Partial differential equations for medical image segmentation

This study is concerned with image segmentation techniques using mathematical models based on elastic curves or surfaces defined within an image domain that can move under the influence of a defined energy. These active contour models use internal and external forces generated from curves or surfaces in 2D and 3D image data. The algorithms that measure these energies must cope with non-homogeneous objects and regions, low contrast boundaries and image noise. It investigates level sets, which employ an energy formulation defined by partial differential equations (PDEs), that are sensitive to weak boundaries yet are robust to noise whilst maintaining computational stability. The methodology is evaluated using medical imagery, which commonly suffer from high levels of noise, blur and exhibit weak boundaries between different types of adjacent tissue. An energy based on PDEs has been used to evolve an image contour from an initial guess using image forces derived from region properties to drive the search to locate the boundaries of the desired objects that includes the maximum and minimum curvature function to enable length shortening in the curve evolution. It is applied to both 2D and 3D CTA datasets for the segmentation of abdominal and thoracic aortic aneurysm (AAA&TAA). For some image data the methodology can be initialised automatically using a contour detected after intensity thresholding. Non-homogeneous regions require a manual initialisation that crosses the boundary between the aorta and thrombus. Sussman's re-initialization has been used in the 3D algorithm to maintain stability in the evolving boundary, as a consequence of the re-formulation from the continuous to the discrete domain. A hybrid method is developed that combines a novel approach using region information (i.e. intensities inside and outside the object) and edge information, computed using a diffusion-based approach integrated into a level set formulation, to guide the initial curve to the object boundary by finding strong edges with local minima. Boundary information supports finding a local minimum length curve on evaluation and only examines data on the contour. Using Green's theorem, region information is be used to address the boundary leakage problem, as it minimizes the energy related to the whole image data and the moving curve is stopped by strong gradients on the borders of objects. Finally, a Gabor filter has been integrated into the hybrid algorithm to enhance the image and support the detection of textured regions of interest. The method is evaluated on both synthetic and real image data and compared with the region-based methods of Chan-Vese and Li et al.