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

Tilings, Patterns and Technology

In this paper we discuss various situations where tilings and patterns, with the aid of technology, facilitate the teaching of mathematics and serve as tools in understanding and developing new mathematical ideas. We also illustrate how technology makes possible cultural connections in the study of mathematics using Islamic tilings and patterns

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

Kinship Matters: Structures of Alliance, Indigenous Foragers, and the Austronesian Diaspora

The study of kinship systems has direct relevance for the field of human genetics and the study of microevolution in human populations. Some types of postmarital residence rules—rules requiring a married couple to live with or near relatives of the husband or wife—will have consequences for the distribution of mitochondrial DNA and Y chromosome lineages. Rules that proscribe or encourage marriage with close kin will also have consequences for allele frequency. A preference for marrying at a distance, both socially and geographically, creates alliances that can have survival value for individuals and groups in an environment of periodic or unpredictable scarcity. This article considers the nature of early contact between the indigenous foraging populations of the Philippines and Austronesian speaking settlers who began arriving ~4,500–4,000 BP. It argues that when the first Austronesians arrived they brought with them a kinship system based on symmetrical exchange between descent-based groups. It considers why such a system was gradually changed into the bilateral kinship systems that characterize the various peoples of the Philippines today, negrito and non-negrito alike