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 latent shape is a three-dimensional data point set
_{X}_{X}

Give the data point _{X}_{0}, _{1},⋯, _{k}_{−1}}⊂ _{X}

Where _{X}

Here the Gauss kernel is used to define the weight factor

Among them _{X}

In the above model,

_{1} ≥ _{2} ≥ _{3}, then there are

We can get the corresponding Frenet frame (^{T}n^{T}b_{X}

The calculation of normal _{3} ≠ −1 according to the obtained tangent direction _{1}, _{2}, _{3})^{T}:

Suppose _{0}) on the curve, the curve has a third-order asymptotic expansion

∀

In this way, the curvature _{X}_{j}_{j}^{3} to give the least-squares fit

In this way, an estimate of the torsion rate _{X}_{X}

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

We choose one endpoint of the point cloud space curve as the initial point _{0}. We use the differential information calculation method to obtain the amount {_{X}_{0};_{0}, _{0},_{0},_{0},_{0}} corresponding to the initial point.

We use relation
_{i}_{+1},_{X}_{i+1};_{i}_{+1}, _{i}_{+1},_{i}_{+1},_{i}_{+1}, _{i}_{+1}} of _{i}_{+1}. If (_{X}_{i+1} − _{x}^{T} _{i}_{+1} < 0, it means that the direction of _{i}_{+1} is opposite to the direction of the point cloud space curve segmentation. Then let _{i}_{+1} = −_{i}_{+1}, _{i}_{+1} =−_{i}_{+1}.

Segment the point cloud space curve in sequence. When the local point _{j}_{i}_{i}_{j}_{Xi}^{T} t_{i}_{1} is the set error value.

Then use the same method to segment the remaining point cloud space curve along the −_{0} direction of the initial point _{0}. Figure 3 shows the effect of segmenting a certain point cloud space curve.

We assume that _{A}_{B}

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

1) First, match the curvature
_{2} is the set error value. Otherwise, the curvature

2) The matching scheme between the Frenet frame

Finally, get the match between {_{1}, _{2},⋯, _{u}_{1}, _{2},⋯, _{u}

After rough global matching, {_{1}, _{2},⋯, _{u}_{1}, _{2},⋯, _{u}_{1}, _{2},⋯, _{u}_{1}, _{2},⋯, _{u}_{0},_{0}) can be determined:

Where _{A}_{B}_{1}, _{2},⋯, _{u}_{1}, _{2},⋯, _{u}

We choose ^{′}_{1}, ^{′}_{2},⋯, ^{′}_{r}} on the movable point cloud space curve _{B}_{0},_{0}) rotation and translation on the new local point set {^{′}_{1}, ^{′}_{2},⋯, ^{′}_{r}} to obtain a new movable point set {_{1}, _{2},⋯, _{r}_{i}_{A}

What is of interest in the local matching of curves is a kind of uniform spatial motion:

Where

Where

Where _{0} = cos(_{0}, _{1},_{2})^{T} _{1}, _{2},⋯_{r}_{1}, _{2},⋯_{r}

This model is about the quadratic function of the unknown _{1},_{1}) 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

Act on the sub-point set {^{′}_{1}, ^{′}_{2},⋯, ^{′}_{r}}on the moving point cloud curve _{B}_{0},_{0}) to obtain {_{1}, _{2},⋯, _{r}

Calculate the corresponding differential information amount
_{1}, _{2},⋯, _{r}_{A}

Solve the fine matching optimization model equation (12) to obtain the corresponding rotation and translation (_{1},_{1}).

Set _{0} = _{1}_{0},_{0} = _{1}_{0} +_{1} and return to step (1).

Take two-point cloud space curves with a common part on the space curve (cos

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 _{2} = 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.

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.

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