Reconstruction of Three-Dimensional Blood Vessel Model Using Fractal Interpolation

Fractal method is used in the image processing and studying the irregular and the complex shapes in the image. It is also used in the reconstruction and smoothing of one-, two-, and three-dimensional data. In this chapter, we present an interpolating fractal algorithm to reconstruct 3D blood vessels. Firstly, the proposed method determines the blood vessel centerline from the 2D retina image, and then it uses the Douglas-Peucker algorithm to detect the control points. Secondly, we use the 3D fractal interpolation and iterated function systems for the visualization and reconstruction of these blood vessels. The results showed that the obtained reduction rate is between 71 and 94% depending on the tolerance value. The 3D blood vessels model can be reconstructed efficiently by using the 3D fractal interpolation method

Fractal interpolation in modeling of 2D contours

The problem of fractal modeling is very simple when we know the mathematical description of a fractal. We just apply one of the well-known algorithms. The inverse problem of finding the mathematical description for given fractal is not so trivial and we do not know any general method to solve this problem. So there are several approaches to this problem e.g. via Bezier curves, fractal compression. In this paper we present automatic method for finding fractal description of 2D contours. Our algorithm uses fractal interpolation for this purpose. We also present some of practical examples

Adaptive rational fractal interpolation function for image super-resolution via local fractal analysis

© 2019 Elsevier B.V. Image super-resolution aims to generate high-resolution image based on the given low-resolution image and to recover the details of the image. The common approaches include reconstruction-based methods and interpolation-based methods. However, these existing methods show difficulty in processing the regions of an image with complicated texture. To tackle such problems, fractal geometry is applied on image super-resolution, which demonstrates its advantages when describing the complicated details in an image. The common fractal-based method regards the whole image as a single fractal set. That is, it does not distinguish the complexity difference of texture across all regions of an image regardless of smooth regions or texture rich regions. Due to such strong presumption, it causes artificial errors while recovering smooth area and texture blurring at the regions with rich texture. In this paper, the proposed method produces rational fractal interpolation model with various setting at different regions to adapt to the local texture complexity. In order to facilitate such mechanism, the proposed method is able to segment the image region according to its complexity which is determined by its local fractal dimension. Thus, the image super-resolution process is cast to an optimization problem where local fractal dimension in each region is further optimized until the optimization convergence is reached. During the optimization (i.e. super-resolution), the overall image complexity (determined by local fractal dimension) is maintained. Compared with state-of-the-art method, the proposed method shows promising performance according to qualitative evaluation and quantitative evaluation

On Self‐Affine and Self‐Similar Graphs of Fractal Interpolation Functions Generated from Iterated Function Systems

This chapter provides a brief and coarse discussion on the theory of fractal interpolation functions and their recent developments including some of the research made by the authors. It focuses on fractal interpolation as well as on recurrent fractal interpolation in one and two dimensions. The resulting self‐affine or self‐similar graphs, which usually have non‐integral dimension, were generated through a family of (discrete) dynamic systems, the iterated function system, by using affine transformations. Specifically, the fractal interpolation surfaces presented here were constructed over triangular as well as over polygonal lattices with triangular subdomains. A further purpose of this chapter is the exploration of the existent breakthroughs and their application to a flexible and integrated software that constructs and visualises the above‐mentioned models. We intent to supply both a panoramic view of interpolating functions and a useful source of links to assist a novice as well as an expert in fractals. The ideas or findings contained in this paper are not claimed to be exhaustive, but are intended to be read before, or in parallel with, technical papers available in the literature on this subject

Basics of Modelling and Visualization

This textbook presents basic concepts related to modelling and visualization tasks. Chapters 1-4 describe transformations in the plane and in the space, and geometrical forms of graphical objects such as curves, patches and fractals. Chapter 5 is about lights, materials, textures, colours that all are needed to enrich a severe appearance of pure geometrical objects leading to their photorealistic visualizations. In Chapter 6 freeware software such as POV Ray, MayaVi and Deep View are described. Using those software one can obtain photorealistic renderings and visualizations. The textbook was prepared for students of the specialization ,,Modelling and Visualization in Bioinformatics'' but it should be helpful to anyone who is interested in computer graphics, modelling techniques, animation and visualization of data. Authors of this textbook believe that information presented in the book will be useful for students and will inspire their imagination in creation of photorealistic static 3D scenes and also will be helpful in creation of animations and visualization of data in an effective and professional way

PlantGL : a Python-based geometric library for 3D plant modelling at different scales

In this paper, we present PlantGL, an open-source graphic toolkit for the creation, simulation and analysis of 3D virtual plants. This C++ geometric library is embedded in the Python language which makes it a powerful user-interactive platform for plant modelling in various biological application domains. PlantGL makes it possible to build and manipulate geometric models of plants or plant parts, ranging from tissues and organs to plant populations. Based on a scene graph augmented with primitives dedicated to plant representation, several methods are provided to create plant architectures from either field measurements or procedural algorithms. Because they reveal particularly useful in plant design and analysis, special attention has been paid to the definition and use of branching system envelopes. Several examples from different modelling applications illustrate how PlantGL can be used to construct, analyse or manipulate geometric models at different scales

Interactive evolutionary 3D fractal modeling.

Pang, Wenjun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 83-88).Abstracts in English and Chinese.ACKNOWLEDGEMENTS --- p.iiABSTRACT --- p.iv摘要 --- p.vCONTENTS --- p.viList of Tables --- p.viiiList of Figures --- p.ixChapter 1. --- INTRODUCTION --- p.1Chapter 1.1 --- Recent research work --- p.4Chapter 1.2 --- Objectives --- p.8Chapter 1.3 --- Thesis Organization --- p.10Chapter 2. --- FRACTAL MODELING --- p.12Chapter 2.1 --- Fractal and Fractal Art --- p.12Chapter 2.2 --- Fractal Geometry --- p.15Chapter 2.3 --- Construction of Fractals --- p.21Chapter 2.4 --- Fractal Measurement and Aesthetics --- p.27Chapter 3. --- OVERVIEW OF EVOLUTIONARY DESIGN --- p.30Chapter 3.1 --- Initialization --- p.33Chapter 3.2 --- Selection --- p.33Chapter 3.3 --- Reproduction --- p.34Chapter 3.4 --- Termination --- p.36Chapter 4. --- EVOLUTIONARY 3D FRACTAL MODELING --- p.38Chapter 4.1 --- Fractal Construction --- p.38Chapter 4.1.1 --- Self-similar Condition of Fractal --- p.38Chapter 4.1.2 --- Fractal Transformation (FT) IFS Formulation --- p.39Chapter 4.1.3 --- IFS Genotype and Phenotype Expression --- p.41Chapter 4.2 --- Evolutionary Algorithm --- p.43Chapter 4.2.1 --- Single-point Crossover --- p.45Chapter 4.2.2 --- Arithmetic Gaussian mutation --- p.45Chapter 4.2.3 --- Inferior Elimination --- p.46Chapter 4.3 --- Interactive Fine-tuning using FT IFS --- p.46Chapter 4.4 --- Gaussian Fitness Function --- p.48Chapter 5. --- GAUSSIAN AESTHETIC FITNESS FUNCTION --- p.49Chapter 5.1 --- Fitness Considerations --- p.50Chapter 5.2 --- Fitness Function Formulation --- p.53Chapter 5.3 --- Results and Discussion on Fitness Function --- p.55Chapter 6. --- EXPERIMENT RESULTS and DISCUSSION --- p.59Chapter 6.1 --- Experiment of Evolutionary Generation --- p.59Chapter 6.2 --- Comparison on Different Methods --- p.60Chapter 7. --- 3D FRACTALS RENDERING and APPLICATION --- p.62Chapter 7.1 --- Transforming Property and User Modification --- p.62Chapter 7.2 --- Visualization and Rendering of 3D Fractals --- p.66Chapter 7.3 --- Applications in Design --- p.74Chapter 8. --- CONCLUSIONS and FUTURE WORK --- p.81Chapter 8.1 --- Conclusions --- p.81Chapter 8.2 --- Future Work --- p.81BIBLIOGRAPHY --- p.83Appendix --- p.89Marching Cubes Method --- p.8

A Survey of Procedural Techniques for City Generation

The computer game industry requires a skilled workforce and this combined with the complexity of modern games, means that production costs are extremely high. One of the most time consuming aspects is the creation of game geometry, the virtual world which the players inhabit. Procedural techniques have been used within computer graphics to create natural textures, simulate special effects and generate complex natural models including trees and waterfalls. It is these procedural techniques that we intend to harness to generate geometry and textures suitable for a game situated in an urban environment. Procedural techniques can provide many benefits for computer graphics applications when the correct algorithm is used. An overview of several commonly used procedural techniques including fractals, L-systems, Perlin noise, tiling systems and cellular basis is provided. The function of each technique and the resulting output they create are discussed to better understand their characteristics, benefits and relevance to the city generation problem. City generation is the creation of an urban area which necessitates the creation of buildings, situated along streets and arranged in appropriate patterns. Some research has already taken place into recreating road network patterns and generating buildings that can vary in function and architectural style. We will study the main body of existing research into procedural city generation and provide an overview of their implementations and a critique of their functionality and results. Finally we present areas in which further research into the generation of cities is required and outline our research goals for city generation