Analytic Solutions of Nonlinear Partial Differential Equations by the Power Index Method*Symmetry* **2018**, *10*(3), 76; doi:10.3390/sym10030076 (registering DOI) - 19 March 2018**Abstract **

An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or $180\xb0$

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An updated Power Index Method is presented for nonlinear differential equations (NLPDEs) with the aim of reducing them to solutions by algebraic equations. The Lie symmetry, translation invariance of independent variables, allows for traveling waves. In addition discrete symmetries, reflection, or $180\xb0$ rotation symmetry, are possible. The method tests whether certain hyperbolic or Jacobian elliptic functions are analytic solutions. The method consists of eight steps. The first six steps are quickly applied; conditions for algebraic equations are more complicated. A few exceptions to the Power Index Method are discussed. The method realizes an aim of Sophus Lie to find analytic solutions of nonlinear differential equations.
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A Robust Image Watermarking Technique Based on DWT, APDCBT, and SVD*Symmetry* **2018**, *10*(3), 77; doi:10.3390/sym10030077 (registering DOI) - 19 March 2018**Abstract **

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Copyright protection for digital multimedia has become a research hotspot in recent years. As an efficient solution, the digital watermarking scheme has emerged at the right moment. In this article, a highly robust and hybrid watermarking method is proposed. The discrete wavelet transform

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Copyright protection for digital multimedia has become a research hotspot in recent years. As an efficient solution, the digital watermarking scheme has emerged at the right moment. In this article, a highly robust and hybrid watermarking method is proposed. The discrete wavelet transform (DWT) and all phase discrete cosine biorthogonal transform (APDCBT) presented in recent years as well as the singular value decomposition (SVD) are adopted in this method to insert and recover the watermark. To enhance the watermark imperceptibility, the direct current (DC) coefficients after block-based APDCBT in high frequency sub-bands (LH and HL) are modified by using the watermark. Compared with the conventional SVD-based watermarking method and another watermarking technique, the watermarked images obtained by the proposed method have higher image quality. In addition, the proposed method achieves high robustness in resisting various image processing attacks.
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Dispersion Forces Between Fields Confined to Half Spaces*Symmetry* **2018**, *10*(3), 74; doi:10.3390/sym10030074 (registering DOI) - 19 March 2018**Abstract **

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We consider the Casimir effect for a scalar field interacting with another scalar field that is confined to two half spaces. This model is aimed to mimic the interaction of the photon field with matter in two slabs. We use Dirichlet boundary conditions

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We consider the Casimir effect for a scalar field interacting with another scalar field that is confined to two half spaces. This model is aimed to mimic the interaction of the photon field with matter in two slabs. We use Dirichlet boundary conditions on the interfaces for the fields in the half spaces and calculate their one-loop contribution to the wave equation for the other field. We perform the ultraviolet renormalization and develop a convenient formalism for the calculation of the vacuum energy in this configuration.
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An Improved Set-Membership Proportionate Adaptive Algorithm for a Block-Sparse System*Symmetry* **2018**, *10*(3), 75; doi:10.3390/sym10030075 (registering DOI) - 19 March 2018**Abstract **

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In this paper, an improved set-membership proportionate normalized least mean square (SM-PNLMS) algorithm is proposed for block-sparse systems. The proposed algorithm, which is named the block-sparse SM-PNLMS (BS-SMPNLMS), is implemented by inserting a penalty of a mixed ${l}_{2,1}$ norm of

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In this paper, an improved set-membership proportionate normalized least mean square (SM-PNLMS) algorithm is proposed for block-sparse systems. The proposed algorithm, which is named the block-sparse SM-PNLMS (BS-SMPNLMS), is implemented by inserting a penalty of a mixed ${l}_{2,1}$ norm of weight-taps into the cost function of the SM-PNLMS. Furthermore, an improved BS-SMPNLMS algorithm (the (BS-SMIPNLMS algorithm) is also derived and analyzed. The proposed algorithms are well investigated in the framework of network echo cancellation. The results of simulations indicate that the devised BS-SMPNLMS and BS-SMIPNLMS algorithms converge faster and have smaller estimation errors compared with related algorithms.
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Ensemble Genetic Fuzzy Neuro Model Applied for the Emergency Medical Service via Unbalanced Data Evaluation*Symmetry* **2018**, *10*(3), 71; doi:10.3390/sym10030071 (registering DOI) - 17 March 2018**Abstract **

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Equally partitioned data are essential for prediction. However, in some important cases, the data distribution is severely unbalanced. In this study, several algorithms are utilized to maximize the learning accuracy when dealing with a highly unbalanced dataset. A linguistic algorithm is applied to

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Equally partitioned data are essential for prediction. However, in some important cases, the data distribution is severely unbalanced. In this study, several algorithms are utilized to maximize the learning accuracy when dealing with a highly unbalanced dataset. A linguistic algorithm is applied to evaluate the input and output relationship, namely Fuzzy c-Means (FCM), which is applied as a clustering algorithm for the majority class to balance the minority class data from about 3 million cases. Each cluster is used to train several artificial neural network (ANN) models. Different techniques are applied to generate an ensemble genetic fuzzy neuro model (EGFNM) in order to select the models. The first ensemble technique, the intra-cluster EGFNM, works by evaluating the best combination from all the models generated by each cluster. Another ensemble technique is the inter-cluster model EGFNM, which is based on selecting the best model from each cluster. The accuracy of these techniques is evaluated using the receiver operating characteristic (ROC) via its area under the curve (*AUC*). Results show that the *AUC* of the unbalanced data is 0.67974. The random cluster and best ANN single model have *AUCs* of 0.7177 and 0.72806, respectively. For the ensemble evaluations, the intra-cluster and the inter-cluster EGFNMs produce 0.7293 and 0.73038, respectively. In conclusion, this study achieved improved results by performing the EGFNM method compared with the unbalanced training. This study concludes that selecting several best models will produce a better result compared with all models combined.
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Nonclassical Symmetry Solutions for Fourth-Order Phase Field Reaction–Diffusion*Symmetry* **2018**, *10*(3), 72; doi:10.3390/sym10030072 (registering DOI) - 17 March 2018**Abstract **

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Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing

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Using the nonclassical symmetry of nonlinear reaction–diffusion equations, some exact multi-dimensional time-dependent solutions are constructed for a fourth-order Allen–Cahn–Hilliard equation. This models a phase field that gives a phenomenological description of a two-phase system near critical temperature. Solutions are given for the changing phase of cylindrical or spherical inclusion, allowing for a “mushy” zone with a mixed state that is controlled by imposing a pure state at the boundary. The diffusion coefficients for transport of one phase through the mixture depend on the phase field value, since the physical structure of the mixture depends on the relative proportions of the two phases. A source term promotes stability of both of the pure phases but this tendency may be controlled or even reversed through the boundary conditions.
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Intelligent Image Processing System for Detection and Segmentation of Regions of Interest in Retinal Images*Symmetry* **2018**, *10*(3), 73; doi:10.3390/sym10030073 (registering DOI) - 17 March 2018**Abstract **

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The automatic detection, segmentation, localization, and evaluation of the optic disc, macula, exudates, and hemorrhages are very important for diagnosing retinal diseases. One of the difficulties in detecting such regions of interest (RoIs) with computer vision is their symmetries, e.g., between the optic

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The automatic detection, segmentation, localization, and evaluation of the optic disc, macula, exudates, and hemorrhages are very important for diagnosing retinal diseases. One of the difficulties in detecting such regions of interest (RoIs) with computer vision is their symmetries, e.g., between the optic disc and exudates and also between exudates and hemorrhages. This paper proposes an original, intelligent, and high-performing image processing system for the simultaneous detection and segmentation of retinal RoIs. The basic principles of the method are image decomposition in small boxes and local texture analysis. The processing flow contains three phases: preprocessing, learning, and operating. As a first novelty, we propose proper feature selection based on statistical analysis in confusion matrices for different feature types (extracted from a co-occurrence matrix, fractal type, and local binary patterns). Mainly, the selected features are chosen to differentiate between similar RoIs. The second novelty consists of local classifier fusion. To this end, the local classifiers associated with features are grouped in global classifiers corresponding to the RoIs. The local classifiers are based on minimum distances to the representatives of classes and the global classifiers are based on confidence intervals, weights, and a voting scheme. A deep convolutional neural network, based on supervised learning, for blood vessel segmentation is proposed in order to improve the RoI detection performance. Finally, the experimental results on real images from different databases demonstrate the rightness of our methodologies and algorithms.
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Casimir Energies for Isorefractive or Diaphanous Balls*Symmetry* **2018**, *10*(3), 68; doi:10.3390/sym10030068 (registering DOI) - 16 March 2018**Abstract **

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It is known that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside

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It is known that the Casimir self-energy of a homogeneous dielectric ball is divergent, although a finite self-energy can be extracted through second order in the deviation of the permittivity from the vacuum value. The exception occurs when the speed of light inside the spherical boundary is the same as that outside, so the self-energy of a perfectly conducting spherical shell is finite, as is the energy of a dielectric-diamagnetic sphere with $\epsilon \mu =1$ , a so-called isorefractive or diaphanous ball. Here we re-examine that example and attempt to extend it to an electromagnetic $\delta $ -function sphere, where the electric and magnetic couplings are equal and opposite. Unfortunately, although the energy expression is superficially ultraviolet finite, additional divergences appear that render it difficult to extract a meaningful result in general, but some limited results are presented.
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Enabling Symmetric Collaboration in Public Spaces through 3D Mobile Interaction*Symmetry* **2018**, *10*(3), 69; doi:10.3390/sym10030069 (registering DOI) - 16 March 2018**Abstract **

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Collaboration has been common in workplaces in various engineering settings and in our daily activities. However, how to effectively engage collaborators with collaborative tasks has long been an issue due to various situational and technical constraints. The research in this paper addresses the

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Collaboration has been common in workplaces in various engineering settings and in our daily activities. However, how to effectively engage collaborators with collaborative tasks has long been an issue due to various situational and technical constraints. The research in this paper addresses the issue in a specific scenario, which is how to enable users to interact with public information from their own perspective. We describe a 3D mobile interaction technique that allows users to collaborate with other people by creating a symmetric and collaborative ambience. This in turn can increase their engagement with public displays. In order to better understand the benefits and limitations of this technique, we conducted a usability study with a total of 40 participants. The results indicate that the 3D mobile interaction technique promotes collaboration between users and also improves their engagement with the public displays.
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Symmetries in Classical and Quantum Treatment of Einstein’s Cosmological Equations and Mini-Superspace Actions*Symmetry* **2018**, *10*(3), 70; doi:10.3390/sym10030070 (registering DOI) - 16 March 2018**Abstract **

The use of automorphisms of the various Bianchi-type Lie algebras as Lie-point symmetries of the corresponding Einstein field equations entails a reduction of their order and ultimately leads to the entire solution space. When a valid reduced action principle exists, the symmetries of

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The use of automorphisms of the various Bianchi-type Lie algebras as Lie-point symmetries of the corresponding Einstein field equations entails a reduction of their order and ultimately leads to the entire solution space. When a valid reduced action principle exists, the symmetries of the configuration mini-supermetric space can also be used, in conjunction with the constraints, to provide local or non-local constants of motion. At the classical level, depending on their number, these integrals can even secure the acquisition of the entire solution space without any further solving of the dynamical equations. At the quantum level, their operator analogues can be used, along with the Wheeler–DeWitt equation, to define unique wave functions that exhibit singularity-free behavior at a semi-classical level.
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Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face*Symmetry* **2018**, *10*(3), 67; doi:10.3390/sym10030067 - 15 March 2018**Abstract **

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A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed

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A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed numbers of *k*-gons, $k\ge 7$ . Any simple polytope with only 5-, 6- and at most one 7-gon is Pogorelov. For any other prescribed numbers of *k*-gons, $k\ge 7$ , we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from *k*-barrels (also known as Löbel polytopes, whose graphs are biladders on $2k$ vertices) can be constructed from the 5- or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the $(5,0)$ -nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6- and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons.
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Carbon Oxides Gases for Occupancy Counting and Emergency Control in Fog Environment*Symmetry* **2018**, *10*(3), 66; doi:10.3390/sym10030066 - 15 March 2018**Abstract **

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The information of human occupancy plays a crucial role in building management. For instance, fewer people, less demand for heat and electricity supply, and vice versa. Moreover, when there is a fire in a building, it is convenient to know how many persons

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The information of human occupancy plays a crucial role in building management. For instance, fewer people, less demand for heat and electricity supply, and vice versa. Moreover, when there is a fire in a building, it is convenient to know how many persons in a single room there are in order to plan a more efficient rescue strategy. However, currently most buildings have not installed adequate devices that can be used to count the number of people, and the most popular embedded fire alarm system triggers a warning only when a fire breaks out with plenty of smoke. In view of this constraint, in this paper we propose a carbon oxides gases based warning system to detect potential fire breakouts and to estimate the number of people in the proximity. In order to validate the efficiency of the devised system, we simulate its application in the Fog Computing environment. Furthermore, we also improve the iFogSim by giving data analytics capacity to it. Based on this framework, energy consumption, latency, and network usage of the designed system obtained from iFogSim are compared with those obtained from Cloud environment.
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Spectrum Graph Coloring and Applications to Wi-Fi Channel Assignment*Symmetry* **2018**, *10*(3), 65; doi:10.3390/sym10030065 - 14 March 2018**Abstract **

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We introduce and explore a family of vertex-coloring problems, which, surprisingly enough, have not been considered before despite stemming from the problem of Wi-Fi channel assignment. Given a spectrum of colors, endowed with a matrix of interferences between each pair of colors, the

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We introduce and explore a family of vertex-coloring problems, which, surprisingly enough, have not been considered before despite stemming from the problem of Wi-Fi channel assignment. Given a spectrum of colors, endowed with a matrix of interferences between each pair of colors, the Threshold Spectrum Coloring problem fixes the number of colors available and aims to minimize the interference threshold, i.e., the maximum of the interferences at the vertices. Conversely, the Chromatic Spectrum Coloring problem fixes a threshold and aims to minimize the number of colors for which respecting that threshold is possible. As the main theoretical results, we prove tight upper bounds for the solutions to each problem. Since both problems turn out to be NP-hard, we complete the scene with experimental results. We propose a DSATUR-based heuristic and study its performance to minimize the maximum vertex interference in Wi-Fi channel assignment, both for randomly-generated graphs and for a real-world scenario. Further, for all these graphs, we experimentally check the goodness of the theoretical bounds.
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Vision-Based Parking-Slot Detection: A Benchmark and A Learning-Based Approach*Symmetry* **2018**, *10*(3), 64; doi:10.3390/sym10030064 - 13 March 2018**Abstract **

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Recent years have witnessed a growing interest in developing automatic parking systems in the field of intelligent vehicles. However, how to effectively and efficiently locating parking-slots using a vision-based system is still an unresolved issue. Even more seriously, there is no publicly available

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Recent years have witnessed a growing interest in developing automatic parking systems in the field of intelligent vehicles. However, how to effectively and efficiently locating parking-slots using a vision-based system is still an unresolved issue. Even more seriously, there is no publicly available labeled benchmark dataset for tuning and testing parking-slot detection algorithms. In this paper, we attempt to fill the above-mentioned research gaps to some extent and our contributions are twofold. Firstly, to facilitate the study of vision-based parking-slot detection, a large-scale parking-slot image database is established. This database comprises 8600 surround-view images collected from typical indoor and outdoor parking sites. For each image in this database, the marking-points and parking-slots are carefully labeled. Such a database can serve as a benchmark to design and validate parking-slot detection algorithms. Secondly, a learning-based parking-slot detection approach, namely $PS{D}_{L}$ , is proposed. Using $PS{D}_{L}$ , given a surround-view image, the marking-points will be detected first and then the valid parking-slots can be inferred. The efficacy and efficiency of $PS{D}_{L}$ have been corroborated on our database. It is expected that $PS{D}_{L}$ can serve as a baseline when the other researchers develop more sophisticated methods.
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Symmetric Agency Graphs Facilitate and Improve the Quality of Virtual Network Embedding*Symmetry* **2018**, *10*(3), 63; doi:10.3390/sym10030063 - 11 March 2018**Abstract **

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Virtual network embedding (VNE) is a key technology in network virtualization. Advantages of network symmetry are well known in the design of load-balanced routing algorithms and in network performance analysis. Our work in this paper shows that benefits of graph symmetry also extend

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Virtual network embedding (VNE) is a key technology in network virtualization. Advantages of network symmetry are well known in the design of load-balanced routing algorithms and in network performance analysis. Our work in this paper shows that benefits of graph symmetry also extend to the domain of network embedding. Specifically, we propose an efficient VNE method based on modular and structured agency guidance, a regular graph function. The proposed method, which is based on symmetric intermediate graphs, offers two main advantages. Firstly, characteristics of the intermediate structures enhance the computational efficiency of the VNE process. Secondly, the static agency network modeled with such intermediate structures improves the quality of the resulting embedding. These two advantages of our method are elaborated upon and verified by examples and simulations, respectively. In addition, we present a theoretical analysis explaining the reasons behind the benefits offered by such middleware.
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The LHC Higgs Boson Discovery: Updated Implications for Finite Unified Theories and the SUSY Breaking Scale*Symmetry* **2018**, *10*(3), 62; doi:10.3390/sym10030062 - 7 March 2018**Abstract **

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Finite Unified Theories (FUTs) are $N=1$ supersymmetric Grand Unified Theories, which can be made finite to all orders in perturbation theory, based on the principle of the reduction of couplings. The latter consists of searching for renormalization group invariant relations among

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Finite Unified Theories (FUTs) are $N=1$ supersymmetric Grand Unified Theories, which can be made finite to all orders in perturbation theory, based on the principle of the reduction of couplings. The latter consists of searching for renormalization group invariant relations among parameters of a renormalizable theory holding to all orders in perturbation theory. FUTs have proven very successful so far. In particular, they predicted the top quark mass one and half years before its experimental discovery, while around five years before the Higgs boson discovery, a particular FUT was predicting the light Higgs boson in the mass range ∼121–126 GeV, in striking agreement with the discovery at LHC. Here, we review the basic properties of the supersymmetric theories and in particular finite theories resulting from the application of the method of reduction of couplings in their dimensionless and dimensionful sectors. Then, we analyze the phenomenologically-favored FUT, based on SU(5). This particular FUT leads to a finiteness constrained version of the Minimal SUSY Standard Model (MSSM), which naturally predicts a relatively heavy spectrum with colored supersymmetric particles above 2.7 TeV, consistent with the non-observation of those particles at the LHC. The electroweak supersymmetric spectrum starts below 1 TeV, and large parts of the allowed spectrum of the lighter might be accessible at CLIC. The FCC-hhwill be able to fully test the predicted spectrum.
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The Impact of the Anisotropy of the Media between Parallel Plates on the Casimir Force*Symmetry* **2018**, *10*(3), 61; doi:10.3390/sym10030061 - 7 March 2018**Abstract **

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Quantum fluctuations of the electromagnetic field give rise to the force between parallel plates: the Casimir force. We theoretically calculate the Casimir force between two parallel isotropic plates when the space between the plates is filled with anisotropic material. Our result shows that

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Quantum fluctuations of the electromagnetic field give rise to the force between parallel plates: the Casimir force. We theoretically calculate the Casimir force between two parallel isotropic plates when the space between the plates is filled with anisotropic material. Our result shows that the Casimir force, especially in the direction of the force, can be significantly affected by the anisotropy of the intervening material. We also discuss the combined influence of dispersion and anisotropy, and analyze the impact of the external electric filed on the Casimir force by affecting the anisotropy of the intervening material by Kerr electro-optical effect.
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Microstructure and Properties of the Ferroelectric-Ferromagnetic PLZT-Ferrite Composites*Symmetry* **2018**, *10*(3), 59; doi:10.3390/sym10030059 - 6 March 2018**Abstract **

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The paper presents the technology of ferroelectric-ferromagnetic ceramic composites obtained from PLZT powder (the chemical formula Pb_{0.98}La_{0.02}(Zr_{0.90}Ti_{0.10})_{0.995}O_{3}) and ferrite powder (Ni_{0.64}Zn_{0.36}Fe_{2}O_{4}), as well

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The paper presents the technology of ferroelectric-ferromagnetic ceramic composites obtained from PLZT powder (the chemical formula Pb_{0.98}La_{0.02}(Zr_{0.90}Ti_{0.10})_{0.995}O_{3}) and ferrite powder (Ni_{0.64}Zn_{0.36}Fe_{2}O_{4}), as well as the results of X-ray powder-diffraction data (XRD) measurement, microstructure, dielectric, ferroelectric, and magnetic properties of the composite samples. The ferroelectric-ferromagnetic composite (P-F) was obtained by mixing and the synthesis of 90% of PLZT and 10% of ferrite powders. The XRD test of the P-F composite shows a two-phase structure derived from the PLZT component (strong peaks) and the ferrite component (weak peaks). The symmetry of PLZT was identified as a rhombohedral ferroelectric phase, while the ferrite was identified as a spinel structure. Scanning electron microscope (SEM) microstructure analysis of the P-F ceramic composites showed that fine grains of the PLZT component surrounded large ferrite grains. At room temperature P-F composites exhibit both ferroelectric and ferromagnetic properties. The P-F composite samples have lower values of the maximum dielectric permittivity at the Curie temperature and a higher dielectric loss compared to the PLZT ceramics, however, the exhibit overall good multiferroic properties.
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Deep Learning as a Tool for Automatic Segmentation of Corneal Endothelium Images*Symmetry* **2018**, *10*(3), 60; doi:10.3390/sym10030060 - 6 March 2018**Abstract **

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The automatic analysis of the state of the corneal endothelium is of much interest in ophthalmology. Up till now, several manual and semi-automatic methods have been introduced, but the need of fully-automatic segmentation of cells in the endothelium is still in search. This

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The automatic analysis of the state of the corneal endothelium is of much interest in ophthalmology. Up till now, several manual and semi-automatic methods have been introduced, but the need of fully-automatic segmentation of cells in the endothelium is still in search. This work addresses the problem of automatic delineation of cells in the corneal endothelium images and suggests to use the convolutional neural network (CNN) to classify between cell center, cell body, and cell border in order to achieve precise segmentation. Additionally, a method to automatically select and split merged cells is given. In order to skeletonize the result, the best-fit method is used. The achieved outcomes are compared to manual annotations in order to define the mutual overlapping. The Dice index, Jaccard coefficient, modified Hausdorff distance, and several other metrics for mosaic overlapping are used. As a final check-up, the visual inspection is shown. The performed experiments revealed the best architecture for CNN. The correctness and precision of the segmentation were evaluated on Endothelial Cell “Alizarine” dataset. According to the Dice index and Jaccard coefficient, the automatically achieved cell delineation overlaps the original one with 93% precision. While modified Hausdorff distance shows 0.14 pixel distance, proving very high accuracy. These findings are confirmed by other metrics and also supported by presented visual inspection of achieved segmentations. To conclude, the methodology to achieve fully-automatic delineation of cell boundaries in the corneal endothelium images was presented. The segmentation obtained as a result of pixel classification with CNN proved very high precision.
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Cuckoo Search Algorithm with Lévy Flights for Global-Support Parametric Surface Approximation in Reverse Engineering*Symmetry* **2018**, *10*(3), 58; doi:10.3390/sym10030058 - 3 March 2018**Abstract **

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This paper concerns several important topics of the *Symmetry* journal, namely, computer-aided design, computational geometry, computer graphics, visualization, and pattern recognition. We also take advantage of the symmetric structure of the tensor-product surfaces, where the parametric variables *u* and *v* play a symmetric

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This paper concerns several important topics of the *Symmetry* journal, namely, computer-aided design, computational geometry, computer graphics, visualization, and pattern recognition. We also take advantage of the symmetric structure of the tensor-product surfaces, where the parametric variables *u* and *v* play a symmetric role in shape reconstruction. In this paper we address the general problem of global-support parametric surface approximation from clouds of data points for reverse engineering applications. Given a set of measured data points, the approximation is formulated as a nonlinear continuous least-squares optimization problem. Then, a recent metaheuristics called *Cuckoo Search Algorithm* (CSA) is applied to compute all relevant free variables of this minimization problem (namely, the data parameters and the surface poles). The method includes the iterative generation of new solutions by using the *Lévy flights* to promote the diversity of solutions and prevent stagnation. A critical advantage of this method is its simplicity: the CSA requires only two parameters, many fewer than any other metaheuristic approach, so the parameter tuning becomes a very easy task. The method is also simple to understand and easy to implement. Our approach has been applied to a benchmark of three illustrative sets of noisy data points corresponding to surfaces exhibiting several challenging features. Our experimental results show that the method performs very well even for the cases of noisy and unorganized data points. Therefore, the method can be directly used for real-world applications for reverse engineering without further pre/post-processing. Comparative work with the most classical mathematical techniques for this problem as well as a recent modification of the CSA called *Improved CSA* (ICSA) is also reported. Two nonparametric statistical tests show that our method outperforms the classical mathematical techniques and provides equivalent results to ICSA for all instances in our benchmark.
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