Closed fractal interpolation surfaces

AbstractBased on the construction of bivariate fractal interpolation surfaces, we introduce closed spherical fractal interpolation surfaces. The interpolation takes place in spherical coordinates and with the transformation to Cartesian coordinates a closed surface arises. We give conditions for this construction to be valid and state some useful relations about the Hausdorff and the Box counting dimension of the closed surface

Fractal Image Compression Using Modified Operator (IFS)

Image data Compression based on fractal theory is fundamentally dierent from conventional compression methods, its idea is to generate a contraction operator whose fixed point approximates the original image in a complete metric space of images. The specication of such operator can be stored as the fractal code for the original image. The contraction mapping principle implies that the iteration of the stored operator starting from arbitrary initial image will recover its xed point which is an approximation for the original image. This Contraction mapping is usually constructed using the partitioned IFS(PIFS) technique which relies on the assertion that parts of the image resemble other parts of the same image. It then, nds the fractal code for each part by searching for another larger similar part. This high costly search makes fractal image compression dicult to be implemented in practice, even it has the advantages of a high compression ratio, a low loss ratio, and the resolution independence of the compression rate. In this paper, we investigate fractal image compression(FIC) using Iterated Function Systems(IFS). After reviewing the standard scheme, we state a mathematical formulation for the practical aspect. We then propose a modied IFS that relies on the fact  that, there are very smooth parts in certain images. From the view point of mathematics, we present the modied operator, proving its properties that make it not only a fractal operator but also more eective than the standard one. The experimental results are presented and the performance of the proposed algorithm is discussed

Image compression using recurrent bivariate fractal interpolation surfaces

A new method for constructing recurrent bivariate fractal interpolation surfaces through points sampled on rectangular lattices is proposed. This offers the advantage of a more flexible fractal modeling compared to previous fractal techniques that used affine transformations. The compression ratio for the above mentioned fractal scheme as applied to real images is higher than other fractal methods or JPEG, though not as high as JPEG2000. Theory, implementation and analytical study are also presented. © World Scientific Publishing Company

The study of chaos theory and information theory in enhancing data standard towards smart infrastructure

This dissertation explores the complex dynamics underlying Building Information Modeling (BIM) data standard development, with the aim of enhancing efficiency and effectiveness in the Architecture, Engineering, Construction and Operations (AECO) industry. The research integrates chaos theory and information theory to elucidate hidden patterns and principles governing BIM standards. This thesis establishes the methodological framework, combining theoretical research and design science approaches. Information theory provides a quantitative lens to analyze BIM information flows, while chaos theory recognizes inherent complexity and unpredictability. Philosophically, the research embraces interdisciplinarity and pragmatism. Through sandpile simulations, the dynamics of BIM standard development are modeled computationally. Innovative mapping techniques connect simulation patterns to actual BIM standards topologies, represented as tree structures. Analyses reveal “similarity cross-scalability,” indicative of chaos and self-organized criticality. This suggests BIM standards evolve akin to a chaotic system, with sensitivity to initial conditions. Mathematical techniques rigorously prove chaotic properties in BIM standard development. Time series data from simulations enable phase space reconstruction. Determining optimal time delay and dimensionality allows creating an accurate phase space capturing system dynamic. Calculation of a positive Lyapunov exponent provides definitive evidence of chaos. New methodologies emerge from the chaos-driven perspective. Information theory and sandpile principles generate novel Model View Definitions (MVDs) for tunnel linings, embracing dynamism while reducing ambiguity. Comparative analysis shows improved consistency over conventional standards. System attractors within reconstructed phase space form the basis for a chaos-informed performance indicator for BIM models, using distance to attractors as a stability metric. In summary, this pioneering research makes significant contributions: It proves, mathematically and empirically, the presence of chaos in BIM standard development related to information flows. Chaos theory and information theory are shown to offer valuable perspectives for enhancing BIM standards. Innovative techniques are proposed for generating adaptable, robust MVDs and evaluating BIM model stability. Philosophy of interdisciplinarity and pragmatism is embraced to integrate diverse concepts. Computational modeling and mapping reveal new insights into complex BIM standard dynamics. The implications are profound. Identifying chaos enables harnessing advanced techniques from disparate disciplines to optimize BIM processes. The proposed methodologies demonstrate enhanced efficiency, consistency, and performance. This research lays the foundations to utilize chaos theory for next-generation innovations in the AECO industry. The transformative potential is to fundamentally evolve BIM standards to be highly adaptive and responsive to the industry’s dynamic needs