Method for simplifying a simple polygon defined by a plurality of vertices

Abstract

The invention relates to a method for simplifying a simple polygon defined by a plurality of vertices by means of an electronic computing device, in which input data describing a polygon in a two-dimensional plane are received by means of the electronic computing device, and in which a simplified polygon is generated by reducing the number of vertices, which represents a geometric approximation of the original polygon, wherein the simplified polygon is used for further technical processing in a downstream processing device, wherein the reduction of the vertices takes place while maintaining the topology of the original polygon, the simplified polygon completely encompasses the original polygon without penetrating or intersecting its surface, the number of vertices of the simplified polygon is limited by a predeterminable target size, and the selection of additional vertices within intermediate regions defined by the convex hull is carried out depending on geometric and / or application-specific criteria.

Description

[0001] The invention relates to a method for simplifying a simple polygon defined by a plurality of vertices by means of an electronic computing device according to the preamble of patent claim 1.

[0002] Such methods for simplifying polygons using electronic computing devices are intended for the processing of geometric data in computer-aided systems, in particular for the representation, recognition or further technical use of polygonal objects in two-dimensional data structures.

[0003] US2013 / 0076732 A1 describes a method in which a polygon is modified through geometric transformations, in particular through expansion or contraction, with subsequent filtering steps being applied to remove irrelevant intermediate points. However, this does not prevent area truncation or the simplified polygon from becoming self-intersecting. Furthermore, there is no explicit limitation on the number of remaining vertices, and regional weighting of the simplification is not considered.

[0004] A polygon is a two-dimensional geometric figure consisting of a finite number of straight line segments connected to form a closed chain. These line segments are called edges or sides, and the points where the edges meet are called vertices. Polygons can be convex or concave and have at least three sides. Simple polygons have no intersecting edges, while complex polygons may have self-intersecting edges.

[0005] Polygons are used in many areas of information technology: in computer graphics, in geographical information systems (e.g., maps), in architecture (CAD programs), and even in driver assistance systems. Polygons can represent, for example, the outlines of objects, areas, countries, rivers, or buildings.

[0006] The number of vertices in a polygon is a relevant parameter for memory, bandwidth, and computing time. The more vertices a polygon has, the more complex it is to store, send, and process the polygon. In a processing chain where a polygon with many vertices is initially present, but later modules (in the processing chain) require fewer vertices, for example, to meet memory and computing time budgets, an algorithm is needed to reduce the number of vertices in the polygon.

[0007] The object of the invention is to provide a method for simplifying a simple polygon defined by a plurality of vertices by means of an electronic computing device, which enables a controlled reduction while maintaining geometric and topological properties. This object is achieved by means of a method for simplifying a simple polygon defined by a plurality of vertices using an electronic computing device having the features of patent claim 1. Furthermore, advantageous developments of the invention are described by the dependent patent claims, the following description, and the figures.

[0008] One aspect of the invention relates to a method for simplifying a simple polygon defined by a plurality of vertices using an electronic computing device. The method comprises receiving input data describing a polygon in a two-dimensional plane using the electronic computing device. By reducing the number of vertices, a simplified polygon is generated that represents a geometric approximation of the original polygon. The simplified polygon is used for further technical processing in a downstream processing device. The method comprises an electronic computing device that evaluates the received input data to generate a simple polygon with a plurality of vertices.The reduction of the vertices occurs in a single computational step, with the resulting polygon then being made available for further technical use, for example, for representation or decision support in an automated system. The relationship between the input data and the generated geometric approximation is determined by computer-aided processing, providing a reduced but topologically consistent representation of the original polygon.

[0009] In order to achieve the object of the invention and thus provide a controlled simplification while simultaneously preserving the geometric and topological properties, the invention provides that the reduction of the vertices takes place while maintaining the topology of the original polygon. This preserves the property of a simple polygon that does not exhibit self-intersections. Furthermore, it is provided that the simplified polygon completely encompasses the original polygon without penetrating or intersecting its surface. This measure ensures that no information necessary for safety-relevant or functional applications is lost. Furthermore, the number of vertices of the simplified polygon is limited by a predeterminable target size, which must at least correspond to the number of vertices of the convex hull.This target value enables a controllable reduction of complexity depending on the system requirements. All vertices of the convex hull are also vertices of the simplified polygon. The selection of additional vertices within intermediate regions defined by the convex hull is carried out depending on geometric and / or application-specific criteria. The intermediate regions, also called pockets, arise between the edges of the convex hull and the interior of the original polygon. Targeted selection within these regions allows for a differentiated simplification depending on the importance of individual polygon sections for the downstream technical application.

[0010] In a particularly advantageous embodiment of the invention, in a limiting case of maximum vertices reduction, the simplified polygon approximates the convex hull of the original polygon, thus ensuring a comprehensive representation even with strong simplification. The convex hull represents a technical limit of the simplification, within which all simplified polygon versions lie.

[0011] In another advantageous embodiment of the invention, intermediate regions are identified between each two consecutive vertices of the convex hull, in which additional points are determined by recursive area maximization to represent the simplified polygon, enabling adaptive detailing of the geometry depending on the local contour. This recursive approach allows for a data-based decision regarding which points should be considered for reasons of shape accuracy.

[0012] In another advantageous embodiment of the invention, the selection of the additional vertices in the intermediate regions is carried out based on a geometric optimization criterion that takes into account the maximum area of ​​a triangle lying entirely within the intermediate region, thus enabling efficient selection of relevant structural points. The area serves as a measure for prioritizing the points that have significant geometric significance.

[0013] In another advantageous embodiment of the invention, the number of additional vertices in an intermediate region is determined depending on an assigned weighting parameter resulting from spatial location, safety-relevant significance, or local point density, which allows context-dependent control of the simplification. In particular, critical regions can be retained with greater accuracy, while less relevant sections are simplified more significantly.

[0014] In another advantageous embodiment of the invention, collinear vertices are removed in a post-processing step. This step ensures that the resulting data structure does not contain any redundant elements that could hinder further processing.

[0015] In other words, it is intended that the method for simplifying a simple polygon defined by a plurality of vertices by means of an electronic computing device not only comprises a controlled reduction while maintaining topological properties, but also enables a differentiated, structurally faithful and lossless simplification through the combination of pocket-based decomposition, geometric selection mechanisms and application-specific weighting.

[0016] Further advantages, features, and details of the invention will become apparent from the following description of a preferred embodiment and from the drawings. The features and combinations of features mentioned above in the description, as well as the features and combinations of features mentioned below in the description of the figures and / or shown alone in the figures, can be used not only in the respective specified combinations, but also in other combinations or on their own, without departing from the scope of the invention.

[0017] Showing: Fig. 1 an example of simplifying a polygon; Fig. 2 an example of a polygon; Fig. 3 an example of a polygon; Fig. 4 an example of a simplification of polygons; and Fig. 5 a flowchart for a method for simplifying polygons.

[0018] In the figures, identical or functionally identical elements are provided with the same reference numerals.

[0019] Fig. 1 shows an example of the simplification of a polygon performed using a method for simplifying a simple polygon defined by a plurality of vertices using an electronic computing device.

[0020] The electronic computing device receives input data describing a polygon in a two-dimensional plane. By reducing the number of vertices, a simplified polygon is generated that represents a geometric approximation of the original polygon. The simplified polygon is used for further technical processing in a downstream processing device.

[0021] The method is characterized in that the reduction of the vertices is carried out while maintaining the topology of the original polygon, the simplified polygon completely encompasses the original polygon without penetrating or intersecting its surface, the number of vertices of the simplified polygon is limited by a predeterminable target size, and the selection of additional vertices within intermediate areas (pockets) defined by the convex hull is carried out depending on geometric and / or application-specific criteria.

[0022] In contrast to known state-of-the-art methods, the new method presented here offers the following advantages: - No area is truncated from the original polygon during or after simplification. The "reduced polygon" is therefore strictly larger than the original polygon. - The simplified polygon has no self-intersections. It is therefore always a "simple polygon." - It is possible to specify how many points the "reduced polygon" should be represented by, whereby at least the vertices of the convex hull of the original polygon will remain in the simplified polygon. - It can be specified in which spatial area the simplification of the polygon should take place and in which area fewer vertices should be simplified. - The method can be implemented on known parallel hardware (e.g. graphics cards or automotive IDCs with GPU) and runs smoothly in real time (e.g. for a specific use case on a modern graphics card, processing in <200 µs for hundreds of polygons with different numbers of vertices). - There is a worst-case estimate for simplification: the simplified polygon (with fewer vertices) approaches the convex hull.

[0023] In Fig. Figure 1 shows an example of the result of such an algorithm. The first line, G, shows the original polygon with 65 vertices (crosses K). The second dashed line, R, is the convex hull of the polygon. The third line, B, which always lies between (or on) the original polygon and the convex hull, represents the result of the new algorithm: The original polygon has been reduced to 27 vertices.

[0024] Thus, Fig. 1 is a simplification. Line G with crosses K: original polygon with many vertices. Dashed line R: convex hull of the polygon. Line B: simplified form of the polygon.

[0025] There are several options for the procedure, one of which is shown here.

[0026] The process consists of several steps, which are described in detail below.

[0027] The algorithm requires three parameters: the polygon P to be simplified, the maximum number of vertices N that the polygon may have after reduction, and optionally a list of criteria L that describes the spatial areas in which reduction should take place.

[0028] To calculate the simplified polygon, the first step is to calculate the convex hull of the polygon. Well-known methods from the literature can be used for this, such as the Melkman method or Jarvis methods suitable for parallelization. The result of this step is a list of indices. These indices indicate which vertices of the original polygon belong to the convex hull.

[0029] If the number of vertices of the convex hull is greater than or equal to the target size N chosen by the user, then the algorithm is terminated at this point because no further simplification is possible.

[0030] To identify the pockets and allocate the additional points, it is planned that in this step the initial areas of the polygon that are to be simplified are identified.

[0031] A pocket refers to all points of the original polygon that lie between two points of the convex hull. Fig. Figure 2 shows a polygon with a convex hull. This results in two pockets.

[0032] Fig. Figure 2 shows: Line B - the original polygon; line R - the convex hull of the polygon. The polygon has two pockets P1 and P2.

[0033] All points of the convex hull are automatically also points of the reduced polygon. If there are M points on the convex hull and the user has selected N points as the reduction target, then K = N - M points are available to select in addition to the M hull points.

[0034] Now we determine how many points from each pocket can be added to the final polygon. Different methods can be chosen depending on the application: 1. Use of criteria defined by the user L 2. Distribution of K points proportional to the area of ​​the pockets: large pockets receive more points than small pockets 3. Distribution of K points proportional to the number of points in the pocket: Pockets with many points receive more points than pockets with few points

[0035] Various combinations of 1., 2. and 3.: For example, pockets in certain spatial areas (defined by L) receive as many points as they could receive proportionally to their area, but not more than the number of points in the pocket.

[0036] The result of this step is a list of P pockets, where each pocket is defined by a starting point (on the convex hull), an end point (also on the convex hull), and the number of additional points that may be added. The sum of the points that may be added from the pockets is at most K.

[0037] To reduce the number of individual pockets, this processing step involves determining additional points from the pockets and adding them to the final polygon.

[0038] Each pocket is processed separately.

[0039] In each pocket, the point is searched for which, together with the start and end points of the pocket, forms the triangle with the largest area, whereby the edges of the generated triangle lie completely within the pocket and, in particular, no edges of the polygon are intersected.

[0040] The area of ​​the generated triangle is chosen as a criterion because the point found in this way “explains” as much of the pocket as possible through the approximation: the larger the triangle area, the more of the entire pocket area is correctly described by the reduced polygon.

[0041] In Fig. Figure 3 shows a polygon with two pockets P1 and P2. In the first pocket, P1, two points are shown as examples, indicating whether they are valid or invalid. The invalid point has an intersection with one of the polygon's edges. At the valid point, the edges of triangle G lie entirely within pocket P1.

[0042] Fig. Figure 3 thus shows a valid and invalid point within a pocket P1. At the invalid point, an edge of the polygon is intersected (marked by an X).

[0043] If the valid point that creates the largest triangular area has been found, two new pockets are created: The found point (“split point”) becomes the start or end point of two new pockets, see Fig. 4.

[0044] Fig. Figure 4 shows a split of pocket P1 into two new pockets: pocket P1a and pocket P1b. The split point is a new point of the reduced polygon. The new pockets are displayed with different backgrounds.

[0045] The two new pockets P1a and P1b are now assigned a number of points to be distributed based on the criteria from the previous section. For example, if the entire pocket had m ​​points, m - 1 points can now be distributed between the two pockets a and b, since one point has already been used.

[0046] In this way, the pockets are repeatedly divided until all the points allocated for that pocket have been used.

[0047] In this step, it is important to correctly identify "valid" and "invalid" points. This requires consideration of various special cases such as parallel edges and points of contact.

[0048] To output the reduced polygon, it is intended that once the allocated number of new points (split points) have been found in each pocket, the main task of the algorithm is completed.

[0049] In a parallel implementation of the algorithm, where multiple processors work on the pockets simultaneously, post-processing steps may need to be performed to put the found points in the correct order in the output container.

[0050] Optionally, collinear points can also be removed in post-processing. These can arise in the algorithm even if the original polygon had no collinear points.

[0051] Fig. Figure 5 shows a flowchart illustrating a possible implementation of the method.

[0052] In a first step S1, an input is provided. A polygon with N_in vertices and a target number N are entered, optionally a list of criteria L.

[0053] In a second step S2 the convex hull is calculated.

[0054] In a third step S3, M indices / points of the convex hull of the input polygon are available. This step also provides an output labeled "write points to output list," which leads directly to the final step S11, where the output is a polygon with at most N vertices.

[0055] In a fourth step S4, pockets are identified and additional points are assigned to them.

[0056] In a fifth step S5, a list of pockets is available.

[0057] In a sixth step S6, the split points are determined for each pocket.

[0058] In a seventh step S7, the split points are determined, whereby an output called “write points to output list” is provided here, which leads directly to step S11, where the output is a polygon with at most N vertices.

[0059] In an eighth step S8, each pocket is divided into two new pockets at the split point and, if necessary, additional points are assigned to them.

[0060] In a ninth step S9, a new list of pockets is created.

[0061] In a tenth step S10, a check is made to determine whether further pocket splits are possible. If the decision is positive, the process returns to step S6. If the decision is negative, the process continues to the final step S11, which produces a polygon with a maximum of N vertices as the output. QUOTES CONTAINED IN THE DESCRIPTION

[0000] This list of documents submitted by the applicant was generated automatically and is included solely for the convenience of the reader. This list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions. Cited patent literature

[0000] US 2013 / 0076732 A1

[0003]

Claims

[1] Method for simplifying a simple polygon defined by a plurality of vertices by means of an electronic computing device, in which input data describing a polygon in a two-dimensional plane are received by means of the electronic computing device, and in which a simplified polygon is generated by reducing the number of vertices, which represents a geometric approximation of the original polygon, wherein the simplified polygon is used for further technical processing in a downstream processing facility, characterized by , that - the reduction of the vertices is carried out while maintaining the topology of the original polygon, - the simplified polygon completely encompasses the original polygon without penetrating or intersecting its surface, - the number of vertices of the simplified polygon is limited by a specified target size, and - the selection of additional vertices within intermediate regions defined by the convex hull is carried out depending on geometric and / or application-specific criteria. [2] Method according to claim 1, characterized by that in a limiting case of maximum reduction of vertices, the simplified polygon approximates the convex hull of the original polygon. [3] Method according to one of the preceding claims, characterized by that between each two consecutive vertices of the convex hull, intermediate regions are identified in which additional points are determined to represent the simplified polygon by recursive area maximization. [4] Method according to one of the preceding claims, characterized bythat the selection of the additional vertices in the intermediate areas is carried out based on a geometric optimization criterion that takes into account a maximum area of ​​a triangle lying entirely in the intermediate area. [5] Method according to one of the preceding claims, characterized by that the number of additional vertices in an intermediate area is determined depending on an assigned weighting parameter which results from spatial location, safety-relevant importance or local point density. [6] Method according to one of the preceding claims, characterized by that collinear vertices are removed in a post-processing step.

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