Objective

2

The subdivision technique has been developing in relevant to the design of efficient

hierarchical

local

and adaptive algorithms for modeling

plotting

and manipulating arbitrary topology-related free-shaped objects beyond the non-uniform rational B-splines (NURBS). The initial subdivision step is oriented at a control polygon or mesh. First

novel vertices can be involved in and the existed vertices can be optimized. Next

a new control polygon or control mesh can be obtained. Finally

the target curves of surfaces can be replicated to produce. Subdivision schemes can be segmented into two categories: 1) linear-based and 2) nonlinear-based because the issue of new points can change the linear combinations of old points in the iterative process. Generally speaking

linear subdivision schemes are easier to be implemented

but there are inflection points plotting on the limit curves and it is challenged to represent precise circle

while nonlinear subdivision schemes can eliminate inflection points and reproduce a circle accurately. The smooth curve-fit point clouds problem is concerned more in the context of computer-aided geometric design (CAGD) and computer graphics (CG). Measurement data can be obtained on real objects via such techniques like laser scanning

structure light source converter

and X-ray tomography. To perform a commonly-used model reconstruction and functional recovery for the original model or product

these discrete data points are scanned and used for data fitting. But

data points are often linked to position information and normal vector information like optical reflector design. To resolve this problem

we develop two schemes of nonlinear subdivision in related to a parameters-dual 4-point binary and a parameter-solo 3-point ternary interpolation in terms of circle average.

Method

2

First

a two points and its normal vector-related binary nonlinear circle average is introduced. This task is targeted on circle average because the new point is on the circle-constructed derived from the original two points and corresponding normal vectors. Next

linear subdivision method is rewritten into a replicated binary average of points. In order to optimize linear subdivision schemes

linear average is replaced by the circle average. Third

the weighted geodesic average is used to calculate the newly vertex-inserted normal vector. The two kinds of circle average-based nonlinear subdivision schemes are obtained through melting the operations mentioned above into linear parameters-dual 4-point binary subdivision and parameter-solo 3-point ternary interpolation subdivision. For circle average-related two-parameter four-point binary subdivision scheme

each subdivision process is composed of two steps of displacement and tension both. For circle average-based single-parameter three-point ternary subdivision method

each subdivision step is reconstructed by left interpolation

interpolation

and right interpolation. In addition

the feasibility of these methods is tested theoretically and numerically. Some theorems of convergence and consistency of two proposed methods are illustrated because normal vectors-proved have factor-contracted

and the data points have factor-contracted and backup-displaced in the subdivision process.

Result

2

These methods are implemented in terms of MATLAB overall. The issue of parameters is studied on the two proposed subdivision schemes. First

for the circle average-based two-parameter 4-point binary subdivision scheme

the smaller of the tension parameter is

the limit curve is closer to the initial control polygon. The smaller of the displacement parameter is the limit curve is near to the initial control vertex. When the parameter-displaced is zero

the subdivision method is transferred to interpolation subdivision. For the circle average-based solo-parameter 3-point ternary subdivision method

the circle average is first to be applied to the linear ternary interpolation subdivision

which makes the vertices-controlling more fast. Then

for the same initial control vertices

the normal vector of one fixed control vertex is changed to produce different limit curves freely. Test results show that the selection of parameters and initial normal vectors can be used to control the shape of limit curves effectively. Finally

our nonlinear subdivision schemes proposed are compared to the corresponding linear subdivision schemes. When the initial control vertices are sampled from the circle

the corresponding normal vectors will be pointed between the center of the circle and the vertex. Test results show that our nonlinear subdivision schemes proposed can reconstruct the circle through the proposed nonlinear subdivision schemes and the corresponding linear subdivision schemes-reconstructed

but the corresponding linear subdivision schemes cannot be used to reconstruct the circle. Furthermore

three sorts of case studies for curve models are selected in comparison with curve reconstruction from multiple subdivision methods. The initial control vertices and their normal vectors are sampled based on curves-consistent

and they are subdivided for 8 times totally. Our nonlinear subdivision-schemed limit curve is much smoother

while the corresponding linear subdivision schemes have their sharp points.

Conclusion

2

Theoretically

it shows that our circle average-based two nonlinear subdivisions proposed are convergent and consistent with C

1

. Experimental results indicate that our nonlinear subdivision schemes can optimize linear subdivision-schemed modeling ability

and it has its circular regenerative potentials. Normal vectors-selected is beneficial to the shapes of limit curves to some extent.