Smoothing of Mesh Data Using Fourth Divided Difference

We propose a smoothing method of mesh data for the reverse engineering that generates a mathematical model in the computer from measured data of a physical model. We assume that measured data are given as a regular mesh. We define the absolute value of fourth divided difference along u and v parametric lines for input points, and multiply it by a scale factor. Then we get local fairness criterion as the summation for both directions. And we define global fairness criterion as the summation of all the mesh points. Our smoothing algorithm is the local and iterative procedure. We find and modify the point with the maximum local criterion, and repeat this procedure until the global criterion does not decrease. To modify the mesh data, we change central 9 points using 16 points from 5×5 mesh points at the same time. For smoothing the points located around boundary, we modify the points that are located in the affected area of the point with the maximum local criterion respectively, and select the point which has the minimum global criterion. For smoothing mesh data which have lack areas, we also modify them similaly. After smoothing mesh data, we calculate surface equations for the mesh using Newton's polynomial whose fourth divided difference is locally zero. This generates a C2 surface.

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