In this letter, we present a novel method for reconstructing continuous data field from scattered point data, which leads to a more characteristic visualization result by volume rendering. The gradient distribution of scattered point data is analyzed for local feature investigation via singular-value decomposition. A data-adaptive ellipsoidal shaped function is constructed as the penalty function to evaluate point weight coefficient in MLS approximation. The experimental results show that the proposed method can reduce the reconstruction error and get a visualization with better feature discrimination.

Assume that 2m red points and 2n blue points are given on the lattice Z2 in the plane R2. We show that if they are in general position, that is, if at most one point lies on each vertical line and horizontal line, then there exists a rectangular cut that bisects both red points and blue points. Moreover, if they are not in general position, namely if some vertical and horizontal lines may contain more than one point, then there exists a semi-rectangular cut that bisects both red points and blue points. We also show that these results are best possible in some sense. Moreover, our proof gives O(N log N), N=2m+2n, time algorithm for finding the desired cut.