CN111914450A - AST spline surface rapid reconstruction algorithm based on local refinement solution - Google Patents

Abstract

The invention belongs to the field of three-dimensional surface topography measurement, and particularly discloses an AST spline surface fast reconstruction algorithm based on local refinement solution. The algorithm comprises the following steps: constructing an initial T grid and converting the initial T grid into an AST grid; solving the control vertex to obtain an AST spline surface and calculating the error of each data point; local refinement and local fitting solution are carried out on an AST grid surface patch where data points with errors not meeting a preset precision threshold are located; judging whether the error of each data point in the updated AST spline surface meets the requirement of a preset precision threshold, if so, finishing the AST spline surface fast reforming based on local refinement solution; if not, repeating the thinning process until the requirement of the preset precision threshold is met. According to the method, the initial T grid is constructed by utilizing the curvature characteristic information, so that the iteration times of later-stage local refinement are effectively reduced, and the reconstruction efficiency is improved; meanwhile, the conventional global fitting is replaced by the local solution fitting, so that the calculation efficiency is obviously improved.

Description

AST spline surface rapid reconstruction algorithm based on local refinement solution

Technical Field

The invention belongs to the field of three-dimensional surface topography measurement, and particularly relates to an AST spline surface fast reconstruction algorithm based on local refinement solution.

Background

Surface reconstruction refers to the inverse derivation of a mathematical expression model of the object from a set of discrete scan data points, which is very important in many fields, such as aviation parts, mold manufacturing and design, 3D printing, cultural relic restoration, and the like. The spline model is a common expression mode of a curved surface, such as B-splines and NURBS, and can represent any complex curved surface only by one control mesh and corresponding control vertexes, so that the spline model is widely applied to the fields of CAD and the like. NURBS is also a standard format for surface representation.

The traditional surface reconstruction algorithm fits point cloud data into a B spline surface or a NURBS surface. However, the B spline and the NURBS spline require that the control grid is a regular topological rectangle, the node vectors are globally shared, local refinement is difficult to realize, new nodes are added in rows and columns to meet the topological requirement when each node is introduced, and the nodes have no significance for controlling the curved surface. Therefore, when expressing complex curved surfaces with rich characteristic information, B-splines and NURBS curved surfaces generate a plurality of redundant nodes, which greatly influences the calculation and storage of the surface design and the surface reconstruction. To solve this problem, Sederberg proposed T-splines in 2003 and perfected the theory in 2004. Unlike B-spline curves and NURBS, T-spline curves allow control of T-nodes in the mesh, which allows the T-spline curve surface to be refined locally. When inserting a control point each time, only a few control points are needed to be introduced, and the row or column of the whole control point does not need to be spread, thereby greatly simplifying the representation of the curved surface and saving the storage space.

The T-spline breaks the topological requirements of B-spline curves and NURBS, realizes local refinement, and brings some problems. The linear independence and the unit decomposability of the T spline mixed function cannot be guaranteed, so that the optimization equation is not solved, and the calculation cost of the T spline surface is increased. Furthermore, in some cases, the local refinement of the T-spline curve is poor and not robust enough. To solve these problems, a T-spline (AST) suitable for analysis is proposed. AST splines are a subclass of T splines, with certain topological constraints. The mixing function of the AST spline has linear independence, and if boundary condition constraint is met, a uniform partition is formed. Then, a scholars gives a local thinning algorithm of the AST spline, which is more stable than the local thinning algorithm of the general T spline. After the AST spline is proposed, the method is applied to the field of isogeometric analysis. Although the AST spline has the advantages of high flexibility, high precision and high numerical stability, as a new technology, a data reconstruction algorithm for the AST spline is currently lacking. The AST spline is more complex than a T spline topological structure, so the reconstruction computing efficiency is lower than that of the T spline.

Disclosure of Invention

Aiming at the defects and/or improvement requirements in the prior art, the invention provides an AST spline surface rapid reconstruction algorithm based on local refinement solution, wherein the algorithm utilizes curvature characteristic information to construct an initial T grid, so that the iteration times of later local refinement can be effectively reduced, and the reconstruction efficiency is improved; meanwhile, the conventional global fitting is replaced by the local solution fitting, so that the calculation efficiency is obviously improved.

In order to achieve the purpose, the invention provides an AST spline surface rapid reconstruction algorithm based on local refinement solution, which comprises the following steps:

s1, performing feature distribution analysis on the surface sampling point cloud on a parameter domain or a sampling domain, thereby constructing an initial T grid;

s2 converting the initial T mesh into an AST mesh;

s3, solving the control vertex to obtain an AST spline surface and calculating the error of each data point in the AST spline surface;

s4, local refining and local fitting solving are carried out on an AST mesh surface patch where the data points with errors not meeting the preset precision threshold value are located, and therefore the control vertex and the AST spline surface are updated;

s5, judging whether the error of each data point in the updated AST spline surface meets the requirement of the preset precision threshold, if so, finishing the AST spline surface fast reforming based on local refining solution; if not, repeating the step S4 until the requirement of the preset precision threshold is met.

As a further preference, step S1 includes the following sub-steps:

s11, calculating the Gaussian curvature of the surface sampling point cloud, and taking the point with the absolute value larger than a preset threshold value as a characteristic point;

and S12, carrying out iterative subdivision on the parameter domain or the sampling domain until the number of the data points and the number of the characteristic points in each patch are smaller than the respective preset threshold value, so as to construct the initial T grid.

Further preferably, in step S12, iterative subdivision is performed by a bisection method or a quartering method.

As a further preference, in step S2, the initial T-grid is converted into an AST grid by performing an approximate minimum search using a greedy algorithm.

As a further preference, step S3 includes the following sub-steps:

s31, setting the weight values of all nodes as 1, and solving a control vertex by using a least square method;

and S32, calculating the error of each data point in the AST spline surface according to the control vertex.

As a further preference, step S4 includes the following sub-steps:

s41, carrying out reconstruction error analysis on the AST spline surface, and if the error of a certain data point exceeds a preset precision threshold, subdividing the local AST grid patch where the data point is located;

s42, converting the subdivided non-AST grids into AST grids;

s43, local fitting solution is carried out on the new control vertexes introduced by AST mesh local refinement and the affected control vertexes, so that the control vertexes and the AST spline surface are updated.

Further preferably, the subdivision in step S41 is performed by a bisection method or a quartering method.

Generally, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages:

1. according to the method, the initial T grid is constructed by utilizing the curvature characteristic information, so that the iteration times of later-stage local refinement can be effectively reduced, and the reconstruction efficiency is improved; meanwhile, the conventional global fitting is replaced by the local solution fitting, so that the calculation efficiency is obviously improved;

2. meanwhile, the linear independence and the unit decomposability of the mixed function are effectively reserved, so that spline expression is changed from a rational polynomial into a simple polynomial, the calculation efficiency and the numerical stability are improved, and the curved surface differential calculation is facilitated;

3. in addition, compared with the common B spline or NURBS curved surface reconstruction technology, for the same group of data, when the number of control vertexes is approximately the same, the method can realize real local refinement based on the topological property, thereby effectively reducing redundant control points, saving data storage space and reconstruction complexity;

4. in the aspect of precision, adaptive local subdivision and local fitting solution is iteratively performed according to surface feature distribution and reconstruction error analysis, AST reconstruction can better express the geometric features of the curved surface when processing complex curved surfaces with rich feature information, and the AST reconstruction has higher reconstruction precision compared with B splines and NURBS.

Drawings

Fig. 1 is a flow chart of an AST spline surface fast reconstruction algorithm based on local refinement solution constructed in accordance with a preferred embodiment of the present invention;

FIG. 2 is an example of T nodes in a T mesh and their node intervals;

fig. 3 is an AST grid example and the T node extensions are marked with dashed lines;

FIG. 4 is a data set and its corresponding parameters, where (A) is the data set and (B) is the corresponding parameters;

FIG. 5 is a comparison between a prior art algorithm and an initial T-grid constructed by the present invention, wherein (A) is an m × n initial T-grid used in the prior art algorithm, and (B) is an initial T-grid adaptively constructed by the present invention;

fig. 6 is a diagram of the conversion of a T mesh into an AST mesh, where (a) is the T mesh, (B) is the intermediate conversion T mesh, and (C) is the AST mesh.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

As shown in fig. 1, an embodiment of the present invention provides an AST spline surface fast reconstruction algorithm based on local refinement solution, which constructs an initial T-grid according to curvature information of an input point cloud and performs AST transformation on the initial T-grid to obtain an initial AST grid, solves a control vertex by using a least square method to obtain an AST spline surface, and continuously refines the AST grid according to an error until an error requirement is met, and finally generates an optimal reconstructed AST spline surface.

For example, given a set of surface sample point clouds V and an error threshold, the AST spline surface reconstruction can be described as: finding an AST spline surface S (u, v) such that the following inequality holds:

dist(S(u,v)-V_i)≤,V_i∈V (1)

for example, the AST spline surface fast reconstruction algorithm based on local refinement solution provided by the present invention may be applied to facial data acquisition, an initial T-mesh is calculated for a surface sampling point cloud V and a parameter domain or a sampling domain U thereof (as shown in fig. 4), and is converted into an AST mesh, then a control vertex is solved by using a least square method, an AST spline surface is obtained and an error is calculated, and local refinement of the AST mesh is performed at a position where the error does not satisfy a threshold until the error requirement is satisfied. The method specifically comprises the following steps:

(a) constructing initial T grid with characteristic adaptability according to characteristic points

Similar to B-splines and NURBS, T-splines are splines defined on a control mesh and its corresponding control vertices, and this control mesh is called a T-mesh. Unlike the control grids of B-splines and NURBS splines, where T-nodes are allowed to exist in the T-grid (as shown in fig. 2), control vertices in three-dimensional space correspond one-to-one to nodes in the T-grid. Given a T-mesh and corresponding control vertices, the specific expression S (u, v) for a T-spline can be written as:

wherein, w_iIs the weight, P_iIs controlling the vertex, and B_i(u, v) is a mixing function for each node, in the specific form:

B_i(u,v)＝N[u_i](u)N[v_i](v) (3)

wherein, N [ u ]_i](u) and N [ v ]_i](v) Are respectively defined in the following two node intervals u_iAnd v_iAbove 3-degree B-spline basis function:

u_i＝[u_i0,u_i1,u_i2,u_i3,u_i4] (4)

v_i＝[v_i0,v_i1,v_i2,v_i3,v_i4] (5)

wherein u is_iAnd v_iIs a node interval, u_i0～u_i4And v_i0～v_i4The u-direction and v-direction node elements, respectively.

Given a T grid, the node interval u of each node_iAnd v_iThe determination is as follows: in the T grid, let the coordinates of the node P be (u)_i2,v_i2) With the point as the origin, a ray is emitted to the right (in the direction in which u increases), and the coordinates of two edges which intersect with the ray first are u_i3And u_i4The remaining coordinates can be derived in a similar way.

By normalizing the parameter or sample domain U, all data points are mapped to [0,1 ]]Within the interval. Selecting one [0,1 ]]×[0,1]Is refined as the initial starting grid. The refinement of the grid is carried out by carrying out curvature analysis according to the quantity and the characteristics of parameter points in the regionThe number of dots. For points on a smooth surface, the maximum curvature is denoted as k₁The minimum curvature is denoted by k₂. These two curvatures are called principal curvatures, gaussian k ═ k₁k₂. And selecting a point with the Gaussian curvature absolute value larger than a preset threshold value as the characteristic point. For example, a data point having an absolute gaussian curvature located at the top 10% may be selected as the feature point.

A preset threshold N for a given number of data points_dAnd a preset threshold N for the number of feature points_fWhen the number of data points in a region patch exceeds N_dOr the number of feature points exceeds N_fThen, subdividing the area patch until the threshold requirement is met. The initial T grid obtained through curvature analysis and feature segmentation is closer to the final T grid in a topological structure, so that the iteration refinement times based on reconstruction errors in the later period can be reduced, the calculation time is shortened, and the purpose of quick fitting is achieved.

The subdivision method of the region mainly includes a dichotomy or a quartering method. The bisection method is to divide a region patch into two parts; the quartering rule divides it into four. For example, the dichotomy is to divide the long edges of the region patches. That is, let the coordinates of the lower left corner node of the region be (u)_min,v_min) The coordinate of the upper right corner node is (u)_max,v_max) If u is_max-u_min<v_max-v_minI.e. the vertical side is the long side, thus in

Dividing the plant; otherwise, it is just at

Is divided. This allows the demarcated regions to be more symmetrical than a flat rectangle (see figure 5). The quarter-division method uniformly divides a region patch into 4 small patches with the same shape and the same area by using a cross line.

(b) AST grid conversion

AST splines are a subclass of T splines, and some topological constraints need to be added, and these constraints are based on the concept of T node expansion. For each T node, the T node expansion refers to a line segment, and the node interval u and v of the T node are set as follows:

u＝[u₀,u₁,u₂,u₃,u₄] (6)

v＝[v₀,v₁,v₂,v₃,v₄] (7)

if the node is missing the right, the T node of the point is expanded into a line segment

If the node lacks the left side, the T node of the node is expanded into a line segment

If the node lacks the upper edge, the T node of the node is expanded into a line segment

If the node lacks the lower edge, the T node of the node is expanded into a line segment

For a given T grid, if the T node expansion in the horizontal direction and the T node expansion in the vertical direction are not intersected pairwise, the T grid is an AST grid; accordingly, the T-spline surface defined on this AST mesh is called an AST spline surface (see fig. 3).

Typically a T-grid can be converted into many AST grids, but it is necessary to find that simplest grid, or AST grid closest to the original grid, i.e. with the least number of nodes inserted for conversion. Here, a greedy strategy is adopted to seek a local optimal solution, and the specific method is as follows:

(b1) finding all crossed T node extensions and corresponding T nodes in the original T grid;

(b2) and performing primary extension on each T node in the last step along the missing direction of the T node to obtain a new T grid, and calculating the number of the T node extensions intersected in the new T grid. Doing so for each T node, and finding the T grid with the minimum number of the crossed T node expansion to replace the previous T grid;

(b3) and (c) repeating the step (b2) until no intersected T node expands, and obtaining the final AST grid (see figure 6).

(c) Least squares control point fitting optimization

After the AST grid is obtained, the weight w of all nodes is made to be 1, P is set to be a corresponding control vertex, and by utilizing the unit decomposability of the AST grid, an AST spline surface can be written into the following simple polynomial form:

the simple expression form brings simplification and improves the numerical stability for the surface fitting calculation; in addition, the simple polynomial has concise differential expression, and can bring convenience to curved surface differential analysis.

Given a set of m data point surface sampling point clouds V and a parameterization result parameter domain U thereof, it is desirable to find a set of control vertexes P so that the distance between the AST spline surface and the surface sampling point cloud V is minimum in the least-squares sense, that is:

in the formula, S (u)_j,v_j) For AST spline surfaces at parameter locations (u)_j,v_j) Coordinate value of (V)_jFor the measured coordinate values at the respective parameter locations, if the control vertices P and the surface sample point cloud V are written as column vectors, the optimization problem can be a least squares solution to solve the following over-determined system of equations:

AP＝V (10)

wherein, the AST spline surface is determined by the formula (8)A matrix of models, and A_i,j＝B_i(u_j,v_j)，B_i(u_j,v_j) A mixing function determined for equation (3); from the linear independence of the AST spline mixture function, the least-squares solution P of this equation^*Is unique, and can be obtained by calculating the MP generalized inverse matrix A of A⁺Obtaining, namely:

P^*＝A⁺V (11)

after the control vertex P is determined, the entire AST spline surface is completely determined, and the error E of all data points can also be calculated by the following formula:

E＝AP^*-V (12)

(d) local refinement and local solution

Assuming that there are m input data, the refined AST grid contains n nodes, where the newly introduced node and the node with changed node interval total r nodes, and the mixing function of the r nodes is nonzero at s data points. After refinement, only the s data points are used as fitting data, the control vertexes of the r nodes in the refined region are used as unknown quantities, and the control vertexes in the rest non-refined regions are used as known quantities. Taking s rows corresponding to the mixed function matrix A, and writing into a block matrix according to corresponding r columns, the equation set to be solved can be expressed as:

wherein, V_sMeasuring a data vector or matrix for s rows, A_s×rFor refining AST spline model matrix of newly added control peak and control peak with changed adjacent node interval in area, A_s×(n-r)Model matrix, P, for unaffected control vertices of the remaining node interval_n-rIs a control vertex of known quantity, P_rIf the control vertex is to be solved, the equation is shifted, and the least square solution of the equation is obtained

Comprises the following steps:

solving the formula only needs to calculate local unknown quantity, and the effect of accelerated fitting is realized. And after the control vertex is obtained, recalculating the error, and continuously repeating the steps until the error meets the requirement, so as to obtain the reconstructed AST spline surface.

The AST grid conversion also needs to be performed repeatedly in each local refinement, since the local refinement will generate a general T grid. And carrying out local fitting solution on new control points introduced by AST grid local refinement and control points with the affected mixing functions nearby. Namely, the control points which are not influenced by the mixing function far away from the newly added control point are used as known quantities, and local solution fitting is carried out on the control points to be updated only by using surface sampling data in the influenced area. The process realizes local subdivision and local solution, avoids global solution, greatly accelerates the calculation efficiency and shortens the reconstruction time.

It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (7)

1. An AST spline surface rapid reconstruction algorithm based on local refinement solution is characterized by comprising the following steps:

s1, performing feature distribution analysis on the surface sampling point cloud on a parameter domain or a sampling domain, thereby constructing an initial T grid;

s2 converting the initial T mesh into an AST mesh;

s3, solving the control vertex to obtain an AST spline surface and calculating the error of each data point in the AST spline surface;

s4, local refining and local fitting solving are carried out on an AST mesh surface patch where the data points with errors not meeting the preset precision threshold value are located, and therefore the control vertex and the AST spline surface are updated;

s5, judging whether the error of each data point in the updated AST spline surface meets the requirement of the preset precision threshold, if so, finishing the AST spline surface fast reforming based on local refining solution; if not, repeating the step S4 until the requirement of the preset precision threshold is met.

2. The AST spline surface fast reconstruction algorithm based on local refinement solution as claimed in claim 1, wherein the step S1 comprises the following sub-steps:

s11, calculating the Gaussian curvature of the surface sampling point cloud, and taking the point with the absolute value larger than a preset threshold value as a characteristic point;

and S12, carrying out iterative subdivision on the parameter domain or the sampling domain until the number of the data points and the number of the characteristic points in each patch are smaller than the respective preset threshold value, so as to construct the initial T grid.

3. The AST spline surface fast reconstruction algorithm based on local refinement solution as claimed in claim 2, wherein in step S12, the iterative refinement is performed by using a bisection method or a quartering method.

4. The AST spline surface fast reconstruction algorithm based on local refinement solution of claim 2, wherein in step S2, a greedy algorithm is used to perform an approximate minimum search, thereby converting the initial T-grid into an AST grid.

5. The AST spline surface fast reconstruction algorithm based on local refinement solution as claimed in claim 1, wherein the step S3 comprises the following sub-steps:

s31, setting the weight values of all nodes as 1, and solving a control vertex by using a least square method;

and S32, calculating the error of each data point in the AST spline surface according to the control vertex.

6. The AST spline surface fast reconstruction algorithm based on local refinement solution as claimed in any one of claims 1 to 5, wherein step S4 includes the following sub-steps:

s41, carrying out reconstruction error analysis on the AST spline surface, and if the error of a certain data point exceeds a preset precision threshold, subdividing the local AST grid patch where the data point is located;

s42, converting the subdivided non-AST grids into AST grids;

s43, local fitting solution is carried out on the new control vertexes introduced by AST mesh local refinement and the affected control vertexes, so that the control vertexes and the AST spline surface are updated.

7. The AST spline surface fast reconstruction algorithm based on local refinement solution as claimed in claim 6, wherein the subdivision is performed by using a dichotomy or a quartering method in step S41.

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