An analytic approach to tensor scale with efficient computational solution and applications to medical imaging

Scale is a widely used notion in medical image analysis that evolved in the form of scale-space theory where the key idea is to represent and analyze an image at various resolutions. Recently, a notion of local morphometric scale referred to as "tensor scale" was introduced using an ellipsoidal model that yields a unified representation of structure size, orientation and anisotropy. In the previous work, tensor scale was described using a 2-D algorithmic approach and a precise analytic definition was missing. Also, with previous framework, 3-D application is not practical due to computational complexity. The overall aim of the Ph.D. research is to establish an analytic definition of tensor scale in n-dimensional (n-D) images, to develop an efficient computational solution for 2- and 3-D images and to investigate its role in various medical imaging applications including image interpolation, filtering, and segmentation. Firstly, an analytic definition of tensor scale for n-D images consisting of objects formed by pseudo-Riemannian partitioning manifolds has been formulated. Tensor scale captures contextual structural information which is useful in local structure-adaptive anisotropic parameter control and local structure description for object/image matching. Therefore, it is helpful in a wide range of medical imaging algorithms and applications. Secondly, an efficient computational solution of tensor scale for 2- and 3-D images has been developed. The algorithm has combined Euclidean distance transform and several novel differential geometric approaches. The accuracy of the algorithm has been verified on both geometric phantoms and real images compared to the theoretical results generated using brute-force method. Also, a matrix representation has been derived facilitating several operations including tensor field smoothing to capture larger contextual knowledge. Thirdly, an inter-slice interpolation algorithm using 2-D tensor scale information of adjacent slices has been developed to determine the interpolation line at each image location in a gray level image. Experimental results have established the superiority of the tensor scale based interpolation method as compared to existing interpolation algorithms. Fourthly, an anisotropic diffusion filtering algorithm based on tensor scale has been developed. The method made use of tensor scale to design the conductance function for diffusion process so that along structure diffusion is encouraged and boundary sharpness is preserved. The performance has been tested on phantoms and medical images at various noise levels and the results were quantitatively compared with conventional gradient and structure tensor based algorithms. The experimental results formed are quite encouraging. Also, a tensor scale based n-linear interpolation method has been developed where the weights of neighbors were locally tuned based on local structure size and orientation. The method has been applied on several phantom and real images and the performance has been evaluated in comparison with standard linear interpolation and windowed Sinc interpolation methods. Experimental results have shown that the method helps to generate more precise structure boundaries without causing ringing artifacts. Finally, a new anisotropic constrained region growing method locally controlled by tensor scale has been developed for vessel segmentation that encourages axial region growing while arresting cross-structure leaking. The method has been successfully applied on several non-contrast pulmonary CT images. The accuracy of the new method has been evaluated using manually selection and the results found are very promising.