Date of Award

5-2013

Level of Access

Campus-Only Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics

Advisor

Andre Khalil

Second Committee Member

William Halteman

Third Committee Member

David Bradley

Abstract

The Wavelet-Transform Modulus Maxima (WTMM) method has been implemented into nearly all fields of applied sciences. In this adaption of the 2D WTMM method, the continuous wavelet transform is the mathematical microscope used to characterize the fractal geometry of clusters of microcalcifications (MC) in human breast tissue and to determine the roughness of the background tissue seen in mammograms. The WTMM method yields the so-called singularity spectrum, D(h), i.e. the fractal dimension D, of points having a Holder exponent of h. The MC are seen as Dirac singularities by the WTMM method, therefore having Holder exponent value of h = -1. This allows the MC with h ~ -1 to be abstracted from the background tissue which has h ~ 0.30 for fatty breast tissue and h ~ 0.65 for dense tissue. Thus, the WTMM method is used to perform a segmentation of the breast tumors based on the strength of the singularities composing the mammogram images, and to simultaneously quantify their fractal dimension. After analyzing several hundred images from a digital databank of mammograms with known radiologist diagnostics, the fractal dimensions of benign and malignant breast lesions are significantly different, with benign having an integer dimension corresponding to a non-invasive Euclidean object and cancer having a non-integer dimension, representing an invasive structure. In addition, the microenvironments characterized by the roughness of the tissue in which the lesions are embedded are different for benign and malignant tumors, and provides an insight into the onset and development of breast cancer.

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