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**1 - 3**of**3**### 3D Curve Evolution Algorithm with Tangential Redistribution for a Fully Automatic Finding of an Ideal Camera Path in Virtual Colonoscopy

"... Abstract. In this paper we develop new method, based on 3D evolving curves, for finding the optimal trajectory of the camera in the virtual colonoscopy -the medical technology dealing with colon diagnoses by computer. The proposed method consists of three steps: 3D segmentation of the colon from CT ..."

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Abstract. In this paper we develop new method, based on 3D evolving curves, for finding the optimal trajectory of the camera in the virtual colonoscopy -the medical technology dealing with colon diagnoses by computer. The proposed method consists of three steps: 3D segmentation of the colon from CT images, finding an initial trajectory guess inside the segmented 3D subvolumes, and driving the initial 3D curve to its optimal position. To that goal, the new fast and stable 3D curve evolution algorithm is developed in which the initial curve is driven by the velocity field in the plane normal to the evolving curve, the evolution is regularized by curvature and accompanied by the suitable choice of tangential velocity. Thanks to the asymtotically uniform tangential redistribution of grid points, originally introduced in this paper for 3D evolving curves, and to the fast and stable semi-implicit scheme for solving our proposed intrinsic advection-diffusion PDE, we end up in fast and robust way with the smooth uniformly discretized 3D curve representing the ideal path of the camera in virtual colonoscopy.

### METHODS FOR SOLVING ADVECTION EQUATIONS

"... Abstract. We introduce a new class of methods for solving non-stationary advection equations. The new methods are based on finite volume space discretizations and a semi-implicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicit ..."

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Abstract. We introduce a new class of methods for solving non-stationary advection equations. The new methods are based on finite volume space discretizations and a semi-implicit discretization in time. Its basic idea is that outflow from a cell is treated explicitly while inflow is treated implicitly. This is natural, since we know what is outflowing from a cell at the old time step but we leave the method to resolve a system of equations determined by the inflows to a cell to obtain the solution values at the new time step. The matrix of the system in our inflow-implicit/outflow-explicit (I2OE) method is determined by the inflow fluxes which results in a M-matrix yielding favourable stability properties for the scheme. Since the explicit (outflow) part is not always dominated by the implicit (inflow) part and thus some oscillations can occur, we build a stabilization based on the upstream weighted averages with coefficients determined by the flux-corrected transport approach [2, 19] yielding high resolution versions, S1I2OE and S2I2OE, of the basic scheme. We prove that our new method is exact for any choice of a discrete time step on uniform rectangular grids in the case of constant velocity transport of quadratic functions in any dimension. We also show its formal second order accuracy in space and time for 1D advection problems with variable velocity. Although designed for non-divergence free velocity fields, we show that the basic I2OE scheme is locally mass