Scattered data interpolation is a fundamental problem in approximation theory and CAGD. It finds applications in a variety of fields such as automotive, aircraft and ship design, architecture, medicine, computer graphics, and more. Recently, the problem has become particularly relevant in bioinformatics and scientific visualization. The interpolation of scattered data in attracted a considerable amount of research. Different methods and approaches were proposed and discussed, excellent surveys are, e. g. (Franke and Nielson, 1991; Lodha and Franke, 1997; Mann et al., 1992), see also (Dey, 2006; Amidror, 2002; Anjyo et al., 2014; Cazals and Giesen, 2006).
Consider the following problem: Given scattered data , , that is points are different and non-collinear, find a bivariate function defined in a certain domain containing points , such that possesses continuous partial derivatives up to a given order and .
Nielson (1983) proposed a three steps method for solving the problem as follows:
Step 1. Triangulation. Construct a triangulation of .
Step 2. Minimum norm network. The interpolant and its first order partial derivatives are defined on the edges of to satisfy an extremal property. The obtained minimum norm network is a cubic curve network, i. e. on every edge of it is a cubic polynomial.
Step 3. Interpolation surface. The obtained network is extended to by an appropriate blending method.
Andersson et al. (1995) paid special attention to Step 2 of the above method, namely the construction of the minimum norm network. Using a different approach, the authors gave a new proof of Nielson’s result. They constructed a system of simple linear curve networks called basic curve networks and then represented the second derivative of the minimum norm network as a linear combination of these basic curve networks.
The problem of interpolation of scattered data by minimum -norms networks for was considered in (Vlachkova, 1992) where sufficient conditions for the solution were formulated without proof.
In this paper we prove the existence and the uniqueness of the solution to the problem for and provide its complete characterization using the basic curve networks defined in (Andersson et al., 1995).
The paper is organized as follows. In Sect. 2 we introduce notation, formulate the extremal problem for interpolation by minimum -norms networks for , and present some related results. In Sect. 3 we prove the existence and the uniqueness of the solution to the problem for . In Sect. 4 we establish a full characterization of the solution. In final Sect. 4 we present the results from our experimental work. Based on numerical solving of nonlinear systems of equations we apply computer modeling and visualization tools to illustrate and support our results.
2 Preliminaries and related results
Let be an integer and be different points in . We call this set of points data. The data are scattered if the projections onto the plane are different and non-collinear. A collection of non-overlapping, non-degenerate triangles in is a triangulation of the points , if the set of the vertices of the triangles coincides with the set of the points . Hereafter we assume that a triangulation of the points , is given and fixed. The union of all triangles in is a polygonal domain which we denote by . In general is a collection of polygons with holes. The set of the edges of the triangles in is denoted by . If there is an edge between and in , it will be referred to by or simply by if no ambiguity arises.
A curve network is a collection of real-valued univariate functions defined on the edges in . With any real-valued bivariate function defined on we naturally associate the curve network defined as the restriction of on the edges in , i. e. for ,
Furthermore, according to the context will denote either a real-valued bivariate function or a curve network defined by (1). For , such that , we introduce the following class of smooth interpolants
where is the class of bivariate continuous functions defined in , is the class of univariate absolutely continuous functions defined in , and is the class of univariate functions defined in whose p-th power of the absolute value is Lebesgue integrable. The restrictions on of the functions in form the corresponding class of so-called smooth interpolation curve networks
We note that the class is nonempty since, e. g. Clough-Tocher Clough and Tocher (1965) and Powell-Sabin Powell and Sabin (1977) interpolants belong to it. Hence is nonempty too. The smoothness of the interpolation curve network geometrically means that if we consider the graphs of functions as curves in then at every point , they have a common tangent plane.
Inner product and -norm are defined in by
where and . We denote the networks of the second derivative of by and consider the following extremal problem:
Problem is a generalization of the classical univariate extremal problem for interpolation of data in by a univariate function with minimal -norm of the second derivative. The latter was studied by Holladay Holladay (1957) for and by de Boor de Boor (1976) in more general settings.111C. de Boor studied the more general problem of minimum -norm of the -th derivative, , . Holladay Holladay (1957) proved that the natural interpolating cubic spline is the unique solution to . Nielson’s approach to construct minimum norm network can be seen as an extention of Holladay’s proof (Holladay, 1957).
For let denote the degree of the vertex , i. e. the number of the edges in incident to . Furthermore, let be the edges incident to listed in clockwise order around . The first edge is chosen so that the coefficient defined below is not zero - this is always possible. A basic curve network is defined on for any pair of indices , such that and , as follows (see Fig. 1):
, are uniquely determined to sum to one and to form a zero linear combination of the three unit vectors along the edgesstarting at .
Note that basic curve networks are associated with points that have at least three edges incident to them. We denote by the set of pairs of indices for which a basic curve network is defined, i. e.,
With each basic curve network for we associate a number defined by
which reflects the position of the data in the supporting set of . The following two lemmas are proved in (Andersson et al., 1995) for but they clearly hold for any , .
Functions , are linearly independent in .
3 Existence and uniqueness of the solution
In the next theorem we prove that problem for always has a unique solution which we call optimal curve network.
The extremal problem for always has a unique solution .
Proof. Let . We recall that . The set of real non-negative numbers is bounded from below and therefore it has a greatest lower bound . Let , where be a minimizing sequence, i. e. . We denote
Next we prove that is a fundamental sequence in , i. e. . For this purpose we use the following Clarkson’s inequalities (see (Hewitt and Stromberg, 1975), pp. 225, 227) which hold for ,
Since then . Hence . Let . From the first inequality in (5) we obtain
For from the second inequality in (5) we obtain
Therefore for every , i. e. is a fundamental sequence. Since is a complete space then there exists curve network such that for every and
From (6) it follows that there exists a subsequence of that converges pointwise almost everywhere (a.e.) to . For simplicity we assume that is that subsequence, i. e. Moreover, from the continuity of the norm we have , and hence .
Let be the unique curve network that satisfies the interpolation conditions and its second derivative coincides a.e. with . To prove the existence of the solution to the problem, next we show that belongs to . We have to show that for every vertex there exists a tangent plane to the curve network . First, we prove that
Since then and since , we have
From (8) after integration we obtain
hence from the interpolation conditions we have
Further on, if are vectors in , we denote by their scalar triple product.
Functions , defined by (1) are parametric curves in represented by Then vector defined by is a tangent vector to curve at point . Let be a vertex in of degree (if then a tangent plane in always exists). Let , , and be three arbitrary edges incident to . Since then has a tangent plane at . A necessary and sufficient condition for the existence of such a plane is
where , and are the three tangent vectors. We take the limit in (9) for , use (7), and obtain that the scalar triple product of the limit vectors is zero too. The three edges , , and are arbitrarily chosen, hence the curve network has a tangent plane at point , which has been arbitrarily chosen too. Therefore belongs to and solves problem .
It remains to prove uniqueness of the solution. Let and be two solutions of . Then . From Minkowski’s inequality it follows
Hence, in (10) we have equality which holds if and only if a.e., where and are non-negative real numbers such that . From the equality of the norms it follows , i. e. a.e. which means a.e. in for every . Since and coincide at the endpoints of the edge then for every . Therefore . ∎
4 Characterization of the solution
In this section we provide a full characterization of the solution to the extremal problem for . Its existence and uniqueness have been already established in Theorem 3. Further, for simplicity we use the notation Next we prove that can be represented as a linear combination of the basis curve networks defined by (3). Finding of reduces to the unique solution of a system of equations. The following theorem holds.
Smooth interpolation curve network is a solution to problem , if and only if
where are real numbers and .
Proof. Let us consider the set of interpolation curve networks
and the mapping
where is such that , and the class is defined by (4). If then obviously . Now let belong to . We integrate twice the function , use the two interpolation conditions and at the end of the interval , and obtain curve network such that and . Therefore the mapping (11) is a bijection. According to Lemma 2, (11) maps the set defined by (2) onto the following subset of :
Thus, problem , is equivalent to the following problem
Using the Lagrange multipliers (see, e.g, Shilov (1965), pp. 113) we obtain that (, respectively) is a solution to problem for if and only if there exist real numbers such that is a solution to the problem
From the last inequality it follows that the solution (, respectively) has the form
where , are real numbers and . Moreover, the representation (14) is unique. ∎
As a consequence, we can formulate the following theorem.
Curve network solves problem for if and only if . The coefficients are the unique solution to the following system of equations
Proof. From Theorem 4 it follows that there exist real numbers such that the second derivative of the unique solution to problem for is . On the other hand, according to Lemma 2, is a smooth interpolation curve network if and only if for every . Therefore numbers are a solution to system (15).
5 Examples and results
To find the minimum -norm networks for we have to solve system (15) which is nonlinear except in the case where when it is linear. We have adopted a Newton’s algorithm Vlachkova (2000) to solve this type of systems. We use Mathematica package to visualize the extremal curve networks. Below we present results of our experiments where solutions of were computed and visualized for different on two small data sets.
We consider data obtained from a regular triangular pyramid. We have , , , , , and , , The set of indices defining the edges of the corresponding triangulation is . We have for and four basic curve networks , , , are defined. The triangulation, the minimum -norms network , and the corresponding -norms of the second derivatives for , and are shown in Fig. 2 (left).
We have and the data are , , , , , , and . The set of indices defining the edges of the corresponding triangulation is . We have , , , and hence, the number of the basic curve networks is fourteen. The triangulation, the minimum -norms network , and the corresponding -norms of the second derivatives for , and are shown in Fig. 2 (right).
6 Conclusions and future work
In this paper we considered the extremal problem of interpolation of scattered data in by smooth minimum -norm networks for . We proved the existence and the uniqueness of the solution for and provided its complete characterization. We presented numerical experiments and gave examples to visualize and support the obtained results.
The case is not completely understood and needs to be studied further. First of all it is known that the solution in this case is not unique. Second, the approach based on Lagrange multipliers can not be applied directly to the case . Another interesting question that arises is whether the sequence of solutions for converges as , and if yes, what is the limit?
This work was supported by Sofia University Science Fund Grant No. 80-10-145/2018, and by European Regional Development Fund and the Operational Program “Science and Education for Smart Growth” under contract № BG05M2OP001-1.001-0004 (2018-2023).
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