- a loss of generality that is
satisfactory in many practical applications if one
judiciously chooses the proper one-dimensional
functions -- and (b) simplifying the one-dimen-
sional operations so that recursive filters may be
used in digital computation, and ordinary electri-
cal filters may be used in the analog case. One of
the important features is the time-reversal step,
accomplished by routine programming (digital) or
by using a reversible storage medium such as a
tape recorder (analog). The time reversal per.
mite impulse responses which are nonrealizable
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in the ordinary sense. In particular symmetrical
impulse responses become possible.
REFERENCES
1. L. S. G. Kovasznay and H. M. Joseph,
"Image Processing, " Proc. I. R. E.,
Volume 43, pp. 560-568, May 1955.
Z. E. E. David, Jr., "Digital Simulation in
Research on Human Communication,"
Proc. I. R. E., Volume 49, pp. 319-329,
January 1961. See especially pp. 327-8.
3. W. D. Fryer and W. C. Schultz, "Digital
Computer Simulation of Transfer Functions,"
Proc. Nat. Elec. Conf., Volume XVII,
pp. 419-420, 1961.
4.
"Digital
Computer Simulation of Transfer Functions,"
To be published, 1962.
6
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Ai ?
3.0-
2.5
2.0
Lb
1.0
0.6
-0.5
-1.0
-2.01
-2.61
-3.0
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-3 0 -2.5 -2.0 -1 61 -1 0 -0 6 0 0.6 1.0 I 5 2.0 2.6
Figure I CONTOURS OF CONSTANT IMPULSE RESPONSE FOR THIRD ORDER
APPROXIMATION TO GAUSSIAN FILTER ?
1.0
a
hit)
I
I 1
1
\
IN.)
,
.
I
.2 .1 i
IMPULSE it SPONSE ALONG THE 01A00 AL
o
I
1 1 .1 ? i
I i i 1 I
4. !
I PULS
1
..,
S.
RESPONSE ?0140
i
1
: ?????.L.
I
1 1
i
I
T CO
-4--
ROMA
E AXE
--4-----
2 t
Figure 2 IMPULSE RESPONSE ALONG THE AXES AND DIAGONALS FOR
THE SAME FILTER AS FIGURE I
7
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3.0