An interactive surgical simulation system needs to meet three main requirements, speed, accuracy, and stability. In this paper, we present a stable and accurate method for animating mass-spring systems in real time. An integration scheme derived from explicit integration is used to obtain interactive realistic animation for a multiobject environment. We explore a predictor-corrector approach by correcting the estimation of the explicit integration in a poststep process. We introduce novel constraints on positions into the mass-spring model (MSM) to model the nonlinearity and preserve volume for the realistic simulation of the incompressibility. We verify the proposed MSM by comparing its deformations with the reference deformations of the nonlinear finite-element method. Moreover, experiments on porcine organs are designed for the evaluation of the multiobject deformation. Using a pair of freshly harvested porcine liver and gallbladder, the real organ deformations are acquired by computed tomography and used as the reference ground truth. Compared to the porcine model, our model achieves a 1.502 mm mean absolute error measured at landmark locations for cases with small deformation (the largest deformation is 49.109 mm) and a 3.639 mm mean absolute error for cases with large deformation (the largest deformation is 83.137 mm). The changes of volume for the two deformations are limited to 0.030% and 0.057%, respectively. Finally, an implementation in a virtual reality environment for laparoscopic cholecystectomy demonstrates that our model is capable to simulate large deformation and preserve volume in real-time calculations.
Modeling energy-minimizing curves have many applications and are a basic problem of Geometric Modeling. In this paper, we propose the method for geometric design of energy-minimizing Bézier curves. Firstly, the necessary and sufficient condition on the control points for Bézier curves to have minimal internal energy is derived. Based on this condition, we propose the geometric constructions of three kinds of Bézier curves with minimal internal energy including stretch energy, strain energy and jerk energy. Given some control points, the other control points can be determined as the linear combination of the given control points. We compare the three kinds of energy-minimizing Bézier curves via curvature combs and curvature plots, and present the collinear properties of quartic energy-minimizing Bézier curves. We also compare the proposed method with previous methods on efficiency and accuracy. Finally, several applications of the curve generation technique, such as curve interpolation with geometric constraints and modeling of circle-like curves are discussed.
This paper is concerned with the problem of constructing an aesthetically pleasing triangular mesh with a given closed polygonal contour in three dimensional space as boundary. Triangular meshes of minimal area from all triangular meshes with the prescribed boundary are suggested as the candidates for this problem. An iterative algorithm of constructing such a triangular mesh from a given polygonal boundary is presented. Experimental examples show that the proposed algorithm is reliable and effective. Some related theoretical issues, possible extensions and applications are also discussed.
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