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Histogram-based method for contrast measurement

Luis Miguel Sanchez-Brea, Juan Antonio Quiroga, Angel Garcia-Botella, and Eusebio Bernabeu

A histogram-based technique for robust contrast measurement is proposed. The method is based on
fitting the histogram of the measured image to the histogram of a model function, and it can be used for
contrast determination in fringe patterns. Simulated and experimental results are presented. © 2000
Optical Society of America

OCIS codes: 100.0100, 100.2650, 100.2960.
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1. Introduction

Contrast measurement is a useful tool for the mea-
surement of the modulation transfer function
~MTF! of an optical system. A direct method for
determining the MTF is by measuring the contrast
of several sinusoidal fringe patterns with different
spatial frequencies imaged by the optical system
under test. However, since sinusoidal patterns
are difficult to obtain, usually the MTF is indirectly
measured with Ronchi patterns. This gives an es-
timation of the contrast transfer function ~CTF!.

y means of a modal expansion of the square pat-
ern, it is possible to obtain a relationship between
he CTF and the MTF, and when this relationship is
nverted, the MTF can be measured in terms of the
TF,1,2

MTF~ f ! 5
p

4 FCTF~ f ! 1
CTF~3f !

3
2

CTF~5f !

5

1
CTF~7f !

7
1 · · ·G , (1)

The authors are with the Departamento de Óptica, Universidad
omplutense de Madrid, Facultad de Ciencias Fı́sicas, Cuidad
niversitaria syn, 28040 Madrid, Spain. When this research was
erformed, J. A. Quiroga was with the Centro de Investigaciones
n Óptica, Apartado Postal 1-948, 37000 Léon GTO, México.
. M. Sanchez-Brea’s e-mail address is optbrea@eucmos.sim.
cm.es.
Received 13 October 1999; revised manuscript received 27
arch 2000.
0003-6935y00y234098-09$15.00y0
© 2000 Optical Society of America
098 APPLIED OPTICS y Vol. 39, No. 23 y 10 August 2000
where f is the frequency. In the absence of noise and
for a periodic signal the straightforward definition of
contrast is

C 5
IMAX 2 IMIN

IMAX 1 IMIN
, (2)

where IMAX and IMIN are the maximum and the min-
imum values of the signal.

When noise or small background variations are
present, it is not possible to apply this definition di-
rectly. If the image from which we wish to deter-
mine the contrast is formed by a straight-line
pattern, one possibility is to align the pattern ~by
digital or optical methods! parallel to one of the ref-
erence axes and then sum along it. In this way the
averaging will reduce the noise, enabling the mea-
surement of IMAX and IMIN. Unfortunately this sim-
ple method has big drawbacks, especially if medium-
low contrasts are to be measured ~C ' 0.5–0.1!. In
addition, the determination of IMAX and IMIN by a
simple averaging is sensitive to noise andyor to small
background variations. Furthermore, this method
cannot be applied when fringes become curved after
passing through the optical system or for circular
fringes, which can be used for the determination of
directional MTF when we analyze a small slit of the
image at the proper direction.

Another way to evaluate the contrast of a fringe
pattern is by use of the histogram, as suggested by
Lai and von Bally.3 However, the algorithm they
propose does not match the definition of contrast @Eq.
~2!# even when no noise is present. In this study we

ropose a histogram-based technique for contrast
easurement of fringe patterns that can be applied

n the presence of additive noise and with patterns
omposed of fringes that are not straight. The
ethod is based on fitting the histogram of the mea-

ured fringe pattern to the histogram of a model func-



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tion that depends on several parameters. The
parameters of the model function provide informa-
tion about contrast and noise level.

By means of histogram analysis we can obtain
more information about fringes, besides contrast.
Generally histograms of fringe patterns present two
lobes, but square and sinusoidal fringe patterns with
the same contrast do not have the same histogram
shape. From this difference, information about the
fringe shape can be obtained.

This paper is organized as follows: In Section 2
we show how to calculate the histogram of the chosen
model function and how to extract the contrast infor-
mation from an experimental fringe pattern by use of
this model function. In Section 3 we apply the algo-
rithm to simulated fringe patterns. In Section 4 ex-
perimental results with real fringe patterns are
obtained. Finally in Section 5 conclusions are given.

2. Contrast Measurement from the Histogram

The continuous histogram h~y! of a continuous func-
tion f ~x! can be defined as the number of points j that
fulfill y # f ~j! , y 1 dy. The mathematical expres-
sion of this definition is

h~y! 5 *
V

d@ f ~x! 2 y#dx, (3)

where d stands for the Dirac-delta function and V is
the set of points x where we are computing the his-
togram. The Dirac-delta function of a continuous
function g~x! is4

d@g~x!# 5 (
i

1
ug9u

d~x 2 xi!, (4)

where g9 represents derivative of g with respect to x,
u z u means absolute value, xi are the roots of g~x!, and

is an index that runs over them. Then, if we con-
ider

g~x! 5 f ~x! 2 y, (5)

by applying Eqs. ~3! and ~4! and taking into account
that the root of Eq. ~5! is x 5 f21~y!, we obtain

h~y! 5 *
V

1
u f9u

d@x 2 f 21~y!#dx, (6)

or

h~y! 5
1

u f9@ f 21~y!#u
. (7)

Expression ~7! can be rewritten, by means of the in-
verse function theorem, as

h~y! 5 Udf 21~y!

dy U . (8)

Equations ~7! and ~8! are a well-known result of the
heory of probability if we interpret the histogram of
function as the probability distribution for the val-
es of y. From Eq. ~6! it is clear that the conditions
for f ~x! to have a histogram are the existence of the
inverse function f21~y! and the absence of extrema of
f ~x! within V. For these reasons, in the case of pe-
riodic functions, we cannot directly apply Eqs. ~7! or
8!. To compute the histograms, we must use only
ne semiperiod between a minimum and a maximum
nd multiply the histogram by the number of semi-
eriods present. For instance, to compute the con-
inuous histogram of f ~x! 5 a 1 b cos~wx!, a . b, we
se the interval V 5 ~0, pyw! where no extrema are
resent and f21~y! is defined. Then, applying Eq.

~7!, we obtain

h~y! 5 H1yuv@b2 2 ~y 2 a!2#1y2u a 2 b , y , b 1 a
0 elsewhere .

(9)

n this way, when our fringe pattern corresponds to
ines of any shape with sinusoidal profile, we can fit
ts histogram with the one given by Eq. ~9!, and from
he fitting parameters we can obtain the contrast by

5 ubyau. However, if we use Ronchi rulings to
etermine the CTF of an optical system, this simple
cheme does not work properly because of noise in the
ringe pattern and discontinuities of the model, as
xplained in Subsections 2.A–2.D.

A. Histogram of a Noisy Signal

When a fringe pattern is obtained by means of a CCD
camera or by other procedures, there always exists
noise that modifies the histogram shape. We will
assume an additive noise n~r! whose probability dis-
tribution is p~y!, where r is the position vector. The
histogram h~y! of the fringe pattern I~r! can be con-
sidered to be the probability distribution of the inten-
sity values. Thus the histogram of a noisy image,

I#~r! 5 I~r! 1 n~r!, (10)

will be the probability distribution of a signal com-
posed by the sum of two signals with probability dis-
tributions h~y! and p~y!. From probability theory5,6

it can be proved that the histogram of I#~r! is

h# ~y! 5 h~y! p p~y!, (11)

where p denotes the convolution product. Then, by
selecting a model for the additive noise, we can de-
termine the histogram of the selected model with
additive noise as the convolution product of the model
histogram with the noise histogram.

B. Elimination of Divergences

Another problem of the continuous histogram @Eq.
~7!# are the divergences that exist near the extrema of
the model, where f9~x! 5 0. The presence of these
divergences destabilizes the minimization algorithm
necessary for the fitting process, making it difficult
and unreliable. We solved this problem by using a
sampled model instead of a continuous one. For a
sampled model with M samples it is clear that the
histogram will never diverge, the fitting making eas-
10 August 2000 y Vol. 39, No. 23 y APPLIED OPTICS 4099



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ier and more reliable. This solution is not really an
approximation, since any digital image acquisition
system imposes a spatial sampling and an intensity
quantization.

The first step to compute the discrete histogram of
a sampled function is to define the range of values for
the intensities and the domain of the sampled func-
tion. The possible values for the intensities are

yn 5 ~n 2 1!Dy, n 5 1, . . . , N, (12)

here Dy is the difference between two adjacent in-
ensity levels and we are assuming that the lowest
alue for the intensity is 0. The domain V, where
he sampled function is defined, is given by a uniform
ampling,

xm 5 x1 1 ~m 2 1!Dx, m 5 1, . . . , M, (13)

here x1, Dx, and M are free parameters that deter-
mine this sampling.

The discrete histogram h@n# ~where the square
brackets indicate that the discrete histogram can
have only positive integer values! of the sampled
b
a

function f ~xm! is defined as the number of points xm
such that

y1 # f ~xm! , y1 1 Dyy2, n 5 1,

yn 2 Dyy2 # f ~xm! , yn 1 Dyy2, 1 , n , N,

yN 2 Dyy2 # f ~xm! , yN, n 5 N. (14)

ith this definition h@n# is basically calculated by
eans of counting the number of points xm in the

interval @ f21~yn 2 Dyy2!, f21~yn 1 Dyy2!#, that is,
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where int~x! denotes the integer part of x and
nMAX and nMIN are the first and the final in-

ices with h@n# different from 0. They are computed
y

nMAX 5 int@ f ~xM!yDy# 1 1,

nMIN 5 intf @~x1!yDy# 1 1. (16)
100 APPLIED OPTICS y Vol. 39, No. 23 y 10 August 2000
The 11 added to h@nMAX# is to take into account the
nal point @xM, f ~xM!#. From Eq. ~15! it is easy to

verify that

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N

h@n# 5 M, (17)

where M is the total number of sampling points.
Then Eqs. ~15! and ~16! define the way a discrete

histogram of a sampled function can be computed.
However, working with integer parts and absolute
values is not comfortable from the viewpoint of the
analysis, so we are going to define a continuous his-
togram of a sampled model function, h~n!, from its
discrete counterpart, h@n#. To eliminate the abso-
lute values, we are going to consider only model func-
tions f ~x! monotonically increasing in the range @x1,
xM#. This is not a lack of generalization, because the

istogram of a monotonically decreasing function is
he same as the corresponding mirror reflection about
he vertical axis. With this consideration and when
he integer parts of Eq. ~15! are eliminated, the con-
inuous histogram of the sampled function, h~n!, is
where nMAX and nMIN are defined by Eq. ~16!.
Again, from Eq. ~18! it is easy to verify that

(
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N

h~n! 5 M, (19)

so no divergences appear as long as f21~yn! exists.
Also, in the limit Dx3 0, Dy3 0, Eq. ~18! is the link
etween the discrete histogram of a sampled function
nd the corresponding continuous histogram of a con-
inuous function given by Eqs. ~7! and ~8!. Thus Eq.
18! is the expression of the histogram we adopted to
ake the calculations. Figure 1 shows the relation

etween h@n# and h~n! for the model function f ~x! 5
a 1 b cos~wx! with b 5 75, a 5 125; x1 5 0, xM 5 pyw,
and M 5 300.

C. Model Function

Once we know how to account for the effect of the
noise in the histogram of the model function @Eq.
h@n# 5 5
uint@ f 21~ynMIN

1 Dyy2!yDx# 2 int~x1yDx!u n 5 nMIN

uint@ f 21~yn 1 Dyy2!yDx# 2 int@ f 21~yn 2 Dyy2!yDx#u nMIN , n , nMAX

uint~xMyDx! 2 int@ f 21~ynMAX
2 Dyy2!yDx#u 1 1 n 5 nMAX

0 elsewhere

, (15)
h~n! 5 5
@ f 21~ynMIN

1 Dyy2! 2 x1#yDx n 5 nMIN

@ f 21~yn 1 Dyy2! 2 f 21~yn 2 Dyy2!#yDx nMIN , n , nMAX

@xM 2 f 21~ynMAX
2 Dyy2!#yDx 1 1 n 5 nMAX

0 elsewhere

, (18)



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~11!#, and how to manage the divergences appearing
in the continuous definition of the histogram @Eq.
18!#, the next step is to select a suitable model func-
ion able to adapt to a variety of profiles, from high-
ontrast square patterns to almost pure sinusoidal
atterns, considering a continuous grading of shapes
rom square to sinusoidal. The simple sinusoidal

odel function of Eq. ~9! is not flexible enough to
ccomplish all this phenomenology. A good candi-
ate for this, and the one we finally adopted, was the
igmoidal function fS~x! given by

fS~x! 5 a 1
b 2 a

1 1 exp@2~x 2 x0!yl#
, (20)

where a and b are parameters that control the max-
imum and the minimum values of fS~x!, x0 is a pa-
rameter that controls the possible lateral shift, and l
is the parameter that controls the shape: As l
grows, fS~x! changes from a step function to a sinu-
soidallike function. Then, as explained above for pe-
riodic signals, fS~x! will represent an approximation
of one semiperiod. The continuous histogram of
fS~x! is, when we apply Eq. ~7! or Eq. ~8!,

hS~y! 5 Hl~b 2 a!y@~b 2 y!~y 2 a!# y [ ~a, b!
0 elsewhere . (21)

s can be seen in Eq. ~21!, divergences appear in y 5
and y 5 b, making it difficult to be used as a fitting

unction ~as explained above!.
When we take into account that the inverse func-

ion of fS~x! is

fS
21~y! 5 x0 2 l lnSy 2 b

a 2 yD , (22)

the continuous histogram of the sampled version of
fS~x! is

where nMIN and nMAX are given by Eq. ~16!.
We have assumed the additive noise to be Gauss-

ian, with zero mean, since it is easily tractable.
Then its probability distribution will be

p~y, s! 5
1

Î2ps
expS2

y2

2s2D , (24)

and then the model histogram for the sigmoidal func-
tion with additive Gaussian noise incorporated in the
model will be, when we apply Eq. ~11!,

h# S~n, a, b, l, x0, s! 5 hS~n! p p~n, s!, (25)

hS~n! 5 5
@ fS

21~ynMIN
1 Dyy2! 2 x1#yDx

l

Dx
ln(H1 1

Dy
@b 2 ~n 2 1y2!Dy#JH1

$xM 2 fS
21~ynMAX

2 Dyy2!%yDx 1 1
0

where we have explicitly written the free parameters
of the model and p~n, s! is the sampled version of Eq.
~24! obtained by changing of the continuous variable
y with yn given by Eq. ~12!.

D. Calculation of Contrast Parameters

The final step in the proposed method for contrast
measurement is the calculation of the parameters a,
b, l, x0, and s that minimize the functional E given by

E 5 (
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$hD@n# 2 h# S~n, a, b, l, x0, s!%2, (26)

where hD@n# is the discrete histogram of the experi-
ental data to be analyzed, with its maximum nor-

alized to 1. From the parameters obtained in the
inimization of Eq. ~26! the contrast of the image
ith the histogram is

C 5
fS~xM! 2 fS~x1!

fS~xM! 1 fS~x1!
. (27)

When the noise is independent of the intensity val-
ues, Eq. ~26! resumes our contrast measurement

ethod. However, in real experiments we observed
n intensity dependence on the noise level. We ex-
ect that the development of a model with multipli-
ative noise will solve this problem. Nonetheless,

n 5 nMIN

Dy
~n 2 3y2!Dy 2 a#J) nMIN , n , nMAX

n 5 nMAX

elsewhere

, (23)
Fig. 1. Relationship between h@n# ~circles! and h~n! ~diamonds!
for f ~x! 5 a 1 b cos~wx!, with b 5 75, a 5 125, x1 5 0, xM 5 pyw,
and M 5 300. Only one of each of the four points is presented to
improve the visibility of the figure.
1
@

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we adopted a strategy based on the additive noise
model, which consists of dividing the histogram of a
noisy model function into two parts and using two
noise levels s1 and s2 for each one. We implemented
his idea by means of a weighting function L~n!, de-
ned as a step function with a linear transition zone
f width Dn 5 int@a~b 2 a!# gray levels and centered

at n 5 int@~a 1 b!y2#. The parameter that controls
the width of the transition zone is typically a 5 0.1.

Finally, the histogram for the sigmoidal function
with two levels of additive noise will be

h# S~n, a, b, l, x0, s1, s2! 5 L~n!@hs~n! p p~n, s1!#

1 @1 2 L~n!#@hs~n! p p~n, s2!#,

(28)

Fig. 2. Simulated one-dimensional fringe patterns with additiv
quasi-sine fringe pattern, ~c! quasi-square fringe pattern. For a
obtained from ~a!, ~b!, ~c!, respectively; curve, fits to sigmoidal his
102 APPLIED OPTICS y Vol. 39, No. 23 y 10 August 2000
and the parameters are determined by the minimi-
zation of

E 5 (
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$hD@n# 2 h# S~n, a, b, l, x0, s1, s2!%
2. (29)

From Eq. ~23! it can be seen that Eqs. ~26! and ~29!
re nonlinear minimization problems. Many algo-
ithms exist to perform this task. In particular we
sed the Nelder–Mead-type simplex algorithm imple-
ented in the optimization toolbox of the Matlab en-

ironment.7 In general the successful minimization
of a nonlinear problem needs good starting values for
the parameters involved. In our case the best re-
sults were obtained with the following rules for se-
lecting the starting values. When hD@n# is bimodal,

se and different fringe profile: ~a! for a sine fringe pattern, ~b!
es a 5 50, b 5 200, s 5 15 g.l. ~d!, ~e!, ~f !: circles, histograms
m by means of minimization of Eq. ~26!.
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togra



G

Table 1. Estimated Parameters for Fringe Profiles of Fig. 2 with a Histogram-Based Method
a and b are initialized as the gray values that corre-
spond to each maximum of hD@n#. If hD@n# presents
only one lobe, whose maximum is located in the gray
value G, the parameters a and b are initialized as

2 DG and G 1 DG, with DG typically 4.

Fig. 3. Simulated ~thin! and estimated ~thick! fringe shape for the
fringe patterns given in Fig. 1. The estimated fringe shape is
obtained from results of Table 1. As we can see, there is a good
agreement between the simulated and the estimated profiles.

Figure a* b* fS~x1!* fS

2~a! 50.67 203.59 54.01 2
2~b! 53.07 204.41 53.36 2
2~c! 52.35 198.50 52.35 1

*Gray levels.
The shape parameter l is initialized as hD@~na 1
nb!y2#, where na and nb are the indices of the gray
values a and b previously initialized. If the histo-
gram hD@n# is too noisy, local averaging centered on
n 5 ~na 1 nb!y2 is done for the calculation of hD@~na 1
nb!y2#.

Finally, the parameter x0 is initialized to 0, and the
parameters s1 and s2 are initialized as the width at
half the height of each of the lobes if the histogram is
bimodal. If the histogram is monomodal s1 and s2
are initialized as the corresponding width of the
unique lobe.

3. Application to Simulated Patterns

The algorithm depicted above was applied to three
simulated fringe patterns with added noise, as shown
in Fig. 2. Since we are working with 256 gray levels
~g.l.! in all these cases, then N 5 256, Dy 5 1, and
thus yMIN 5 0 and yMAX 5 255. We arbitrarily fixed
x1 5 21, xM 5 1, Dx 5 1023, M 5 2001, for all our
measurements.

Figures 2~a!–2~c! show the fringe patterns that
range from a sinusoidal to a square profile shape. In
the three cases the actual contrast is 0.6 ~a 5 50, b 5
200!, x0 is zero, and the noise standard deviation is
15 g.l. The corresponding histograms and the result
of the minimization of Eq. ~26! are shown in Figs.

Fig. 4. Estimated uncertainty for contrast estimation in terms of
Gaussian noise for several fringe shapes: ~inverted triangles! l 5
0.263; ~asterisks! l 5 0.160; ~circles!, l 5 0.0308. In this case the
contrast is C 5 0.6 ~a 5 50, b 5 200!.

Parameter

* C x0* l s*

0.575 0.06 0.263 14.77
0.586 20.07 0.160 12.42
0.583 0.03 0.0308 15.67
~xM!

00.25
04.12
98.50
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2~d!–2~f !. Visually the fit is good; and also numeri-
cally, as shown in Table 1.

Figure 3 represents half a period of the simulated
profiles of Fig. 2 together with the sigmoidal function
~20! evaluated for the corresponding parameters
shown in Table 1. The profiles depicted in Figs. 3~a!,
3~b!, and 3~c! correspond to the simulated profiles of

igs. 2~a!, 2~b!, and 2~c!, respectively. In this exam-
le we can see that the sigmoidal model function
dapts well from sine to square fringes.
To determine the performance of the algorithm we

stimated the contrast in terms of the noise level for
hree different shapes: sinusoidal, quasi sinusoidal,
nd almost square. For each level of noise and pro-
le shape we estimated the contrast ten times and
alculated the mean relative error. In Fig. 4 the
ean relative error is represented in terms of the

elative error. As can be seen, the error for the
quare-shaped profile is less that the one correspond-
ng to the sinusoidal and the quasi-sinusoidal ones.
he reason for that behavior is that the square profile
as well-defined populations that produce two well-

Fig. 5. ~a! Estimation of l in terms of the number of fringes per
frame for three different shape profiles, ~b! estimation of contrast
for the same cases. The theoretical contrast is C 5 0.6, and a
noise of 10 g.l. has been added.
104 APPLIED OPTICS y Vol. 39, No. 23 y 10 August 2000
eparated lobes in the histogram. As a consequence
t is easier for the minimization algorithm to detect
hem correctly. In the case of the sinusoidal and the
uasi-sinusoidal patterns the lobes are not so well
esolved, making it more difficult to determination
heir positions correctly as the noise increases.

We also carried out an experiment that consists of
odifying the number of fringes per frame to show

hat the fitting parameters do not depend on the spa-
ial frequency of the output signal, especially the pa-
ameter l that accounts for the fringe shape. We
btained that the spatial frequency of the signal af-
ects only the histogram as a scaling factor. In Fig.
~a! we show the parameter l in terms of the number
f fringes per frame for three different fringe profiles.
s we can see, l is constant for the three cases, except

or a small fluctuation that is due to noise. A con-
rast estimation in terms of the number of fringes
as also carried out @Fig. 5~b!#. For the three sim-
lations the contrast is approximately the same.

4. Experimental Results

In this section we show the results of the application
of the algorithm depicted above to determine the CTF
of an optical system. Our particular interest was
the measurement of the MTF of translucent rough
screens. The first step of the method we are using is
the measurement of the CTF and afterward estima-
tion of the MTF1 from the CTF measurement.

The optical setup for the measurement of the CTF
is shown in Fig. 6. The light produced by a halogen
lamp is directed to the input port of an integrating
sphere such that at the output port we have a nearly
uniform white-light beam that is localized on a dia-
phragm by means of an achromatic doublet. The
light emerging from the diaphragm is collimated by
means of another achromatic doublet and directed to
a Ronchi bar test that is projected on a rough trans-
lucent screen. The transmitted pattern is imaged
on a bidimensional CCD camera by means of a third
achromatic doublet. The CCD performs an 8-bit
quantization ~256 g.l.!, and then N 5 256, Dy 5 1,

MIN 5 0, yMAX 5 255. For the sampling of the
sigmoidal function we used x1 5 21, xM 5 1, Dx 5
023, M 5 2001.
As an example, in Fig. 7 we can see the application

of the algorithm to three real images. Figures 7~a!–
~c! are profiles from the images recorded by the CCD

Fig. 6. Sketch of the experimental setup for contrast measure-
ment of bar tests when a translucent rough sheet is interposed
between the bar test and the CCD camera.



t
t

camera, and figures 7~d!–7~f ! are the experimental
histograms of the corresponding images together
with the results of the minimization of Eq. ~29!. Es-
pecially remarkable is the example of Figs. 7~a! and
7~d!; in this case the maxima of the contrast pattern
are modulated, producing a lobe whose width is not
due to the noise but to the intensity modulation of the
maxima. Even in this case the algorithm works well
and correctly locates the lobes. This shows that the
parameters associated with the noise, s1 and s2, can
be interpreted as high-frequency additive noise or as
a low-frequency modulation present in the image, or,
in general, as a mix of both effects. Table 2 shows
the parameters obtained by the fits.

Finally we measured the CTF of a translucent
rough sheet. The contrast of the observed pattern
depends on the period of the bar test h, the separation

Fig. 7. Profiles from two-dimensional real fringe patterns and fit
and ~e! sinusoidal high-contrast pattern, and ~c! and ~f ! square hig
improve the visibility of ~d!–~f !. Circles, experimental histogram
between the bar test and the translucent rough sheet
d, and its roughness tys, as shown by Garcia-Botella
et al.,8 according to

CTF~h, d, syt! 5
4
p (

k50

`

~21!k S 1
2k 1 1D

3 expH2F~2k 1 1!
2p

h
sd~n 2 1!

t G2J .

(30)

In Fig. 8 we can see the fit of our experimental
measurements of the CTF ~obtained by the
histogram-based method! to Eq. ~30! in terms of spa-
ial frequency for two values of the separation be-
ween the bar test and the sheet: d1 5 5 cm ~circles!

their histograms: ~a! and ~d! sinusoidal low-contrast pattern, ~b!
trast pattern. Only one of each of the four points is presented to
e, the fit.
s for
h-con
; curv
10 August 2000 y Vol. 39, No. 23 y APPLIED OPTICS 4105



2

3

4

5

6

7

8

b

4

and d2 5 10 cm ~squares!. The roughness parame-
ter for the sample was tys 5 165 6 5, measured by
profilometry and reflectogoniometry. As can be
seen, the agreement between the model and the ex-
perimental measurements is good. Especially re-
markable is the behavior of the experimental
measurements in the tails, where a small peak can be
observed ~Fig. 8, circles, at 2 cyclesymm!. This be-
havior is reproducible, so we think that it corresponds
to diffraction effects that the geometric model @Eq.
~30!# does not consider.

5. Conclusions

In this study we have presented a histogram-based
technique for robust contrast measurement. The

Fig. 8. Measurements of contrast transfer function by means of
histogram-based method, for a dielectric rough sheet with n 5
1.523 and roughness parameter of tys 5 165 6 4, at distances
etween bar test and sheet of d1 5 5 cm ~squares! and d2 5 10 cm

~circles!, compared with the model proposed by Garcia-Botella et
al.8

Table 2. Estimated Parameters for Measurements of Real Fringe
Patterns by Means of a Histogram-Based Method

Figure

Parameter

a* b* fS~x1!* fS~xM!* C l s1* s2*

~a! 47.25 77.25 50.65 73.85 0.186 0.486 2.11 1.26
~b! 5.64 122.47 6.61 121.50 0.897 0.209 2.25 24.59
~c! 0.10 177.35 0.10 177.35 0.999 0.0026 0.55 8.35

*Gray levels.
106 APPLIED OPTICS y Vol. 39, No. 23 y 10 August 2000
method is based on the fitting of the histogram of the
measured image with the histogram of a model func-
tion. Analytical expressions for the histogram of
continuous as well as sampled functions were ob-
tained. The selected model function was the sigmoi-
dal function, which was shown to be flexible enough
to accommodate a great variety of cases. With this
technique contrast measurement can be performed
by means of patterns of almost any shape. Finally,
experimental measurements of the CTF of an optical
system were made in good agreement with the theo-
retical model.

Juan Antonio Quiroga acknowledges the support
of a postdoctoral grant of the Universidad Com-
plutense de Madrid and the Becas Internacionales
Flores-Valles program, Spain, and Centro de Inves-
tigaciones en Óptica, León, México. Luis Miguel
Sanchez-Brea acknowledges the support of a predoc-
toral grant of the Ministerio de Educación y Cultura
~Intercambio de Personal Investigador entre Indus-
trias y Centros Públicos de Investigación! with Ki-
nossel S.L.

This study was supported in part by Sainco Tráfico,
S.A., in the framework of Proyecto de Estı́mulo a
la Transferencia de Resultados de Investigación
~PETRI! program, Baliza Luminosa para el Guiado
de Tráfico Vial con Tecnologı́a LED project of the
Ministerio de Educación y Cultura.

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