Mathematical tools for computer-generated ornamental patterns

This article presents mathematical tools for computer-generated ornamental patterns, with a particular attention payed to Islamic patterns. The article shows how, starting from a photo or a sketch of an ornamental figure, the designer analyzes its structure and produces the analytical representation of the pattern. This analytical representation in turn is used to produce a drawing which is integrated into a computer-generated virtual scene. The mathematical tools for analysis of ornamental patterns consist of a subset of tools usually used in the mathematical theory of tilings such as planar symmetry groups and Cayley diagrams. A simple and intuitive step-by-step guide is provided

Image Sampling with Quasicrystals

We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique.Comment: For a full resolution version of this paper, along with supplementary materials, please visit at http://www.Eyemaginary.com/Portfolio/Publications.htm

A new method to analyse mosaics based on Symmetry Group theory applied to Islamic Geometric Patterns

[EN] This article presents a new method for analysing mosaics based on the mathematical principles of Symmetry Groups. This method has been developed to get the understanding present in patterns by extracting the objects that form them, their lattice, and the Wallpaper Group. The main novelty of this method resides in the creation of a higher level of knowledge based on objects, which makes it possible to classify the objects, to extract their main features (Point Group, principal axes, etc.), and the relationships between them. In order to validate the method, several tests were carried out on a set of Islamic Geometric Patterns from different sources, for which the Wallpaper Group has been successfully obtained in 85% of the cases. This method can be applied to any kind of pattern that presents a Wallpaper Group. Possible applications of this computational method include pattern classification, cataloguing of ceramic coatings, creating databases of decorative patterns, creating pattern designs, pattern comparison between different cultures, tile cataloguing, and so on.The authors wish to thank the Patronato de la Alhambra y Generalife (Granada, Spain) and the Patronato del Real Alcázar de Sevilla (Seville, Spain) for their valuable collaboration in this research work.Albert Gil, FE.; Gomis Martí, JM.; Blasco, J.; Valiente González, JM.; Aleixos Borrás, MN. (2015). A new method to analyse mosaics based on Symmetry Group theory applied to Islamic Geometric Patterns. Computer Vision and Image Understanding. 130:54-70. doi:10.1016/j.cviu.2014.09.002S547013

La progettazione parametrica come strumento di analisi: dai pattern algoritmici decorativi ai pattern “performanti”, esempi nei Beni Culturali

La ricerca si incentra sulle relazioni intercorrenti tra i concetti/sistemi di “decorazione” e “modellazione parametrica” per sottolinearne i punti di contatto, i parallelismi e i margini di applicazione ai beni culturali e alla progettazione. L’originalità del lavoro consiste nel coniugare conoscenze storiche, acquisite all’interno della II Facoltà di Architettura, e informazioni tecniche sviluppabili nei laboratori di ricerca statunitensi. La prima parte della tesi analizzerà le affinità tra i due ambiti (decorazione e parametricità), evidenziandone le somiglianze. L’introduzione del concetto di pattern prevede la descrizione del suo approccio algoritmico con le possibilità che offre per la comprensione della decorazione all’interno dei beni culturali e l’esame di pattern decorativi applicati ad architetture contemporanee criticamente selezionate. Il discorso sarà poi ampliato ai pattern performanti dal punto di vista strutturale. Partendo dalla constatazione che nell’architettura forte è l’interesse per la geometria, a tal punto che spesso struttura e decorazione si fondono, verranno descritti e analizzati esempi di architetture storiche in cui la struttura assume un alto valore decorativo, con particolare attenzione alle ricerche già svolte su Bernini e Borromini. La trattazione si concluderà descrivendo come i pattern abbiano concorso a cambiare il concetto di modularità e contribuito a creare numerose icone architettoniche della fine del XX secolo. Paragrafi appositi saranno dedicati al ruolo svolto dalle tecniche di prototipazione rapida nella diffusione di questa nuova estetica. La parte finale sarà dedicata alla sperimentazione e proporrà l’utilizzo di pattern decorativi performanti applicati al benessere degli utenti sotto il profilo delle prestazioni acustiche, del controllo dell’irraggiamento solare, della trasparenza o opacità degli ambienti e delle prestazioni dei materiali

Image Sampling with Quasicrystals

We investigate the use of quasicrystals in image sampling. Quasicrystals produce space-filling, non-periodic point sets that are uniformly discrete and relatively dense, thereby ensuring the sample sites are evenly spread out throughout the sampled image. Their self-similar structure can be attractive for creating sampling patterns endowed with a decorative symmetry. We present a brief general overview of the algebraic theory of cut-and-project quasicrystals based on the geometry of the golden ratio. To assess the practical utility of quasicrystal sampling, we evaluate the visual effects of a variety of non-adaptive image sampling strategies on photorealistic image reconstruction and non-photorealistic image rendering used in multiresolution image representations. For computer visualization of point sets used in image sampling, we introduce a mosaic rendering technique

Development and evaluation of a digital tool for virtual reconstruction of historic Islamic geometric patterns

For the purpose of cultural heritage preservation, the task of recording and reconstructing visually complicated architectural geometrical patterns is facing many practical challenges. Existing traditional technologies rely heavily on the subjective nature of our perceptual power in understanding its complexity and depicting its color differences. This study explores one possible solution, through utilizing digital techniques for reconstructing detailed historical Islamic geometric patterns. Its main hypothesis is that digital techniques offer many advantages over the human eye in terms of recognizing subtle differences in light and color. The objective of the study is to design, test and evaluate an automatic visual tool for identifying deteriorated or incomplete archaeological Islamic geometrical patterns captured in digital images, and then restoring them digitally, for the purpose of producing accurate 2D reconstructed metric models. An experimental approach is used to develop, test and evaluate the specialized software. The goal of the experiment is to analyze the output reconstructed patterns for the purpose of evaluating the digital tool in respect to reliability and structural accuracy, from the point of view of the researcher in the context of historic preservation. The research encapsulates two approaches within its methodology; Qualitative approach is evident in the process of program design, algorithm selection, and evaluation. Quantitative approach is manifested through using mathematical knowledge of pattern generation to interpret available data and to simulate the rest based on it. The reconstruction process involves induction, deduction and analogy. The proposed method was proven to be successful in capturing the accurate structural geometry of the deteriorated straight-lines patterns generated based on the octagon-square basic grid. This research also concluded that it is possible to apply the same conceptual method to reconstruct all two-dimensional Islamic geometric patterns. Moreover, the same methodology can be applied to reconstruct many other pattern systems. The conceptual framework proposed by this study can serve as a platform for developing professional softwares related to historic documentation. Future research should be directed more towards developing artificial intelligence and pattern recognition techniques that have the ability to suplement human power in accomplishing difficult tasks

Development and evaluation of a digital tool for virtual reconstruction of historic Islamic geometric patterns

For the purpose of cultural heritage preservation, the task of recording and reconstructing visually complicated architectural geometrical patterns is facing many practical challenges. Existing traditional technologies rely heavily on the subjective nature of our perceptual power in understanding its complexity and depicting its color differences. This study explores one possible solution, through utilizing digital techniques for reconstructing detailed historical Islamic geometric patterns. Its main hypothesis is that digital techniques offer many advantages over the human eye in terms of recognizing subtle differences in light and color. The objective of the study is to design, test and evaluate an automatic visual tool for identifying deteriorated or incomplete archaeological Islamic geometrical patterns captured in digital images, and then restoring them digitally, for the purpose of producing accurate 2D reconstructed metric models. An experimental approach is used to develop, test and evaluate the specialized software. The goal of the experiment is to analyze the output reconstructed patterns for the purpose of evaluating the digital tool in respect to reliability and structural accuracy, from the point of view of the researcher in the context of historic preservation. The research encapsulates two approaches within its methodology; Qualitative approach is evident in the process of program design, algorithm selection, and evaluation. Quantitative approach is manifested through using mathematical knowledge of pattern generation to interpret available data and to simulate the rest based on it. The reconstruction process involves induction, deduction and analogy. The proposed method was proven to be successful in capturing the accurate structural geometry of the deteriorated straight-lines patterns generated based on the octagon-square basic grid. This research also concluded that it is possible to apply the same conceptual method to reconstruct all two-dimensional Islamic geometric patterns. Moreover, the same methodology can be applied to reconstruct many other pattern systems. The conceptual framework proposed by this study can serve as a platform for developing professional softwares related to historic documentation. Future research should be directed more towards developing artificial intelligence and pattern recognition techniques that have the ability to suplement human power in accomplishing difficult tasks