27. Fractal Models for Image Synthesis, Compression, and Analysis

Full Day / Intermediate

This course consists of two sections: linear fractals (morning sessions) and random fractals including multi-fractals (afternoon lectures). Linear fractals hold the most promise for fractal modeling in computer graphics and for fractal image coding, and we have devoted the entire morning to their study. Random fractal models were the first and remain the most prominent fractal models for computer graphics. Important recent applications of random fractals can be found in image analysis where concepts from fractal dimensions and multi-fractals lead to implementations e.g. in computer assisted medical diagnosis.

Who Should Attend
Students and professionals who want a comprehensive view of fractals in computer graphics, with an emphasis on recent results.

Dietmar Saupe
Universität Freiburg

John Hart
Washington State University

Richard Voss
Florida Atlantic University

F. Kenton Musgrave
The George Washington University

Carl J.G. Evertz
Universität Bremen

Przemyslaw Prusinkiewicz
University of Calgary


Linear Fractal Models

8:30 am: Recurrent Iterated Function Systems - Hart

Dr. Hart introduces iterated function systems, recurrent iterated function systems, and fractal interpolation functions, providing a theoretical basis for the rest of the morning's talks.

9:00 am: Fractal Image Compression - Saupe

Dr. Saupe will walk through the algorithm behind fractal image compression aided by an interactive computer demo whichg exemplifies the concept in the more intuitive one-dimensional case. Techniques from the prior talk will show that the basic data structure behind the compression algorithm is a particular iterated function system. Enhancements to the basic algorithms will be given, leading to state-of-the-art fractal compressors, and solutions for problem of the time-complexity in the encoder will be presented.

10:00 am: Break

10:15 am: Modeling with L-systems - Prusinkiewicz

Dr. Prusinkiewicz will outline the fundamentals of the L-system formalism, and present some extensions motivated by graphical applications, such as stochastic and environmentally-sensitive L-systems. In addition to the "standard" turtle interpretation, several alternative interpretations of L-systems will be introduced and compared. These comparisons shed new light on the relationships between different methods of generating fractals, and facilitate the understanding and modeling of complex plant structures. The presentation will be illustrated using numerous examples, both mathematical and botanical.

11:15 am: Recurrent Modeling - Hart

Dr. Hart will wrap up the morning's presentations on linear fractals with an overview of the Recurrent Modeling Project. The goal of the project is to elevate fractal representations to the level of sophistications other smooth-surface representations enjoy in computer graphics and computer-aided geometric design. The presentation will explore subtle differences between the RIFS, fractal image compression and L-systems. It will also demonstrate new linear fractal modeling tools such as blending and direct manipulation. The presentation will conclude with a new technique for detecting and extracting morphological self-similarity from images and objects.

12:00 noon: Break

Random and Multi-Fractals: Synthesis and Analysis

1:30 pm: Random Fractals - Voss

Dr. Voss will introduce the theory of random fractals: fractal dimension, self-similarity, self-affinity, and fractional Brownian motion. The theoretical concepts will be visualized with applications such as self-affine landscapes, musical note sequences, and economic time series.

2:30 pm: Multi-Fractals - Voss

Multi-fractals are one of the recent advanced fractal models. They extend the original concept of fractal dimension, developed for "sets" (black and white images) to "measures" (grey scale images) that are characterized by a distribution of dimensions with rich modeling and characterization possibilities. The foundations are explained and illustrated with examples from the analysis of ancient Chinese paintings and x-ray mammograms.

3:00 pm: Break

3:15 pm: Fractal Image Analysis - Evertsz

Dr. Evertsz will demonstrate how methods based on fractals are used in medical imaging and image analysis. The concepts will be highlighted by examples of detection of microcalcifications for the diagnosis of breast cancer, the reconstruction and segmentation of vascular systems in the liver, a model for the growth of vascular systems, the quantization of liver fattiness based on sonography, and the quantization of osteoporosis. All applications are chosen from ongoing research at MeVis (Center for Medical Diagnostic Systems at the University of Bremen) headed by the lecturer.

4:15 pm: Multi-Fractal Terrain Modeling - Musgrave

Dr. Musgrave will demonstrate random fractals and multi-fractals in the context of his realistic and artistic fractal models of natural landscapes and planets. Moreover, fractional Brownian motion will be extended to develop a theory of multi-fractals and their application in fractal modeling for natural scene synthesis.

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