Surface reconstruction method for unordered point cloud based on adaptive learning neural network

Abstract

The application relates to an unordered point cloud surface reconstruction method based on an adaptive learning neural network, which is applied to the fields of geometric modeling and reverse engineering and the like, and comprises the following steps: mapping unordered space point clouds to a two-dimensional surface, determining point cloud parameter values by constructing a two-dimensional parameter surface; taking a bivariate B-spline base function as an activation function, and constructing a neural network representing a tensor product B-spline surface; according to the local support property of the B-spline, applying a Lagrange multiplier method to superimpose a tangent constraint condition into a corresponding item of a loss function; and iteratively updating to obtain B-spline surface control points. The application realizes point cloud parameterization by constructing a two-dimensional parameter surface, and reconstructs a B-spline surface based on a neural network, so that the tangent constraint condition can be met while the overall reconstruction precision is ensured.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of curved surface reconstruction, and in particular to a method for reconstructing a curved surface from unordered point cloud based on an adaptive learning neural network. BACKGROUND

[0002] In the fields of geometric modeling, reverse engineering and computer vision, reconstructing a free-form surface from scattered data points is a basic problem. Free-form surfaces can be roughly divided into implicit surfaces, subdivision surfaces and parametric surfaces according to their representation forms. Among them, parametric spline surfaces are widely welcomed due to their explicit mathematical expressions, easy derivation and integration, strong geometric intuition and other characteristics. Among them, B-spline has become one of the industry standards for geometric modeling due to its local control, convex hull and optimal continuity. Therefore, B-spline surface reconstruction is still an important research content of computer-aided geometric design.

[0003] Generally, B-spline surface reconstruction is divided into two tasks: surface parameterization and surface fitting. Both of them are crucial in order to obtain appropriate geometric figures and topological structures. Traditional surface reconstruction can be carried out in two ways: interpolation and approximation. When considering interpolation, the fitted surface will pass through the target points exactly. In this case, although the error of each target point (i.e. the distance between the target point and the fitted surface) will be greatly reduced, the surface shape between the target points cannot be well maintained. The purpose of the approximation method is to ensure the overall shape of the target surface and minimize the overall deviation between the target point set and the fitted surface.

[0004] In recent years, with the development of computer technology and the development of various advanced optimization algorithms, neural networks have also been used for surface reconstruction. By using the nonlinear approximation ability of neural networks, high-precision B-spline surface reconstruction can be achieved. However, the current method for reconstructing B-spline surfaces based on neural networks is not suitable for surface reconstruction problems with end point and tangent constraints due to the lack of constraint conditions. SUMMARY

[0005] The present application provides a method for reconstructing a curved surface from unordered point cloud based on an adaptive learning neural network, which aims to achieve high-precision surface reconstruction with end point and tangent constraints.

[0006] The technical solution adopted by the present application is as follows:

[0007] A method for reconstructing a curved surface from unordered point cloud based on an adaptive learning neural network, comprising:

[0008] Step one: obtaining an unordered spatial point cloud of a target object, mapping the unordered spatial point cloud to a two-dimensional curved surface, constructing a parametric curved surface that exactly matches the two-dimensional point cloud mapped to the two-dimensional curved surface, discretizing the parametric curved surface, determining the parameter values of the two-dimensional point cloud by finding the closest discrete points to the two-dimensional point cloud, performing sample sampling, and recording the parameter values and corresponding actual coordinates of the sample points;

[0009] Step two: constructing a three-layer neural network to represent the tensor product B-spline surface, including:

[0010] Taking the two-variable B-spline basis function as the activation function of the hidden layer of the neural network, taking the control point set of the tensor product B-spline surface as the weight coefficient of the neural network, taking the number of control points as the number of nodes of the hidden layer of the neural network, taking the parameter values of the sample points as the input of the neural network, and taking the point coordinates on the tensor product B-spline surface corresponding to the parameter values as the output of the neural network; introducing the unit tangent constraint of the tensor product B-spline surface at a specific parameter value into the square loss function as the loss function of the neural network;

[0011] Step three: training the neural network by inputting the sample points to minimize the loss function, to obtain the optimal weight coefficient, and thus obtain the optimal control point of the tensor product B-spline surface, realizing the reconstruction of the curved surface based on the tensor product B-spline surface that satisfies the tangent constraint.

[0012] Further technical solutions are:

[0013] In step two, the loss function of the neural network is constructed by using the Lagrange multiplier method, and the loss function also satisfies the end point constraint condition of the tensor product B-spline surface. For the x, y, z components of the actual coordinates corresponding to the sample point parameter values, the loss function is calculated independently, wherein the loss function L w is defined as follows:

[0014]

[0015] In the formula, Ω is the set of all sample points, Ω1 is the set of four end points in the sample points, Ω2 is the set of sample points with tangent constraints, κ is the penalty factor, and β is the weight of the tangent constraint part; the subscripts w represent one of x, y, and z, and x, y, and z are three coordinate axes corresponding to the actual coordinates;

[0016] w k represents the w coordinate component of the kth sample point;

[0017] s w (ξ k ,η k ) is the w coordinate component of the output S(ξ k ,η k ) of the kth sample point, and S(ξk ,η k ) is the parameter value of the kth sample point (ξ k ,η k ) corresponding to the tensor product B-surface point coordinates, and ξ and η are two parameter axes of the parameter surface;

[0018] s ξ w (ξ k ,η k ) represents the w component of the ξ directional partial derivative vector of the tensor product B-surface at the parameter value (ξ k ,η k ), S ξ (ξ k ,η k ) represents the ξ directional partial derivative vector of the tensor product B-surface at the parameter value (ξ k ,η k ), that is ||·|| represents the modulus;

[0019] u k,w is the w component of the unit tangent vector u k of the tensor product B-surface at the parameter value (ξ k ,η k ) of the kth sample point, and u k is a constant vector, and the expression is

[0020] In step one, the unordered space point cloud is mapped to a two-dimensional surface, and a parameter surface that accurately matches the two-dimensional point cloud mapped to the two-dimensional surface is constructed, comprising:

[0021] A least squares plane is fitted according to a plurality of points closest to the center of gravity in the unordered space point cloud, and the normal vector of the least squares plane is used as the projection direction for point cloud projection. A rotation matrix is used to convert three-dimensional coordinates of the projection plane into two-dimensional coordinates.

[0022] Boundary points of the two-dimensional point cloud are extracted, and the boundary points are respectively divided into four boundary lines in order. Control points on the four boundary lines are fitted as boundary control points of the surface, and internal control points are uniformly selected to construct a parameter surface that accurately matches the two-dimensional point cloud.

[0023] The boundary points of the two-dimensional point cloud are extracted, comprising:

[0024] Each point in the two-dimensional point cloud is traversed, and all points within a search circle having a specific radius value and centered on each point are found.

[0025] The angle α formed by the center of the circle and other points within the search circle is calculated.

[0026] The search circle is evenly divided into a plurality of first sector search areas with a central angle of theta, and the first sector search areas are sequentially searched along the circumference, if there is another point in any first sector search area, that is, the angle alpha falls into the first sector search area, a judgment value f1=1 is given, otherwise, if there is no other point in any sector search area, a judgment value f1=0 is given;

[0027] The search circle is evenly divided into a plurality of second sector search areas with a central angle of theta+delta, and the second sector search areas are sequentially searched along the circumference, if there is another point in any second sector search area, that is, the angle alpha falls into the second sector search area, a judgment value f2=1 is given, otherwise, if there is no other point in any sector search area, a judgment value f2=0 is given;

[0028] When f1*f2=0, the point is determined as a boundary point, the index position and the corresponding actual coordinates of the point in the two-dimensional point cloud are recorded, all points are traversed, and the two-dimensional point cloud boundary line is extracted.

[0029] The boundary points are sequentially divided into four boundary lines, including:

[0030] After obtaining the two-dimensional point cloud boundary line, a reference angle is set, the boundary points are traversed, points satisfying that the angle formed by adjacent two points is less than the reference angle are found, that is, end points, a reference point is set, the reference point is connected with the four end points respectively, and the boundary points are divided into four boundary lines.

[0031] The beneficial effects of the present application are as follows:

[0032] The present application meets the tangential constraint condition while maintaining the overall reconstruction accuracy of the B-spline surface, and is suitable for surface reconstruction with end point and tangential constraint.

[0033] The present application can better reflect the position relationship of adjacent points by constructing a two-dimensional parametric surface to determine the point cloud parameter value.

[0034] Compared with other B-spline surface reconstruction methods based on neural networks, the present application introduces the constraint condition into the loss function by the Lagrange multiplier method, so that the tangential and end point constraint conditions are met while ensuring the overall reconstruction accuracy.

[0035] The present application applies a square loss function and uses the gradient descent method for parameter updating, which is essentially also the principle of least squares fitting, and when the number of iterations is sufficient, the surface reconstruction accuracy is higher. The traditional least squares surface approximation method adopts the average standard accumulation chord length parameter method, and there is an overall error in the calculation of the parameter value of the point cloud. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 The flowchart of the method of the present application.

[0037] Figure 2A neural network structure for representing a tensor product B-spline surface of the application.

[0038] Figure 3 A schematic diagram of the principle of point cloud boundary extraction of an embodiment of the application.

[0039] Figure 4 A schematic diagram of the point cloud shape in the parameter surface construction process of an embodiment of the application.

[0040] Figure 5 A schematic diagram of boundary extraction and classification of an embodiment of the application.

[0041] Figure 6 A surface reconstruction error cloud chart of the method (method 2) of the application and the traditional method (method 3) in the unconstrained case of an embodiment of the application.

[0042] Figure 7 A schematic diagram of the tangential constraint position of the point cloud model of an embodiment of the application.

[0043] Figure 8 A surface reconstruction error cloud chart of method 1 of an embodiment of the application. DETAILED DESCRIPTION

[0044] The specific embodiments of the application are described below with reference to the accompanying drawings.

[0045] The method for reconstructing a surface of unordered point cloud based on an adaptive learning neural network of the application comprises:

[0046] Step 1: Obtain the unordered space point cloud of the target object, map the unordered space point cloud to a two-dimensional surface, construct a parameter surface that accurately matches the two-dimensional point cloud mapped to the two-dimensional surface, discretize the parameter surface, determine the parameter value of the two-dimensional point cloud by finding the nearest discrete point to the two-dimensional point cloud, sample the samples, and record the parameter value and the corresponding actual coordinates of the sample points;

[0047] Step 2: Construct a three-layer neural network to represent the tensor product B-spline surface, comprising:

[0048] Use the bivariate B-spline basis function as the activation function of the hidden layer of the neural network, use the control point set of the tensor product B-spline surface as the weight coefficient of the neural network, use the number of control points as the number of nodes of the hidden layer of the neural network, use the parameter value of the sample points as the input of the neural network, and use the point coordinates on the tensor product B-spline surface corresponding to the parameter value as the output of the neural network; introduce the unit tangent vector of the tensor product B-spline surface at a specific parameter value as the tangential constraint into the square loss function as the loss function of the neural network;

[0049] Step three: training the neural network by inputting sample points to minimize the loss function to obtain the optimal weight coefficient, so as to obtain the optimal control point of the tensor product B-spline surface, and realize the surface reconstruction based on the tensor product B-spline surface meeting the tangent constraint.

[0050] The unordered point cloud surface reconstruction method based on the adaptive learning neural network has high overall reconstruction accuracy.

[0051] In step two, the loss function of the neural network is constructed by using the Lagrange multiplier method, and the loss function also meets the end point constraint condition of the tensor product B-spline surface. For the x, y and z components of the actual coordinates corresponding to the sample point parameter value, the loss function is calculated independently, wherein the loss function L of the w component w The definitions are as follows:

[0052]

[0053] In the formula, Ω is the set of all sample points, Ω1 is the set of four end points in the sample points, Ω2 is the set of sample points with tangent constraint, κ is the penalty factor, β is the weight of the tangent constraint part; the upper and lower indexes w represent one of x, y and z, x, y and z are three coordinate axes corresponding to the actual coordinates;

[0054] w k represents the w coordinate component of the kth sample point;

[0055] s w (ξ k ,η k ) is the w coordinate component of the output S(ξ k ,η k ) of the kth sample point, S(ξ k ,η k ) is the point coordinate of the tensor product B-spline surface corresponding to the parameter value (ξ k ,η k ) of the kth sample point, ξ and η are two parameter axes of the parameter surface;

[0056] represents the w component of the ξ directional partial derivative vector of the tensor product B-spline surface at the parameter value (ξ k ,η k ), S ξ (ξ k ,η k ) represents the ξ directional partial derivative vector of the tensor product B-spline surface at the parameter value (ξ k ,η k ), that is ||·|| represents the modulus;

[0057] uk,w The w component of the unit tangent vector u k at the kth sample point of the tensor product B-spline surface with parameter value (ξ k ,η k ) is u k , and u i,p is a constant vector, and the expression is

[0058] The concept and principle of constructing the above loss function are as follows:

[0059] According to the definition of the tensor product B-spline surface:

[0060]

[0061] In formula (2), N j,q (ξ) and M i,j (η) are the i-th p-th univariate B-spline basis function on the node vector Ξ1 and the j-th q-th univariate B-spline basis function on Ξ2, respectively, m and n are the number of control points of the parameter axis ξ and the parameter axis η, respectively, P c is the control point of the tensor product B-spline surface;

[0062] Using the global index A = n(j-1) + i, the above formula can be written as:

[0063]

[0064] In formula (3), B is a bivariate B-spline basis function, n k = m x n is the total number of control points of the tensor product B-spline surface;

[0065] The tensor product B-spline surface can be regarded as a linear combination of bivariate B-spline basis functions B , and the combination coefficients can be regarded as the control point set P Each set of parameter values (ξ, η) corresponds to a point S(ξ, η) on the tensor product B-spline surface.

[0066] Formula (3) is represented by a three-layer neural network as shown in formula (4): The control point set of the tensor product B-spline surface can be used as the weight coefficient of the neural network, the number of control points can be used as the number of nodes of the hidden layer of the neural network, the parameter values of the sample points can be used as the input of the neural network, and the coordinates of the points on the tensor product B-spline surface corresponding to the parameter values can be used as the output of the neural network. Figure 2

[0067] Specifically, the input is the kth sample point parameter value (ξ k ,η k ), the weight from the input layer to the hidden layer is fixed as 1, the activation function in the hidden layer is the bivariate B-spline basis function, and the output is the point in the B-spline surface corresponding to the parameter value (ξ k ,ηk ) corresponding point coordinates S(ξ k ,η k ).

[0068] Based on the local support of B-spline, the Lagrange multiplier method can be applied to superimpose the tangent constraint condition into the corresponding term of the loss function of the neural network, including:

[0069] B-spline has local support:

[0070]

[0071] In formula (4), the interval [u i , u i+p+1 ] is the support interval of the i-th p-th single variable B-spline basis function N i,p (u); a point on the p-th B-spline curve is related to at most p+1 control points, and by analogy, a point on the B-spline surface with p-th and q-th parameters is related to at most (p+1)×(q+1) control points, therefore, when ξ∈[ξ i ,ξ i+1 ],η∈[η j ,η j+1 ], the ξ-direction partial derivative vector of the B-spline surface can be expressed as:

[0072]

[0073] Wherein, Ψ represents the serial number set of the two-variable B-spline basis functions (same control points) that have an effect at the parameter value, which can be expressed as:

[0074]

[0075] The number of serial numbers in Ψ is (p+1)(q+1), which represents the number of two-variable B-spline basis functions (the number of control points) that affect a point on the surface.

[0076] For the tangent constraint at the parameter value (ξ k ,η k ), the tensor product B-spline surface at the parameter value can be expressed as the unit tangent vector of the surface equal to a constant vector u k =(u k,x ,u k,y ,u k,z ), which can be expressed as:

[0077]

[0078] For the B-spline surface, the end point constraint also needs to be satisfied, that is, at the four end points of the surface, the control points are interpolated to the data points, as shown in the following formula:

[0079]

[0080] The m*n control points of the B-spline surface constitute a control grid, P1, P m 、 respectively represent the control point of the first row and the first column, the control point of the first row and the last column, the control point of the last row and the first column, and the control point of the last row and the last column.

[0081] For any point of the B-spline surface, the calculation of the coordinate components is relatively independent. In the process of updating the neural network parameters, different coordinates are also independent of each other. The present application adopts a square loss function and uses the gradient descent method to update the parameters. For the unconstrained case, the loss function of the w component (w represents the x, y, and z components) is defined as follows:

[0082]

[0083] where s w (ξ k ,η k ) are the w coordinate components of the output layer S(ξ k ,η k ) corresponding to the kth sample, w k represents the w coordinate component of the kth sample point, p j,w represents the w coordinate component of the jth control point, represents the bivariate B-spline basis function of the jth control point corresponding to the kth sample parameter.

[0084] By adding a penalty factor κ (10-20) to the corresponding term in the loss function, the influence of the endpoint error is amplified, and the purpose of endpoint interpolation is achieved to some extent. According to the principle of Lagrange multiplier method, the endpoint interpolation constraint and the tangential constraint condition are introduced into equation (9), and the modified loss function as shown in equation (1) is obtained. In equation (1), β represents the weight of the tangential constraint part in the loss function, which is used to modify the influence of different dimensions. Adjust β based on different size point cloud models to balance the distance error and the tangential constraint.

[0085] By minimizing the loss function shown in equation (1), the weight coefficients (i.e. control points) are obtained, so that the B-spline surface satisfying the tangential constraint condition is reconstructed.

[0086] In step one, the unordered space point cloud is mapped to a two-dimensional surface, and a parameter surface that accurately matches the two-dimensional point cloud mapped to the two-dimensional surface is constructed, comprising:

[0087] A least square plane is fitted according to a plurality of points closest to the center of gravity in the disordered space point cloud, a normal vector of the least square plane is taken as a projection direction for point cloud projection, and a rotation matrix is used to convert three-dimensional coordinates of the projection plane into two-dimensional coordinates.

[0088] Boundary points of the two-dimensional point cloud are extracted, the boundary points are respectively divided into four boundary lines in sequence, control points on the four boundary lines are fitted as boundary control points of the curved surface, internal control points are uniformly selected, and a parameter curved surface accurately matched with the two-dimensional point cloud is constructed.

[0089] The extracting the boundary points of the two-dimensional point cloud comprises:

[0090] (1) each point in the two-dimensional point cloud is traversed, and all points in a search circle with a specific radius value and each point as a center are found;

[0091] (2) the angle α formed by the center and other points in the search circle is calculated;

[0092] (3) the search circle is equally divided into a plurality of first sector search areas with a center angle θ, the first sector search areas are sequentially searched along the circumference, if there are other points in any first sector search area, that is, the angle α falls into the first sector search area, a judgment value f1 = 1 is given, otherwise, if there are no other points in any sector search area, a judgment value f1 = 0 is given;

[0093] (4) the search circle is equally divided into a plurality of second sector search areas with a center angle θ + △δ, the second sector search areas are sequentially searched along the circumference, if there are other points in any second sector search area, that is, the angle α falls into the second sector search area, a judgment value f2 = 1 is given, otherwise, if there are no other points in any sector search area, a judgment value f2 = 0 is given;

[0094] (5) when f1 * f2 = 0, the point is determined as a boundary point, the index position and the corresponding actual coordinates of the point in the two-dimensional point cloud are recorded, all points are traversed, and the boundary line of the two-dimensional point cloud is extracted.

[0095] The above method for extracting the boundary points of the two-dimensional point cloud can obtain the principle of the boundary points, as shown in Figure 3 It is observed that a circle is constructed by taking an arbitrary point on the boundary as a center and taking an appropriate radius, and there is a part of continuous area in which there is no other point, and the internal point does not have the feature, and the feature is taken as a boundary condition.

[0096] Specifically, θ is 90°, that is, the search circle is divided into four parts, and each time a sector with a central angle of 90° is searched. As long as there are other points, for any angle α, it can fall into the search area. If there are no other points, it cannot be searched. However, there may be a continuous region without other points inside, but it spans two first sector search areas, and there are other points in both areas. Therefore, a deflection angle △δ is set, and the second sector search area is searched again as a unit to ensure comprehensive search information.

[0097] Those skilled in the art can understand that the values of △δ and θ can be set according to the shape of the unordered spatial point cloud of the target object.

[0098] The dividing the boundary points into four boundary lines in sequence respectively includes:

[0099] After obtaining the boundary lines of the two-dimensional point cloud, a reference angle is set, the boundary points are traversed, and points that satisfy that the angle formed with adjacent two points is less than the reference angle are found, that is, end points. A reference point is set, the reference point is connected with the four end points respectively, and the boundary points are divided into four boundary lines.

[0100] Those skilled in the art can understand that the reference angle and the reference point can be set according to the shape of the unordered spatial point cloud of the target object.

[0101] The fitting four boundary lines respectively, taking the control points on the four boundary lines as the boundary control points of the curved surface, and uniformly selecting internal control points to construct a parameter curved surface that accurately matches the two-dimensional point cloud includes:

[0102] According to the traditional least square curve approximation algorithm, the four boundary lines are fitted respectively: the parameter value of the data point is determined by the accumulated chord length parameterization method; the node vector is determined by the average technique; all unknown control points are obtained by solving the least square minimization;

[0103] Taking the control points of the four boundary lines as the boundary control points of the parameter curved surface, uniformly selecting internal control points, and determining the node vector by the average technique, the parameter curved surface constructed in this way can accurately match the point cloud; the parameter curved surface is discretized, and the parameter value of the point cloud is determined by finding the nearest discrete point to the original point cloud; a suitable sampling interval is selected, and uniform sampling is performed to record the parameter value and actual coordinates of the sampling points.

[0104] The technical solutions of the present application are further illustrated by specific embodiments.

[0105] Step 1: Map the unordered spatial point cloud to a two-dimensional curved surface through projection and rotation, and realize point cloud parameterization by constructing a two-dimensional parameter curved surface. For example, Figure 4As shown, find the N points in the point cloud closest to the centroid, and fit a least-squares plane based on these N points. Use the normal vector of this plane as the projection direction. Figure 4 As shown in (a); by selecting a point on the projection plane, the projection plane can be determined and the spatial point cloud can be projected, as shown in (a). Figure 4 As shown in (b); for ease of analysis, a rotation matrix is ​​used to convert the three-dimensional coordinates of the projection plane into two-dimensional coordinates, as follows. Figure 4 As shown in (c).

[0106] After completing the two-dimensional mapping of the spatial point cloud, the next step is to construct a parametric surface. To ensure an accurate match between the parametric surface and the point cloud, the point cloud boundary must first be extracted, i.e., the data corresponding to the parameters (ξ=0 or 1,η) and (ξ,η=0 or 1).

[0107] Boundary points are extracted based on the boundary features of the point cloud. The results of the boundary point extraction are as follows: Figure 5 As shown in (a).

[0108] After obtaining all boundary points, the next step is to extract the four endpoints. Compared to other boundary points, endpoints form smaller angles with their adjacent points. We define an angle ρ; any point whose angle with its two adjacent points is less than ρ is considered an endpoint. Figure 5 The endpoint extraction result is shown in (b).

[0109] After extracting the four endpoints, the boundary points need to be classified into top, bottom, left, and right edges, such as... Figure 5 As shown in (c): by determining a reference point C and connecting point C with the four endpoints, the boundary points can be divided into four parts; by obtaining the vector formed by point C and all the boundary points, the classification of the boundary points can be completed.

[0110] To ensure the point cloud boundaries match the parametric surface boundaries, firstly, using the traditional least-squares curve approximation algorithm, curve fitting is performed on the four boundaries of the point cloud, and these are used as boundary control points of the parametric surface. Internal control points are then uniformly selected, and an averaging technique is employed to determine the node vectors. Figure 5 The control points of the parametric surface are shown in (d).

[0111] Once the parametric surface is constructed, the parameter value corresponding to each data point in the point cloud can be determined. The specific method is as follows: First, discretize the parametric surface and find the discrete point closest to the original data points to determine the parameter value of the point cloud. Then, select an appropriate sampling interval, perform uniform sampling, and record the parameter value and actual coordinates of the sampled points.

[0112] Step 2: Using bivariate B-spline basis functions as activation functions, construct a ternary neural network to represent the B-spline surface:

[0113] For the point cloud model used in the present application, the number of sampling points is 51x51, and uniform sampling is performed along two parameter directions, i.e., in the neural network structure representing the B-spline surface, the number of training samples is 2601; the activation function in the neural network is a double 3rd order double variable B-spline basis function, the number of control points is set to 11x11, and the number of neurons in the hidden layer is 11x11. Based on the local support of the B-spline, the Lagrange multiplier method is applied to introduce the tangential constraint condition into the corresponding term of the loss function, and the loss function shown in formula (1) is constructed, and the loss function is minimized by the training data to obtain the weight coefficient (i.e., the control point), so as to reconstruct the B-spline surface satisfying the tangential constraint condition.

[0114] In order to verify the feasibility of the method of the present application, the method of the present application (denoted as method 1), and the method of the present application using the loss function without tangential constraint (denoted as method 2), and the traditional least square surface approximation algorithm (first least square curve fitting along one parameter direction, then fitting the generated intermediate control points along the other parameter direction, denoted as method 3) are used to perform surface reconstruction on the point cloud model, and the influence of the weight coefficient of the constraint part in the loss function is analyzed. The results are as follows:

[0115] The surface reconstruction results obtained by applying methods 2 and 3 respectively are shown in Figure 6 Figure 6 Fig. (a) is a point cloud model, Figure 6 Fig. (b) is an error cloud diagram of surface reconstruction using method 3, Figure 6 Fig. (c) is an error cloud diagram of surface reconstruction using method 2. The error analysis of the reconstruction results is performed, the iteration number is set to 100000 times, and the minimum, maximum and average distance errors are counted, as shown in Table 1:

[0116] Table 1 Surface reconstruction error

[0117]

[0118] From the maximum distance error and the average distance error, it can be seen that the reconstructed surface obtained by applying method 2 has high overall reconstruction accuracy.

[0119] The point cloud model with tangential constraint of method 1 is reconstructed and analyzed, as shown in Figure 7 Fig. (a) is a point cloud model, Fig. (b) is an error cloud diagram of surface reconstruction using method 3,

[0120] Fig. (c) is an error cloud diagram of surface reconstruction using method 2. The constraint part weight coefficient β is adjusted, and the average distance error of the reconstructed surface and the average error angle of the unit tangent vector are observed, as shown in Table 2:

[0121]

[0122] The weight coefficient is 0.3, and the specific constraints and errors are shown in Tables 3 and 4:

[0123] According to Tables 3 and 4, at the specified parameter position, the tangential constraint condition can be fully met, and the maximum error angle in the tangent direction is not more than 0.2 degrees.

[0124] The error cloud chart of the surface reconstruction of method 1 is shown in Figure 8 Compared with the unconstrained case, the maximum, minimum and average error distances are increased to a certain extent, but the overall still has high accuracy.

[0125] Table 3 ξ-tangent vector constraint

[0126]

[0127] Table 4 ηξ-tangent vector constraint

[0128]

[0129] The present application determines the point cloud parameter value by constructing a two-dimensional parameter surface, which better reflects the position relationship of adjacent points; a neural network representing the B-spline surface is constructed, and based on the local support of the B-spline, the Lagrange multiplier method is applied to introduce the tangential constraint condition into the loss function. The results show that the tangential constraint condition can be met while ensuring the overall reconstruction accuracy.

Claims

1. A method for reconstructing unordered point cloud surfaces based on adaptive learning neural networks, characterized in that, include: Step 1: Obtain the disordered spatial point cloud of the target object, map the disordered spatial point cloud onto a two-dimensional surface, construct a parametric surface that precisely matches the two-dimensional point cloud mapped onto the two-dimensional surface, discretize the parametric surface, determine the parameter values ​​of the two-dimensional point cloud by finding the discrete point closest to the two-dimensional point cloud, perform sample sampling, and record the parameter values ​​and corresponding actual coordinates of the sample points. Step 2: Construct a three-layer neural network to represent the tensor product B-spline surface, including: The bivariate B-spline basis function is used as the activation function of the hidden layer of the neural network. The control point set of the tensor product B-spline surface is used as the weight coefficient of the neural network. The number of control points is used as the number of nodes in the hidden layer of the neural network. The parameter value of the sample point is used as the input of the neural network. The coordinates of the point on the tensor product B-spline surface corresponding to the parameter value are used as the output of the neural network. The unit tangent vector of the tensor product B-spline surface at a specific parameter value is used as the tangential constraint and introduced into the squared loss function as the loss function of the neural network. Step 3: Train the neural network by inputting sample points to minimize the loss function and obtain the optimal weight coefficients, thereby obtaining the optimal control points of the tensor product B-spline surface and realizing the surface reconstruction based on the tensor product B-spline surface that satisfies the tangential constraints.

2. The method for reconstructing unordered point cloud surfaces based on adaptive learning neural networks according to claim 1, characterized in that, In step two, the loss function of the neural network is constructed using the Lagrange multiplier method. This loss function also satisfies the endpoint constraints of the tensor product B-spline surface. For the x, y, and z components of the actual coordinates corresponding to the sample point parameter values, the loss function is calculated independently. The loss function L for the w component... w The definition is as follows: In the formula, Ω is the set of all sample points, Ω1 is the set of four endpoints among the sample points, Ω2 is the set of sample points with tangential constraints, κ is the penalty factor, and β is the weight of the tangential constraint part; the superscript and subscript w both represent one of x, y, z, where x, y, z are the three coordinate axes corresponding to the actual coordinates; w k This represents the w-coordinate component of the k-th sample point; s w (ξ k ,η k S(ξ) is the output of the k-th sample point. k ,η k The w-coordinate component of ξ, S(ξ) k ,η k ) is the parameter value (ξ) of the kth sample point. k ,η k The coordinates of the points on the tensor product B-spline surface corresponding to ξ and η are the two parametric axes of the parametric surface, respectively. This indicates that the tensor product B-spline surface is in the range of parameter values ​​(ξ). k ,η k The w component of the ξ-directed partial derivative vector, S ξ (ξ k ,η k ) represents the tensor product B-spline surface at parameter value (ξ) k ,η k The partial derivative vector of ξ at point ) is, i.e. ||·|| indicates modulo; u k,w Let ξ be the parameter value of the tensor product B-spline surface at the k-th sample point. k ,η k The unit tangent vector u at ) k The w component, u k For a constant vector, the expression is:

3. The method for reconstructing unordered point cloud surfaces based on adaptive learning neural networks according to claim 1, characterized in that, In step one, mapping the disordered spatial point cloud onto a two-dimensional surface and constructing a parametric surface that precisely matches the two-dimensional point cloud mapped onto the two-dimensional surface includes: A least squares plane is fitted based on the multiple points closest to the centroid in the unordered spatial point cloud. The normal vector of the least squares plane is used as the projection direction to project the point cloud. The three-dimensional coordinates of the projection plane are converted into two-dimensional coordinates using a rotation matrix. The boundary points of the 2D point cloud are extracted, and the boundary points are divided into four boundaries in sequence. The four boundary lines are fitted respectively, and the control points on the four boundary lines are used as the boundary control points of the surface. The internal control points are selected uniformly to construct a parametric surface that accurately matches the 2D point cloud.

4. The method for reconstructing unordered point cloud surfaces based on adaptive learning neural networks according to claim 3, characterized in that, The extraction of boundary points from the two-dimensional point cloud includes: Traverse each point in the 2D point cloud and find all points within a search circle centered at each point and having a specific radius value. Calculate the angle α formed by the center of the circle and other points within the search circle; Divide the search circle into several first sector search areas with a central angle of θ. Search the first sector search areas sequentially along the circumference. If there are other points in any first sector search area, i.e., angle α falls into the first sector search area, then assign a judgment value f1 = 1; otherwise, there are no other points in any sector search area, and assign a judgment value f1 = 0. Divide the search circle into several second sector search areas with a central angle of θ + Δδ. Search the second sector search areas sequentially along the circumference. If there are other points in any second sector search area, i.e., angle α falls into the second sector search area, then assign a judgment value f2 = 1; otherwise, there are no other points in any sector search area, and assign a judgment value f2 = 0. When f1*f2=0, the point is identified as a boundary point. The index position and corresponding actual coordinates of the point in the two-dimensional point cloud are recorded. All points are traversed to extract the boundary line of the two-dimensional point cloud.

5. The method for reconstructing unordered point cloud surfaces based on adaptive learning neural networks according to claim 4, characterized in that, The step of dividing the boundary points into four boundary lines in sequence includes: After obtaining the boundary line of the two-dimensional point cloud, a reference angle is set, and the boundary points are traversed to find the point that satisfies the condition that the angle formed by the two adjacent points is less than the reference angle. This point is the endpoint. Another reference point is set, and the reference point is connected to the four endpoints respectively to complete the division of the boundary points into four boundary lines.

Images