Much research was done on the treatment of cancer. It is a fact that there are many more chances of recovery with early detection of malignant tumors.
Abstract
Much research was done on the treatment of cancer. It is a fact that there are many more chances of recovery with early detection of malignant tumors. CT and MRI are examples of noninvasive tests to identify tumors. In these tests there is great importance to guide the doctor or technician to regions with similar properties to malignant tumors.
In this project the implementation of Ricci curvature equation as formulated by Forman was examined and an algorithm that will use them to detect cancerous tumors in the human body was proposed. The formulas used in the work are based on calculation of the Ricci curvature which is tailored to a discrete world. Curvature was tested in third-order and second-order three-dimensional space. The effectiveness of these curvatures was tested to detect edges of organs and segmentation, the measuring technique was also tested to retrieve the curvature of surfaces.
The problem and basic approach
The first step in the process of the project is the investigation of the general algorithm of Ricci curvature as formulated by Forman with the hope that this investigation will benefit the continued work in the field.
In particular, two different curvatures were tested: third-order and second order in a three – dimensional space.
The effectiveness of these curvatures were tested for segmentation (separation of the organ environment) and in addition the measurement of surface deformations (cancerous regions are characterized by hight surface curvature).
The project is based on two main premises:
- Each segment contains relatively uniform curvature and therefore we can identify transition between segments using 3-dimensional curvature
- The topology of cancers is characterized by a perimeter that is of higher curvature than its surroundings
Solution Flowchart
Results on Stanford Bunny
Conclusions
Curvature calculation of the Foreman Ricci formulas yields reasonable results in a three-dimensional space. In the first part of the project, we examined the curvature of three-dimensional organs, marking success in definition of of various organs edges, but these surfaces were not continuous enough to preform segmentation for medical images (noisy environment).
The second part of the project tested Ricci curvature of two-dimensional surfaces in a 3D space. Satisfactory results were obtained for the independent topographic weights. However, the triangulation process has a disadvantage because it takes precious computer resources and information is lost in the process.
This initial work provides a proof of concept for Forman’s Ricci curvature in a 3D space.