Subdivision Depth Computation for Tensor Product n-Ary Volumetric Models

and Applied Analysis 3 P 1 2i P 1 2i 1 P 1 2i 2 P i 1 P 1 2i 3 P 1 2i 4 P k i 2 P k i a P 1 3i P 1 3i 1 P 1 3i 2 P i 1 P 1 3i 3 P 1 3i 4 P 1 3i 5 P 1 3i 6 P k i 2 P k i b P 1 4i P 1 4i 1 P 1 4i 2 P k i 1 P 1 4i 3 P 1 4i 4 P 1 4i 5 P 1 4i 6 P 1 4i 7 P 1 4i 8 P k i 2 P k i c Figure 1: Solid lines show coarse polygons whereas doted lines are refined polygons. a – c represent binary, ternary, and quaternary refinement of coarse polygon of scheme 2.1 for n 2, 3, 4, respectively. 2. Preliminaries In this section, first we list all the basic facts about subdivision curve, surface and volumetric models needed in this paper. Then we settle some notations for fair reading and better understanding of Section 3. 2.1. Concepts 2.1.1. n-Ary Subdivision Curve Given a sequence of control points p i ∈ N , i ∈ , N 2, where the upper index k 0 indicates the subdivision level, an n-ary subdivision curve 5 is defined by p 1 ni α m ∑ j 0 aα,jp k i j , α 0, 1, . . . , n − 1, 2.1 where m > 0 and m ∑ j 0 aα,j 1, α 0, 1, . . . , n − 1. 2.2 The set of coefficients {aα,j , α 0, 1, . . . , n − 1}mj 0 is called subdivision mask. Given initial values p0 i ∈ N , i ∈ , then in the limit k → ∞, the process 2.1 defines an infinite set of points in N . The sequence of control points {p i } is related, in a natural way, with the dyadic mesh points tki i/n k , i ∈ . The process then defines a scheme whereby p 1 ni α replaces the value p i α/n for α ∈ {0, n}. Here p k 1 ni α is inserted at the mesh point t k 1 ni α 1/n n−α t k i αt k i 1 for α 0, 1, . . . , n. Labelling of old and new points is shown in Figure 1 which illustrates subdivision scheme 2.1 . 2.1.2. Tensor Product n-Ary Subdivision Surface Given a sequence of control points p i,j ∈ N , i, j ∈ , N 2, where the upper index k 0 indicates the subdivision level, a tensor product n-ary surface is a tensor product of 2.1 defined by p 1 ni α,nj β m ∑ r 0 m ∑ s 0 aα,raβ,sp k i r,j s, α, β 0, 1, . . . , n − 1, 2.3 4 Abstract and Applied Analysis P 1 2i,2j 2


Introduction
Subdivision is a simple and elegant method to describe smooth curves and surfaces. The approach of subdivision schemes is simple and efficient due to its mathematical formulation. Its application ranges from industrial design and animation to scientific visualization and simulation. Due to this, subdivision method is becoming a standard technique and now wellunderstood by both academic and industrial communities. It is an algorithm to generate smooth curves and surfaces as a sequence of successively refined control polygons. At each subdivision level, the subdivision scheme describe the source grid maps to the subdivided grid, which results in increase in the number of points. The number of points inserted at level k 1 between two consecutive points from level k is called arity of the scheme. In the case when number of points inserted are 2, 3, . . . , n the subdivision schemes are called binary, ternary, . . ., n-ary, respectively. For more details on n-ary subdivision schemes, we may refer to 1-4 thesis of Aspert 5 and Ko 6 . However tensor product trivariate schemes obtained from above schemes have been proven themselves an excellent tool for the modeling of largely regular volumetric/solid models over hexahedron lattice, for example, manufacturing of industrial regular block, font animation and garment pressures for the biomechanical design of functional apparel products, and so forth.

Abstract and Applied Analysis
Although subdivision has many attractive advantages for modeling purposes, it has received far less attention in volumetric modeling. One of the reason is the topological complexity of domain meshes in higher-dimensional spaces. Another reason is lack of proper mathematical tools for extraordinary analysis. Even so, it is clear that subdivision is slowly gaining interest in solid modeling community. MacCracken and Joy 7 proposed a tensor product extension of the Catmull-Clark scheme in the volumetric setting over hexahedron lattice. Later on, Bajaj  Subdivision schemes have become important in recent years because they provide a precise and efficient way to describe smooth curves/surfaces/volumetric models, however the little have been done in the area of error control for tensor product n-ary volumetric models. The investigation of error control for volumetric models arises two questions in mind.
i How well the regular hexahedron lattice approximate to the limit volumetric model?
ii How many subdivision steps are needed to satisfy a user-specified error tolerance?
For given error tolerance, the subdivision levels performed on the initial control polygon, so that the error/distance between the resulting control polygon and the limit volumetric models would be less than the error tolerance is called subdivision depth.
A subdivision depth and error bound based on forward differences of control points have been presented by 11, 13-18 , while the methods 19-22 are based on eigenanalysis. But nothing in this area has been done for more general tensor product n-ary volumetric models yet. In this paper, we will answer-above-said questions and present a subdivision depth computation technique based on error bounds for tensor product n-ary volumetric models.
It is notified that the increase in arity offers greater freedom than offered by low arity subdivision volumetric scheme in term of coefficients. Higher arity volumetric schemes allow a range of different behaviors than the lower arity volumetric schemes. Ko 6 notified that subdivision curves/surfaces with higher arity results in higher smoothness and approximation order but smaller in support, which make it more practical in use. It is also noticed that higher arity volumetric models have slightly lower computational cost than lower arity volumetric models. This discussion motivate us to calculate error bound and depth for higher arity subdivision volumetric models, that is, in general for tensor product n-ary subdivision volumetric models. Our method is generalization of Mustafa et al. 13,[15][16][17] . The paper is arranged as follows.
Section 2 is devoted for basic definitions and notations. In Section 3, we have computed subdivision depth for tensor product n-ary volumetric models. Section 4 presents applications of our results for tensor product n-ary volumetric models. Conclusion and future research directions are given in Section 5. The typical mathematical proofs are placed in Appendices A and B for improved presentation of the paper.  Figure 1: Solid lines show coarse polygons whereas doted lines are refined polygons. a -c represent binary, ternary, and quaternary refinement of coarse polygon of scheme 2.1 for n 2, 3, 4, respectively.

Preliminaries
In this section, first we list all the basic facts about subdivision curve, surface and volumetric models needed in this paper. Then we settle some notations for fair reading and better understanding of Section 3.

n-Ary Subdivision Curve
Given a sequence of control points p k The set of coefficients {a α,j , α 0, 1, . . . , n − 1} m j 0 is called subdivision mask. Given initial values p 0 i ∈ Ê N , i ∈ , then in the limit k → ∞, the process 2.1 defines an infinite set of points in Ê N . The sequence of control points {p k i } is related, in a natural way, with the dyadic mesh points t k i i/n k , i ∈ . The process then defines a scheme whereby p k 1 ni α replaces the value p k i α/n for α ∈ {0, n}. Here p k 1 ni α is inserted at the mesh point t k 1 Labelling of old and new points is shown in Figure 1 which illustrates subdivision scheme 2.1 .

Tensor Product n-Ary Subdivision Surface
Given a sequence of control points p k Abstract and Applied Analysis where a α,r satisfies 2.2 . Given initial values p 0 i,j ∈ Ê N , i, j ∈ , then in the limit k → ∞, the process 2.3 defines an infinite set of points in Ê AE . The sequence of values {p k i,j } is related, in a natural way, with the dyadic mesh points i/n k , j/n k , i, j ∈ . The process then defines a scheme whereby p k 1 ni α,nj β replaces the values p k i α/n,j β/n for α, β ∈ {0, n}. Here the values p k 1 ni α,nj β are inserted at the mesh points ni α /n k 1 , nj β /n k 1 for α, β 0, 1, . . . , n. Labelling of old and new points is shown in Figure 2 which illustrates subdivision scheme 2.3 .

Tensor Product n-Ary Volumetric Model
Given a sequence of control points p k   in a natural way, with the dyadic mesh points i/n k , j/n k , l/n k , i, j, l ∈ . The process then defines a scheme whereby p k 1 ni α,nj β,nl γ replaces the values p k i α/n,j β/n,l γ/n for α, β, γ ∈ {0, n}. Here the values p k 1 ni α,nj β,nl γ are inserted at the mesh points ni α /n k 1 , nj β /n k 1 , nl γ /n k 1 for α, β, γ 0, 1, . . . , n. Labelling of old and new points is shown in Figure 3 which illustrates subdivision scheme 2.4 .

Subdivision Depth
Given control polygon of tensor product n-ary volumetric model and an error tolerance , if we subdivide control polygon k times so that the error between resulting polygon and volumetric model is smaller than , then k is called subdivision depth of tensor product nary volumetric model with respect to .

Notations
Here, we assume

Depth for Tensor Product n-Ary Volumetric Models
In this paragraph, we compute subdivision depth for tensor product n-ary volumetric model. Moreover, we show that error bound for binary subdivision volumetric models 17 is special case of our bound. Here we need following lemmas for Theorem 3.5. The proof of first two lemmas are shown in Appendices A and B, respectively. i,j,l and P ∞ is the limit volumetric model of 2.4 . If δ < 1, then error bound between tensor product n-ary volumetric model and its control polygon after k-fold subdivision is where δ and ϑ are defined by 2.5 and 2.18 , respectively.
Proof. Let · ∞ denote the uniform norm. Since the maximum difference between P k 1 and P k is attained at a point on the k 1 th mesh, we have where M k α,β,γ is defined by 2.7 . By 3.2 and 3.4 , we get where δ and ϑ are defined by 2.5 and 2.18 , respectively. By triangle inequality we get 3.3 . This completes the proof.
Proof. From 3.3 , we have This implies, for arbitrary given > 0, when subdivision depth k satisfies the following inequality: This completes the proof.
The error bounds and subdivision depth of 4.1 are shown in Tables 1 and 2, respectively. In these tables, we have shown the error bounds and depth of different arity interpolating subdivision volumetric models by using Lemma 3.3 and Theorem 3.5 with χ 0.1.

Error Bound and Subdivision Depth of Tensor Product n-Ary Approximating Volumetric Models
In this section, we estimate the error bound and subdivision depth of tensor product 2b 2point n-ary approximating volumetric models. By taking the tensor product of 2b 2 -point n-ary scheme of 4 , we get the following:  where  Tables 3 and 4, respectively. In these tables, we have shown the error bounds and depth of different arity approximating subdivision volumetric by using Lemma 3.3 and Theorem 3.5 with χ 0.1.

Conclusion and Future Work
We have computed subdivision depth based on error bounds for tensor product n-ary volumetric models. Furthermore, we have shown that error bounds for binary subdivision volumetric model 17 is special case of our bounds. It is noticed that the increase in arity results gradually decrease in error, which is shown in Tables 1, 3 and graphically in Figure 4. It is noticed from Tables 2 and 4 that higher arity subdivision volumetric models need less number of subdivision steps than lower arity to satisfy user-specified error tolerance.
The authors are looking, as a future work, to extend the computational techniques of subdivision depth for n-ary arbitrary subdivision volumetric models over rectangular/triangular hexahedron lattice. we will discuss them elsewhere.