Shape-Preserving Properties of a Relaxed Four-Point Interpolating Subdivision Scheme

: In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating subdivision scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed scheme.


Introduction
Subdivision scheme is the technique of generating curves and surfaces by iterative refinement of initial control polygon/mesh accordingly some refinement rules. The implementation of subdivision scheme can be visualized much better by analyzing its shape-preserving properties that can be considered to be geometrical properties of a subdivision scheme. The attribute of shape preservation is of great prominence in medical imaging, ship hulls and airplane designing. Shape preservation is always worthwhile in surgery, meteorology, designing pipe system, designing car bodies, in chemical engineering, sectional drawing, geometric modeling and visualization.
The paper is organized as follows: In Section 2, we discuss positivity preservation property of the FP-scheme. The conditions of preserving monotonicity and convexity of the FP-scheme are given in Sections 3 and 4. In Section 5, we present some numerical examples to show shape-preserving behavior of the scheme and conclude our work with a summary in this section.

Positivity Preservation
In this section, we show that the limit curve generated by the FP-scheme preserves positivity of initial data. Subdivision scheme is said to preserve positivity, if starting from a positive control polygon, the limit curves produced by the scheme preserve the positivity of the initial data.
Positivity preservation of FP-scheme (1) can be analyzed by choosing q k j = g k j+1 g k j and Q k = max{q k j , 1 q k j }, j ∈ Z, k ∈ N 0 . In the following theorem, we give a result which plays a vital role to prove positivity preservation of limit curve of the FP-scheme. Theorem 1. Assume the set of initial control points {(x 0 j , g 0 j ) : j ∈ Z}, is positive, i.e., g 0 j > 0, ∀ j ∈ Z. Furthermore, let ω be such that 1 < ω < √ 5 − 1, if 1 ω ≤ Q 0 ≤ ω and {g k j } j∈Z is defined by the FP-scheme, then, that is the limit function generated by the FP-scheme is positive.
Proof. We prove the theorem by induction. By given condition, it is easy to see that (2) is valid for k = 0. Assume that (2) is satisfied for some k ≥ 1. Now we prove that (2) is also satisfied for k + 1. We first prove that g k+1 Thus, by combining (3) and (4), we have g k+1 j > 0, ∀ j ∈ Z. Induction shows that g k j > 0, ∀ j ∈ Z, k ∈ N 0 . Now, we prove that 1 The denominator of above expression is greater than zero by (3) and the numerator N 1 satisfies The denominator of above expression is greater than zero by (4) and the numerator N 2 satisfies Therefore q k+1 2j+1 ≤ ω. In the same way, we can get 1 (2) is satisfied. Therefore, FP-scheme preserves positivity. This completes the proof.

Monotonicity Preservation
This section examines monotonicity preservation of FP-scheme. Monotonicity preservation is achieved by generating first-order divided differences (DD). Subdivision scheme holds property of monotonicity preservation if starting from a monotone control points, the limit curves produced by the scheme preserves the monotonicity of the initial data. First-order DD can be examined by applying d k+1 So FP-scheme in the form of first-order DD is given by In the following theorem, we derive some conditions on initial control points which guarantee monotonicity preservation of limit curve of the FP-scheme.

Theorem 2. Assume the set of strictly monotone increasing initial control points
Therefore, the limit curves generated by the FP-scheme are strictly monotonically increasing.
Proof. We use induction to prove the theorem. From assumption it is clear that (5) is satisfied for k = 0. Suppose (5) holds for some k ≥ 1 and we show that it also holds for k + 1. We first prove that d k+1 Therefore, we have d k+1 . Thus, we have . The denominator of above expression is greater than zero by (6) and the numerator N 3 satisfies Thus, we have The denominator of above expression is greater than zero by (7) and the numerator N 4 satisfies Therefore r k+1 2j+1 ≤ µ. In the same way, we can get 1 r k+1 2j ≤ µ and 1 r k+1 2j+1 ≤ µ. Therefore, 1 µ ≤ R k+1 ≤ µ and induction leads to 1 µ ≤ R k ≤ µ, ∀ k ∈ N 0 , thus (5) is satisfied. Therefore, the FP-scheme preserves monotonicity. This completes the proof.

Convexity Preservation
In this section, we show that the limit curve generated by the FP-scheme preserves convexity of initial data. A subdivision scheme enjoys convexity-preserving property, if starting from a convex control polygon, the limit curves produced by the scheme preserves the convexity of the initial data.
Convexity preservation can be examined by applying second-order DD, i.e., D k j = 2 2k−1 (g k j−1 − 2g k j + g k j+1 ). So the FP-scheme in the form of second-order DD is given by In the following theorem, we derive some conditions on initial control points which guarantee convexity preservation of limit curve of the FP-scheme. . If 1 ν ≤ S 0 ≤ ν and {g k j } j∈Z is defined by the FP-scheme, then: Specifically, the limit curves generated by the FP-scheme preserve convexity.

Proof.
To prove the result, we use induction. Since it is given that D 0 (8) is true for k = 0. Suppose (8) holds for some k ≥ 1. We will verify it also holds for k + 1. We first prove that D k+1 Therefore, we have D k+1 The denominator of above expression is greater than zero by (9) and the numerator N 5 satisfies The denominator of above expression is greater than zero by (10) and the numerator N 6 satisfies Therefore s k+1 2j+1 ≤ ν. In the same way, we can get 1 s k+1 2j ≤ ν and 1 s k+1 2j+1 ≤ ν. Therefore, 1 ν ≤ S k+1 ≤ ν and induction leads to 1 ν ≤ S k ≤ ν, ∀ j ≥ Z, k ∈ N 0 , thus (8) is satisfied. Therefore, FP-scheme preserves convexity. This completes the proof.

Numerical Examples and Conclusions
In this section, we present some numerical examples to show shape-preserving behavior of the FP-scheme. At the end of the section, we conclude the work done so far.

Numerical Examples
Example 1. In this example we choose a positive data which is given in Table 1, that fulfill the derived condition of positivity, i.e., it is easy to get that Q 0 = 1.14. We apply FP-scheme on this positive data five times. Graphical representation of this application is given in the Figure 1a. In this figure dotted line shows the initial positive data and the solid line represents the limit curve generated by FP-scheme. From the Figure 1a, it is clear that FP-scheme preserves positivity of initial data.

Example 2.
In this example we consider another positive data which is given in Table 2, that fulfill the derived condition of positivity, i.e., it is easy to get that Q 0 = 1.019. We apply FP-scheme on this positive data five times. Graphical representation of this application is given in the Figure 1b. In this figure dotted line shows the initial positive data and the solid line represents the limit curve generated by FP-scheme. From the Figure 1b, it is clear that FP-scheme preserves positivity of initial data. Example 3. In this example we consider a monotonically increasing data which is given in Table 3, that fulfill the derived condition of monotonicity, i.e., it is easy to get that R 0 = 1. We apply FP-scheme on this monotone data five times. Graphical representation of this application is given in the Figure 2a. In this figure dotted line shows the initial positive data and the solid line represents the limit curve generated by FP-scheme. From the Figure 2a, it is clear that the limit curve generated by the FP-scheme is also monotonically increasing.

Example 4.
In this example we consider another monotonically increasing data which is given in Table 4, that fulfill the derived condition of monotonicity, i.e., it is easy to get that R 0 = 1. We apply FP-scheme on this monotone data five times . Graphical representation of this application is given in the Figure 2b. In this figure dotted line shows the initial positive data and the solid line represents the limit curve generated by FP-scheme. From the Figure 2b, it is clear that the FP-scheme is capable of producing monotonically increasing limit curves.

Example 5.
In this example we consider convex data from a convex function which is given in Table 5, that fulfill the derived condition of convexity, i.e., it is easy to get that S 0 = 1. We apply FP-scheme on this convex data five times . Graphical representation of this application is given in the Figure 3a. In this figure dotted line shows the initial positive data and the solid line represents the limit curve generated by FP-scheme. From the Figure 3a, it is clear that the FP-scheme is capable of producing convex limit curves. Example 6. In this example we consider convex data from another convex function which is given in Table 6, that fulfill the derived condition of convexity, i.e., it is easy to get that S 0 = 1. We apply FP-scheme on this convex data five times . Graphical representation of this application is given in the Figure 3b. In this figure dotted line shows the initial positive data and the solid line represents the limit curve generated by FP-scheme. From the Figure 3b, it is clear that the FP-scheme is capable of producing convex limit curves.   Table 4. Monotone data set of values.  Table 5. Convex data set of values.  Table 6. Convex data set of values.

Conclusions
An approximating subdivision scheme with cubic precision and satisfying shape-preserving properties is a charming scheme for designers. We have presented analysis of some important shape-preserving properties of the FP-scheme, which make the scheme more efficient for application in geometric modeling. These properties assure that the shape preservation of the limit curve is an effective tool for modifying the FP-scheme for different requirements. We have shown that by taking initial control data positive, monotone and convex, the limit curves generated by the FP-scheme are also positive, monotone and convex. Also, we support our findings through several numerical examples. In future work, we are interested to analyze these shape-preserving properties in geometric notion. Extension of this work to case of surface is another future direction.