Method for calculating effective volume of fruit tree canopy

Abstract

A fruit tree canopy effective volume calculation method relates to the technical field of volume calculation, and comprises the following steps: S1, collecting original point cloud data of an orchard, and carrying out preprocessing to obtain point cloud data of a fruit tree canopy; s2, performing canopy reconstruction region segmentation on the fruit tree canopy point cloud data; s3, reconstructing a canopy model based on the fruit tree canopy point cloud data, and calculating the volume of the canopy model; s4, calculating a canopy effective volume coefficient based on the fruit tree canopy point cloud data; and S5, multiplying the canopy model volume by the canopy effective volume coefficient to obtain the canopy effective volume. According to the scheme, the canopy volume is calculated through the dynamic parameter optimized Alpha-Shape algorithm, the effective volume coefficient calculation model is constructed, the influence of the pores on the canopy volume calculation is quantified, the problem of volume overestimation caused by the fact that an existing canopy volume calculation method contains the pores can be effectively solved, and decision support is provided for an orchard precise pesticide application technology.

Description

Technical Field

[0001] This invention relates to the field of volume calculation technology, specifically a method for calculating the effective volume of a fruit tree canopy. Background Technology

[0002] In the entire life cycle management of orchards, pest and disease control is a core component, accounting for approximately 30% of the overall workload. Spraying chemical pesticides is the main method of pest and disease control in orchards, but under conventional application methods, the actual utilization rate of pesticides is less than 40%, which not only wastes resources but also poses a threat to the ecological environment and food safety.

[0003] Against this backdrop, precision variable application technology has become a crucial breakthrough in green plant protection for orchards, and the core decision-making process of this technology relies on the accurate perception of fruit tree canopy characteristics. Canopy volume, as a key feature characterizing the growth status of fruit trees, is the core data for constructing precision variable application models; therefore, accurate measurement of canopy volume is particularly important.

[0004] Patent CN116704005A discloses a method for extracting fruit tree canopy volume based on vehicle-mounted LiDAR point cloud data, calculating the canopy volume using a frustum-cone formula. Patent CN118429412B discloses a method for calculating canopy volume using an improved Alpha Shape algorithm. These patents, when calculating canopy volume, ignore or simplify the asymmetrical, porous, and irregular structure of the canopy, resulting in a calculated volume that includes numerous pores between branches and leaves, leading to an overestimation of the volume. If variable-rate pesticide application decisions are made based on this volume containing pores, it will inevitably lead to excessive pesticide application, causing pesticide waste and environmental pollution, making it difficult to achieve truly precise pesticide application. Summary of the Invention

[0005] The purpose of this invention is to provide a method for calculating the effective volume of fruit tree canopy, which can solve the technical problem that existing canopy volume calculation methods are inaccurate in calculating the porosity in the canopy, resulting in overestimation of the volume.

[0006] To achieve the above objectives, the present invention adopts the following technical solution.

[0007] A method for calculating the effective volume of a fruit tree canopy includes the following steps: S1. Collect raw point cloud data of the orchard and preprocess it to obtain point cloud data of the fruit tree canopy; S2. Perform canopy reconstruction region segmentation on the fruit tree canopy point cloud data; S3. Reconstruct the canopy model based on the fruit tree canopy point cloud data and calculate the volume of the canopy model; S4. Calculate the effective volume coefficient of the canopy based on the point cloud data of the fruit tree canopy; S5. Multiply the canopy model volume by the canopy effective volume coefficient to obtain the canopy effective volume.

[0008] Furthermore, in step S1, the preprocessing of the original point cloud data of the orchard includes radius filtering, horizontal plane calibration, and separation of ground point cloud.

[0009] Furthermore, in step S2, region segmentation includes row segmentation and column segmentation.

[0010] Further, in step S3, the canopy 3D model is reconstructed based on the Alpha Shape algorithm with dynamic parameter optimization, including the following steps: S31. Define the benchmark based on the average neighborhood distance of point clouds. value; S32, at the reference Based on the value, iterative optimization is adopted. The value increment strategy continues until the reconstructed canopy model is a closed mesh.

[0011] Furthermore, in step S31, the reference... The value is: , in, The average point spacing, This is an empirical scaling factor; In step S32, The value increment strategy is: , in, For this iteration value, For the previous iteration value, The increment for each iteration, This is the step size coefficient.

[0012] Furthermore, the volume of the canopy model is: , in, To reconstruct the number of triangular meshes contained in the model, , , These are the vertex vectors of each triangle in the reconstructed mesh.

[0013] Further, in step S4, the effective volume factor of the canopy is calculated, including the following steps: S41. Project the canopy point cloud data to... , , For the plane, the projection surface is voxelized, and the Alpha Shape algorithm is used to reconstruct the two-dimensional boundary contours of the projection point cloud on the projection surface. The voxels contained in the two-dimensional boundary contours are retained, and the voxels containing the projection point cloud are marked as valid voxels. S42. Divide the projection surface into partitions and calculate the range of the projection point cloud set in the vertical direction of the projection surface in each partition, which is used as the weight of the corresponding partition. S43. Calculate the total number of voxels and the effective number of voxels contained in each partition of the projection plane. Then the effective coefficient of the projection plane is: , in, This represents the total number of projection plane partitions. For the projection plane The weight of each partition, For the projection plane Number of valid voxels per partition For the projection plane The total number of elements in each partition; S44, calculate separately , , The effective coefficient of the projected surface is then the effective volume coefficient of the canopy is: , in, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane.

[0014] Further, in step S42, for a zero-weight partition, its eight neighboring partitions are retrieved, the arithmetic mean of the non-zero-weight partitions is calculated and assigned as the weight of the current partition, and this process is iteratively executed until the zero-weight partitions within the two-dimensional boundary contour are eliminated.

[0015] By adopting the above technical solution, the present invention has the following beneficial effects: 1. This scheme is based on the Alpha Shape canopy reconstruction method with dynamic parameter optimization, through iterative optimization. This value allows the reconstructed canopy model to retain details in dense point cloud regions and avoid surface fragmentation in sparse regions, resulting in more accurate canopy volume calculations. 2. This solution constructs an effective volume coefficient calculation model, quantifies the impact of porosity on canopy volume calculation, and effectively solves the problem of overestimation of canopy volume caused by the inclusion of porosity in existing methods, providing decision support for precision pesticide application technology in orchards. Attached Figure Description

[0016] Figure 1 This is a flowchart illustrating the preprocessing process of raw point cloud data.

[0017] Figure 2 This is a schematic diagram of the probability density distribution and row segmentation points of the canopy point cloud data along the Y-axis.

[0018] Figure 3 This is a schematic diagram of the probability density distribution and column segmentation points of the canopy point cloud data along the X-axis.

[0019] Figure 4 This is a flowchart illustrating the calculation of the effective volume of the canopy.

[0020] Figure 5 This is a schematic diagram of a three-dimensional reconstruction model of the canopy.

[0021] Figure 6 This is a schematic diagram of the method for calculating the effective volume factor of the canopy.

[0022] Figure 7 This is a schematic diagram of the voxelization of the point cloud projected onto the yoz plane.

[0023] Figure 8 This is a schematic diagram of the weighted partitioning of the yoz plane projection point cloud.

[0024] Figure 9 This is a schematic diagram for calculating the effective coefficient of the yoz plane. Detailed Implementation

[0025] To make the objectives, technical solutions, and advantages of this invention clearer, the features and performance of a method for calculating the effective volume of a fruit tree canopy according to this invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0026] Please see the appendix Figures 1-9 A method for calculating the effective volume of fruit tree canopy includes the following steps.

[0027] S1. Collect raw point cloud data of the orchard and perform preprocessing to obtain point cloud data of the fruit tree canopy. Specifically, the preprocessing of the raw point cloud data of the orchard includes radius filtering, horizontal plane calibration, and separation of ground point cloud.

[0028] S2. Perform canopy reconstruction region segmentation on the fruit tree canopy point cloud data, including row segmentation and column segmentation.

[0029] S3. Reconstruct the canopy model based on the fruit tree canopy point cloud data and calculate the volume of the canopy model.

[0030] Specifically, when reconstructing the canopy model, the Alpha Shape algorithm based on dynamic parameter optimization is used to reconstruct the three-dimensional model of the canopy, including the following steps.

[0031] S31. Define the benchmark based on the average neighborhood distance of point clouds. Value, benchmark The value is: , in, The average point spacing, This is an empirical scaling factor.

[0032] S32, at the reference Based on the value, iterative optimization is adopted. The value increment strategy continues until the reconstructed canopy model is a closed mesh. The value increment strategy is: , in, For this iteration value, For the previous iteration value, The increment for each iteration, This is the step size coefficient.

[0033] The volume of the canopy model is: , in, To reconstruct the number of triangular meshes contained in the model, , , These are the vertex vectors of each triangle in the reconstructed mesh.

[0034] S4. Calculate the effective volume coefficient of the canopy based on the canopy point cloud data. Specifically, the calculation of the effective volume coefficient of the canopy includes the following steps.

[0035] S41. Project the canopy point cloud data to... , , For the plane, the projection surface is voxelized, and the Alpha Shape algorithm is used to reconstruct the two-dimensional boundary contours of the projection point cloud on the projection surface. The voxels contained in the two-dimensional boundary contours are retained, and the voxels containing the projection point cloud are marked as valid voxels.

[0036] S42. Divide the projection surface into partitions and calculate the range of the projection point cloud set in each partition in the vertical direction of the projection surface, which is used as the weight of the corresponding partition.

[0037] Specifically, for a zero-weight partition, its eight neighboring partitions are retrieved, the arithmetic mean of the non-zero-weight partitions is calculated and assigned as the weight of the current partition, and this process is iteratively executed until the zero-weight partitions within the two-dimensional boundary contour are eliminated.

[0038] S43. Calculate the total number of voxels and the effective number of voxels contained in each partition of the projection plane. Then the effective coefficient of the projection plane is: , in, This represents the total number of projection plane partitions. For the projection plane The weight of each partition, For the projection plane Number of valid voxels per partition For the projection plane The total number of elements in each partition.

[0039] S44, calculate separately , , The effective coefficient of the projected surface is then the effective volume coefficient of the canopy is: , in, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane.

[0040] S5. Multiply the canopy model volume by the canopy effective volume coefficient to obtain the canopy effective volume.

[0041] In practical implementation, the effective volume of the canopy layer was calculated using this scheme, including the following steps.

[0042] I. Data Preprocessing.

[0043] The raw point cloud data obtained by scanning orchards using lidar often contains outliers and ground points caused by sensor errors, environmental interference, and other factors. These noise points can affect the calculation of the effective volume of the fruit tree canopy, causing errors. To reduce these errors, this solution designs a four-step preprocessing workflow, such as... Figure 1 As shown.

[0044] First, a radius filtering algorithm is used to eliminate outliers caused by sensor errors and environmental interference. This algorithm is a commonly used denoising method in point cloud data processing. Its core idea is to remove outliers that do not meet the criteria by analyzing the point density in the neighborhood of each point. In this embodiment, the search radius and the minimum number of neighborhood points are set to 0.2 meters and 20, respectively.

[0045] Then, the point cloud data is horizontally calibrated. A random sampling consensus algorithm is used to coarsely extract the ground plane. Next, the normal vector of the fitted plane is calculated, which is the eigenvector corresponding to the smallest eigenvalue of the covariance matrix. Finally, the point cloud is rotated so that this normal vector aligns with the ground plane. Axis alignment.

[0046] After horizontal plane calibration, a cloth simulation filtering algorithm is used to separate the ground point cloud. The cloth simulation filtering algorithm is a point cloud filtering method based on physical simulation. The core idea is to simulate a virtual cloth covering an inverted terrain surface under the action of gravity, and to distinguish the ground from ground features by the settlement process of the cloth nodes.

[0047] This invention uses the Cloth Simulation Filter (CSF) package in Python to remove ground points from point cloud data. The algorithm involves six key parameters: BSloopSmooth determines whether to smooth the slope of cloth nodes; enabling this parameter can significantly reduce misclassification for terrain with slopes greater than 30°; Class_Threshold sets the maximum distance threshold from the point cloud to the cloth node; points smaller than this threshold are classified as ground points; Cloth_Resolution controls the cell size of the cloth mesh; a smaller value results in a more detailed map model; Iterations determines the maximum number of iterations for cloth settlement; a higher value increases accuracy but also increases computation time; Rigidness reflects the physical property of the cloth's resistance to deformation; a lower level indicates a softer cloth, better adaptable to rugged terrain, but slower settlement speed, requiring more iterations; and Time_Step controls the time increment of the simulated physical process, affecting the settlement speed. In this scheme, the values ​​of the above parameters are, in order: False, 0.3, 0.1, 500, 3, and 0.65.

[0048] After filtering by the fabric simulation filtering algorithm, some scattered weeds or ground points may remain and not be completely removed. Therefore, a secondary radius filtering is used for refinement to eliminate local outliers and provide accurate fruit tree canopy point cloud data for calculating the effective canopy volume.

[0049] II. Division of the canopy reconstruction area.

[0050] To ensure the accuracy of subsequent 3D reconstruction and effective volume calculation of the fruit tree canopy, it is necessary to segment the canopy reconstruction region from the fruit tree canopy point cloud data.

[0051] The first step is to segment the fruit tree rows and extract the 3D point cloud data of each row of fruit trees in the study area. For standardized orchards where the fruit trees are neatly arranged in rows and columns, a row detection method based on point cloud coordinate probability density estimation is used for segmentation. Gaussian kernel density estimation (KDE) is used to calculate... The probability density distribution of the coordinates, and based on the trough positions corresponding to the distribution. Tree row segmentation is performed using values ​​and boundary points, such as... Figure 2 As shown.

[0052] The second step is to divide the fruit trees into rows, such as... Figure 3 As shown. Because the canopies of trees in standardized orchards are closely connected, forming a continuous "tree wall," it is impossible to effectively separate individual fruit trees. Directly reconstructing the entire row of trees would result in significant errors. Therefore, this solution proposes a method for dividing the three-dimensional reconstruction area based on the distribution of tree wall height troughs, including the following steps.

[0053] S21. Height distribution extraction: extract point cloud data along... The axis is divided into several partitions, and the maximum value of each partition is calculated. The value (i.e., the height of the top of the canopy in the partition) forms the maximum. Value follows A sequence of axis partition changes.

[0054] S22, trough detection, at the maximum Identify the location of troughs in the value sequence.

[0055] S23. Effective trough filtering: Set the trough and adjacent peak values. Value difference threshold. If a trough is significantly different from any adjacent peak... If the difference between the values ​​is less than the threshold, it is determined to be a false valley and removed, which is used to filter out invalid valleys with gentle height changes.

[0056] S24. Valley Spacing Constraint: Set a horizontal distance threshold between adjacent valleys. If the horizontal distance between two valleys is less than this threshold, then retain one of them. The lower value of the trough is used to ensure that the segmented region has a reasonable horizontal scale.

[0057] S25. Add significant troughs, and set the trough and adjacent peak values. Value difference threshold, if a trough is different from any adjacent peak If the difference in values ​​is greater than this threshold, it is determined to be a significant valley and retained to supplement effective valleys with significant height differences that may be removed in step S24.

[0058] S26. Determine the dividing point and divide the column. Combine all the valleys retained after filtering in steps S23, S24 and S25, and use them as the final fruit tree column dividing point to complete the fruit tree column division.

[0059] III. Canopy Reconstruction and Reconstruction Model Volume Calculation.

[0060] First, the preprocessed fruit tree canopy point cloud data was used to perform 3D reconstruction using the Alpha Shape algorithm based on dynamic parameter optimization. Alpha Shape is a computational geometry-based surface reconstruction method that can extract 3D surface models with complex topological structures from discrete point cloud data. Its core idea is to use an adjustable parameter... It allows for precise control over the level of detail on surfaces, making it suitable for reconstructing non-uniformly distributed point clouds and open / closed surfaces.

[0061] This paper proposes an Alpha Shape algorithm based on dynamic parameter optimization. By introducing an adaptive parameter adjustment mechanism, it selects a smaller α value in dense point cloud regions to preserve details and increases the α value in sparse regions to avoid surface fragmentation. This significantly improves the applicability and reconstruction quality of the Alpha Shape algorithm in complex canopy scenes. The adaptive parameter adjustment mechanism uses a density-aware benchmark. Computational and iterative optimization Incremental strategy implementation.

[0062] Initial values ​​are defined based on the average neighborhood distance of the point cloud. value: , in, The average point spacing, This is an empirical scaling factor, with an initial value set to 5.

[0063] The goal of the iterative optimization strategy is to improve upon the benchmark. On this basis, gradually increase The value is maintained until the reconstructed canopy model is a closed mesh, mathematically described as follows: , in, For this iteration value, For the previous iteration value, The increment for each iteration, To control the magnitude of each adjustment using the step size factor, this invention sets it to 0.5.

[0064] When using this scheme for canopy reconstruction, the appropriate parameters are first calculated for the divided three-dimensional reconstruction region. The value, and then based on that Values ​​are used for surface reconstruction, such as Figure 5 As shown.

[0065] Finally, the volume of the closed mesh of the reconstructed model was calculated. The calculation formula is as follows: , in, To reconstruct the number of triangular meshes contained in the model, , , These are the vertex vectors of each triangle in the reconstructed mesh.

[0066] IV. Calculate the effective volume coefficient.

[0067] This invention constructs an effective volume factor This is used to measure the proportion of the actual canopy volume in the reconstructed model volume, effectively removing the influence of porosity. For example... Figure 6 As shown, the effective volume coefficient constructed by this scheme is... From the point clouds of the fruit tree canopy , , The effective coefficients of the three projection planes are constituted, and the calculation method is as follows: , in, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane.

[0068] , , The calculation method is similar, and the present invention uses Taking the projection plane as an example, let's explain. The calculation method.

[0069] like Figure 7 As shown, the three-dimensional point cloud data of the fruit tree canopy is first projected onto... Plane, by ignoring Axial coordinates achieve dimensionality reduction while preserving the point cloud in the horizontal direction. (axis) and vertical direction ( The spatial distribution characteristics of the axes are then analyzed. Next, the projected point cloud is voxelized, spatially discretized. During voxelization, the spatial distribution characteristics of the projected point cloud are analyzed. The plane is divided into uniform square voxels, and voxels containing the projected point cloud are marked as valid voxels (red voxels in the figure), forming a discretized spatial representation of the point cloud. This scheme uses the average nearest neighbor distance of the projected point cloud as the optimal size for the key voxelization parameter, voxel size (VS). Finally, the Alpha Shape algorithm is used to reconstruct the 2D boundary contour of the projected point cloud, and all voxels contained within this contour are extracted.

[0070] like Figure 8 As shown, a square mesh with a side length of 5 times the voxel size is used to partition the projected point cloud. This size design ensures that the voxels generated in the previous step can be completely divided into each partition. For each partition, the set of projected point clouds it contains is calculated. The range along the axial direction, i.e., the canopy thickness of the canopy reconstruction model in the partition of this projection plane, is used as the weight of the partition. The calculation method is as follows: , in: For partition weights, The largest in the partitioned point cloud set Coordinate values The smallest in the set of partitioned point clouds Coordinate values.

[0071] During the weight calculation process, some partitions within the projected point cloud contour have a weight of 0 due to the lack of point cloud data, which contradicts the physical meaning that the reconstructed canopy model should have a non-zero thickness within the contour partitions. To address this, an iterative interpolation strategy based on eight-neighbor average is proposed. For a zero-weight partition, its eight neighboring partitions are retrieved, and the arithmetic mean of the non-zero-weight partitions among them is calculated and assigned as the weight of the current partition. This process is iteratively executed until all zero-weight partitions within the contour are eliminated.

[0072] like Figure 9 As shown, all voxels contained in the boundary contour of the projected point cloud are mapped to the divided partitions, and the total number of voxels contained in each partition is calculated. and effective voxel count . Effective coefficient of the projection plane That is, the number of effective voxels in all partitions. With partition weight The cumulative sum of the products and the total prime number of all partitions With partition weight The ratio of the cumulative sum of products is expressed mathematically as follows: , in, for Total number of projection plane partitions for Projection plane The weight of each partition, for Projection plane The number of valid grids in each partition for Projection plane The total number of grids in each partition.

[0073] Similarly, we can calculate that... , for: , in, for Total number of projection plane partitions for Total number of projection plane partitions; for Projection plane The weight of each partition, for Projection plane The weight of each partition; for Projection plane The number of valid grids in each partition for Projection plane The number of valid grids in each partition; for Projection plane The total number of grids in each partition for Projection plane The total number of grids in each partition.

[0074] V. Calculate the effective volume of the canopy.

[0075] effective volume of the canopy Defined as the volume of the reconstructed model With effective volume factor The product of these two products is expressed mathematically as follows: , Introducing effective volume of the canopy , Effective coefficient of the projection plane , Effective coefficient of the projection plane and Effective coefficient of the projection plane ,have to: .

[0076] It should be noted that the parts not described in detail in this solution are all prior art. The above embodiments are only used to illustrate the present invention, but the present invention is not limited to the above embodiments. Any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention shall fall within the protection scope of the present invention.

Claims

1. A method for calculating the effective volume of a fruit tree canopy, characterized in that: Includes the following steps, S1. Collect raw point cloud data of the orchard and preprocess it to obtain point cloud data of the fruit tree canopy; S2. Perform canopy reconstruction region segmentation on the fruit tree canopy point cloud data; S3. Reconstruct the canopy model based on the fruit tree canopy point cloud data and calculate the volume of the canopy model; S4. Calculate the effective volume coefficient of the canopy based on the point cloud data of the fruit tree canopy; S5. Multiply the canopy model volume by the canopy effective volume coefficient to obtain the canopy effective volume.

2. The method for calculating the effective volume of a fruit tree canopy as described in claim 1, characterized in that: In step S1, the preprocessing of the original point cloud data of the orchard includes radius filtering, horizontal plane calibration, and separation of ground point cloud.

3. The method for calculating the effective volume of a fruit tree canopy as described in claim 1, characterized in that: In step S2, region segmentation includes row segmentation and column segmentation.

4. The method for calculating the effective volume of a fruit tree canopy as described in claim 1, characterized in that: In step S3, the canopy 3D model is reconstructed based on the Alpha Shape algorithm with dynamic parameter optimization, including the following steps: S31. Define the benchmark based on the average neighborhood distance of point clouds. value; S32, at the reference Based on the value, iterative optimization is adopted. The value increment strategy continues until the reconstructed canopy model is a closed mesh.

5. The method for calculating the effective volume of a fruit tree canopy as described in claim 4, characterized in that: In step S31, the reference The value is, ， in, The average point spacing, This is an empirical scaling factor; In step S32, The value increment strategy is as follows: ， in, For this iteration value, For the previous iteration value, The increment for each iteration, This is the step size coefficient.

6. The method for calculating the effective volume of a fruit tree canopy as described in claim 5, characterized in that: The volume of the canopy model is, ， in, To reconstruct the number of triangular meshes contained in the model, , , These are the vertex vectors of each triangle in the reconstructed mesh.

7. The method for calculating the effective volume of a fruit tree canopy as described in claim 1, characterized in that: Step S4 involves calculating the effective volume factor of the canopy, including the following steps: S41. Project the canopy point cloud data to... , , For the plane, the projection surface is voxelized, and the Alpha Shape algorithm is used to reconstruct the two-dimensional boundary contours of the projection point cloud on the projection surface. The voxels contained in the two-dimensional boundary contours are retained, and the voxels containing the projection point cloud are marked as valid voxels. S42. Divide the projection surface into partitions and calculate the range of the projection point cloud set in the vertical direction of the projection surface in each partition, which is used as the weight of the corresponding partition. S43. Calculate the total number of voxels and the effective number of voxels contained in each partition of the projection plane. Then the effective coefficient of the projection plane is: ， in, This represents the total number of projection plane partitions. For the projection plane The weight of each partition, For the projection plane Number of valid voxels per partition For the projection plane The total number of elements in each partition; S44, calculate separately , , If the effective coefficient of the projected surface is given, then the effective volume coefficient of the canopy is given. ， in, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane, for Effective coefficient of the projection plane.

8. The method for calculating the effective volume of a fruit tree canopy as described in claim 7, characterized in that: In step S42, for a zero-weight partition, its eight neighboring partitions are retrieved, the arithmetic mean of the non-zero-weight partitions is calculated and assigned as the weight of the current partition, and this process is iteratively executed until the zero-weight partitions within the two-dimensional boundary contour are eliminated.

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