Ad-nilpotent ideals and The Shi arrangement

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the $I$-deleted Shi arrangement $\texttt{Shi}(I)$ naturally emerges. This arrangement interpolates between the Coxeter arrangement $\texttt{Cox}$ and the Shi arrangement $\texttt{Shi}$, and breaks the symmetry of $\texttt{Shi}$ in a certain symmetrical way. Among other things, we determine the characteristic polynomial $\chi(\texttt{Shi}(I), t)$ of $\texttt{Shi}(I)$ explicitly for $A_{n-1}$ and $C_n$. More generally, let $\texttt{Shi}(G)$ be an arbitrary arrangement between $\texttt{Cox}$ and $\texttt{Shi}$. Armstrong and Rhoades recently gave a formula for $\chi(\texttt{Shi}(G), t)$ for $A_{n-1}$. Inspired by their result, we obtain formulae for $\chi(\texttt{Shi}(G), t)$ for $B_n$, $C_n$ and $D_n$.Comment: The third version, quasi-antichains are shown to be in bijection with elements of L(Cox). arXiv admin note: text overlap with arXiv:1009.1655 by other author

Coronary Artery Segmentation and Motion Modelling

Conventional coronary artery bypass surgery requires invasive sternotomy and the use of a cardiopulmonary bypass, which leads to long recovery period and has high infectious potential. Totally endoscopic coronary artery bypass (TECAB) surgery based on image guided robotic surgical approaches have been developed to allow the clinicians to conduct the bypass surgery off-pump with only three pin holes incisions in the chest cavity, through which two robotic arms and one stereo endoscopic camera are inserted. However, the restricted field of view of the stereo endoscopic images leads to possible vessel misidentification and coronary artery mis-localization. This results in 20-30% conversion rates from TECAB surgery to the conventional approach. We have constructed patient-specific 3D + time coronary artery and left ventricle motion models from preoperative 4D Computed Tomography Angiography (CTA) scans. Through temporally and spatially aligning this model with the intraoperative endoscopic views of the patient's beating heart, this work assists the surgeon to identify and locate the correct coronaries during the TECAB precedures. Thus this work has the prospect of reducing the conversion rate from TECAB to conventional coronary bypass procedures. This thesis mainly focus on designing segmentation and motion tracking methods of the coronary arteries in order to build pre-operative patient-specific motion models. Various vessel centreline extraction and lumen segmentation algorithms are presented, including intensity based approaches, geometric model matching method and morphology-based method. A probabilistic atlas of the coronary arteries is formed from a group of subjects to facilitate the vascular segmentation and registration procedures. Non-rigid registration framework based on a free-form deformation model and multi-level multi-channel large deformation diffeomorphic metric mapping are proposed to track the coronary motion. The methods are applied to 4D CTA images acquired from various groups of patients and quantitatively evaluated