Nonnegativity Preserving Interpolation by C¹ Bivariate Rational Spline Surface

1. Introduction

Interpolation to the scientific data is of great significance in the area of computer-aided geometric design. Particularly, there is often some property inherent in the data which one wishes to preserve when the interpolant is visualized. One useful shape property is nonnegativity: one may have all data values nonnegative and seek an interpolant that is everywhere nonnegative. In this paper, we are concerned with the nonnegativity preserving bivariate interpolation to data on a rectangular grid.

Several kinds of surfaces are concerned with nonnegativity preserving interpolation on rectangular grid. For example, C¹ biquadratic splines on a refined rectangular grid have been considered in paper [1]. In paper [2, 3], Brodlie et al. followed the same approach (but for bicubic interpolation) of maintaining nonnegativity by modifying estimated slopes at data points. In [4], the interpolant is piecewise an average of two blending surfaces. In [5], C¹ interpolating surface is constructed piecewise as a convex combination of two bicubic Bézier patches with the same set of boundary Bézier ordinates. Sufficient nonnegativity conditions on the Bézier ordinates are derived to ensure the nonnegativity of a bicubic Bézier patch. The Bézier ordinates are modified locally to fulfill the sufficient nonnegativity conditions.

The surface we consider here is rational spline. Rational spline with parameters has been considered in recent years. Those kinds of interpolation splines have simple and explicit mathematical representation. In paper [6], a bivariate rational interpolation is constructed using both function values and partial derivatives of the function being interpolated as the interpolation data. The convexity control method and point control method have been studied in [7, 8]. In paper [9], a rational cubic function is extended to a rational bicubic partially blended functions (Coons-patches) and the constraints on parameters are derived to visualize the shape of nonnegative surface data. In this paper, we consider the rational spline from different points of view. The sufficient nonnegativity conditions of the bicubic Bézier patch are introduced to consider this problem. We get the sufficient nonnegativity conditions of the rational spline. Gradients at data sites are modified if necessary to ensure that the nonnegativity conditions are fulfilled. It is designed in such a way that no additional points need to be supplied and the sufficient conditions are simple and explicit.

The paper is organized as follows. In Section 2, the bivariate rational spline is introduced. Section 3 deals with the smoothness of the interpolating surface. In Section 4, the sufficient nonnegativity conditions of the rational spline are derived and a local scheme for C¹ nonnegativity preserving interpolation is described. We conclude in Section 5 by illustrating the method with some graphical examples.

2. Rational Interpolation Spline

In this section, the univariate rational cubic spline is introduced which was developed by Hussain and Sarfraz [10]. We extend it to a bivariate rational interpolation spline function.

Let Ω : [a, b; c, d] be a planar region, {(x_i, y_j, f_i,j, ∂f_i,j/∂x, ∂f_i,j/∂y),   (i = 1,2, …, n; j = 1,2, …, m)} a given set of data points, where a = x₁ < x₂ < ···<x_n = b and c = y₁ < y₂ < ···<y_m = d are the knot sequences. And let f_i,j, ∂f_i,j/∂x, ∂f_i,j/∂y represent f(x_i, y_j), ∂f(x_i, y_j)/∂x, ∂f(x_i, y_j)/∂y, respectively. ∂f(x_i, y_j)/∂x and ∂f(x_i, y_j)/∂y are not always given; they can be estimated by the method in [11].

It is obvious that d_i,s(x, y_s) satisfies d_i,s(x_r, y_s) = ∂f_r,s/∂y,   r = i, i + 1,   s = j, j + 1. And the bivariate rational function S_i,j(x, y) satisfies the interpolation conditions S_i,j(x_r, y_s) = f(x_r, y_s), ∂S_i,j(x_r, y_s)/  ∂x = ∂f_r,s/∂x, ∂S_i,j(x_r, y_j)/∂y = ∂f_r,s/∂y, r = i, i + 1 and s = j, j + 1.

3. The Smoothing Conditions

In this section, the smoothing conditions of the rational spline S_i,j(x, y) defined by (2.4) are derived. The rational interpolating function $$ defined by (2.1) is a piecewise Hermite interpolant, and it has continuous first-order derivative when x ∈ [x₁, x_n]. So the bivariate interpolating function S_i,j defined by (2.4) has continuous first-order partial derivatives ∂S_i,j(x, y)/∂x and ∂S_i,j(x, y)/∂y in the interpolating region [x₁, x_n; y₁, y_m] except ∂S_i,j(x, y)/∂x at the points (x_i, y),   i = 1,2, …, n − 1 for every y ∈ [y_j, y_j+1], j = 1,2, …, m − 1. So it is sufficient for S_i,j(x, y) ∈ C¹ in the whole interpolating region [x₁, x_n; y₁, y_m] if ∂S_i,j(x_i+, y)/∂x = ∂S_i,j  (x_i−, y)/∂x holds. This leads to the following theorem.

The interpolating scheme above begins in x-direction first. If the interpolation begins with y-direction first, we would get a restriction on the data in the y-direction.

4. Construction of Nonnegativity Preserving Interpolating Surface

In this section we first introduce the following theorem in paper [5], which is the basis of our discussion below.

It can be seen that $$ is positive for α_i,j > 0 and β_i,j > 0. The function t_i,j is a bicubic Bézier patch. It is nonnegative if the Bézier ordinates of it satisfy the conditions in Theorem 4.1.

Bézier ordinates of t_i,j need not to satisfy the nonnegative conditions. To ensure this, we shall impose upon these Bézier ordinates the conditions b_p,q ≥ −ι/3a  (p, q = 0,1, 2,3) according to Theorem 4.1, where ι = min {f_i,j, f_i,j+1, f_i+1,j, f_i+1,j+1} and a is determined by (4.1). This can be achieved by modifying if necessary the gradients at vertices V_i,j = (x_i, y_j). The modification of derivatives ∂f_i,j/∂x and ∂f_i,j/∂y at a vertex is performed by scaling each of them with a positive factor α< 1. The scaling factor α is obtained by taking into account all the rectangular patches sharing that vertex. For the four rectangles which share vertex V_i,j, denote vertex V_i,j as O and its adjacent vertices as A, B, C, D, E, F, G, H, respectively (see Figure 1).

Consider rectangle 1 and the lower bound −ι₁/3a₁ where ι₁ = min {S(O), S(A), S(B), S(C)} and a₁ is obtained by solving (4.1) in Theorem 4.1. Scalar $$ is defined as follows. If μ_i,jf_i,j + l_j(∂f_i,j/∂y)≥−ι₁/3a₁, then $$. Otherwise, $$ is defined by the equation $$. Similarly for the scalar $$, if α_i,jf_i,j + h_i(∂f_i,j/∂x)≥−ι₁/3a₁, then $$; otherwise $$ is given by the equation $$. For the scalar $$, if μ_i,jα_i,jf_i,j + μ_i,jh_i(∂f_i,j/∂x) + l_j(∂f_i,j/∂y)≥−ι₁/3a₁, then $$; otherwise $$ is determined by the equation $$. Then define $$. By using the same argument above, we get α₂₁ and α₂₂, α₃₁ and α₃₂, α₄₁ and α₄₂ for rectangles 2, 3, 4, respectively. In order for all the Bézier ordinates adjacent to O to fulfill the positivity preserving conditions stated in Theorem 4.1, set α_O1 = min {α₁₁, α₂₁, α₃₁, α₄₁}, α_O2 = min {α₁₂, α₂₂, α₃₂, α₄₂}.

If α_O1 < 1 (or α_O2 < 1), the x (or y) partial derivatives at O are redefined as α_O1(∂f_O/∂x) and α_O2(∂f_O/∂y). Then the Bézier ordinates adjacent to O in each rectangle are redetermined. The above process is repeated at all the nodes V_i,j.

For the boundary node O, which belongs to one or two rectangles of the rectangular grid, α_O is defined in a similar way. The only difference is that we consider only one or two rectangles instead of four.

Now all the Bézier ordinates are determined, so t_i,j is nonnegative. Thus we can get the C¹ rational interpolant S, which is nonnegative.

5. Numerical Examples

We shall illustrate our discussion with the following examples.

6. Conclusion

In this paper, we construct a nonnegativity preserving interpolant of data on rectangular grids by using the rational spline. The sufficient nonnegativity conditions are derived, which are simple and explicit. And the experimental results illustrate the feasibility and validity of our method.