The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change

Retrieving patient orientation from previous images

The present invention relates to re-positioning in medical imaging. In order to facilitate the re-positioning of a patient in relation to an X-ray imaging system, it is proposed to provide (12) previously acquired first image data (14) of a patient in a first spatial position in relation to an X-ray image acquisition device. Current second image data (18) with the X- ray image acquisition device of the patient in a second spatial position in relation to the X-ray image acquisition device is acquired (16). One of the first or second image data is a 3D volume data of the patient, and the other one of the second or first image data is 2D or 3D image data of the patient. Further, the first image data and the second image data are registered (20). Next, a relative movement factor (24) between the first and the second spatial position is determined (22). Then, the patient and the X-ray image acquisition device in relation to each other are re-positioned (26) in a re-positioned relative spatial position based on the determined relative movement factor for movement compensation to achieve a predetermined matching degree of the re-positioned spatial position with the first spatial position for the acquisition of further image data.</p

Quantification of arterial flow using digital subtraction angiography

PURPOSE: In this paper, a method for the estimation of arterial hemodynamic flow from x-ray video densitometry data is proposed and validated using an in vitro setup. METHODS: The method is based on the acquisition of three-dimensional rotational angiography and digital subtraction angiography sequences. A modest contrast injection rate (between 1 and 4 ml/s) leads to a contrast density that is modulated by the cardiac cycle, which can be measured in the x-ray signal. An optical flow based approach is used to estimate the blood flow velocities from the cyclic phases in the x-ray signal. RESULTS: The authors have validated this method in vitro, and present three clinical cases. The in vitro experiments compared the x-ray video densitometry results with the gold standard delivered by a flow meter. Linear correlation analysis and regression fitting showed that the ideal slope of 1 and intercept of 0 were contained within the 95 percentile confidence interval. The results show that a frame rate higher than 50 Hz allows measuring flows in the range of 2 ml/s to 6 ml/s within an accuracy of 5%. CONCLUSIONS: The in vitro and clinical results indicate that it is feasible to estimate blood flow in routine interventional procedures. The availability of an x-ray based method for quantitative flow estimation is particularly clinically useful for intra-cranial applications, where other methods, such as ultrasound Doppler, are not available

Quantification of arterial flow using digital subtraction angiography

International audiencePURPOSE: In this paper, a method for the estimation of arterial hemodynamic flow from x-ray video densitometry data is proposed and validated using an in vitro setup. METHODS: The method is based on the acquisition of three-dimensional rotational angiography and digital subtraction angiography sequences. A modest contrast injection rate (between 1 and 4 ml/s) leads to a contrast density that is modulated by the cardiac cycle, which can be measured in the x-ray signal. An optical flow based approach is used to estimate the blood flow velocities from the cyclic phases in the x-ray signal. RESULTS: The authors have validated this method in vitro, and present three clinical cases. The in vitro experiments compared the x-ray video densitometry results with the gold standard delivered by a flow meter. Linear correlation analysis and regression fitting showed that the ideal slope of 1 and intercept of 0 were contained within the 95 percentile confidence interval. The results show that a frame rate higher than 50 Hz allows measuring flows in the range of 2 ml/s to 6 ml/s within an accuracy of 5%. CONCLUSIONS: The in vitro and clinical results indicate that it is feasible to estimate blood flow in routine interventional procedures. The availability of an x-ray based method for quantitative flow estimation is particularly clinically useful for intra-cranial applications, where other methods, such as ultrasound Doppler, are not available