Automatic liver vessel segmentation using 3D region growing and hybrid active contour model

This paper proposes a new automatic method for liver vessel segmentation by exploiting intensity and shape constraints of 3D vessels. The core of the proposed method is to apply two different strategies: 3D region growing facilitated by bi-Gaussian filter for thin vessel segmentation, and hybrid active contour model combined with K-means clustering for thick vessel segmentation. They are then integrated to generate final segmentation results. The proposed method is validated on abdominal computed tomography angiography (CTA) images, and obtains an average accuracy, sensitivity, specificity, Dice, Jaccard, and RMSD of 98.2%, 68.3%, 99.2%, 73.0%, 66.1%, and 2.56 mm, respectively. Experimental results show that our method is capable of segmenting complex liver vessels with more continuous and complete thin vessel details, and outperforms several existing 3D vessel segmentation algorithms

Constant Approximation for kk-Median and kk-Means with Outliers via Iterative Rounding

In this paper, we present a new iterative rounding framework for many clustering problems. Using this, we obtain an $(\alpha_1 + \epsilon \leq 7.081 + \epsilon)$-approximation algorithm for $k$-median with outliers, greatly improving upon the large implicit constant approximation ratio of Chen [Chen, SODA 2018]. For $k$-means with outliers, we give an $(\alpha_2+\epsilon \leq 53.002 + \epsilon)$-approximation, which is the first $O(1)$-approximation for this problem. The iterative algorithm framework is very versatile; we show how it can be used to give $\alpha_1$- and $(\alpha_1 + \epsilon)$-approximation algorithms for matroid and knapsack median problems respectively, improving upon the previous best approximations ratios of $8$ [Swamy, ACM Trans. Algorithms] and $17.46$ [Byrka et al, ESA 2015]. The natural LP relaxation for the $k$-median/$k$-means with outliers problem has an unbounded integrality gap. In spite of this negative result, our iterative rounding framework shows that we can round an LP solution to an almost-integral solution of small cost, in which we have at most two fractionally open facilities. Thus, the LP integrality gap arises due to the gap between almost-integral and fully-integral solutions. Then, using a pre-processing procedure, we show how to convert an almost-integral solution to a fully-integral solution losing only a constant-factor in the approximation ratio. By further using a sparsification technique, the additive factor loss incurred by the conversion can be reduced to any $\epsilon > 0$