Construction of C1 Rational Bi-Quartic Spline With Positivity-Preserving Interpolation: Numerical Results and Analysis

From the observed datasets, we should be able to produce curve surfaces that have the same characteristics as the original datasets. For instance, if the given data are positive, then the resulting curve or surface must be positive on entire given intervals, i.e., everywhere. In this study, a new partial blended rational bi-quartic spline with C1 continuity is constructed through the partially blended scheme. This rational spline is defined on four corners of the rectangular meshes. The sufficient condition for the positivity of rational bi-quartic spline is derived on four boundary curve networks. There are eight free parameters that can be used for shape modification. The first-order partial derivatives are estimated by using numerical techniques. We also show that the proposed scheme is local quadratic reproducing such that it can exactly reproduce the quadratic surface. We test the proposed scheme to interpolate various types of positive surface data. Based on statistical indicators such as the root mean square error (RMSE) and coefficient of determination (R2), we found that the proposed scheme is on par with some established schemes. In fact, it requires less CPU times (in seconds) to generate the interpolating surface on rectangular meshes. Furthermore, by combining the statistical indicators’ result and graphically visualizing the test functions, the proposed method has the capability to reconstruct very comparable smoothing interpolating positive surfaces compared to some existing schemes. This finding is significant in producing a better interpolating surface for computer graphics applications since the proposed scheme has a smaller error compared with existing schemes.

Reconstruction of Curve Networks from Unorganized Spatial Points

Curve network reconstruction from a set of unorganized points is an important problem in reverse engineering and computer graphics. In this paper, we propose an automatic method to extract curve segments and reconstruct curve networks from unorganized spatial points. Our proposed method divides reconstruction of curve networks into two steps: 1) detecting nodes of curve segments and 2) reconstructing curve segments. For detection of nodes of curve segments, we present a principal component analysis-based algorithm to obtain candidate nodes from unorganized spatial points and a Euclidean distance-based iterative algorithm to remove peripheral nodes and find the actual nodes. For reconstruction of curve segments, we propose an extraction algorithm to obtain the points on each of curve segments. We present quite a number of examples which use our proposed method to reconstruct curve networks from unorganized spatial points. The results demonstrate the effectiveness of our proposed method and its advantages of good automation and high reconstruction efficiency.