A Polynomial Splines Identification Method Based on Control Nets

In this study, based on Polynomial Splines with control nets, an identification method is investigated. We introduce polynomial splines with control nets defined over T-mesh. 1e basic idea is to extend T-vertices such that those T-vertices become interior cross vertices or boundary vertices. To this end, we introduce the design-suitable T-mesh for constructing polynomial splines with control net. In design-suitable T-meshes, there are no extra basis vertices produced by an appropriate extension of T-vertices. 1e basis functions are defined over each vertex in a design-suitable T-mesh by the means of constructing PHT-splines basis functions.

PHT-splines are introduced in [15] as bicubic C 1 continuous polynomial splines spaces defined over hierarchical T-meshes. PHT-splines possess a set of nonnegative and linearly independent basis functions and allow for very efficient local refinement. PHT-splines have been widely applied in geometric processing and analysis. e finite element discretization of elliptic equations based on PHTsplines was discussed in [16], where numerical solutions are refined adaptively and have the optimal convergence rate. PHT-splines are also favored in isogeometric analysis for solving elastic problems [3,17,18] and adaptive isogeometrican analysis [19]. PHT-splines are also applied in reconstructing surface models efficiently. PHT-splines were used in stitching several surface patches to construct complex models in paper [20] and PHT-splines were also applied in surface reconstruction from a very large set of point clouds in implicit form [21]. IGA collocation approaches are introduced in paper [22], and PHT-splines have been used as basis functions for the adaptive collocation method [23].
In order to make PHT-splines better suited for analysis and geometric processing, some improvements and extensions have been made to PHT splines in recent years. In [24], the authors discussed the decay phenomenon of PHTsplines basis functions. ey found some of the basis functions will tend to zero as the refinement level increases under certain types of refinement. Such a decay makes the stiffness matrix in isogeometric analysis be ill-conditioned, which is not expected by analysis.
us a new basis construction method is proposed for PHT-splines to avoid the decay. e extension of PHT-splines to general T-meshes is considered in [25], where the bicubic C 1 continuous polynomial splines are defined over general T-meshes. e basis functions are constructed by computing the geometric information at basis vertices such that the resulting basis functions are nonnegative and linearly independent and form a partition of unity. PHT splines were defined over hierarchical T-meshes without irregular vertex in [15], and then the bicubic C 1 continuous splines construction based on irregular quad layout were discussed in [26]. Current implementation of PHT-splines stores the basis functions in Bézier forms, which saves some computational costs but consumes a lot of memories. In [27], an algorithm to evaluate PHT-splines is provided where only the information about the control coefficients and the hierarchical mesh structure is given. e evaluation algorithm takes about the same computational costs while requiring much less amount of memory compared with the Bézier representations.
For PHT-splines, there is no one-to-one correspondence between the vertices in the underlying T-mesh and PHTsplines basis functions. e boundary vertices and interior cross vertices are called basis vertices. Each basis vertex is associated with four basis functions. e control net concept is essential in computer aided geometric design (CAGD), because the control net makes editing models easily and intuitively. In this paper, we introduce polynomial splines with control polygon to complement PHT-splines. e proposed splines defined over all the vertices of the underlying T-meshes, such that there is a one-to-one correspondence between control points and vertices in T-meshes. e basis idea is to extend T-vertices to be basis vertices. In order to avoid generating extra basis vertices when extending T-vertices, we introduce a subset of T-meshes, called design-suitable T-meshes. Over design-suitable T-meshes, the T-vertices can be extended appropriately without generating extra basis vertices. For a given T-mesh, it needs to connect some T-vertices to be a design-suitable T-mesh. We define polynomial splines basis functions over the design-suitable T-mesh. e resulting splines not only inherit the nice properties of PHT-splines, but also have control nets. e paper is organized as follows. In Section 2, we review some preliminary knowledge about polynomial splines over T-meshes. In Section 3, we proposed a subset of T-mesh called design-suitable T-mesh and prove that there are no extra basis vertices produced by an appropriate extension. In Section 4, we present an algorithm of local refinement of the proposed splines. Finally, we give a conclusion of this paper in Section 5.

T-Meshes.
A T-mesh is a rectangular grid with T-junctions. e cells in a T-mesh must be rectangles. A grid point in a T-mesh is called a vertex. And the vertex on a boundary grid line is called a boundary vertex; otherwise, it is called an interior vertex. Interior vertices consist of cross vertices and T-vertices. We adopt the notations ⊤, ⊥, ⊢, and ⊣ to indicate the four possible orientations of T-vertices. e T-vertices of type ⊢ and ⊣ are called horizontal T-vertices and the T-vertices of type ⊤ and ⊥ are called vertical T-vertices.

Bicubic
be a knot vector with interior knots of multiplicity two. en each knot t j is associated with two cubic B-splines N 1 and and N 2 j (t) together with their derivatives have the same support [t j−1 , t j+1 ]. Except for N 1 j (t) and N 2 j (t), all the other cubic B-splines and their derivatives vanish at t j .
For given a tensor product mesh T, the two associated global knot vectors in s-direction and t-direction are respectively. e bicubic C 1 continuous B-splines surface For a function f(x, y), the function value, the first partial derivatives, and the mixed partial derivative are called the geometric information of f(x, y), denoted by For the basis function N k i,j defined above, it has GN k i,j (s p , t q ) � δ ip δ jq and δ ip � 1, if i � p; otherwise, δ ip � 0. at is the geometric information of N k i,j vanishes at other vertices in T except its associated vertex (s i , t j ).

Polynomial Splines with Control Nets Defined over T-Meshes
PHT-splines span the polynomial splines space S(3, 3, 1, 1) over a T-mesh and are defined over cross interior vertices and boundary vertices (called basis vertices), resulting in PHTsplines lacking control net. Our aim is to equip PHT-splines with control net. For any given control mesh, we parameterize the control mesh into a T-mesh in 2D domain and then extend all T-vertices in the T-mesh such that they become basis vertices and no extra basis vertices is introduced, and finally construct basis functions of S(3, 3, 1, 1) over the extended T-mesh. us, there is a one-to-one correspondence between the control mesh and parameter T-mesh. For arbitrary T-mesh, it is inevitable that extra basis vertices are produced by directly extending T-vertices. erefore, in order to avoid extra basis vertices, we introduce design-suitable T-meshes for defining basis functions.

Design-Suitable T-Meshes.
For a given T-mesh T, we shall adopt some definitions and notations ( Figure 1).

(i) Extension of T-vertex.
e closed line segment created by extending a T-vertex in the missing direction until it intersects with an edge is called the extension of this T-vertex. Particularly, if the edge is an edge in T, then the extension is called a full-extension. e extension of a horizontal (vertical) T-vertex is called the horizontal (vertical) extension, respectively. In Figure 1 Figure 1(b) shows the full-extension mesh of the T-mesh in Figure 1(a). (iii) Edge-Type Intersections. e intersection between two same type extensions and the intersection between an extension and a boundary edge is called an edge-type intersection (E-intersection for short). In Figure 1 Intersections. e intersection between a horizontal extension and a vertical extension is called a face-type intersection (F-intersection for short). In Figure 1 then T is called a design-suitable T-mesh. e diagonal T-mesh (the diagonal elements are refined in the diagonal direction) shown in Figure 2(a) is a designsuitable T-mesh. e T-mesh shown in Figure 1(b) is not a design-suitable T-mesh. But if we connect all the CT-vertices and ECT-vertices, then the resulting mesh is a designsuitable T-mesh which is shown in Figure 3(a).
For a design-suitable T-mesh, there are no extra basis vertices produced by extending T-vertices appropriately. e intuitive way is to extend a T-vertex in the missing direction, until it intersects with an edge in the underlying mesh instead of the original T-mesh. We call such a way of extending T-vertices as p-extending to distinguish from the extension mentioned above. For the design-suitable T-mesh shown in Figure 2(a), the new T-mesh produced by p-extending T-vertices is shown in Figure 2(b). We also plot the fullextension T-mesh in Figure 2(c) for comparison. is p-extending way is not enough for the T-meshes containing ECT-vertices. It still requires an order of extending T-vertices. us, we introduce the following algorithm to extend T-vertices in a given T-mesh, and the resulting T-mesh is denoted by T e . We take Figure 3 as an example to explain the above algorithm. In the T-mesh shown in Figure 3(a), F 1 , F 2 , . . . , F 6 are T-elements. We need to extend the T-vertices in these elements. e element F 3 is an S-element. We start from extending T-vertices in this element. e vertex v 5 is a ST-vertex, and it is extended until it intersects with an existing edge. en we extend the vertex v 4 , and the first edge it intersects with is the extension of v 5 . e element F 6 is the adjacent element of F 3 . e vertex v 8 is p-extended. For the rest T-elements, if there are horizontal T-vertices, then extend them first. us, in F 3 , v 3 is extended first and Complexity then v 2 is extended. e resulting extended T-mesh is shown Figure 3(b).

Theorem 1. For a design-suitable T-mesh T, extend the T-vertices in T as above. en there is a one-to-one correspondence between basis vertices in T e and vertices in T,
which means there are no extra basis vertices produced in T e by extending T-vertices in T appropriately.
Proof. We only need to deal with T-elements. We extend T-vertices element by element. We need to prove there are no edge-type intersections and face-type intersections in T e . In a       t)) be the basis functions associated with V i needed to be constructed. ere are mainly three steps for constructing B i (s, t) in [25]:  Figure 4 shows the support meshes of four basis vertices v 13 , v 17 , and v 4 in the extended T-mesh, where the support meshes are shaded. In Figure 4(a), v 13 is a basis vertex and among the neighbors of v 13 , v 48 is a T-vertex; thus, it is put into K i . en we check the neighbors of v 48 , and v 49 is a T-vertex which is put into K i . For v 49 , there are no more vertices needed to be added into K i . us, the set K i � v 13 , v 48 , v 49 . Similarly for v 17 in Figure 4 by N k (s, t), k � 0, 1, 2, 3, and en the geometric information of four basis functions B i (s, t) at V i is defined as the geometric information of N k (s, t), k � 0, 1, 2, 3. e geometric information of B i (s, t) at the other basis vertices in M v is set as zero. And the geometric information of B i (s, t) at T-vertices in M v is computed by C 1 constraints. According to the known geometric information of B i (s, t) at the vertices in M v , then B i (s, t) can be easily represented in Bézier form or B-splines form.

Polynomial Splines with Control Net Defined on T-Meshes.
With the help of index T-mesh, there is a one-to-one correspondence between vertices and basis functions. e index T-mesh is adopted in [3] for ease of constructing T-splines. It is constructed by plotting the knots at equally spaced intervals regardless of their actual spacing and labeling each knot line with integer values [3]. Figure 5(b) shows the index T-mesh of the mesh shown in Figure 5(a). Figure 5(c) shows a control net in R 3 corresponding to the T-mesh shown in Figure 5(a). Figure 5(d) shows the polynomial splines surface with control net Figure 5(c) defined on the design-suitable T-mesh Figure 5(a). Notice the knots in this paper are of multiplicity two; thus, if we view the T-mesh in index space, each vertex in T-meshes corresponds to four vertices in index T-mesh. In a design-suitable T-mesh, it happens that each vertex is associated with four basis functions. us, there is a one-to-one correspondence between vertices and basis functions exactly in a design-suitable T-mesh. For convenience, in this paper, we still use T-meshes for constructing T-meshes and local refinement. e index T-mesh is used only to explain the control net. For a design-suitable T-mesh T, we construct basis functions on the extended T-mesh T e as stated in section 3.2. e polynomial splines surface defined over T is defined as follows: where B k i (u, v) are basis functions constructed in section 3.2 and P k i ∈ R 3 are the corresponding control points. Figure 6 presents the polynomial splines model with control nets. e construction method is based on the above polynomial splines with control nets defined over T-meshes; we design a goblet model and a dolphin model; see Figures 7 and 8. For a design-suitable T-mesh, we define four basis functions at each vertex. If we need to adjust the surface part corresponding to a T-vertex, we can adjust the control point corresponding to the basis functions defined at the T-vertex to modify the surface directly. But for PHT-spline surface, it can only be adjusted indirectly by adjusting the control points corresponding to the basis functions defined at other basis vertices. Figure 9 shows the modification of the surface by adjusting the position of the control points corresponding to the T-vertices and the normal basis vertices.

Local Refinement
Given a design-suitable T-mesh T 1 , denote the extended T-mesh as T 1 e . e edge insertion is to insert edges into T 1 , the resulting T-mesh is denoted by T 2 . e local refinement is attributed to how to construct the extended T-mesh T 2 e such that S(3, 3, 1, 1, T 1 e ) ⊆ S(3, 3, 1, 1, T 2 e ), which is equivalent to T 1 e ⊆T 2 e . Suppose a vertical edge e � v 0 v 1 is inserted into a element F, and the 2-neighboring elements of F in vertical direction are denoted by F i , i � 1, 2, . . . , l (if existing). e local refinement algorithm is described as follows: is a CT-vertex or a ECT-vertex, then extend them until it is either a CT-vertex or ECT-vertex. We use Figure 10 to demonstrate the local refinement algorithm as proposed above. A design-suitable T-mesh T 1 is shown in Figure 10(a) and the corresponding extended

Complexity
T-mesh T 1 e is shown in Figure 10(b). A vertical edge v 0 v 1 (marked by red line) is inserted in T 1 , which is shown in Figure 10(c). Now v 0 is a CT-vertex and v 1 is a ECT-vertex; then they are extended as shown in Figure 10   e resulting T-mesh is a design-suitable T-mesh, denoted by T 2 . Finally, we construct the extended T-mesh corresponding to T 2 . We only need to update the extension of the new T-vertices. Figure 10(f) shows the extended T-mesh T 2 e . Figure 11 shows the local refinement of a diagonal T-mesh. e inserted edges are marked by red line segments and the additional inserted edges are marked by blue line

Conclusions
In this paper, we present an algorithm for extending bicubic C 1 polynomial spline spaces defined over any given T-meshes such that there is a one-to-one correspondence between basis functions and vertices in the T-meshes. e key idea is to extend the T-junctions such that the T-junctions become basis vertices. us how to extend these T-junctions to avoid extra basis vertices produced is the main challenge. In this paper, we proposed a subset of T-meshes called design-suitable T-meshes. For designsuitable T-meshes, we first extend T-vertices through an appropriate order and then construct basis functions on the extended T-meshes. ere is a one-to-one correspondence between basis functions and vertices in the T-mesh. Furthermore, it is easy to modify a given T-mesh such that it becomes a design-suitable T-mesh.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.