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The Optimization Model of Public Space Design Teaching Reform Based on Fractional Differential Equations

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
Páginas: -
Recibido: 08 Feb 2022
Aceptado: 18 Apr 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

Point cloud data often implies potential shapes. Its geometric differential information is the basis of point cloud geometric modeling and complex applications. Research the calculation of geometric differential information of point cloud curve and the matching between point cloud space curves. Here the point cloud space curve matching refers to the given two curves of the point set to be matched [1]. The rigid body (transformation) makes a certain part of one piece coincide with a certain part of the other one. Point cloud curve matching is widely used in pattern recognition, computer vision, intelligent detection of graphics and images, and cultural relics restoration.

Some scholars perform matching by extracting feature points and interpolation. Some scholars used arc length and curvature to match the curve but did not fully consider the directional characteristics of the curve. Some scholars detect the characteristic points of the curve according to the curvature and torsion and divide them into several segments for matching [2]. Some scholars first smoothly fit the dispersion curve. The sample on the obtained fitting curve calculates the sampling point's curvature and the Frenet frame and then aligns the Frenet frame. Calculate the sum of the squared distances of the matching points as the error to select the best match. The implementation of the above matching method requires obtaining the expression of the curve in advance. If the input data is a point set, it is generally necessary to obtain the curve expression through fitting or interpolation. However, it is difficult to fit the spatial curve of disordered point sets (especially point cloud data with noise). The fitting curve obtained often has lost a considerable part of the characteristic information. Therefore, subsequent analysis and matching will be distorted.

We establish the relevant theoretical framework for calculating the differential information of the point cloud space curve and propose a new point cloud space curve part matching method based on this. This method does not need to obtain the curve expression in advance. This method can avoid curve fitting errors and greatly improve the matching accuracy. Numerical examples verify the point cloud curve differential information calculation [3]. Numerical experiments also show that the differential information calculation and curve matching methods can be well applied to point cloud data with noise. This method effectively realizes the high-precision matching of the point cloud space curve.

The theoretical framework of point cloud space curve differential information calculation

The latent shape is a three-dimensional data point set P={pj}j=1N P = \left\{{{p_j}} \right\}_{j = 1}^N of a certain spatial curve. Suppose RxP is the data point closest to X. By establishing a local moving least squares model, the vertical foot of point Rx on the potential curve and its differential information (OX;t,n,b,κ,τ) can be obtained. Where (t, n, b) is the Frenet frame with the potential curve at OX. κ is the curvature and τ is the torsion.

Calculation of Frenet frame

Give the data point RxP closest to X obtained by the ANN search method. Take Rx k neighboring points from near to far as N(RX) ={p0, p1,⋯, pk−1}⊂ P. We assume that OX is the foot of Rx on the latent curve, and the Frenet frame at that place is (t, n, b), then the expression OX=RX+αn+βb {O_X} = {R_X} + \alpha n + \beta b

Where α, β is a real number. We need to request OX and its frame (t, n, b). Considering the weighted sum of the squared distances from the local neighbors of Rx to the close plane, the following moving least squares model is established minPjN(RX)[(pjOX)Tb]2θjs.t.bTb=1 \eqalign{& \min \sum\limits_{{P_j} \in N({R_X})} {{{\left[{{{\left({{p_j} - {O_X}} \right)}^T}b} \right]}^2}{\theta _j}} \cr & s.t.\,\,\,{b^T}b = 1 \cr}

Here the Gauss kernel is used to define the weight factor θj=θ(pjRX)=exp(pjRX2h2) {\theta _j} = \theta \left({\left\| {{p_j} - {R_X}} \right\|} \right) = \exp \left({- {{{{\left\| {{p_j} - {R_X}} \right\|}^2}} \over {{h^2}}}} \right)

Among them h is the local parameter, let h=p[k3]RX h = \left\| {{p_{\left[{{k \over 3}} \right]}} - {R_X}} \right\| generally. If the expression (1) of the foot OX is substituted into the model expression (2), we get: minPjN(RX)[(pjRX)Tbβ]2θjs.t.bTb=1 \eqalign{& \min \sum\limits_{{P_j} \in N({R_X})} {{{\left[{{{\left({{p_j} - {R_X}} \right)}^T}b - \beta} \right]}^2}{\theta _j}} \cr & s.t.\,\,\,{b^T}b = 1 \cr}

In the above model, β, b is an unknown variable, which is easily obtained by the Lagrange multiplier method β=(PjN(RX)θj(pjRX)Tb)PjN(RX)θj=LTb \beta = {{\left({\sum\limits_{{P_j} \in N({R_X})} {{\theta _j}{{\left({{p_j} - {R_X}} \right)}^T}b}} \right)} \over {\sum\limits_{{P_j} \in N({R_X})} {{\theta _j}}}} = {L^T}b . Then we can get the minimum eigenvalue problem minbAbs.t.bTb=1 \eqalign{& \min bAb \cr & s.t.\,\,\,\,{b^T}b = 1 \cr}

A=PjN(RX)(pjRXL)(pjRXL)Tθj A = \sum\limits_{{P_j} \in N({R_X})} {({p_j} - {R_X} - L){{\left({{p_j} - {R_X} - L} \right)}^T}{\theta _j}} . Assuming that the three characteristic values of A are λ1λ2λ3, then there are {At=λ1tAn=λ2nAb=λ3b \left\{\matrix{At = {\lambda _1}t \hfill \cr An = {\lambda _2}n \hfill \cr Ab = {\lambda _3}b \hfill \cr} \right.

We can get the corresponding Frenet frame (t, n, b) by performing SVD decomposition on the 3×3 matrix A. We further calculate α= LTn, β= LTb to get the vertical foot OX on the potential curve [4]. Figure 1 shows an example of Frenet frame calculation for the point cloud space curve.

Figure 1

Calculation of the Frenet frame

The calculation of normal n and subnormal b may be numerically unstable. Now assume t3 ≠ −1 according to the obtained tangent direction t = (t1, t2, t3)T: n=(t121+t31t1t21+t3t1) n = \left({\matrix{{{{t_1^2} \over {1 + {t_3}}} - 1} \cr {{{{t_1}{t_2}} \over {1 + {t_3}}}} \cr {{t_1}} \cr}} \right) , b=(t1t21+t3t221+t31t2) b = \left({\matrix{{{{{t_1}{t_2}} \over {1 + {t_3}}}} \cr {{{t_2^2} \over {1 + {t_3}}} - 1} \cr {{t_2}} \cr}} \right) .

Estimation of curvature and torsion

Suppose r(s) is the arc length parameter representation of the potential curve of the point cloud. At a point r(s0) on the curve, the curve has a third-order asymptotic expansion r(s)=r(0)sr(0)+s22r(0)+s36r(0)+o(s3) r(s) = r(0)sr'(0) + {{{s^2}} \over 2}r''(0) + {{{s^3}} \over 6}r''(0) + o({s^3}) . We use the Frenet formula and take the standard orthogonal frame of the three-dimensional space as the Frenet frame {r (0); t, n, b} of the curve, then the components of r(s) are: x(s)=sκ26s3y(s)=κ2s2+κ6s3z(s)=κτ6s3 \eqalign{& x(s) = s - {{{\kappa ^2}} \over 6}{s^3} \cr & y(s) = {\kappa \over 2}{s^2} + {\kappa \over 6}{s^3} \cr & z(s) = {{\kappa \tau} \over 6}{s^3} \cr}

pN(R). We assume {xj=(pjOX)Ttyj=(pjOX)Tnzj=(pjOX)Tbωj=θ(pjOX) \left\{{\matrix{{{x_j} = {{\left({{p_j} - {O_X}} \right)}^T}t} \cr {{y_j} = {{({p_j} - {O_X})}^T}n} \cr {{z_j} = {{\left({{p_j} - {O_X}} \right)}^T}b} \cr {{\omega _j} = \theta \left({\left\| {{p_j} - {O_X}} \right\|} \right)} \cr}} \right. , and the weighted least squares model can be obtained from the approximate relation {x(s)=sκ26s3sy(s)=κ2s2+κ6s3κ2s2 \left\{\matrix{x(s) = s - {{{\kappa ^2}} \over 6}{s^3} \approx s \hfill \cr y(s) = {\kappa \over 2}{s^2} + {\kappa \over 6}{s^3} \approx {\kappa \over 2}{s^2} \hfill \cr} \right. : minj[yj12κxj2]2ωj \min \sum\limits_j {{{\left[{{y_j} - {1 \over 2}\kappa x_j^2} \right]}^2}{\omega _j}}

In this way, the curvature κ of the potential curve of the point cloud at OX is obtained. We can use the obtained κ to solve the sj corresponding to xj from the relation x=sκ6s3 x = s - {\kappa \over 6}{s^3} . Further apply the relation z(s) = (κτ/6)s3 to give the least-squares fit minj[zjκτ6sj3]2 \min \sum\limits_j {{{\left[{{z_j} - {{\kappa \tau} \over 6}s_j^3} \right]}^2}}

In this way, an estimate of the torsion rate τ of the point cloud potential curve at OX is obtained [5]. We give the theoretical framework of geometric analysis of point cloud space curve and the calculation method of differential information. Obtain the position point OX of the potential curve of the point cloud and its differential information (t, n, b, κ, τ) by the above calculation method.

Global rough matching
Simplification of point cloud space curve

We segment the point cloud space curve with a fixed arc length to obtain segmented points. The rough global matching is determined by the matching situation of the differential information of each segment point [6]. The specific algorithm is divided into the following steps:

ANN searches for the function and calculates the maximum diagonal length D of the point cloud space curve border. We use arc length s = D/n as the unit length to segment the point cloud space curve. Where n is the number of segments divided into the point cloud space curve.

We choose one endpoint of the point cloud space curve as the initial point X0. We use the differential information calculation method to obtain the amount {OX0;t0, n0,b0,κ0,τ0} corresponding to the initial point.

We use relation Xi+1=OXi+sti+12κis2ni+κiτi6s3bi {X_{i + 1}} = {O_{{X_i}}} + s{t_i} + {1 \over 2}{\kappa _i}{s^2}{n_i} + {{{\kappa _i}{\tau _i}} \over 6}{s^3}{b_i} under the Frenet frame of the local coordinate system. The value of Xi+1,i = 0,1,⋯,n−1 can be obtained sequentially [7]. We use the differential information calculation method to get the differential information {OXi+1;ti+1, ni+1,bi+1,κi+1, τi+1} of Xi+1. If (OXi+1Ox)T ti+1 < 0, it means that the direction of ti+1 is opposite to the direction of the point cloud space curve segmentation. Then let ti+1 = −ti+1, ni+1 =−ni+1.

Segment the point cloud space curve in sequence. When the local point pjN(Xi) of Xi satisfies max(pjOXi)T ti < 0 or max(pjOXi)Ttimax(pjOXi)Tti<ε1 {{\max {{\left({{p_j} - {O_{{X_i}}}} \right)}^T}{t_i}} \over {\left\| {\max {{\left({{p_j} - {O_{{X_i}}}} \right)}^T}{t_i}} \right\|}} < {\varepsilon _1} , j = 0,1,⋯, k −1, it means that the point cloud space curve has been segmented to the endpoint. Where ε1 is the set error value.

Then use the same method to segment the remaining point cloud space curve along the −t0 direction of the initial point X0. Figure 3 shows the effect of segmenting a certain point cloud space curve.

Figure 2

Simplification of point cloud space curve

Global rough matching scheme

We assume that CA is a fixed point cloud space curve, and CB is a movable point cloud space curve [8]. The two-segmented point sets obtained after simplified segmentation are {Ai(1)} \left\{{A_i^{(1)}} \right\} , {Bj(2)} \left\{{B_j^{(2)}} \right\} respectively. Since the differential information of the segment points at both ends of the curve is often inaccurate, the segment points at both ends are not considered in the global rough matching scheme. Because the number of segment points of the point cloud space curve is relatively small, we adopt a traversal method to match the two segment point sets globally roughly.

{Ai(1)} \left\{{A_i^{(1)}} \right\} is a fixed point set, and {Bj(2)} \left\{{B_j^{(2)}} \right\} is a movable point set. Use the traversal matching method to make Bj2 B_j^2 , j = 1, 2,⋯, n −1 match each point in the point set {Ai(1)} \left\{{A_i^{(1)}} \right\} respectively.

Repeat the following iterative process, including double judgment conditions, until all segment points are matched.

1) First, match the curvature κj(2) \kappa _j^{(2)} of the point Bj2 B_j^2 , j = 1, 2,⋯, n −1 with the curvature κi(1) \kappa _i^{(1)} of point Ai(1) A_i^{(1)} , i =1,2,⋯,m−1. When κj(2)κi(1)<ε2 \left\| {\kappa _j^{(2)} - \kappa _i^{(1)}} \right\| < {\varepsilon _2} is satisfied, then go to 2) Perform Frenet frame matching. Where ε2 is the set error value. Otherwise, the curvature κj(2) \kappa _j^{(2)} of point Bj(2) B_j^{(2)} and the curvature κi+1(1) \kappa _{i + 1}^{(1)} of point Ai+1(1) A_{i + 1}^{(1)} are matched.

2) The matching scheme between the Frenet frame {tj(2),nj(2),bj(2)} \left\{{t_j^{(2)},n_j^{(2)},b_j^{(2)}} \right\} at point Bj(2) B_j^{(2)} and Frenet frame {ti(1),ni(1),bi(1)} \left\{{t_i^{(1)},n_i^{(1)},b_i^{(1)}} \right\} at point Ai(1) A_i^{(1)} is as follows: Record the matching degree value M (i, j) = 0 when κj(2)κi(1)<ε2 \left\| {\kappa _j^{(2)} - \kappa _i^{(1)}} \right\| < {\varepsilon _2} is satisfied. Where (i, j) is the position of the matching point and the rotation matrix R satisfying R(tj2nj(2),bj(2))T=(ti(1),ni(1),bi(1)) R{\left({t_j^2 - n_j^{(2)},b_j^{(2)}} \right)^T} = \left({t_i^{(1)},n_i^{(1)},b_i^{(1)}} \right) is calculated. If κj+1(2)κi+1(1)<ε2 \left\| {\kappa _{j + 1}^{(2)} - \kappa _{i + 1}^{(1)}} \right\| < {\varepsilon _2} is satisfied, use the obtained R. Rotate the frame (tj+1(2),nj+1(2),bj+1(2)) \left({t_{j + 1}^{(2)},n_{j + 1}^{(2)},b_{j + 1}^{(2)}} \right) of Bj(2) B_j^{(2)} to obtain (tj+1(2),nj+1(2),bj+1(2)) \left({t_{j + 1}^{(2)'},n_{j + 1}^{(2)'},b_{j + 1}^{(2)'}} \right) . If the absolute value of the diagonal data of the matrix (tj+1(2)',nj+1(2)',bj+1(2)')T(ti+1(1),ni+1(1),bi+1(1)) {\left({t_{j + 1}^{(2)'},n_{j + 1}^{(2)'},b_{j + 1}^{(2)'}} \right)^T}\left({t_{i + 1}^{(1)},n_{i + 1}^{(1)},b_{i + 1}^{(1)}} \right) satisfies the Frenet judgment condition, the matching degree value is increased by 1. The case of (i + l, j + l), l = 2,⋯ is judged one by one until the double judgment condition is not satisfied and returns to 1).

Finally, get the match between {y1, y2,⋯, yu} in point set {Ai(1)} \left\{{A_i^{(1)}} \right\} and {x1, x2,⋯, xu} in point set {Bj(2)} \left\{{B_j^{(2)}} \right\} and the maximum matching degree u = M (s, t). The s point of the fixed point set Ai(1) A_i^{(1)} matches the t point of the moving point set Bj(2) B_j^{(2)} . The number of consecutive matching segment points is M (s, t). Figure 3 shows the corresponding points of the curve obtained by rough global matching of certain two-segmented point sets.

Figure 3

Segmentation points that match globally

Local fine matching
Determination of fine matching point set

After rough global matching, {y1, y2,⋯, yu} in the fixed point set Ai(1) A_i^{(1)} is matched with {x1, x2,⋯, xu} in the moving point set Bj(2) B_j^{(2)} . We take the middle point of the point set {x1, x2,⋯, xu} and {y1, y2,⋯, yu} to determine the initial rotation and translation transformation. The purpose is to make the initial rotation and translation transformation more accurate [9]. From the Frenet frame (t[u2](2),n[u2](2),b[u2](2)) \left({t_{\left[{{u \over 2}} \right]}^{(2)},n_{\left[{{u \over 2}} \right]}^{(2)},b_{\left[{{u \over 2}} \right]}^{(2)}} \right) at point x[u2] {x_{\left[{{u \over 2}} \right]}} and the Frenet frame (t[u2](1),n[u2](1),b[u2](1)) \left({t_{\left[{{u \over 2}} \right]}^{(1)},n_{\left[{{u \over 2}} \right]}^{(1)},b_{\left[{{u \over 2}} \right]}^{(1)}} \right) at point y[u2] {y_{\left[{{u \over 2}} \right]}} , the initial rotation and translation amount (R0,T0) can be determined: R0=(t[u2](1),n[u2](1),b[u2](1))(t[u2](2),n[u2](2),b[u2](2))TT0=R0x[u2]+y[u2] \eqalign{& {R_0} = \left({t_{\left[{{u \over 2}} \right]}^{(1)},n_{\left[{{u \over 2}} \right]}^{(1)},b_{\left[{{u \over 2}} \right]}^{(1)}} \right)\,\,\,\,{\left({t_{\left[{{u \over 2}} \right]}^{(2)},n_{\left[{{u \over 2}} \right]}^{(2)},b_{\left[{{u \over 2}} \right]}^{(2)}} \right)^T} \cr & {T_0} = - {R_0}\,\,{x_{\left[{{u \over 2}} \right]}} + {y_{\left[{{u \over 2}} \right]}} \cr}

Where u = M (s, t), [u2] \left[{{u \over 2}} \right] means u2 {u \over 2} is an integer. Global rough matching is to find the global matching segment between the two-point cloud space curves CA, CB through the segmented point differential information. If only {x1, x2,⋯, xu} and {y1, y2,⋯, yu} are used to perform local rotation and translation transformation, the effect is not ideal. This needs to be based on the result of the rough global matching [10]. Increase the data points of local matching to build a fine matching optimization model. The method of adding data points for partial matching is as follows:

We choose ANN to search for radius d = s / 2. Where s is the arc length when simplifying the point cloud space curve. Call ANN annkFRSearch function to get the point x[u2] {x_{\left[{{u \over 2}} \right]}} as the center. We use d as the radius of the point set {p1, p2,⋯, pr} on the movable point cloud space curve CB. We perform (R0,T0) rotation and translation on the new local point set {p1, p2,⋯, pr} to obtain a new movable point set {p1, p2,⋯, pr}. pi, i =1,2,⋯,r is calculated from the differential information. The corresponding differential information set Q={qi;ti,ni,bi,κi,τi}i=1r Q = \left\{{{q_i};{t_i},{n_i},{b_i},{\kappa _i},{\tau _i}} \right\}_{i = 1}^r on the fixed point cloud space curve CA.

Fine matching model based on spatial kinematics

What is of interest in the local matching of curves is a kind of uniform spatial motion: Φ(x)=x+ν(x)=x+c×x+c¯,xR3 \Phi (x) = x + \nu (x) = x + c \times x + \bar c,\forall x \in {R^3}

Where ν(x) represents the instantaneous speed of x movement. (c,c¯) \left({c,\bar c} \right) describes the velocity field of motion. c represents the angular velocity vector [11]. By calculating the instantaneous velocity vector ν(x), the rigid body transformation that approximates the motion can be obtained. And have an explicit expression x¯=Φ(x)=B(xg×g¯)+ρθg+g×g¯ \bar x = \Phi (x) = B(x - g \times \bar g) + \rho \theta g + g \times \bar g

Where (g,g¯) \left({g,\bar g} \right) is the axis coordinate. If c ≠ 0 has θ= ||c||, ρ=cc¯c2 \rho = {{c\,\bar c} \over {{{\left\| c \right\|}^2}}} , (g,g¯)=(cθ,c¯ρcθ) \left({g,\bar g} \right) = \left({{c \over \theta},{{\bar c - \rho c} \over \theta}} \right) . The rotation matrix B is (b02+b12b22b322(b1b2b0b3)2(b1b3b0b2)2(b1b2b0b3)b02b12+b22b322(b2b3b0b1)2(b1b3b0b2)2(b2b3b0b1)b02b12b22b32) \left({\matrix{{b_0^2 + b_1^2 - b_2^2 - b_3^2} & {2\left({{b_1}{b_2} - {b_0}{b_3}} \right)} & {2\left({{b_1}{b_3} - {b_0}{b_2}} \right)} \cr {2\left({{b_1}{b_2} - {b_0}{b_3}} \right)} & {b_0^2 - b_1^2 + b_2^2 - b_3^2} & {2\left({{b_2}{b_3} - {b_0}{b_1}} \right)} \cr {2\left({{b_1}{b_3} - {b_0}{b_2}} \right)} & {2\left({{b_2}{b_3} - {b_0}{b_1}} \right)} & {b_0^2 - b_1^2 - b_2^2 - b_3^2} \cr}} \right) .

Where b0 = cos(θ/2),(b0, b1,b2)T g sin(θ/2). It is easy to know that rigid body motion is the superposition of rotation around the axis at an angle θ and parallel movement from the axis ρθ. From the moving point set {p1, p2,⋯pr} and its corresponding {q1, q2,⋯qr} on the fixed curve, the following fine matching optimization model is established: mink=1r(Φ(pk)sk)×tk2 \min \sum\limits_{k = 1}^r {{{\left\| {\left({\Phi ({p_k}) - {s_k}} \right) \times {t_k}} \right\|}^2}}

This model is about the quadratic function of the unknown c, c¯ \bar c . Its minimization means solving linear equations [12]. After obtaining c, c¯ \bar c , the corresponding rotation and translation (R1,T1) can be obtained from the space as mentioned above dynamics principle. Here is a description of the local fine matching algorithm. Repeat the following process until the change of (c,c¯) \left({c,\bar c} \right) between the two iterations is less than the given threshold or the number of iterations reaches the preset maximum value.

Act on the sub-point set {p1, p2,⋯, pr}on the moving point cloud curve CB to rotate and translate (R0,T0) to obtain {p1, p2,⋯, pr}.

Calculate the corresponding differential information amount {Oi;ti,ni,bi,κi,τi}i=1r \left\{{{O_i};{t_i},{n_i},{b_i},{\kappa _i},{\tau _i}} \right\}_{i = 1}^r of {p1, p2,⋯, pr} on the fixed point cloud curve CA.

Solve the fine matching optimization model equation (12) to obtain the corresponding rotation and translation (R1,T1).

Set R0 = R1R0,T0 = R1T0 +T1 and return to step (1).

Examples

Take two-point cloud space curves with a common part on the space curve (cos t, 2 sin t, 2t). We set the end of the fixed point cloud space curve to match the movable point cloud space curve [13]. The total number of point sets is 100. First, use an arbitrary rotation matrix to rotate and translate the movable point cloud curve. Let us set the rotation matrix as [cos(π3)sin(π3)0sin(π3)cos(π3)0001] \left[\left[{\matrix{{\cos \left({{\pi \over 3}} \right)} & {- \sin \left({{\pi \over 3}} \right)} & 0 \cr {\sin \left({{\pi \over 3}} \right)} & {\cos \left({{\pi \over 3}} \right)} & 0 \cr 0 & 0 & 1 \cr}} \right] .

We make the movable point cloud space curve, and the fixed point cloud space curve reaches an initial mismatch state and then perform the global coarse matching and local fine matching process. Set the number of local points k = 15 during the calculation of the differential information. Assuming the judgment condition ε2 = 0.1 of curvature during global matching, the judgment condition of the Frenet frame is set to [0.9, 1.1]. In this way, we can get a global matching degree of 4. The fine matching iteration ends 2 times. The total running time is 0.063 seconds.

Conclusion

Combining the principle of differential geometry and the technique of moving least squares, we have established a theoretical framework for calculating differential information of the point cloud empty curve. Calculate the derivative directly on the point cloud to obtain the corresponding curve characteristic information. Based on this, a global rough matching scheme of the point cloud space curve is constructed. We give a fine matching optimization model based on spatial kinematics and the corresponding solution algorithm. Numerical experimental examples can achieve good matching effects. The feasibility and stability of this differential information calculation and the effectiveness of the point cloud space curve matching method are verified. Differential information calculation and matching methods can be well applied to point cloud data with noise. This algorithm can effectively achieve high-precision matching of point cloud space curves.

Figure 1

Calculation of the Frenet frame
Calculation of the Frenet frame

Figure 2

Simplification of point cloud space curve
Simplification of point cloud space curve

Figure 3

Segmentation points that match globally
Segmentation points that match globally

Liu, M. RETRACTED ARTICLE: Optimization of marine biological sediment and aerobics training mode based on SVM. Arabian Journal of Geosciences., 2021; 14(15): 1–17 LiuM. RETRACTED ARTICLE: Optimization of marine biological sediment and aerobics training mode based on SVM Arabian Journal of Geosciences 2021 14 15 1 17 10.1007/s12517-021-07838-1 Search in Google Scholar

Ostad-Ali-Askari, K., & Shayan, M. Subsurface drain spacing in the unsteady conditions by HYDRUS-3D and artificial neural networks. Arabian Journal of Geosciences., 2021; 14(18): 1–14 Ostad-Ali-AskariK. ShayanM. Subsurface drain spacing in the unsteady conditions by HYDRUS-3D and artificial neural networks Arabian Journal of Geosciences 2021 14 18 1 14 10.1007/s12517-021-08336-0 Search in Google Scholar

Sandi-Urena, S. Experimentation skills away from the chemistry laboratory: Emergency remote teaching of multimodal laboratories. Journal of Chemical Education., 2020; 97(9): 3011–3017 Sandi-UrenaS. Experimentation skills away from the chemistry laboratory: Emergency remote teaching of multimodal laboratories Journal of Chemical Education 2020 97 9 3011 3017 10.1021/acs.jchemed.0c00803 Search in Google Scholar

Wang, H., Su, Y., Wang, D., Jin, S., Wei, S. A., & Shen, W. Optimal design and energy-saving investigation of the triple CO2 feeds for methanol production system by combining steam and dry methane reforming. Industrial & Engineering Chemistry Research., 2020; 59(4): 1596–1606 WangH. SuY. WangD. JinS. WeiS. A. ShenW. Optimal design and energy-saving investigation of the triple CO2 feeds for methanol production system by combining steam and dry methane reforming Industrial & Engineering Chemistry Research 2020 59 4 1596 1606 10.1021/acs.iecr.9b05296 Search in Google Scholar

Liashkevych, A., Babyshena, M., Vorokhaev, O., Pylypiv, V., Oliinyk, O., & Kinakh, N. Theoretical Aspects of Blockchain Technologies in The Sphere of Education. International Journal of Computer Science & Network Security., 2021; 21(3): 185–190 LiashkevychA. BabyshenaM. VorokhaevO. PylypivV. OliinykO. KinakhN. Theoretical Aspects of Blockchain Technologies in The Sphere of Education International Journal of Computer Science & Network Security 2021 21 3 185 190 Search in Google Scholar

Tekkumru-Kisa, M., Stein, M. K., & Doyle, W. Theory and research on tasks revisited: Task as a context for students' thinking in the era of ambitious reforms in mathematics and science. Educational Researcher., 2020; 49(8): 606–617 Tekkumru-KisaM. SteinM. K. DoyleW. Theory and research on tasks revisited: Task as a context for students' thinking in the era of ambitious reforms in mathematics and science Educational Researcher 2020 49 8 606 617 10.3102/0013189X20932480 Search in Google Scholar

Zong, X., & Zhang, W. Establishing world-class universities in China: deploying a quasi-experimental design to evaluate the net effects of Project 985. Studies in Higher Education., 2019; 44(3): 417–431 ZongX. ZhangW. Establishing world-class universities in China: deploying a quasi-experimental design to evaluate the net effects of Project 985 Studies in Higher Education 2019 44 3 417 431 10.1080/03075079.2017.1368475 Search in Google Scholar

Hernandez-Perez, L. G., Alsuhaibani, A. S., Radwan, N., El-Halwagi, M. M., & Ponce-Ortega, J. M. Structural and operating optimization of the methanol process using a metaheuristic technique. ACS Sustainable Chemistry & Engineering., 2020; 8(8): 3135–3150 Hernandez-PerezL. G. AlsuhaibaniA. S. RadwanN. El-HalwagiM. M. Ponce-OrtegaJ. M. Structural and operating optimization of the methanol process using a metaheuristic technique ACS Sustainable Chemistry & Engineering 2020 8 8 3135 3150 10.1021/acssuschemeng.9b05981 Search in Google Scholar

Chen, B., & Wang, F. Numerical simulation of heat-pipe and folded reformers for efficient hydrogen production through methane autothermal reforming. International Journal of Energy Research., 2020;44(13): 10430–10441 ChenB. WangF. Numerical simulation of heat-pipe and folded reformers for efficient hydrogen production through methane autothermal reforming International Journal of Energy Research 2020 44 13 10430 10441 10.1002/er.5668 Search in Google Scholar

Bibri, S. E. Data-driven smart eco-cities and sustainable integrated districts: A best-evidence synthesis approach to an extensive literature review. European Journal of Futures Research., 2021; 9(1): 1–43 BibriS. E. Data-driven smart eco-cities and sustainable integrated districts: A best-evidence synthesis approach to an extensive literature review European Journal of Futures Research 2021 9 1 1 43 10.1186/s40309-021-00181-4 Search in Google Scholar

Einav, L., Finkelstein, A., Ji, Y., & Mahoney, N. Randomized trial shows healthcare payment reform has equal-sized spillover effects on patients not targeted by reform. Proceedings of the National Academy of Sciences., 2020; 117(32): 18939–18947 EinavL. FinkelsteinA. JiY. MahoneyN. Randomized trial shows healthcare payment reform has equal-sized spillover effects on patients not targeted by reform Proceedings of the National Academy of Sciences 2020 117 32 18939 18947 10.1073/pnas.2004759117743105232719129 Search in Google Scholar

Aghili, A. Complete Solution For The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method. Applied Mathematics and Nonlinear Sciences., 2021; 6(1): 9–20 AghiliA. Complete Solution For The Time Fractional Diffusion Problem With Mixed Boundary Conditions by Operational Method Applied Mathematics and Nonlinear Sciences 2021 6 1 9 20 10.2478/amns.2020.2.00002 Search in Google Scholar

Wang, Y. & Chen, Y. Evaluation Method of Traffic Safety Maintenance of High-Grade Highway. Applied Mathematics and Nonlinear Sciences., 2021;6(1): 65–80 WangY. ChenY. Evaluation Method of Traffic Safety Maintenance of High-Grade Highway Applied Mathematics and Nonlinear Sciences 2021 6 1 65 80 10.2478/amns.2021.1.00007 Search in Google Scholar

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