Snake-Based Approach for Accurate Contact Angle Determination EPFL

This report presents a snake-based method utilizing B-spline snakes (active contours) for high-accuracy contact angle measurement, avoiding physical assumptions by defining the drop's contour as a versatile B-spline curve. The method extends the curve by mirror symmetry to leverage the drop's reflection for contact point detection. It optimizes an advanced image-energy term with directional gradient and region-based components to drive curve evolution without discretizing the contour. The approach is implemented as a plugin for ImageJ, named DropSnake, and is effective for asymmetric drops on uneven or tilted surfaces. The report details the spline-based illustration of drop contours, image energy evaluation, and the B-Snake algorithm, providing a comprehensive solution for accurate contact angle determination.

A snake-based approach to accurate determination of both contact points and contact angles
1. Introduction
Over the last 200 years, wetting phenomenon has been carried out with different interests [1].
The concept was introduced by Thomas Y. in 1805 through a simple equation which was
equating the application force at interaction point of the drop of the liquid in solid surface. The
equation presented was;
γl,g cos θ = γl,g − γs,l,
Whereby, γ = excess energy per unit area of the boundary as indicated by different indices of g, l,
and s, represent the matter phases. The above equation stays known as Young’s Equation which
is the most used equation to study surface wetting [2]. The surface tension is tabulated by the
well-known tabulated values of free energy γl,g and resemble to surface tension of liquid air. The
contact angle θ is also being analyzed in this paper in relation to the Young’s Equation. Many
questions have been left unattended to in the recent studies. The static measurement angle is
considered to be between ±3◦ which results from residual experiments.
Different studies have shown improvement of the fabrication of micro-arrays. This is one of the
ways of studying the dynamics of liquids in wetting. Another key way used is the tilting of
substrate or even decreasing the volume [3]. The liquid molecules might spread but fail to wet
the surface this is considered as wetting hysteresis. Nevertheless, the determination of the contact
angle is not easy. In addition, rapid and robust methods are needed in determining the interaction
angle for wetting liquids for both micro- as well as nano-structured surfaces which do have
similar surface empathy [4]. Sessile technique is widely used today. The area has a interesting
progress, for estimating the contact angle. Other methods used include use of goniometer on
telescope and protractor on images. Additionally the contact angle can be approximated from
measuring the contour of the sphere on different locations of the profile. Another important
method used is the ADSA [5]. This uses the Laplace equation and numerical integration. The
Laplace profile is usually corresponded to the contour and uses the capillary constant to obtain
the contact angle.
The global models have key limitations in relation to their validity and thus local models are
considered. These include the polynomial-fitting approach, which uses coordinates to determine
the contact angle. The polynomial degree depends on coordinate points. This paper looks a new
approach which looks at both indigenous and international model aspects. This method is based
on snakes taking into consideration that the shape of drop represent international and angle of
contact is local [6]. The forces in this approach are limited but it helps to keep the shape global
since is largely depends on elasticity constraint. This method in addition is advanced since is
helps to reduce the position changes when measuring the angle. This method has an automatic
detection of drop and substrate level. The reflection of the drop is used to identify different
contact points. The international shape of the snake is lastly used to determine the shape of the

drop plugs. In this method, the snake is usually related to the polynomial fitting approach in the
fortitude of contact angle.
The images retrieved, image characterization and segmentation method is applied and the corner,
end of the drop and used to determine in various gradations. The drop in contours in some
methods such as ADSA is defined through theoretical image fitting analysis (TIFA), especially
when the image is not clear [7]. The method applies gradient based error function when the
images are clear and ADSA fails. The image first is created using numerical solution which
interacts with Laplace equation. The definition of an error function is seen as “sum of the square
of the difference which exists between experimental gradient image and theoretical”. The
optimization is able to take care of the defined gradient values.
Another key segment of research is the base segmentation. This can be used to exploit the
direction of gradients and thus help in formation of gradient based image energies. In this
method, the cubic-spline interpolation is used to sample the contours for drop [8]. This
exclamation plays a critical role to allow the sub-pixel to be refined in the contours as well as
determination of the context of the gradient energies. This paper also considers that the image
pixels are continuously defined in terms of images. Additionally, cubic-spline image
interpolation is proposed to help find the sub-pixel resolution used for defining the contour of
drop.
2. Analysis of Spline-based illustration of drop contours
The spline curves parametric are of order d1 when classified in the polynomial function and
consist of concatenated segments. This helps to join the breakpoints of the curves. The
parametric curves are able to symbolize shapes with different spans [9]. B-spline is attained as a
result of summing of a finite number of basic tasks.
2.1. Parametric spline representation
The parametric spline has minimal property of curvature, as well as computational efficiency and
easiness. The cubic B-spline is selected to interpolate the basic function. To represent well drop
the B-spline is able to produce smooth curves which help to represent a continuous regular
contours. Moreover, the curvature constrain of snake is curvilinear cubic spline. When external
forces are absent, B-spline snake is able to ideally represent the resting drop. Under this formula,
when the minimum curvature deviates, the knot also deviates from the ideal least curvature
location [10]. If the contour is able to deviate much from the minimum curvature, suggestion to
use to the non-curvilinear splines according to 4.1section is recommended. Additionally, cubic-
spline parametric is exposed curve happening in x-y plane and usually represented as;
x(t) = M+1 k=−1 cx,kβ3(t − k)
∀t ∈ [0, M] : ∑

y(t) = M+1 k=−1 cy,kβ3(t − k)
The derivation of the above equation is;
The β3 as seen is cube of B-spline, while (cx,k, cy,k) = coordinates of the kth which the control
point among M =control points ck, and where D = differential operator d dt .
2.2. Boundary conditions
A continuity of C2 is able to exist in cubic spline at regular breakpoints for both principal and
subsequent derivatives, which are constant. Nevertheless, the drop curve must be discontinuous
for its first derivative. This will helps to represent the angles [11]. At the contact point, the
border conditions should be used to help achieve triple control points. Phantom point is usually
further to one end point of spline to achieve continuation of a spline by equilibrium. This can be
illustrated as the figure below;
Figure 1: interpolation vertex via phantom vertices.
2.3. Drop reflection of a symmetric model of drops
Then existence of replication drop in substrate image makes contact points to be identified
automatically. Drop spline must be increased using mirror equilibrium at the points of control.
The non-reflected drop profile can be defined as;
The cx, k, cy, k are coordinate control points of the (cx,−1, cy,−1) and (cx,M+1, cy,M+1) represent the
phantoms edge control points. The replicated profile drop is defined by the following evenness.

Where (cr x,k, cr y,k) are coordinates of the points of control of the outline of the (cx,k, cy,k) while
(crx,M+1, cry,M+1) are the coordinates of the phantom control points of the replicated contour.
3. Image energy Evaluation
3.1. Interpolation of the image
The interpolation helps to reduce image drop impact of the discretization. B-spline helps to
achieve good quality interpolation. Cubic-spline is also used to achieve the same result.
Interpolation coefficients are first computed.
3.2. Analysis of unified image energy
3.2.1. Formulation
The major limitation of the optimization schemes is the convergence radius. Smoothing filters to
the image can be applied to increase the radius. New gradient energy is used by this method to
take care of account of the gradient direction and merit of parameterization-invariant. The
gradient energy of a simple surface S which has contour delimited by C is represented as;
Eedge =∫ C k · (∇f (r) × dr)
K denotes unit orthogonal trajectory related to the image plane and ∇f (r) is the slope of the
shape f at the plug r of the curve. Integrating using Green theorem, the superficial integral is;
Eedge = ∫∫ S ∇·∇f˜ (s) ds,
Te{f }
The ∇· = divergence operator
The region ops energy is given as
Eregion = ∫∫ S Tr{f }(s) ds
Tr{f } represents the probability-distribution image
Establishment of statistical value is important to use region energies for the drop of background.
Estimation of probability distribution can be done from temporal contour during optimization.
The unified image energy can found as
Eimage = ∫∫S fu(s) ds, where fu = α Te{f } + (1 − α) Tr{f }. Additionally the unified energy can be
expressed as following using Green’s theorem.

Eimage =∫ C f y u (x, y) dx = − ∫C f x u (x, y) dy,
Whereby,
3.2.2. drops of Spline parameterization
Defining Csup and Cinf for them to signify non-reflected outline and replicated shape
correspondingly, where we Have C = Csup ∪ Cinf, then energy image is;
Or simply as below when using parametric representation
3.3. Derivation of Energy
3.3.1. Using normal control points Derivation
The derivation of the energy image can also be considered when looking at the horizontal
position for a controller as follows;
3.3.2. Axis of symmetry Derivation

Additionally, locations of c0 and cM are able to find the location of the replicated profile. The
location of the plugs highly affects the angle of contact. The position (xh, yh) represent the
internal of the control points of C0 and CM. The axis angle is represented as
Additionally, the image energy considering the position of the boundary control can be
represented as;
4. B-Snake
Snakes or simply lively delineations are used when applying computer-assisted tool, meant for
segmentation. They are widely applied in medical images examination, or piece pursuing when
applied in video arrangements. The snakes are defines as “spline energy minimization happening
under internal and external forces”. The powers when provided at time are able to ensure
evenness of curves [12]. B-spline is able to apply the use of parametric B-spline representation of
the curve. In order to keep the smoothness.
4.1. Re-parameterization energy
B-spline is just a piecewise of polynomial fitting approach which is used for determination of
contact angle. When used, the contact angle will depend on B-spline and length of contours. The
curvature is minimized when a drop of external forces is experienced. On the other hand, the
curvature of the snake can be increased when the detachment between the bumps is reduced. The
contact angle measurement need proper follow of the drop of contours at different contact points
than in the apex. Also, the progressive repartition of points of control and arc speed with linear
differences can be represented as;

4.2. Optimization
Lastly the energy snake which needs to be diminished is given as
E = Eimage + Eint.
Lengthy steps are adjusted during optimization process. This is done with veneration to the
difference movement. Having the step length below the threshold, convergence is achieved. The
typical value of the convergence is usually at 0.010 but it can be adjusted. The curve may be
physically well-defined through placing a few bumps or use of automatic approach. A finer
spline is then evolved to detect the contours effectively. Distance between the knots is an
important parameter of the algorithm.
5. Implementation and application
5.1. Software
This method herein is encoded as a plugin for ImageJ, which is a Java image-processing
program. The plugin are depended on image hardware, which is known as DropSnake.
5.2. Application examples
Contact angle has been measures in application using energy of image basing it on gradient.
Relative good difference of images was not a justification of district constituent. Contact angle
was measured in automatic interface detection and projected drop.
5.3. Robustness experimentations
Various filters were applied to assess the sturdiness of image-energy and contact angle capacity.
All drops were between 300 and 400 pixel. Smoothness dependence is a Gaussian filter method
which was applied to synthetic the data. We measured the contact angle of the resulting images
and then compared to the measurements of the initial angle. Robustness to noise was used to
synthetic the data and accesses the effect of noise levels to the measured contact angle. The
co9ntact angle was found to increase as the noise influence increased.
5.4. Inter-knot detachment
Angle of Contact is addiction on aloofness between the bumps has been accessed on drop.
Resizing of the image was carried to evaluate the effect of lacking control points. Additionally,
supplying small inter-knot distance leads to rejection of both contact angles. Large inter-knot
distances also are seen to increase the contact angle variability. Also, the maximum limit for the

inter-knot distance has direct relation to the minimum number of knots needed to perfectly
represent a drop contour. Reasonable inter-knot distance is required to achieve limited
dependence of contact angle on inter-knot distance.
5.5. Software evaluation
Different methods and software have different assumption and this makes comparison of angle
off contact and different measurement methods challenges. Comparison to other methods was
carried out when this method was carried out. Plugin ImageJ was utilized in this method and also
segmentation method was critical.
5.6. Experiments conclusion
Contours of drop and interface may be automatically detected as in this method. The method was
robust and had little dependence on parameters. Partial interaction angle necessity when
flattening filters was achieved. Excellent noise inherent robustness was able presented in this
method.
6. Conclusion
In the analysis of domain of drop, new image function energy was applied. The energy function
was able to provide a framework robust and accurate detection of contours together with cubic-
spline and led to comprehensive variety of imageries. First outcomes achieved were highly
promising. Snake-based approach is considered to be a novel based for measuring the contact
angle of drops. The method can be used for drops which fails to follow any global model. This
gives this technique a wide variety of application.
References

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