Tabu search-based method for bézier curve parameterization

A very important issue in many applied fields is to construct the fitting curve that approximates a given set of data points optimally in the sense of least-squares. This problem arises in a number of areas, such as computer-aided design & manufacturing (CAD/CAM), virtual reality, medical imaging, computer graphics, computer animation, and many others. This is also a hard problem, because it is highly nonlinear, over-determined and typically involves a large number of unknown variables. A critical step in this process is to obtain a suitable parameterization of the data points. In this context, this paper introduces a new method to obtain an optimal solution for the parameterization problem of the least-squares fitting Bézier curve. Our method is based on a local search metaheuristic approach for optimization problems called tabu search. The method is applied to some simple yet illustrative examples for the cases of 2D and 3D curves. The proposed method is simple to understand, easy to implement and can be applied to any kind of smooth data points. Our experimental results show that the presented method performs very well, being able to fit the data points with a high degree of accuracy.This research has been financially supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project Ref. #TIN2012-30768, Toho University, the University of Cantabria, and the Instituto de Física de Cantabria, a mixed research center of the University of Cantabria and CSIC-Consejo Superior de Investigaciones Científicas.Peer Reviewe

Firefly algorithm for polynomial Bézier surface parameterization

A classical issue in many applied fields is to obtain an approximating surface to a given set of data points. This problem arises in Computer-Aided Design and Manufacturing (CAD/CAM), virtual reality, medical imaging, computer graphics, computer animation, and many others. Very often, the preferred approximating surface is polynomial, usually described in parametric form. This leads to the problem of determining suitable parametric values for the data points, the so-called surface parameterization. In real-world settings, data points are generally irregularly sampled and subjected to measurement noise, leading to a very difficult nonlinear continuous optimization problem, unsolvable with standard optimization techniques. This paper solves the parameterization problem for polynomial Bézier surfaces by applying the firefly algorithm, a powerful nature-inspired metaheuristic algorithm introduced recently to address difficult optimization problems. The method has been successfully applied to some illustrative examples of open and closed surfaces, including shapes with singularities. Our results show that the method performs very well, being able to yield the best approximating surface with a high degree of accuracy

Firefly algorithm for explicit B-spline curve fitting to data points

ABSTRACT. This paper introduces a new method to compute the approximating explicit B-spline curve to a given set of noisy data points.The proposed method computes all parameters of the B-spline fitting curve of a given order.This requires to solve a difficult continuous, multimodal, and multivariate nonlinear least-squares optimization problem. In our approach, this optimization problem is solved by applying the firefly algorithm, a powerful metaheuristic nature-inspired algorithm well suited for optimization. The method has been applied to three illustrative real-world engineering examples from different fields. Our experimental results show that the presented method performs very well, being able to fit the data points with a high degree of accuracy. Furthermore, our scheme outperforms some popular previous approaches in terms of different fitting error criteria

Two simulated annealing optimization schemas for rational bézier curve fitting in the presence of noise

Fitting curves to noisy data points is a difficult problem arising in many scientific and industrial domains. Although polynomial functions are usually applied to this task, there are many shapes that cannot be properly fitted by using this approach. In this paper, we tackle this issue by using rational Bézier curves. This is a very difficult problem that requires computing four different sets of unknowns (data parameters, poles, weights, and the curve degree) strongly related to each other in a highly nonlinear way. This leads to a difficult continuous nonlinear optimization problem. In this paper, we propose two simulated annealing schemas (the all-in-one schema and the sequential schema) to determine the data parameterization and the weights of the poles of the fitting curve. These schemas are combined with least-squares minimization and the Bayesian Information Criterion to calculate the poles and the optimal degree of the best fitting Bézier rational curve, respectively. We apply our methods to a benchmark of three carefully chosen examples of 2D and 3D noisy data points. Our experimental results show that this methodology (particularly, the sequential schema) outperforms previous polynomial-based approaches for our data fitting problem, even in the presence of noise of low-medium intensity.This research has been kindly supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project Ref. #TIN2012-30768, Toho University (Funabashi, Japan), and the University of Cantabria (Santander, Spain)

From Nonlinear Optimization to Convex Optimization through Firefly Algorithm and Indirect Approach with Applications to CAD/CAM

ABSTRACT. Fitting spline curves to data points is a very important issue in many applied fields. It is also challenging, because these curves typically depend on many continuous variables in a highly interrelated nonlinear way. In general, it is not possible to compute these parameters analytically, so the problem is formulated as a continuous nonlinear optimization problem, for which traditional optimization techniques usually fail.This paper presents a new bioinspired method to tackle this issue. In this method, optimization is performed through a combination of two techniques. Firstly, we apply the indirect approach to the knots, in which they are not initially the subject of optimization but precomputed with a coarse approximation scheme. Secondly, a powerful bioinspired metaheuristic technique, the firefly algorithm, is applied to optimization of data parameterization; then, the knot vector is refined by using De Boor’s method, thus yielding a better approximation to the optimal knot vector. This scheme converts the original nonlinear continuous optimization problem into a convex optimization problem, solved by singular value decomposition. Our method is applied to some illustrative real-world examples from the CAD/CAM field. Our experimental results show that the proposed scheme can solve the original continuous nonlinear optimization problem very efficiently

Hybrid Functional-Neural Approach for Surface Reconstruction

ABSTRACT. This paper introduces a new hybrid functional-neural approach for surface reconstruction. Our approach is based on the combination of two powerful artificial intelligence paradigms: on one hand, we apply the popular Kohonen neural network to address the data parameterization problem. On the other hand, we introduce a new functional network, called NURBS functional network, whose topology is aimed at reproducing faithfully the functional structure of the NURBS surfaces. These neural and functional networks are applied in an iterative fashion for further surface refinement. The hybridization of these two networks provides us with a powerful computational approach to obtain a NURBS fitting surface to a set of irregularly sampled noisy data points within a prescribed error threshold. The method has been applied to two illustrative examples. The experimental results confirm the good performance of our approach.This research has been kindly supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project ref. no. TIN2012-30768, Toho University (Funabashi, Japan), and the University of Cantabria (Santander, Spain)

Make robots Be Bats: Specializing robotic swarms to the Bat algorithm

Bat algorithm is a powerful nature-inspired swarm intelligence method proposed by Prof. Xin-She Yang in 2010, with remarkable applications in industrial and scientific domains. However, to the best of authors' knowledge, this algorithm has never been applied so far in the context of swarm robotics. With the aim to fill this gap, this paper introduces the first practical implementation of the bat algorithm in swarm robotics. Our implementation is performed at two levels: a physical level, where we design and build a real robotic prototype; and a computational level, where we develop a robotic simulation framework. A very important feature of our implementation is its high specialization: all (physical and logical) components are fully optimized to replicate the most relevant features of the real microbats and the bat algorithm as faithfully as possible. Our implementation has been tested by its application to the problem of finding a target location within unknown static indoor 3D environments. Our experimental results show that the behavioral patterns observed in the real and the simulated robotic swarms are very similar. This makes our robotic swarm implementation an ideal tool to explore the potential and limitations of the bat algorithm for real-world practical applications and their computer simulations.This research has been kindly supported by the Computer Science National Program of the Spanish Research Agency (Agencia Estatal de Investigación) and European Funds, Project #TIN2017-89275-R (AEI/FEDER, UE), the project EVOLFORMAS Ref. #JU12, jointly supported by public body SODERCAN of the Regional Government of Cantabria and the European funds FEDER, the project PDE-GIR of the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Actions grant agreement #778035, Toho University (Funabashi, Japan), and the University of Cantabria (Santander, Spain). The authors are particularly grateful to the Department of Information Science of Toho University for all the facilities given to carry out this work. Special thanks are also due to the Editors and the three anonymous reviewers for their encouraging and constructive comments and very helpful feedback that allowed us to improve our paper signi cantly

Rank-based ant system with originality reinforcement and pheromone smoothing

Ant Colony Optimization (ACO) encompasses a family of metaheuristics inspired by the foraging behaviour of ants. Since the introduction of the first ACO algorithm, called Ant System (AS), several ACO variants have been proposed in the literature. Owing to their superior performance over other alternatives, the most popular ACO algorithms are Rank-based Ant System (ASRank), Max-Min Ant System (MMAS) and Ant Colony System (ACS). While ASRank shows a fast convergence to high-quality solutions, its performance is improved by other more widely used ACO variants such as MMAS and ACS, which are currently considered the state-of-the-art ACO algorithms for static combinatorial optimization problems. With the purpose of diversifying the search process and avoiding early convergence to a local optimal, the proposed approach extends ASRank with an originality reinforcement strategy of the top-ranked solutions and a pheromone smoothing mechanism that is triggered before the algorithm reaches stagnation. The approach is tested on several symmetric and asymmetric Traveling Salesman Problem and Sequential Ordering Problem instances from TSPLIB benchmark. Our experimental results show that the proposed method achieves fast convergence to high-quality solutions and outperforms the current state-of-the-art ACO algorithms ASRank, MMAS and ACS, for most instances of the benchmark.This research work was funded by the European project PDE-GIR of the European Union’s Horizon 2020 research & innovation program (Marie Sklodowska-Curie action, grant agreement No 778035), and by the Spanish government project #PID2021-127073OB-I00 of the MCIN/AEI/10.13039/501100011033/FEDER, EU “Una manera de hacer Europa”

Immunological-based approach for accurate fitting of 3D noisy data points with Bézier surfaces

Free-form parametric surfaces are common tools nowadays in many applied fields, such as Computer-Aided Design & Manufacturing (CAD/CAM), virtual reality, medical imaging, and many others. A typical problem in this setting is to fit surfaces to 3D noisy data points obtained through either laser scanning or other digitizing methods, so that the real data from a physical object are transformed back into a fully usable digital model. In this context, the present paper describes an immunologicalbased approach to perform this process accurately by using the classical free-form Bézier surfaces. Our method applies a powerful bio-inspired paradigm called Artificial Immune Systems (AIS), which is receiving increasing attention from the scientific community during the last few years because of its appealing computational features. The AIS can be understood as a computational methodology based upon metaphors of the biological immune system of humans and other mammals. As such, there is not one but several AIS algorithms. In this chapter we focus on the clonal selection algorithm (CSA), which explicitly takes into account the affinity maturation of the immune response. The paper describes how the CSA algorithm can be effectively applied to the accurate fitting of 3D noisy data points with Bézier surfaces. To this aim, the problem to be solved as well as the main steps of our solving method are described in detail. Some simple yet illustrative examples show the good performance of our approach. Our method is conceptually simple to understand, easy to implement, and very general, since no assumption is made on the set of data points or on the underlying function beyond its continuity. As a consequence, it can be successfully applied even under challenging situations, such as the absence of any kind of information regarding the underlying function of data

Cuckoo Search with Lévy Flights for Weighted Bayesian Energy Functional Optimization in Global-Support Curve Data Fitting

ABSTRACT. The problem of data fitting is very important in many theoretical and applied fields. In this paper, we consider the problem of optimizing a weighted Bayesian energy functional for data fitting by using global-support approximating curves. By global-support curves we mean curves expressed as a linear combination of basis functions whose support is the whole domain of the problem, as opposed to other common approaches in CAD/CAM and computer graphics driven by piecewise functions (such as B-splines and NURBS) that provide local control of the shape of the curve. Our method applies a powerful nature-inspired metaheuristic algorithm called cuckoo search, introduced recently to solve optimization problems. A major advantage of this method is its simplicity: cuckoo search requires only two parameters, many fewer than other metaheuristic approaches, so the parameter tuning becomes a very simple task.The paper shows that this new approach can be successfully used to solve our optimization problem. To check the performance of our approach, it has been applied to five illustrative examples of different types, including open and closed 2D and 3D curves that exhibit challenging features, such as cusps and self-intersections. Our results show that the method performs pretty well, being able to solve our minimization problem in an astonishingly straightforward way.This research has been kindly supported by the Computer Science National Program of the Spanish Ministry of Economy and Competitiveness, Project Ref. no. TIN2012-30768, Toho University (Funabashi, Japan), and the University of Cantabria (Santander, Spain)