Measurement of surface tension and contact angle using entropic

INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 12 (2001) 288–298
www.iop.org/Journals/mt
PII: S0957-0233(01)17393-5
Measurement of surface tension and
contact angle using entropic edge
detection
C Atae-Allah1 , M Cabrerizo-Vı́lchez1 , J F Gómez-Lopera1 ,
J A Holgado-Terriza1 , R Román-Roldán1 and
P L Luque-Escamilla2,3
1
Departamento Fı́sica Aplicada, Universidad de Granada, Campus Fuente Nueva,
18071 Granada, Spain
2
Departamento Ingenierı́as Mecánica y Minera, EUP Linares, Universidad de Jaén,
C/Alfonso X el Sabio, 28, 23700 Linares (Jaén), Spain
E-mail: [email protected] (C Atae-Allah), [email protected] (M Cabrerizo-Vı́lchez),
[email protected] (J F Gómez-Lopera), [email protected] (J A Holgado-Terriza),
[email protected] (R Román-Roldán) and [email protected] (P L Luque-Escamilla)
Received 25 September 2000, in final form 21 December 2000, accepted for
publication 3 January 2001
Abstract
This paper presents a new method to measure the surface tension and the
contact angle of a liquid. The measurement procedure comprises three
steps: acquisition of the liquid drop image, image segmentation to obtain the
contour of the drop and surface-tension and contact-angle calculation by the
ADSA method. In the second step a new segmentation method is used based
on the Jensen–Shannon divergence, an entropic measurement of coherence
among distribution probabilities. The advantages of using this entropic
edge-detection method are shown; it is especially suitable when the source
image of the drop is affected by any kind of noise, blur or low-contrast effect.
Results reveal a better performance than other methods used in this field.
Keywords: surface tension, contact angle, Jensen–Shannon divergence, edge
detection, edge linking
1. Introduction
Surface tension is one of the most accessible experimental
parameters that describes the thermodynamic state and
structure of an interface. It is also of special interest in many
different fields in physics and engineering, such as lubrication
in machinery, diffusion and migration of liquids through
porous media—very important in the search for oilfields,
coatings, dispersions, adhesion, membranes etc. It is also very
important in the drop-formation process, which is fundamental
in industrial applications such as mixing, chemical processing,
fibre spinning, silicon-chip technology and spraying (in ink-jet
printers, diesel motors and irrigation).
The Wilhelmy plate and the Nouy ring methods have
traditionally been used for surface-tension measurement [1, 2].
These procedures are accurate but difficult to use, and much
care must been taken in the measurement process. A new
3
All correspondence to be submitted to Pedro Luis Luque-Escamilla.
0957-0233/01/030288+11$30.00
© 2001 IOP Publishing Ltd
technique, called the drop-shape method, has been developed
in recent years [3] taking advantage of new advances in image
processing. The drop-shape method operates by adjusting
a theoretical contour to the drop border obtained by image
segmentation. Commonly used drops are the sessile drop
(formed on a surface) and the pendant drop (hanging from
a capillary tube). This technique has some advantages
in comparison to traditional methods. First, only small
quantities of liquid are needed. Second, it can be applied
to liquid–vapour and liquid–liquid interfaces. Third, it can
be used in extreme conditions of temperature and pressure.
Moreover, the interface is not contaminated or interfered with
by the system. In addition, the technique makes it possible
to measure many parameters that cannot be obtained with
traditional methods, such as the dynamic interfacial tension,
dynamic contact angle [3], film balance [4] etc.
The first numerical method to measure surface tension
was established by Bashforth and Adams [5]. Since then,
Printed in the UK
288
Image-based surface tension measurement
great improvements in the speed and accuracy of the method
have been obtained with the introduction of computer image
analysis. There is a vast bibliography that covers all the
different aspects of this topic: data acquisition, different
edge-detection methods to detect the drop profile, different
algorithms of integration and optimization schemes [6–13].
However, none of these methods is robust against noise in
the source-drop image. Moreover, the drop image must be
well focused and images obtained in practice are unfortunately
frequently noisy and out of focus, fundamentally due to
the acquisition procedure (CCD or photographic devices, for
example) [14]. The advantage of using the proposed entropic
edge-detector method is its robustness when managing images
affected by these defects.
The paper is structured as follows. Section 1 introduces
the method. Section 2 reveals the technique of obtaining the
surface-tension value using the novel edge-detection method
proposed and an optimization algorithm to fit the detected drop
shape to the theoretical one. Results are presented in section 3
and the conclusions in section 4.
2. Surface-tension calculation method
The procedure for measuring the surface tension and contact
angle of a liquid is a three-step one, described as follows.
Step 1. Image acquisition. In this step an image of the
liquid drop is acquired from a CCD or photographic device
and stored in digital form in a computer.
Step 2. Edge detection. In this stage an edge-detection
method is applied to find the drop contour in the image.
It is usually recommendable to apply an edge-linking
algorithm to ensure the closeness of the detected borders.
Step 3. Interfacial parameter determination. Once
the experimental drop profile is performed, a numerical
method is used to obtain the theoretical one that best
fits it. The interfacial parameters of the liquid drop can
be obtained from the knowledge of the fitted theoretical
profile.
2.1. Image acquisition
The experimental apparatus for surface-tension measurement
is shown in figure 1. As can be seen, it is comprised of two
main devices: the image-acquisition and drop-control systems
(which controls the drop and environmental conditions).
A Leika Apozoom microscope coupled with a Sony
CCD B&W video camera—SSC-M370CE with 752 × 582
resolution—was used for the image acquisition. The camera
is connected to a video frame-grabber card (DT 2855) that
has a resolution of 768 × 512 pixels with 256 grey levels.
The frame-grabber card has a similar resolution to the video
camera, which prevents loss of accuracy in the light-intensity
detection of each photocell of the video camera. This feature is
important for detecting the drop profile with the best possible
precision. The frame-grabber board is mounted on a Pentium
computer and to a separate RGB monitor to display the drops.
The light source is placed behind a diffuser that produces a
uniform light on the drop, which is controlled by a variac
power supply. To avoid vibration, all the instrument devices
are placed on an antivibratory table from Kinetic System Inc.
Vibraplane. The drop is put into a glass cuvette, inserted into
a thermostatted cell, which maintains a constant temperature
value by water circulation through a jacket. The cell is on a
three-axis micropositioner that allows the drops to be handled
in any direction.
In pendant experiments, the drop is inserted into the cell
with a syringe (Hamilton Microlab 500 microinjector) that
pulls the liquid into a Teflon capillary 0.5 mm in diameter
that prevents wetting of the liquid. This allows the formation
of stable drops automatically at low injection rate to prevent
drop vibration. In sessile experiments, the drop is placed on the
surface with a microsyringe that controls the amount of liquid
inserted. For sessile experiments a solid surface with a low
surface tension was chosen (Teflon (FEP)). The microsyringe
is attached to a stand adapter to minimize vibration. All
glassware and Teflon ware were cleaned in chromic sulphuric
acid.
First of all, instrument calibration is necessary to obtain
accurate and precise surface-tension values. A plumb bob is
used to align the vertical axis with the vertical axis of the video
camera because the drop must be axisymmetric in order to use
the optimization method in section 2.3. The plumb consists of a
weight hanging from a fine copper wire submerged in a beaker
of water to dampen oscillations. An image of the plumb wire
is captured with the CCD camera and a program determines
the vertical axis. This process continues until the user corrects
the vertical axis of the video camera. A grid with an array
of squares of 0.25 mm per side (Graticules Ltd Tonbridge) is
used to determine the picture magnification, horizontal/vertical
aspect ratio and geometric distortion due to the optical system.
The program uses the grid to calculate the co-ordinates of
the drop profile extracted with the edge-detection method in
millimetres.
2.2. Edge detection
In the specialized literature, detection of drop profiles is
usually performed using both global and adaptive thresholding
methods [6–8]. These techniques require highly contrasted
images to select the right threshold value, so they are only
adequate for pendant drops in a liquid–vapour interface.
Sessile and pendant drops in liquid–liquid interfaces have such
poor contrast in general that it is not easy to obtain the drop
profile with these traditional methods. This is why [9, 13] use
more powerful edge detectors, based on gradient magnitudes,
such as Sobel or five-level Robinson detectors [14].
In addition, many causes can degrade the image during
the acquisition process:
(a) uncertainties in the sensor, fluctuations in the light
intensity, and other similar error sources;
(b) photoelectronic, Gaussian noise appearing in the
conversion of photons to electrons;
(c) thermal noise in the signal amplification process—usually
this kind of noise is modelled as Gaussian type, with zero
mean;
(d) impulsive type noise (salt and pepper) appearing in signal
transmission processes and
(e) other causes of error (blur due to drop vaporization, out
of focus etc).
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C Atae-Allah et al
Figure 1. Schematic illustration of drop-shape method.
Unfortunately, none of the edge-detection techniques
commonly used in the specialized literature is robust against
noise or blurring. The method presented in this paper,
in contrast, is applicable in any practical situation, even
when the above-mentioned defects appear. It is based
on the Jensen–Shannon divergence between the normalized
histograms of two samples taken from the image.
2.2.1. The Jensen–Shannon divergence. Jensen–Shannon
divergence (hereafter JS), proposed by Lin [15], has proved to
be a powerful tool in the segmentation of digital images [16].
It is a measurement of the inverse cohesion of a set of
probability distributions having the same number of possible
realizations:
r
i
π i Pi −
πi H (Pi ) (1)
JSπ (P1 , P2 , . . . , Pr ) ≡ H
i=1
i=1
where
P1 , P2 , . . . , Pr are discrete probability distributions
Pi = {Pi,j /j = 1, . . . , n}
i = 1, . . . , r
π1 , π2 , . . . , πr are the distribution weights for Pi ;
r
π ≡ π1 , π2 , . . . , πr /πi > 0,
πi = 1
H (Pi ) = −
n
i=1
Pi,j log Pi,j is the Shannon entropy.
j =1
Divergence grows as the differences between its arguments (the probability distributions involved) increase, and
vanishes when all the probability distributions are identical.
As only two probability distributions are used
here, the final expression of the Jensen–Shannon divergence
is
P1 + P2
1
− [H (P1 ) + H (P2 )]. (2)
JS(P1 , P2 ) ≡ H
2
2
The application of JS to edge detection is based on a three-step
structured procedure, as follows [17].
(a) Calculation of divergence and direction matrices. In
this step the divergence and direction matrices associated
with the image are calculated. The divergence matrix is
composed of real numbers and is similar to that obtained
with the gradient operator for edge detection. The
direction matrix contains the estimated edge direction for
all image pixels.
290
Figure 2. A window sliding across a perfect edge.
(b) Edge-pixel selection. Edge pixels are chosen by means of
a local maximum selection criterion from the divergence
matrix, resulting in a binary image with the image edges.
(c) Edge linking. The final stage is an edge-prolongation
procedure that attempts to connect sets of unconnected
edge pixels in the previously obtained binary image.
2.2.2. Calculation of divergence and direction matrices. Let
us consider a window made up of two identical subwindows
and sliding down over a straight edge between two different
textures (see figure 2). It has been shown [16] that in
such conditions the JS between the normalized histograms
of the subwindows reaches its maximum value when each
subwindow lies completely within one texture.
In accordance with the above procedure, it is possible to
assign a JS value to each pixel in the image. Hence, pixels
with a high JS have a high probability of being edge pixels,
and vice versa. If, unlike the example shown in figure 2,
the window-to-edge angle is not 90◦ , the JS maximum will
be low or even undetectable, while the JS inside a given
texture will be close to zero or to the base value. This then
means trying several window orientations for each pixel. Only
four orientations are, however, technically possible: vertical,
horizontal, and two diagonals. Thus, the values JS1 , JS2 , JS3
and JS4 are calculated for the fixed window orientations 0, π/4,
π/2 and 3π/4. In this work we have used a square sliding
window, with user-defined size.
Now the question is how to obtain an estimate of the
direction from these four values that maximizes the JS and then
the value of this maximum, JSmax . For a given pixel, the JS
value is a π -periodic function of window orientation over the
image. It reaches its maximum value for a given orientation,
β, and a minimum in β + π. A theoretical model describing
this periodic function can thus be expressed as follows:
JS(x) = a + b cos(β + 2πx)
x ∈ [0, 1]
(3)
Image-based surface tension measurement
where a and b are constants determining the amplitude of JS
oscillation and β ∈ [0, π) is the edge direction in this pixel.
The JS direction, x, is normalized in the interval [0, 1] to
simplify calculations. According to the trigonometric relation,
a theoretical model equivalent to (3) is
JS(x) = c + msen(2πx) + n cos(2πx)


16x 2 − 16x + 3
−16x + 32x − 15
2
x ∈ [ 41 , 43 ]
x∈
Estimated edge
direction
x ∈ [0, 1] (4)
where c, m and n are constants. Nevertheless, due to the
computational effort required for trigonometric functions, they
can be replaced by other functions with similar properties, such
as quadratic splines:
x ∈ [0, 21 ]
−16x 2 + 8x
sen(2π x) ≈ f (x) ≡
16x 2 − 24x + 8
x ∈ [ 21 , 1]
cos(2π x) ≈ g(x)

2
x ∈ [0, 41 ]

 −16x + 1
≡
Pixel
under study
(5)
[ 43 , 1].
Then, f (x) is obtained as a quadratic spline of class 1, with
nodes at points {0, 41 , 21 , 43 , 1}, interpolating to sen(2πx) at
points {f (0), f ( 41 ), f ( 21 ), f ( 43 ), f (1), f ( 41 ), f ( 43 )}. In the
same way, g(x) is obtained as a quadratic spline of class 1
with the same nodes as f (x), interpolating to cos(2π x) at
the points {g(0), g( 41 ), g( 21 ), g( 43 ), g(1), g (0), g ( 21 ), g (1)}.
With a least-squares fit of the divergence model (4) and the
modification (5) for points JS1 , JS2 , JS3 and JS4 , the solution
is
JS1 + JS2 + JS3 + JS4 JS2 − JS4
JS(x) =
+
f (x)
4
2
JS1 − JS3
g(x).
(6)
+
2
The direction, x, having the maximum JS values can be
obtained from the equations
if JS1 − JS3 0, JS2 − JS4 0 ⇒
JS2 − JS4
∈ [0, 41 ]
x=
4[(JS1 − JS3 ) − (JS2 − JS4 )]
if JS1 − JS3 0, JS2 − JS4 0 ⇒
4(JS1 − JS3 ) − 3(JS2 − JS4 )
∈ [ 43 , 1]
x=
4[(JS1 − JS3 ) − (JS2 − JS4 )]
if JS1 − JS3 0, JS2 − JS4 0 ⇒
2(JS1 − JS3 ) − (JS2 − JS4 )
∈ [ 41 , 21 ]
x=
4[(JS1 − JS3 ) − (JS2 − JS4 )]
if JS1 − JS3 0, JS2 − JS4 0 ⇒
2(JS1 − JS3 ) + 3(JS2 − JS4 )
(7)
∈ [ 21 , 43 ].
x=
4[(JS1 − JS3 ) − (JS2 − JS4 )]
Finally, defining δ = πx ∈ [0, π) as the estimated edge
direction, the method described above calculates x from (7)
(i.e., the estimated direction that maximizes the JS) and the
estimated JSmax from (6).
In this way, each image pixel is labelled with a pair of
values: the estimated edge direction and the estimated JSmax
placing the sliding window in accordance with the estimated
edge direction. Thus, two matrices are built: the divergence
matrix (which indicates the probability of a pixel belonging to
the image edge) and the direction matrix (which estimates the
edge direction for the edge pixels).
Sliding monodimensional
window
Figure 3. Monodimensional window, perpendicular to estimated
direction in each image pixel.
However, direct application of the above method does
not provide good results for some kinds of image—corrupted
by Gaussian noise, or with regions having small fluctuations
in grey levels—because the JS could be too sensitive to any
change in grey levels between regions. It is therefore better
to construct the divergence matrix including extra information
in addition to the histogram information using the following
expression:
(8)
JS∗i,j = JSi,j (1 − α + αWi,j )
where Wi,j = |Nw1 − Nw2 |/Nw , Nw1 and Nw2 being the
average grey levels of subwindows W1 and W2 , and Nw the
maximum grey level inside the window (normalization factor).
α ∈ [0, 1] is the attenuation factor, which determines the
weights of JS and the grey levels inside the window. This
modification makes the JS suitable for different kinds of image,
thus transforming our algorithm into a hybrid among texturebased algorithms [18], Jensen–Shannon divergence [16] and
grey-level based algorithms (gradient, Laplacian, Laplacian
and gradient of the Gaussian etc) [19–21].
2.2.3. Edge-pixel selection. In this step the procedure
selects which pixels from the divergence matrix are edge
pixels. Thresholding the divergence matrix [22] is not always
useful, since maximum JS values depend on the composition
of adjacent textures, and will thus vary according to texture.
Consequently, it would seem more appropriate to use a local
criterion. Accordingly, each edge-pixel candidate is the
centre of an odd-length monodimensional window, placed
perpendicular to the estimated edge direction in that pixel
(figure 3).
Thus, every edge pixel has to satisfy
JScentre − JSj Td
(9)
for any other pixel j in that particular monodimensional
window, where Td is a threshold. Pixels marked as edge pixels
are then outstanding local maxima of the divergence matrix.
Obviously, detection results depend directly on the parameter
Td , which can be modified by the user if necessary.
This local edge-pixel detection method requires simple
divergence matrix pre-processing. Small fluctuations, often
due to noise in the original image or to texture regularity,
can introduce a great number of false maxima, although they
are usually fairly low. The divergence matrix is therefore
smoothed out by repeatedly applying a 3 × 3 mean filter.
Selection of a local maximum is, in a sense, a thinning
procedure since just one pixel will usually be detected as an
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C Atae-Allah et al
C
C C
C C C
E C
E C
E
Table 1. Typical parameter values.
Parameter value
C
Figure 4. End points and neighbour candidates for edge
prolongation. E, end point; C, neighbour candidates. The remaining
grey pixels are edge pixels.
edge pixel within the neighbourhood, as determined by the size
of the monodimensional window. In fact, rarely would more
than one pixel share the same maximum JS.
2.2.4. Edge linking. The two steps described above make
it possible to extract the image edge pixels. However, it is
not always feasible to establish a good compromise between
the quality of the binary image obtained and the desired
connectivity of the edge pixels, possibly due to the presence
of noise in the original image, and the texture composition of
the image regions.
In order to deal with these problems, a third step can
be added: edge-pixel linking [23]. This step attempts to
join the various sets of edge pixels using information from
the divergence matrix associated with the image, together
with knowledge of the direction in which maximum JS is
produced. In broad terms, the linking procedure consists in
extracting edge pixels unmarked since they did not satisfy
the condition (9), but nearly did. Not all the pixels in the
image are candidates for filling the gaps, only those classified
as neighbour candidates of end pixels.
The definition of end pixel in [24] includes several variants
that may influence the result of the linking process. The
present paper uses the definition of end pixel as a pixel
having one or two marked pixels joined together. Thus, only
certain neighbour end pixels are candidates for prolonging
image edges. In figure 4 we present the candidate pixels for
continuing a given edge.
For a given neighbour candidate to be marked as an edge
pixel, it must satisfy two conditions:
(a) Its associated JS must be reasonably high.
prolongation condition is then
JSend − JSneighbour candidate τd
The first
(10)
where τd is a threshold, which has at first no relation with
parameter Td in step (2) of the procedure.
(b) The estimated edge direction of the end pixel (Dir end ), the
edge-direction neighbour candidate (Dir neighbour candidate )
and the direction of the physical line joining them (Dir
(end, neighbour candidate)) must not differ by more than
a specified amount. The second prolongation condition is
then
(Dir (end, neighbour candidate))-Dir end )
2
+(Dir (end, neighbour candidate)
−Dir neighbour candidate )2 τθ
where τθ is another threshold.
292
(11)
Typical value
3×3
0.75
Step 1
1. Sliding window size
2. Attenuation coefficient
Step 2
1. Monodimensional window size
2. Mean filter iterations
3. Local maximum selection threshold
Step 3
1. Divergence threshold
2. Direction threshold
11
8
0.7
0.1
0.5
The two foregoing conditions are used in an attempt to
extract as edge pixels those pixels lying next to end pixels and
whose JS and direction are sufficiently close to those of the end
pixel to be extended. It should be borne in mind that when a
new pixel is marked as an edge, other adjacent pixels can then
become end pixels. So, the algorithm must foresee this event
in order to continue the search for links.
2.2.5. Final considerations about edge-detection procedure.
This section briefly summarizes all the parameters used by the
edge-detection procedure. Initially, the proposed segmentation
procedure may seem difficult to use due to the elevated number
of parameters that user can vary. But in practice the procedure
is easy to use because all the images are similar. In table 1 we
present the typical parameter values used in this work.
2.3. Interfacial-parameter determination
Once the profile of the drop is obtained, a numerical method is
used for obtaining the theoretical profile of the drop that best
fits it by adjusting the interfacial parameters, which are then
determined.
The theoretical profile determination of a liquid drop is
a complex problem. From equilibrium considerations, Young
and Laplace concluded that
1
1
(12)
+
P = γ
R1 R 2
where P is the pressure difference across the interface
between the two phases (liquid–gas or liquid–liquid), γ is the
surface tension and R1 and R2 are the two principal radii of
the curvature of the drop surface. If the drop is axisymmetric
with respect to the vertical axis, the pressure difference in the
apex of the drop Pa can be calculated from equation (12) by
taking R1 = R2 = R0 :
Pa =
2γ
.
R0
(13)
Thus, P in equation (12) may be put in terms of Pa by
using the hydrostatic fundamental equation
P = Pa ± ρgz =
2γ
± ρgz
R
(14)
where ± means that the sum is carried out when the drop is
lying on a horizontal surface (sessile drop) and the difference is
taken when the drop is falling from a dropper (pendant drop);
ρ is the density difference between liquid and surrounding
Image-based surface tension measurement
y
0
xo
s
φ
so
yo
y
s
0
xo
s
φ
yo
φ
Φ
x
R1
R2
x
Figure 5. Definition of co-ordinate systems in pendant and sessile drops.
gas, to take into account Archimedes’ buoyancy; g is the
gravity acceleration where the measurements were performed
and z is the height above (or below) the apex.
In order to obtain the differential equations describing
the drop profile, hydrostatic equilibrium considerations, or a
variational approach—such as a minimization of the sum of
the gravitational and surface potential energies [25]—can be
taken. These equations are [10]
dx
= cos φ
ds
dy
= sin φ
ds
dφ
2
sin φ
ρgy
= −
±
ds
R
r
γ
(15)
where the angles and lengths are shown in figure 5.
This set of equations can be integrated simultaneously, in
function of parameter s, using a numerical procedure such
as a fourth-order Runge–Kutta scheme [12] or the secondorder implicit Euler method [10]. The theoretical profile
of the drop for a given R0 , g, p and γ is then obtained.
The initial values required to start the integration process are
x(s = 0) = y(s = 0) = φ(s = 0) = 0.
Several optimization methods have been developed to fit
the theoretical drop profile to the experimental one [8, 12]. In
this work, the ADSA (axisymmetric drop shape analysis) [10]
method has been chosen, because it offers better solutions over
a wide range of situations, in addition to being used by some
laboratories [26–28]. This algorithm is described below.
A number of points in the drop-image border are taken and
then the theoretical drop shape is fitted to them by minimizing
the Euclidean distance, d(·, ·), between the empirical points
un and the theoretical profile points, v, obtained from
equation (15). The objective function to optimize is
E=
N
1
[d(un , v)]2
2 n=1
(16)
where N is the number of empirical points taken from
the border image of the drop. The change in d(·, ·) is
achieved by modifying the interface parameters and then
the theoretical profile. Minimization is performed using the
Newton–Rhapson method, which converges to the optimal
value. However, initial values are required to use the method.
3. Experimental results
2-propanol (grade 99.5+%) from Sigma Aldrich and
formamide (grade 99+%) from Carlo Erba Reagents were used
to carry out the surface-tension calculation with the proposed
method. Robinson and Sobel edge-detector results, which are
commonly employed in this kind of problem [9, 13], are also
included for comparison.
Some reference values of the calculated parameters are
given to show the accuracy of the results. The reference
value for surface tension of 2-propanol is 23.32 mJ m−2 at
25 ◦ C [29]. The formamide surface-tension reference value is
57.49 mJ m−2 , and the contact angle is 95.38◦ [30], both at
20 ◦ C.
3.1. Experimental data and analysis
In this section the results of surface-tension and contactangle determination for real drop images are presented. The
interfacial parameters were calculated by the ADSA method
in all cases.
The first experiments correspond to the interfacial
parameter determination of drop images unaffected by blur
or noise. From a set of eight 2-propanol pendant drops,
we obtained surface-tension values of 24.06 ± 0.33 mJ m−2
(Sobel), 24.17 ± 0.26 mJ m−2 (Robinson) and 24.07 ±
0.27 mJ m−2 (JS), whereas from six sessile drops of formamide
the results are 57.1 ± 4.5 mJ m−2 (Sobel), 58.0 ± 4.2 mJ m−2
(Robinson) and 57.3 ± 4.9 mJ m−2 (JS). The contact angle
values in this later case are 96.8 ± 1.8 (Sobel), 96.8 ± 4.6
(Robinson) and 97.4±2.2 (JS). Error estimation√in all the above
data is the 95% confidence interval (1.96σ/ N) assuming
the errors follow a Gaussian model. From the experimental
results we can observe that the ADSA method using JS can
provide results as accurate as those using traditional Sobel and
Robinson edge detectors when applied to clean—unaffected
by noise or blur—drop images.
With respect to formamide, note that the experimental
mean values are slightly different to the reference values
(shown in table 3), although the error intervals include the
latter. A high discrepancy in the error interval of the interfacial
parameters was obtained for the sessile drops compared with
the pendant experiment due to several reasons [3]. First,
the sessile drops are very sensitive to the roughness and
heterogeneity of the Teflon, thus causing a lack of vertical
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a)
b)
c)
d)
e)
f)
g)
Figure 6. Results of edge-detection in blurred images: (a) image of 2-propanol, blurred eight times; (b) the same as (a) for formamide; (c)
and (d) results of segmenting images (a) and (b) with the Sobel edge-detector; (e) and (f ) the same as (c) and (d), applying the Robinson
edge-detector; (g) and (h) the same as (c) and (d), applying the JS edge-detector.
symmetry, which has a negative impact on the ADSA method.
Second, a minimal variation in the drop size causes a change
in the curvature of the drop that produces a variation in the
interfacial measurements. Third, the accurate determination
of the contact point can be difficult when the image has a low
294
contrast, such as in this case. A good contrast between the drop
and the surface is extremely important for precise measurement
of the interfacial parameters.
However, the real advantage of the proposed edge-detector
method occurs in the application to noisy or blurred drop
Image-based surface tension measurement
a)
b)
c)
d)
e)
f)
g)
h)
Figure 7. Results of edge-detection with Gaussian noisy images (zero mean, standard deviation 15): (a) image of 2-propanol with Gaussian
noise; (b) the same as (a) for formamide; (c) and (d) results of segmenting images (a) and (b) with the Sobel edge-detector; (e) and (f ) the
same as (c) and (d), applying the Robinson edge-detector; (g) and (h) the same as (c) and (d), applying the JS edge-detector.
images. Thus, a set of real drop images, into which various
degrees and types of noise have been introduced, are considered in a second group of experiments. These experiments
are important for two reasons: they can simulate experimental
conditions that commonly appear in practice and they allow us
to evaluate the robustness of these edge-detection methods.
Table 2 shows the numerical results of surface tension
using the Sobel, Robinson and JS methods. The original
2-propanol pendant drop is contaminated with three synthetic
distortions (blur, and Gaussian and impulsive noise) to simulate
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C Atae-Allah et al
(a)
(b)
(c)
(d)
(e)
(f )
(g)
(h)
Figure 8. Results of edge-detection with 15% impulsive salt-and-pepper noise: (a) image of 2-propanol with impulsive noise; (b) the same
as (a) for formamide; (c) and (d) results of segmenting images (a) and (b) with the Sobel edge detector; (e) and (f ) the same as (c) and (d),
applying the Robinson edge-detector; (g) and (h) the same as (c) and (d), applying the JS edge-detector.
experimental problems that can appear in practice. Blur, which
suitably simulates lack of focus in image-acquisition systems,
is achieved by recursively applying a 3×3 mask to the original
image (mean filter). Gaussian noise has zero mean and a
296
typical deviation of 5, 10 and 15. Impulsive noise is salt-andpepper noise in amounts of 5%, 10% and 15%. Gaussian and
impulsive noise are both very common in image acquisition
and transmission.
Image-based surface tension measurement
Table 2. Surface-tension calculation of 2-propanol pendant drops with noise. γ = surface-tension value in mJ m−2 . RE = relative error in
%. Reference surface-tension value: 23.32 mJ m−2 .
Noise type
Blur
Original
5
Gaussian
8
5
Impulsive
10
15
5%
10%
15%
Sobel
γ
RE
23.62
0
23.70
0.34
—
—
23.53
0.38
23.18
1.86
—
—
—
—
—
—
—
—
Robinson
γ
RE
23.66
0
23.58
0.34
—
—
23.69
0.13
23.94
1.18
—
—
—
—
—
—
—
—
JS
γ
RE
23.56
0
23.62
0.25
23.42
0.59
23.65
0.38
23.63
0.30
23.69
0.55
23.34
0.93
23.42
0.59
23.43
0.55
Table 3. Surface-tension calculation of formamide sessile drops with noise. γ = surface-tension value in mJ m−2 . φ = contact angle in
sexagesimal degrees. RE = relative error as a percentage. Reference surface-tension value: 57.49 mJ m−2 . Reference contact-angle value:
95.38◦ .
Noise type
Blur
Original
5
Gaussian
8
5
10
Impulsive
15
5%
10%
15%
Sobel
γ
RE
φ
RE
58.32
0
98.49
0
56.54
3.05
97.58
0.92
—
—
—
—
54.58
6.41
100.50
2.04
42.25
27.55
89.16
9.47
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
Robinson
γ
RE
φ
RE
58.97
0
100.71
0
61.62
4.49
100.16
0.54
—
—
—
—
47.33
19.73
98.17
2.52
52.98
10.16
86.73
13.88
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
JS
γ
RE
φ
RE
57.94
0
98.69
0
59.50
2.69
99.73
1.05
55.52
4.18
99.52
0.847
59.89
3.37
99.64
0.96
56.91
1.78
99.84
1.17
56.39
2.68
100.82
2.16
54.12
6.59
102.58
3.94
54.28
6.32
101.54
2.89
57.24
1.21
101.82
3.17
Table 3 presents the numerical results of the same
experiment as in table 2, but using sessile formamide drops.
In both tables the relative error is calculated with respect to the
reference value and is expressed as a percentage.
Figures 6–8 show images corresponding to experiments
in tables 2 and 3.
From tables 2 and 3, and figures 6–8, we can conclude that
JS is a much better edge detector than Sobel and Robinson. In
fact, in tables 2 and 3 the symbol ‘—’ means that the edgedetector cannot even segment this image. Furthermore, JS is
more robust against noise than Sobel and Robinson, as can be
seen in the relative error values given in tables 2 and 3. In fact,
relative error in 2-propanol experiments is always less than 1%,
and smaller than the relative errors of the Sobel and Robinson
methods. The formamide results are similar to the 2-propanol
ones, obtaining a relative error of less than 6.6% in surface
tension and 4% in contact angle. The better performance of JS
can be seen by simply noting that Sobel produces relative errors
above 27% and 9%, respectively, while Robinson produces
relative errors over 19% and 13% respectively.
In figures 6–8 we can observe the superior JS performance.
In fact, JS segmented images are the only ones that can be
processed to obtain surface-tension and contact-angle values.
Moreover, JS images do not need pre-processing (filtering to
eliminate noise but displacing edges) or post-processing (edge
thinning, necessary to apply the ADSA method).
4. Conclusions
In recent years, significant advances in signal-processing
techniques have enabled image-based methods to attain
sufficient accuracy in surface tension measurements. These
approaches (drop-shape methods, such as ADSA) are as good
as traditional ones, without their disadvantages, such as the
extreme care needed in the processing, the relatively high
quantity of liquid used or the interference with the liquid
to measure. In addition, these techniques may only be
useful when studying problematic cases such as interfaces
between liquids.
Moreover, it provides measurements
of additional parameters—i.e. contact angle—in the same
processing. These approaches function as follows: the
interfacial parameters are measured by fitting the theoretical
contour to the real one, obtained by using an edge
detector.
In this context, the authors present an entropic method
of edge detection, based on the Jensen–Shannon divergence,
that is of advantage in the measurement of surface tension
and contact angle in pendant or sessile liquid drops. When it is
used, the contour is appropriately detected, even when noise or
blurring is present, or contrast is very low. In laboratory work,
these problems can easily arise due to the use of electronic
devices (thus generating noise) and to the evaporation of the
liquid when the image is being acquired (thus giving rise to
an out-of-focus, blurred image). Traditional detectors failed
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C Atae-Allah et al
when trying to provide any measurement of the interfacial
parameters in such situations.
The advantages of the Jensen–Shannon divergence
detector are due to its intrinsic properties of noise robustness
[17]. When contrast is low, it is possible to include a refinement
in the algorithm, named attenuation, thus making it capable of
distinguishing the edges even in such a situation [31].
The Jensen–Shannon method provides the best results
when it is used with small windows since the observation level
is the more accurate [17]. It must be said that the edge linking
is essential in the results obtained, because only a few edge
pixels are detected directly, due to the extreme conditions in
which the algorithm has to work. This makes it necessary
to adjust the parameters in the detector to obtain a few edge
pixels, but with a very high confidence level.
To sum up, the proposed Jensen–Shannon divergence edge
detector is very suitable for use in drop-shape methods for determining surface tension or contact angle in liquids, especially
when the quality of the drop images is relatively poor.
Acknowledgments
This work has been partially supported by grants MAR-970464 and MAT-98-0937-C02-C01 of the Spanish Government.
Our thanks to Christine Laurin for revising the English.
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