Contrasts in the basins of attraction of structurally identical iterative root finding methods

A numerical comparison is performed between three methods of third order with the same structure, namely BSC, Halley’s and Euler–Chebyshev’s methods. As the behavior of an iterative method applied to a nonlinear equation can be highly sensitive to the starting points, the numerical comparison is carried out, allowing for complex starting points and for complex roots, on the basins of attraction in the complex plane. Several examples of algebraic and transcendental equations are presented.

Improving The Robustness Of Interventional 4D Ultrasound Segmentation Through The Use Of Personalized Shape Priors

While fluoroscopy is still the most widely used imaging modality to guide cardiac interventions, the fusion of pre-operative Magnetic Resonance Imaging (MRI) with real-time intra-operative ultrasound (US) is rapidly gaining clinical acceptance as a viable, radiation-free alternative. In order to improve the detection of the left ventricular (LV) surface in 4D ultrasound, we propose to take advantage of the pre-operative MRI scans to extract a realistic geometrical model representing the patients cardiac anatomy. This could serve as prior information in the interventional setting, allowing to increase the accuracy of the anatomy extraction step in US data. We have made use of a real-time 3D segmentation framework used in the recent past to solve the LV segmentation problem in MR and US data independently and we take advantage of this common link to introduce the prior information as a soft penalty term in the ultrasound segmentation algorithm. We tested the proposed algorithm in a clinical dataset of 38 patients undergoing both MR and US scans. The introduction of the personalized shape prior improves the accuracy and robustness of the LV segmentation, as supported by the error reduction when compared to core lab manual segmentation of the same US sequences.