The year 2008 is a memorial year for Georgiy Vorono (1868-1908), with a number of events in the scientific community commemorating his tremendous contribution to the area of mathematics, especially number theory, through conferences and scientific gatherings in his honor. A notable event taking place in September 2008 a joint c- ference: the 5th Annual International Symposium on Voronoi Diagrams (ISVD) and the 4th International Conference on Analytic Number Theory and Spatial Tessel- tions held in Kyiv, Georgiy Vorono ’s native land. The main ideas expressed by G. Vorono ’s through his fundamental works have influenced and shaped the key dev- opments in computation geometry, image recognition, artificial intelligence, robotics, computational science, navigation and obstacle avoidance, geographical information systems, molecular modeling, astrology, physics, quantum computing, chemical en- neering, material sciences, terrain modeling, biometrics and other domains. This book is intended to provide the reader with in-depth overview and analysis of the fundamental methods and techniques developed following G. Voronoi ideas, in the context of the vast and increasingly growing area of computational intelligence. It represents the collection of state-of-the art research methods merging the bridges between two areas: geometric computing through Voronoi diagrams and intelligent computation techniques, pushing the limits of current knowledge in the area, impr- ing on previous solutions, merging sciences together, and inventing new ways of approaching difficult applied problems.

The year 2008 is a memorial year for Georgiy Vorono (1868-1908), with a number of events in the scientific community commemorating his tremendous contribution to the area of mathematics, especially number theory, through conferences and scientific gatherings in his honor. A notable event taking place in September 2008 a joint c- ference: the 5th Annual International Symposium on Voronoi Diagrams (ISVD) and the 4th International Conference on Analytic Number Theory and Spatial Tessel- tions held in Kyiv, Georgiy Vorono ’s native land. The main ideas expressed by G. Vorono ’s through his fundamental works have influenced and shaped the key dev- opments in computation geometry, image recognition, artificial intelligence, robotics, computational science, navigation and obstacle avoidance, geographical information systems, molecular modeling, astrology, physics, quantum computing, chemical en- neering, material sciences, terrain modeling, biometrics and other domains. This book is intended to provide the reader with in-depth overview and analysis of the fundamental methods and techniques developed following G. Voronoi ideas, in the context of the vast and increasingly growing area of computational intelligence. It represents the collection of state-of-the art research methods merging the bridges between two areas: geometric computing through Voronoi diagrams and intelligent computation techniques, pushing the limits of current knowledge in the area, impr- ing on previous solutions, merging sciences together, and inventing new ways of approaching difficult applied problems.

The purpose of this volume is to present current work of the Intelligent Computer Graphics community, a community growing up year after year. Indeed, if at the beg- ning of Computer Graphics the use of Artificial Intelligence techniques was quite unknown, more and more researchers all over the world are nowadays interested in intelligent techniques allowing substantial improvements of traditional Computer Graphics methods. The other main contribution of intelligent techniques in Computer Graphics is to allow invention of completely new methods, often based on automation of a lot of tasks assumed in the past by the user in an imprecise and (human) time consuming manner. The history of research in Computer Graphics is very edifying. At the beginning, due to the slowness of computers in the years 1960, the unique research concern was visualisation. The purpose of Computer Graphics researchers was to find new visua- sation algorithms, less and less time consuming, in order to reduce the enormous time required for visualisation. A lot of interesting algorithms were invented during these first years of research in Computer Graphics. The scenes to be displayed were very simple because the computing power of computers was very low. So, scene modelling was not necessary and scenes were designed directly by the user, who had to give co-ordinates of vertices of scene polygons.

Voronoi diagrams partition space according to the influence certain sites exert on their environment. Since the 17th century, such structures play an important role in many areas like Astronomy, Physics, Chemistry, Biology, Ecology, Economics, Mathematics and Computer Science. They help to describe zones of political influence, to determine the hospital nearest to an accident site, to compute collision-free paths for mobile robots, to reconstruct curves and surfaces from sample points, to refine triangular meshes, and to design location strategies for competing markets.This unique book offers a state-of-the-art view of Voronoi diagrams and their structure, and it provides efficient algorithms towards their computation.Readers with an entry-level background in algorithms can enjoy a guided tour of gently increasing difficulty through a fascinating area. Lecturers might find this volume a welcome source for their courses on computational geometry. Experts are offered a broader view, including many alternative solutions, and up-to-date references to the existing literature; they might benefit in their own research or application development.

Biologically inspiredcomputing isdi?erentfromconventionalcomputing.Ithas adi?erentfeel; often the terminology does notsound like it’stalkingabout machines.The activities ofthiscomputingsoundmorehumanthanmechanistic as peoplespeak ofmachines that behave, react, self-organize,learn, generalize, remember andeven to forget.Much ofthistechnology tries to mimic nature’s approach in orderto mimicsome of nature’s capabilities.They havearigorous, mathematical basisand neuralnetworks forexamplehaveastatistically valid set on which the network istrained. Twooutlinesaresuggestedasthepossibletracksforpatternrecognition.They are neuralnetworks andfunctionalnetworks.NeuralNetworks (many interc- nected elements operating in parallel) carryout tasks that are not only beyond the scope ofconventionalprocessing but also cannotbeunderstood in the same terms.Imagingapplicationsfor neuralnetworksseemtobea natural?t.Neural networks loveto do pattern recognition. A new approachto pattern recognition usingmicroARTMAP together with wavelet transforms in the context ofhand written characters,gestures andsignatures havebeen dealt.The KohonenN- work,Back Propagation Networks andCompetitive Hop?eld NeuralNetwork havebeen considered for various applications. Functionalnetworks,beingageneralizedformofNeuralNetworkswherefu- tionsarelearnedratherthanweightsiscomparedwithMultipleRegressionAn- ysisforsome applicationsandtheresults are seen to be coincident. New kinds of intelligence can be added to machines, and we will havethe possibilityof learningmore about learning.Thus our imaginationsand options are beingstretched.These new machines will be fault-tolerant,intelligentand self-programmingthustryingtomakethemachinessmarter.Soastomakethose who use the techniques even smarter. Chapter1 isabrief introduction toNeural and Functionalnetworks in the context of Patternrecognitionusing these disciplinesChapter2 givesa review ofthearchitectures relevantto the investigation andthedevelopment ofthese technologies in the past few decades. Retracted VIII Preface Chapter3begins with the lookattherecognition ofhandwritten alphabets usingthealgorithm for ordered list ofboundary pixelsas well as the Ko- nenSelf-Organizing Map (SOM).Chapter 4 describes the architecture ofthe MicroARTMAP and its capability.

The vehicle routing problem (VRP) is one of the most famous combinatorial optimization problems. In simple terms, the goal is to determine a set of routes with overall minimum cost that can satisfy several geographical scattered - mands. A ?eet of vehicles located in one or more depots is available to ful?ll the requests. A large number of variants exist, adding di?erent constraints to the original de?nition. Some examples are related to the number of depots, the ordering for visiting the customers or to time windows specifying a desirable period to arrive to a given location. The original version of this problem was proposed by Dantzig and Ramser in 1959 [1]. In their seminal paper, the authors address the calculation of a set of optimal routes for a ?eet of gasoline delivery trucks. Since then, the VRP has attractedtheattentionofalargenumberofresearchers.Aconsiderablepartofits success is a consequence of its practical interest, as it resembles many real-world problems faced everyday by distribution and transportation companies, just to mention a few applications areas. In this context, the development of e?cient optimization techniques is crucial. They are able to provide new and enhanced solutionstologisticoperations,andmaythereforeleadtoasubstantialreduction in costs for companies. Additionally, and from a research oriented perspective, the VRP is a challenging NP-hard problem providing excellent benchmarks to access the e?ciency of new global optimization algorithms.

This, the 20th issue of the Transactions on Computational Science journal, edited by Bahman Kalantari, is devoted to the topic of Voronoi Diagrams and their applications. The 10 full papers included in the volume are revised and extended versions of a selection of papers presented at the International Symposium on Voronoi Diagrams 2012, held in Rutgers, NJ, USA, in June 2012. They provide an in-depth overview of current research on topological data structures and a comprehensive evaluation of their applications in the fields of cartography, physics, material modeling, chemistry, GIS, motion planning and computer graphics.