Dr. Muhammad Maqbool
MAT 121, Research Project 2
January 29, 2008
Fractal Geometry A fractal is generally “a rough or fragmented geometric shape that can be subdivided into parts.” One of the ways that fractal geometry is used is in the area of medicine. In the past few years fractal analysis techniques have gained increasing attention in signal and image processing, especially in medical sciences such as Pathology. Fractal analysis has found applications in the detection of coding regions in DNA and measurement of the space-filling properties of tumors, blood vessels and neurons. Fractal concepts have also been usefully incorporated into models of biological processes, including cell growth, blood vessel growth, periodontal disease and viral infections. Other very interesting applications are founded in medical imaging Fractal analysis is widely used in image processing, both in characterizing shapes of objects and in assessing texture. Breast masses present shape and texture characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses. The fractal dimension of the contour of a mass may be computed either directly from the two dimensional contour or from one-dimensional signatures derived from the contour. Other ways that fractal geometry is use is in biology with different applications and techniques use to classify and distinguish various types of cells. The use of fractal dimension can distinguish between malignant and benign cells in cancer patients. It is also helpful in the heart beat interval after a patient has had a heart transplant. Nature is another way we can find fractal applications. For example, select a tree and pick a particular branch and study it closely. Choose a bundle of leaves on that branch. To chaologists, all three of the objects described the tree, the branch, and the leaves are identical. To many, the word chaos suggests randomness, unpredictability and perhaps even messiness. Chaos is actually very organized and follows certain patterns. The problem arises in finding these elusive and intricate patterns. One purpose of studying chaos through fractals is to predict patterns in dynamical systems that on the surface seem unpredictable. A system is a set of things, an area of study a set of equations is a system, as well as more tangible things such as cloud formations, the changing weather, the movement of water currents, or animal migration patterns. Weather is a favorite example for many people. Forecasts are never totally accurate, and long-term forecasts, even for one week, can be totally wrong. This is due to minor disturbances in airflow, solar heating, etc. Each disturbance may be minor, but the change it creates will increase geometrically with time. Soon, the weather will be far different than what was expected. With fractal geometry we can visually model much of what we witness in nature, the most recognized being coastlines and mountains. Fractals are used to model soil erosion and to analyze seismic patterns as well. Seeing that so many facets of Mother Nature exhibit fractal properties, maybe the whole world around us is a fractal after all.