The Mathematical Foundation of Fractal Image Compression

We will discuss the search for more efficient methods of image 
compression. The objective in image compression is to efficiently produce 
graphics files stored with the least amount of data without compromising 
image quality. Efficient data storage is a priority for any large database 
of images. Intelligent image storage is imperative for medical 
applications such as digital X-Ray, CT, MR, mammography, PET, ultrasound, 
and angiography. In addition, the FBI fingerprint database, other forensic 
imaging applications, digital aerial photos, and high-resolution satellite 
systems all utilize image compression. A recent approach to image 
compression, pioneered by Michael Barnsley, is to use the similarities on 
different scales throughout an image to assist in compression. Fractal 
Image Compression enables an incredible amount of data to be stored in 
highly compressed data files. We will explore the mathematical theory 
which supports fractal image compression. One of the most important 
foundations for fractal image compression is the concept of Iterated 
Function Systems (IFS). Through IFS we are able to systematically 
reproduce fractals which occur in nature. With the theory that will be 
presented, we will explore the development of an IFS and how one can apply 
IFS to obtain fractal image compression. Future goals of this project 
would be to continue research to develop a more efficient fractal image 
compression method that can be applied to images used in a variety of 
industries. Another goal would be to combine other forms of image 
compression, such as wavelet compression methods, with fractal compression 
to further improve the compression rates.