1524 J. Opt. Soc. Am. A/Vol. 19, No. 8 /August 2002 Quiroga et al.
Modulo 2p fringe orientation angle estimation
by phase unwrapping with a

regularized phase tracking algorithm

Juan Antonio Quiroga

Departamento de Óptica, Universidad Complutense de Madrid, Ciudad Universitarias S/N, 28040 Madrid, Spain

Manuel Servin and Francisco Cuevas

Centro de Investigaciones en Optica AC, Apartado Postal 1-948, 37150 León, Guanajuato, México

Received November 30, 2001; revised manuscript received February 13, 2002; accepted February 25, 2002

The fringe orientation angle provides useful information for many fringe-pattern-processing techniques.
From a single normalized fringe pattern (background suppressed and modulation normalized), the fringe ori-
entation angle can be obtained by computing the irradiance gradient and performing a further arctangent com-
putation. Because of the 180° ambiguity of the fringe direction, the orientation angle computed from the gra-
dient of a single fringe pattern can be determined only modulo p. Recently, several studies have shown that
a reliable determination of the fringe orientation angle modulo 2p is a key point for a robust demodulation of
the phase from a single fringe pattern. We present an algorithm for the computation of the modulo 2p fringe
orientation angle by unwrapping the orientation angle obtained from the gradient computation with a regu-
larized phase tracking method. Simulated as well as experimental results are presented. © 2002 Optical
Society of America

OCIS codes: 100.2650, 100.2960, 120.3180, 120.5050.
1. INTRODUCTION
In many fringe-processing techniques a fringe pattern can
be modeled in a first approximation as a phase-modulated
sinusoidal irradiance distribution:

I~r! 5 B~r! 1 M~r!cos f~r!, (1)

where I is the irradiance, B is the background, M is the
modulation, f is the modulating phase, and r denotes the
spatial dependence of these quantities. The fringe orien-
tation angle can be defined as the angular direction of the
gradient of the phase, that is,

b2p~r! 5 arctanF ]f~r!/]y

]f~r!/]xG ; (2)

because of the arctangent function, this angle is computed
modulo 2p, and it will be denoted by b2p . From the
point of view of the fringe pattern, this orientation angle
determines the perpendicular orientation to the fringe at
each point. If the fringe pattern is normalized (the back-
ground is suppressed and the modulation is
normalized),1–3 the fringe orientation angle can be com-
puted from the normalized irradiance. The normalized
version of the fringe pattern of Eq. (1) is

IN~r! 5 cos f~r!, (3)

and from it the gradient is computed as

¹IN~r! 5 @sin f~r!#¹f~r!; (4)

in consequence, for a normalized fringe pattern, the gra-
dients of the phase and the irradiance are parallel but
point in opposite directions every time sin f changes its
0740-3232/2002/081524-08$15.00 ©
sign. This change of sign occurs at the zero crossings of
sin f, that is, at locations with f 5 kp, k being an inte-
ger. Also, the modulus of the irradiance gradient is
modulated by sin f. In other words, for normalized
fringe patterns the gradient of the irradiance is parallel to
the gradient of the phase, points toward the maxima of
the fringe pattern, and at the maxima, the minima, and
the stationary points of the phase is not well defined.
When only one fringe pattern is available, Eq. (2) is not
applicable and the fringe orientation angle must be com-
puted from the gradient of the normalized irradiance [Eq.
(4)]:

bp~r! 5 arctanF ]IN~r!/]y

]IN~r!/]xG ; (5)

because of the sign flips of the gradient’s phase, this ori-
entation angle is computed modulo p, and it will be de-
noted by bp . In the general case of a fringe pattern with
space-dependent background and modulation, it is still
possible to compute the orientation angle by using Eq. (5),
but the gradient of the irradiance will be, in general, not
parallel to the gradient of the phase.4

Phase demodulation from single fringe-pattern images
is important, for example, in the study of fast transient
phenomena. In the case of fringe patterns with closed
fringes, it is not possible to obtain good results with ordi-
nary algorithms such as Fourier transform or spatial
phase sampling methods. Recently, several studies that
demonstrate the importance of the fringe orientation
angle in reliable phase demodulation from a single fringe
pattern have been published.5–7
2002 Optical Society of America



Quiroga et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. A 1525
In the work of Servin et al.,5 a fringe-following scan-
ning strategy is employed together with a regularized
phase tracking (RPT) technique to demodulate in a robust
way speckle interferograms. In the RPT method the in-
tensity of the fringe pattern is locally considered spatially
monochromatic, so its irradiance may be modeled as a
sinusoidal function phase modulated by a plane p(•).
Thus, by minimizing an appropriate cost function that
takes into account the differences between the measured
irradiance and the proposed model, one can demodulate
the phase f(r) from the irradiance I(r). In this case it is
necessary to solve a nonlinear set of equations for every
position r for the three parameters of the plane p(•); the
phase f and its local spatial frequencies vx 5 ]f/]x and
vy 5 ]f/]y. In a general scanning strategy, these pa-
rameters change from point to point, and thus the local
phase and its spatial frequencies (or, equivalently, the
modulo 2p fringe orientation angle) are computed simul-
taneously. However, if a fringe-following scanning strat-
egy is used, that is, if the fringe pattern is processed as
much as possible along the isophasic lines, the value of
the local phase f(r) will change slowly from point to
point, decoupling in some way its computation from that
of the local spatial frequencies, making easier the de-
modulation.

In Ref. 6 Marroquin et al. adopt a different strategy
but one with a similar philosophy. In this work the de-
modulation of a single fringe pattern is based on the con-
struction of the analytic signal8 associated with the nor-
malized fringe pattern IN(r), given by A(r) 5 IN(r)
1 i sin f(r). From it we demodulate the wrapped
phase, computing it as W( f ) 5 arctan@Im(A)/Re(A)#,
where Im( ) and Re( ) denote the imaginary and real parts,
respectively, and W denotes the wrapping operator9; that
is, for any f, W( f ) 5 f 1 2kp, k being an integer such
that 2p , W(f ) , p. The analytic signal is computed
by minimizing a global cost function that forces it to sat-
isfy the following constraints: (1) The real part of A(r)
should be approximately proportional to the observed
fringe pattern and should be locally monochromatic, and
(2) the imaginary part of A(r) should be close to the cor-
responding quadrature image of IN(r). As above, the lo-
cal monochromatic character of the model implies that lo-
cally the phase field can be described by the value of the
phase and its local spatial frequencies. As Marroquin
et al. note, the simultaneous computation of the three pa-
rameters is computationally expensive and makes the
minimization of the global cost function difficult and un-
reliable. The solution proposed in Ref. 6 is to decouple
the computation of the local spatial frequencies from that
of the local phase. In this approach the first step is the
computation of the modulo 2p fringe orientation angle.
In their work Marroquin et al. propose a global regulariz-
ing scheme for this purpose, consisting of several global
cost function minimization steps, which are difficult to
implement.

Larkin et al.7 propose a totally different approach for
phase demodulation from a single fringe pattern. In this
work the so-called spiral phase quadrature (SPQ) trans-
form is presented. The main advantage of the SPQ
transform is that it can be implemented as a standard lin-
ear filter and that it can be applied by using techniques
such as a convolution or a Fourier transform. Given a
normalized fringe pattern (3), the output of the SPQ
transform applied to it is

SPQ@IN~r!# > i@sin f~r!#exp~ib2p!. (6)

From relation (6) the result of applying the SPQ trans-
form to a normalized fringe pattern is the corresponding
quadrature term out of a phase factor given by the
modulo 2p fringe orientation angle. In consequence, to
demodulate the phase by using an arctangent computa-
tion, it is necessary to cancel out this phase term by esti-
mating it in some way.

The above discussion indicates that a reliable determi-
nation of the modulo 2p fringe orientation is important
for the direct demodulation of the phase from a single
fringe pattern. In this work we present a new algorithm
for the computation of the modulo 2p fringe orientation
angle by unwrapping the orientation angle obtained from
the gradient computation. The unwrapping process is
carried out by means of a modified RPT method, together
with a fringe-following scanning strategy.

2. THEORETICAL FOUNDATIONS
A. Phase Unwrapping Based on Regularized Phase
Tracking
In a general case a phase map W(f ) can be unwrapped,
using the RPT procedure of Servin et al.,10 by computing
two signals in quadrature given by

fC~r! 5 cos W~f ! 5 cos f;

fS~r! 5 sin W~f ! 5 sin f; (7)

afterward, the phase of these signals is demodulated by
minimizing at each point the cost function

Ur~f, vx , vy!

5 (
~j, h! P N ù L

@ u fC~j, h! 2 cos p~x, y, j, h!u2

1 u fS~j, h! 2 sin p~x, y, j, h!u2

1 muf~j, h! 2 p~x, y, j, h!u2m~j, h!#, (8)

with

p~x, y, j, h! 5 f~x, y ! 1 vx~x, y !~x 2 j!

1 vy~x, y !~ y 2 h!, (9)

where L is a two-dimensional lattice with valid data and
N is a neighborhood region around the coordinate r in the
image where the phase is being determined. The field
m(r) is an indicator that equals 1 if the site has already
been estimated and 0 otherwise, and, finally, m is the so-
called regularization parameter.

The first two terms of Eq. (8) correspond to the so-
called fidelity term of the RPT method, and they reflect
that the observations fS and fC are modeled as sinusoidal
functions phase modulated by a plane p(•). This means
that fS and fC are considered locally monochromatic; thus
the parameters vx and vy can be interpreted as the local
spatial frequencies of the phase f. The last part of Eq.
(8) is the denominated regularization term, and it en-
forces smoothness and continuity of the estimated phase,



1526 J. Opt. Soc. Am. A/Vol. 19, No. 8 /August 2002 Quiroga et al.
making possible the computation of the continuous phase
f. The smoothness of the result is controlled by the size
of the neighborhood N and the value of m. Typical values
for N and m are (5 3 5) –(13 3 13) and 1–10, respec-
tively.

To unwrap the phase map W(f ) and obtain the con-
tinuous phase f by means of Eq. (9), we use the following
method: Starting at a given location r0 , are minimize Ur
with respect to f, vx , and vy , obtaining f(r0), vx(r0),
and vy(r0); in our case the minimization is performed by
using a gradient descent algorithm. Once the minimiza-
tion is performed, the location r0 is marked as processed,
we set m(r0) 5 1 [initially, m(r0) 5 0 ;r], and we may
proceed with the rest of the sites in L, repeating this op-
eration from pixel to pixel following a given path.

B. Phase Unwrapping of the Modulo p Fringe
Orientation Angle
As explained in Section 1, from the normalized version of
a fringe pattern, it is possible to obtain, by Eq. (5), the
modulo p fringe orientation angle bp . The relation of bp

to the modulo 2p orientation angle is bp 5 b2p 1 kp, k
being an integer such that 0 < bp < p; in consequence,

W~2bp! 5 W~2b2p 1 2kp! 5 W~2b2p!. (10)

Equation (10) states that the wrapped version of 2bp is
equal to the wrapped version of 2b2p . Thus, by unwrap-
ping W(2bp), one can obtain 2b2p and, simply by divid-
ing this result by 2, get the modulo 2p fringe orientation.

The unwrapping of W(2bp) has several problems. The
first is that in the presence of closed fringes, b2p is not a
continuous function: If the center of a closed fringe is en-
circled, the fringe angle will present a continuous varia-
tion from 0 to 2p rad, presenting a 2p discontinuity at the
starting point. These discontinuities start and finish at
the maxima and the minima of the modulating phase and
propagate across the fringe pattern, generating disloca-
tions (with an amplitude of 2p rad) in the fringe orienta-
tion angle. This means that in the presence of various
extrema of the modulating phase, the fringe orientation
angle will have several 2p dislocations. In consequence,
the b2p angle can be considered a piecewise-continuous
function, and any algorithm for its processing must take
this into account. As we propose to unwrap W(2bp), the
natural 2p dislocations are translated into 4p ones, which
must be preserved by the unwrapping algorithm. As an
example, Fig. 1(a) shows a circular fringe pattern, and
Fig. 1(b) shows the phase map W(2bp); as indicated in
Fig. 1(b), if the phase map is unwrapped along a circle
containing the center of a closed fringe, the unwrapped
phase will present a 4p dislocation. Figure 1(c) shows a
detail of the central region, indicating the location of the
discontinuity sources and the 4p dislocation.

The second problem is that because of the modulo p
computation of bp [Eq. (5)], two close discontinuity
sources9 of the same sign appear in W(2bp), located at
the centers of the closed fringes, at the start of the 4p dis-
locations, as shown in Fig. 1(c). These discontinuity
sources can be considered the sources of the dislocations.
It is well-known that the discontinuity sources make the
unwrapping path dependent. Also, they distort the cor-
Fig. 1. (a) Noisy circular fringe pattern and (b) associated fringe
orientation phase map W(2bp), where black represents 0 rad
and white represents 2p rad. As indicated in (b), if a circular
path is followed, encircling the discontinuity sources, a total
variation of 4p rad is found. (c) Amplification of the central white
square of W(2bp). The positions of the two positive discontinu-
ity sources as well as the 4p dislocation are marked.



Quiroga et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. A 1527
responding phase map in their neighborhood, making the
phase unwrapping unreliable in their surroundings.

Finally, as bp is computed from the gradient of the ir-
radiance, the noise present will be amplified by the de-
rivative operation. Thus, in general, the phase map
W(2bp) will be noisy. This noise amplification is espe-
cially noticeable at those regions with low spatial fre-
quencies, where both terms of the quotient necessary for
the computation of bp are small. Then, at the centers of
the closed fringes (maxima or minima of the modulating
phase), superposed on the distortion effect of the discon-
tinuity sources, big amounts of noise can be expected.

These three problems together make the phase map
W(2bp) logically inconsistent or, equivalently, its phase
unwrapping path dependent; in consequence, a standard
phase-unwrapping algorithm is precluded.

In this work we propose the use of a modified RPT-
based phase-unwrapping algorithm. In this setting the
phase unwrapping of W(2bp) and the estimation of the
modulo 2p fringe orientation angle will consist in the de-
modulation of the quadrature signals given by

fbC 5 cos W~2bp!, fbS 5 sin W~2bp!, (11)

Fig. 2. (a) Computer-generated fringe pattern and (b) associ-
ated modulo p fringe orientation phase map bp , where black
represents-p/2 rad and white represents p/2 rad.
which, according to Eq. (10), corresponds to

fbC 5 cos 2b2p , fbS 5 sin 2b2p ; (12)

then 2b2p is the modulating phase to be demodulated
from the quadrature images given by Eq. (11). In this
case the three parameters of Eq. (8)—f, vx , and
vy—correspond to 2b2p , 2]b2p /]x, and 2]b2p /]y, re-
spectively.

With respect to the problem associated with the pos-
sible noise present in fbS and fbC , it is well-known that
RPT has a high signal-to-noise ratio in the estimated con-
tinuous phase. The reason for this is that the RPT sys-
tem behaves as an extremely narrow-bandpass filter
tracking a wideband fringe pattern signal. This property
is especially remarkable in the case of the demodulation
of two fringe patterns in quadrature, as it is in our case.

For the processing of the discontinuity sources of
W(2bp), it is possible to design an adequate scanning
strategy. As mentioned above, the RPT demodulates in a
sequential way the fringe pattern, starting at one seed
pixel and then propagating from it until all of the region
of interest has been processed. The way in which this se-

Fig. 3. (a) fbC and (b) fbS signals [Eqs. (11)] associated with the
fringe orientation map bp shown in Fig. 2(b).



1528 J. Opt. Soc. Am. A/Vol. 19, No. 8 /August 2002 Quiroga et al.
Fig. 4. (a) Modulo 2p fringe orientation angle b2p obtained with the modified RPT method for the fringe pattern of Fig. 2(a); in this case
the fringe pattern was corrupted with additive noise normally distributed of standard deviation 1. (b) Modulo 2p fringe orientation angle
b2p ; in this case the fringe pattern was corrupted with phase noise normally distributed of standard deviation 1 rad. (c) Actual b2p map
corresponding to Fig. 2(a). (d) Histogram of the difference divided by p between the estimated b2p and the actual one for the two cases
of additive noise (solid curve) and phase noise (dashed curve). In the phase maps of this figure, black represents 0 rad, and white
represents 2p rad.
quential scanning is performed has been demonstrated to
be a key point in the demodulation process by means of
RPT (actually, for any sequential image processing tech-
nique). The best choice that we have found is the use of
a quality-map guided scanning strategy. Ströbel11 first
proposed this in the context of phase unwrapping. The
objective of the quality map is to guide RPT in such a way
that it should first process locally the highest-quality ar-
eas, leaving low-quality regions to be processed last. In
this way the possible errors that occur in the areas of low
quality will be confined to these areas. In the case of
W(2bp), the discontinuity sources are located at the cen-
ters of the closed fringes; thus a natural choice for the
quality map is the fringe pattern from which W(2bp) is
generated. In this case the quality map will guide the
RPT unwrapping along the fringes encircling the discon-
tinuity sources and always process these problematic
points last; this procedure can be called isophasic scan-
ning. For this reason, in the end, we adopted isophasic
scanning for the processing of W(2bp).
In spite of its advantages, the RPT-based phase-
unwrapping method has a problem with respect to the
discontinuous character of b2p (the 2p dislocations men-
tioned above). This is an important problem because we
know that the orientation angle that we wish to recover
can be discontinuous (if a closed fringe is present); on the
other hand, the regularization term of the local cost func-
tion (8) imposes continuity. This problem can be solved
by introducing a modification in the cost function (8). We
know that if there are closed fringes, the phase 2b2p is
going to have 4p jumps, so if we calculate the differences
appearing in the regularization term of Eq. (8) modulo 4p,
we are going to recover the continuous phase modulo 4p
associated with 2b2p . Then, for our modified functional,
4pk and 4pl represent the same value for all k and l, k
and l being integers. The basic assumption of this modi-
fication is that no noise is present with amplitude of 4p
rad or greater. In general, this is feasible, even for
speckle interferograms, where phase-noise amplitudes of
1 rad are expected.



Quiroga et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. A 1529
Finally, the modified local cost functional becomes

Ur~f, vx , vy!

5 (
~j,h!NPÞL

@ u fbC~j, h! 2 cos p~x, y, j, h!u2

1 u fbS~j, h! 2 sin p~x, y, j, h!u2

1 muW4p@ f~j, h! 2 p~x, y, j, h!#u2m~j, h!#, (13)

where W4p denotes the modulo 4p operation.
This procedure, a modified RPT-based phase unwrap-

ping (including isophasic scanning) of the modulo p fringe
orientation angle, permits a robust method for the deter-
mination of the modulo 2p orientation angle, as shown in
Section 3.

3. EXPERIMENTAL RESULTS
The first example is a computer-generated fringe pattern.
In this case the phase was computed as f(i, j)
5 15 sin(2pi/128)cos(2pj/128), i, j 5 1,..., 128, and from
it a fringe pattern I(r) 5 cos f(r) 1 n(r) was generated,

Fig. 5. (a) Shadow moiré fringe pattern. A clear variation from
left to right of the background and the modulation is clearly vis-
ible. (b) Associated normalized fringe pattern.
n(r) being a normally distributed random variable with
standard deviation 1. For uncooled CCD cameras elec-
tronic additive noise with a normal distribution is the
most important source of noise, and for this reason we
have used it for the simulation. Figure 2(a) shows the
noisy fringe pattern, and Fig. 2(b) shows the orientation
angle modulo p, bp , computed with Eq. (5). Figures 3(a)
and 3(b) show the quadrature signals fbC and fbS , respec-
tively, associated with W(2bp), which have been com-
puted with Eq. (11). Note the noise amplification in bp

and, consequently, in fbC and fbS . Figure 4(a) shows the
modulo 2p orientation angle b2p computed by our
method. To test the method with phase noise, we gener-
ated a fringe pattern I(r) 5 cos@ f(r) 1 n(r)#, n(r) being
a normally distributed random variable with standard de-
viation 1 rad. Figure 4(b) shows the computed modulo
2p orientation angle b2p . For comparison purposes Fig.
4(c) shows the actual fringe orientation angle computed
with Eq. (2). In Fig. 4 one can clearly see the 2p disloca-
tions beginning at the centers of the closed fringes. Fi-
nally, Fig. 4(d) shows the histogram of the difference be-
tween the estimated b2p and the actual one for the two
cases of additive noise (solid curve) and phase noise
(dashed curve). For the additive noise the mean value of
the difference is 20.002 rad, and the standard deviation
is 0.04 rad. For the phase noise the mean value of the
difference is 20.004 rad, and the standard deviation is
0.12 rad.

The second example is an experimental shadow moiré
fringe pattern obtained with a surface under deformation.
Figures 5(a) and 5(b) show the fringe pattern and its nor-
malized version, respectively. In Fig. 5(a) a clear back-
ground and modulation variation can be observed from
left to right. For the normalization we used the algo-
rithm described in Ref. 3. Figure 6 shows the W(2bp)
phase map computed from the normalized fringe pattern
shown in Fig. 5(b). Low-modulation regions associated
with fringe maxima and minima are clearly visible; also,
the center of the closed fringes (low-spatial-frequency re-

Fig. 6. (a) Fringe orientation phase map W(2bp) computed
from Fig. 5(b). Low-modulation regions associated with fringe
maxima and minima are clearly visible; also, the center of the
closed fringes (low-spatial-frequency region) shows a low-
modulation area. In this figure black represents 0 rad, and
white represents 2p rad.



1530 J. Opt. Soc. Am. A/Vol. 19, No. 8 /August 2002 Quiroga et al.
gion) shows a low-modulation area, source of the 2p dis-
location in b2p . Figures 7(a) and 7(b) show the quadra-
ture signals fbC and fbS , respectively, associated with the
W(2bp) phase map shown in Fig. 6. Figure 8(a) shows
the modulo 2p fringe orientation angle computed by our
method. Again, the 2p dislocation is visible, beginning at
the center of the closed fringes. To validate this compu-
tation, we have demodulated the phase of the shadow
moiré fringe pattern by using the SPQ transform [relation
(6)] reviewed in Section 1, the result is shown in Fig. 8(b).

Finally, we have demodulated the phase from a single
electronic speckle pattern interferometry (ESPI) fringe
pattern by using the modulo 2p fringe orientation angle
obtained by our technique and the SPQ transform. Fig-
ure 9(a) shows the ESPI fringe pattern. Figure 9(b)
shows the modulo 2p fringe orientation angle b2p com-
puted by our method. Figure 9(c) shows the demodu-
lated phase with use of the SPQ transform [relation (6)].
For comparison purposes Fig. 9(d) shows the cosine of the
phase map shown in Fig. 9(c) as can be seen, the result is
close to the fringe pattern shown in Fig. 9(a).

Fig. 7. (a) fbC and (b) fbS signals [Eqs. (11)] associated with the
fringe orientation phase map W(2bp) shown in Fig. 6.
4. CONCLUSIONS
We have presented a new method for the computation of
the modulo 2p fringe orientation angle of a single fringe
pattern by unwrapping the orientation angle modulo p
obtained from the gradient computation. The unwrap-
ping process is carried out by means of a modified RPT
method, together with a fringe-following scanning strat-
egy. The modified RPT phase-unwrapping process takes
into account the 2pI discontinuities that appear in the
modulo 2p fringe orientation angle when closed fringes
are present. In the presence of closed fringes, the phase
dislocations of the orientation angle modulo 2p preclude
the use of conventional phase-unwrapping algorithms
such as the methods described in Refs. 10 and 11. This
technique, together with the SPQ transform of Larkin
et al.,7 permits a robust demodulation of the phase from a
single fringe image with closed fringes. Simulated as
well as experimental results have been presented with
good results.

Fig. 8. (a) Modulo 2p fringe orientation angle b2p obtained by
the modified RPT method for the fringe pattern of Fig. 5, (b) de-
modulated phase map obtained by using the SPQ transform [re-
lation (6)] applied to the fringe pattern of Fig. 5 with the orien-
tation angle in (a).



Quiroga et al. Vol. 19, No. 8 /August 2002 /J. Opt. Soc. Am. A 1531
Fig. 9. (a) ESPI fringe pattern obtained in a deformation measurement experiment. (b) Modulo 2p fringe orientation angle b2p obtained
by the modified RPT method for the fringe pattern of (a). (c) Demodulated phase map obtained by using the SPQ transform [relation (6)]
applied to the fringe pattern of (a) with the orientation angle in (b). (d) Cosine of the phase map shown in (c); note the closeness with the
fringe pattern shown in (a).
ACKNOWLEDGMENTS
We are grateful for the financial support of this work
given by the European Union under project INDUCE,
Contract BRPR-CT97-0805, and by the National Council
for Science and Technology (CONACYT), Mexico, under
project PROSUVE, contract PR48/01-9858.

The corresponding author, Juan Antonio Quiroga Mel-
lado, may be reached by e-mail at aq@fis.ucm.es.

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