An efficient approach for solving the straightness and the flatness problems at large number of data points

Abstract

The construction of the convex hull as a critical step of solving the straightness and the flatness errors needs a great amount of computation, especially, when the number of data point is large, that limits the computational efficiency. To establish an efficient algorithm to solve these problems, three theorems are developed in this paper to show that the straightness and the flatness errors can be obtained without the construction of the whole convex hull. Theorem 1 shows how to identify the redundant data points that gives an efficient way to reduce the unnecessary computations. The optimum criterion defined in Theorem 2 shows that the optimum solution can be obtained by a small number of data points if these points meet the criterion. Theorem 3 offers an easy way to identify the potential candidates of the solution holders, which are data points on the vertices of the convex hull. The efficiency of the proposed algorithm is validated at the end of this paper. The validation results show that the computational efficiency of solving the straightness and the flatness problems is improved significantly through the proposed algorithm, especially, when the number of data point is large.