PnS: a PerspectivenSpheres Algorithm for Laparoscope Calibration in Minimally Invasive Surgery Accurate endoscope localization is a key element for ComputerAssisted Laparoscopic Surgery (CALS). External localizers track the endoscope pose and reconstruct the camera geometry through a calibration matrix. PerspectivenPoints (PnP) methods estimate this matrix from a set of paired point coordinates, given by the simultaneous position of a spherical marker on the image (2D coordinates) and in the external localizer frame (3D coordinates). In this paper, the PnP calibration is restated as an iterative PerspectivenSpheres (PnS) problem: the spherical geometry of the marker is taken into account when computing the projection of its 3D center into the 2D image plane. Using calibration residuals as evaluation matrix, the proposed PnS provides better results compared to the StateoftheArt (SoA) PnP methods. Furthermore, results show that the calibration yields different estimates due to different camera poses in the working volume. This is assumed to result from a lack of absolute precision in the marker localization across the external localizer workspace.

An Efficient and Accurate Algorithm for the PerspecitvenPoint Problem In this paper, we address the problem of pose estimation from N 2D3D point correspondences, known as the PerspectivenPoint (PnP) problem. Although many solutions have been proposed, it is hard to optimize both computational complexity and accuracy at the same time. In this paper, we propose an accurate and simultaneously efficient solution to the PnP problem. Previous PnP algorithms generally involve two sets of unknowns including the depth of each pixel and the pose of the camera. Our formulation does not involve the depth of each pixel. By introducing some intermediate variables, this formulation leads to a fourth degree polynomial cost function with 3 unknowns that only involves the rotation. In contrast to previous works, we do not address this minimization problem by solving the firstorder optimality conditions using the offtheshelf Grobner basis method, as the Grobner basis method may encounter numeric problems. Instead, we present a method based on linear system null space analysis to provide a robust initial estimation for a Newton iteration. Experimental results demonstrate that our algorithm is comparable to the startoftheart algorithms in terms of accuracy, and the speed of our algorithm is among the fastest algorithms.