AdaBoost triangle partitioning algorithm

Source: Internet
Author: User

 

1. Definition of triangular segmentation and AdaBoost Segmentation
How to split a scatter set into an uneven triangle mesh is the problem of the triangle division of the scatter set. The triangle division of the scatter set is for Numerical Analysis and graphics, is an extremely important preprocessing technology. The problem is illustrated as follows:

 

 1. Definition of triangle partitioning
[Definition] triangle partitioning: Suppose V is a finite vertex set in two-dimensional real number field, edge e is a closed line segment composed of vertices in the vertex set as endpoints, and E is a set of E. Then, a triangle partition t = (V, E) of the vertex set V is a plan G, which meets the following conditions:
1. Except the endpoint, the edges in the plan do not contain any vertices in the vertex set.
2. No intersection edge.
3. All faces in the plot are triangular surfaces, and the collection of all triangular surfaces is the convex hull of the scatter v.
1.2. Definition of AdaBoost triangle partitioning
In practice, the most widely used triangle division is the Adaboost triangle, which is a special triangle. Let's start with the Dirichlet edge:
[Definition] Dirichlet edge: assume that an edge e in E (two endpoints are a and B). If e meets the following conditions, it is called a Dirichlet edge: There is a circle passing through, B Two points, inside the circle (note that it is inside the circle, the circle can be at most three points in total) does not include any other points in the point set V, this feature is also called the null circle feature.
[Definition] gridded: If a triangle segment t of a point set V only contains a Dirichlet edge, the triangle segment is called a gridded triangle.
1.3.rule for the triangle partitioning of table store
In order to meet the definition of the Cartesian structure, two important criteria must be met:
1. Empty circle feature: the gridded area network is unique (any four points cannot circle in total), and no other points exist in the outer circle range of any triangle in the gridded area. As shown in:

2. maximization and least Angle Feature: in a triangle that may be formed by a scatter set, the smallest angle of a triangle formed by a Dirichlet triangle is the maximum. In this sense, the gridded network is the "closest to the regular" network. Specifically, two adjacent triangles form the diagonal line of a convex quadrilateral. After interchange, the minimum angle of the six inner angles does not increase. As shown in:

1.4.unique features
The following table lists the outstanding features of node partitioning:
1. Closest: a triangle is formed from the nearest three points, and each line segment (the side of the triangle) does not intersection.
2. uniqueness: No matter where the building starts from the region, the results will eventually be consistent.
3. Optimum: If the diagonal lines of the convex quadrilateral formed by any two adjacent triangles can be swapped, the smallest angle of the six inner angles of the two triangles will not become larger.
4. Most rule: if the smallest angle of each triangle in the Triangle network is sorted in ascending order, the maximum value obtained by the arrangement of the Triangle network is obtained.
5. Culture: When a vertex is added, deleted, or moved, only the adjacent triangle is affected.
6. Case with convex polygon: the outer boundary of the Triangle network forms a convex polygon shell.
1. 5. Local Optimization
In theory, in order to construct a gridded network, Lawson proposed the Local Optimization process LOP (Local Optimization Procedure). Generally, the gridded network can be processed by LOP, which can be used to generate a gridded network. The basic practice is as follows:
1. Combine Two triangles with common edges into a polygon.
2. Check whether the fourth vertex is within the outer circle of the triangle by using the maximum null circle criterion.
3. If the diagonal line is correct, the diagonal line will be reversed, that is, the local optimization process will be completed.
Shows the Lop processing process:

2. The algorithm for segmentation
It is a standard for triangle partitioning. There are multiple algorithms to implement it.
2.1.Lawson Algorithm
The point-by-point Lawson algorithm was proposed by Lawson in 1977. It has a simple idea and is easy to program. The basic principle is: first create a large triangle or polygon, enclose all data points and insert a point into it. The point is connected to three vertices of the triangle that contains it, create three new triangles, and then perform the null Outer Circle Detection on them one by one. At the same time, optimize the Lop with the local optimization process designed by lawson, that is to say, by exchanging the diagonal lines, we can ensure that the formed Triangle network is a Cartesian network.
The above scatter-Based Network Construction Algorithm is rigorous in theory and has good uniqueness. The grid satisfies the characteristics of an empty circle and is ideal. It can be seen from its point-by-point insertion of the network construction process that, in the case of non-Dirichlet edge, the deletion adjustment can be used to construct a new Dirichlet edge. After creating a network, you do not need to reconstruct the network of all points when adding new points. Instead, you only need to connect the affected triangle ranges of the new points to the local network, and the local network method is simple and easy. Similarly, you can quickly and dynamically delete or move a vertex. However, in practical applications, this network construction algorithm is slow when the point set is large. If the Point Set range is not convex or there is an inner ring, an invalid triangle is generated.
2.2.Bowyer-Watson Algorithm 
The basic steps of the Lawson algorithm are as follows:
1. Construct a super triangle that contains all the scattered points and put them into the triangle linked list.
2. Insert the scattered points in the vertex set in sequence, find the triangle with the inserted points in the outer circle of the triangle linked list (called the triangle affected by the point), and delete the public edge that affects the triangle, concatenates the insertion point with all vertices that affect the triangle to complete insertion of a vertex in the table of the triangle.
3. Optimize the newly formed triangle according to the optimization criterion. Place the formed triangle into the table list of the gridded triangle.
4. Execute the preceding steps cyclically until all scatter plots are inserted.
The following figure shows the key step 1 of this algorithm:

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