Piecewise Polynomial Monotonic Interpolation of 2D Gridded Data

Abstract

A method for interpolating monotone increasing 2D scalar data with a monotone piecewise cubic C¹-continuous surface is presented. Monotonicity is a sufficient condition for a function to be free of critical points inside its domain. The standard axial monotonicity for tensor-product surfaces is however too restrictive. We therefore introduce a more relaxed monotonicity constraint. We derive sufficient conditions on the partial derivatives of the interpolating function to ensure its monotonicity. We then develop two algorithms to effectively construct a monotone C¹ surface composed of cubic triangular Bézier surfaces interpolating a monotone gridded data set. Our method enables to interpolate given topological data such as minima, maxima and saddle points at the corners of a rectangular domain without adding spurious extrema inside the function domain. Numerical examples are given to illustrate the performance of the algorithm.

Appendix—Sibson Split Interpolant

Let D be a rectangular domain in $\mathbb{R}^{2}$, regularly subdivided into rectangles $D_{\mathit{ij}} = [x_{i},x_{i+1}] \times [y_{j},y_{j+1}]$, $1 \leq i < n_{x},1 \leq j < n_{y}$ and the following ordinates z _ij and gradients $z_{\mathit{ij}}^{x},z_{\mathit{ij}}^{y}$ given at the grid points. Let $h^{x} = x_{i+1} - x_{i},h^{y} = y_{j+1} - y_{j}$. Each rectangle is splitted into four sub-triangles by drawing both diagonals.

The Sibson split (cf. [7]) is a cubic C¹-continuous function $f: D \rightarrow \mathbb{R}$ interpolating the input data with

$$\displaystyle{f(x_{i},y_{i}) = z_{\mathit{ij}},\qquad f_{x}(x_{i},y_{i}) = z_{\mathit{ij}}^{x},\qquad f_{ y}(x_{i},y_{i}) = z_{\mathit{ij}}^{y},}$$

where each patch defined on D _ij is composed of four cubic polynomials with in total 25 Bézier coefficients (see Fig. 4), computed uniquely as follows:

$$\displaystyle\begin{array}{rcl} c_{1}& =& z_{\mathit{ij}}\quad c_{2} = z_{i+1,j}\quad c_{3} = z_{i+1,j+1} {}\\ c_{4}& =& z_{i,j+1}\quad c_{5} = c_{1} + \frac{h^{x}} {3} z_{\mathit{ij}}^{x}\quad c_{ 6} = c_{2} -\frac{h^{x}} {3} z_{i+1,j}^{x} {}\\ c_{9}& =& c_{3} -\frac{h^{x}} {3} z_{i+1,j+1}^{x}\quad c_{ 10} = c_{4} + \frac{h^{x}} {3} z_{i,j+1}^{x}\quad c_{ 12} = c_{1} + \frac{h^{y}} {3} z_{\mathit{ij}}^{y} {}\\ c_{7}& =& c_{2} + \frac{h^{y}} {3} z_{i+1,j}^{y}\quad c_{ 8} = c_{3} -\frac{h^{y}} {3} z_{i+1,j+1}^{y}\quad c_{ 11} = c_{4} -\frac{h^{y}} {3} z_{i,j+1}^{x} {}\\ c_{13}& =& \frac{1} {2}(c_{5} + c_{12})\quad c_{14} = \frac{1} {2}(c_{6} + c_{7})\quad c_{15} = \frac{1} {2}(c_{8} + c_{9}) {}\\ c_{16}& =& \frac{1} {2}(c_{11} + c_{10})\quad c_{21} = \frac{1} {2}(c_{20} + c_{17})\quad c_{22} = \frac{1} {2}(c_{17} + c_{18}) {}\\ c_{23}& =& \frac{1} {2}(c_{18} + c_{19})\quad c_{24} = \frac{1} {2}(c_{19} + c_{20})\quad c_{25} = \frac{1} {2}(c_{21} + c_{23}) = \frac{1} {2}(c_{22} + c_{24}). {}\\ \end{array}$$