A time-varying body data global feature tracking method based on unsupervised segmentation

Abstract

This invention relates to a global feature tracking method for time-varying data based on unsupervised segmentation. It constructs a volume data segmentation network to segment the input volume data and achieve global tracking. This invention introduces deep learning to segment volume data, separating feature domains from the background. This enables automatic and accurate tracking of specific features in complex data without any manual annotation, reducing tracking complexity and improving tracking accuracy. It introduces a global feature tracking method; by selecting a target feature, it can track the trajectories of features similar to the target feature across all time steps. Users can select any feature at any time step to track features of interest in time-varying data. It can track extracted features from a global perspective, avoiding tracking errors and defects caused by local tracking methods. Furthermore, it adds the tracking of spatiotemporally similar features, simultaneously tracking the complete paths of spatially similar features of the target feature, improving feature tracking accuracy.

Description

Technical Field

[0001] This invention relates to the technical field of general image data processing or generation, and particularly to a method for tracking global features of temporal variant data based on unsupervised segmentation in the fields of visualization and visual analysis. Background Technology

[0002] In many scientific fields, simulations generate a large amount of time-dependent volumetric data every day, such as cosmological simulations, fluid dynamics simulations, and biomolecular simulations. This data is often highly complex, and tracking and analyzing some of its characteristics is quite difficult.

[0003] In current research, spatially prioritized feature tracking strategies have been widely applied to feature tracking in time-varying data. The underlying idea is to independently extract features from each time step and then determine the correspondence between adjacent time steps according to defined matching rules. While many methods have been proposed to determine these correspondences, such as volume overlap and its extension, optical flow, path coherence, and distribution, the most widely used method is based on volume overlap. In biomolecular simulations, Krone et al. used the intersection of molecular surfaces of features in adjacent time steps to track feature evolution, a variation of the volume overlap-based method. Based on overlapping voxels, Saikia et al. introduced a global feature tracking method to escape local optima. Facing the challenge of not knowing the precise definition of features, Dutta et al. introduced a distribution-driven feature extraction and tracking method. This method, after selecting the target region of interest, models the features as a Gaussian mixture model (GMM), then calculates the probability of a block becoming part of the target feature and the probability of a block containing foreground objects using incremental Gaussian mixture models, constructing a feature-aware classification domain. Next, a region growing algorithm is applied to this classification field based on a specified threshold to extract candidate features. Finally, tracking is completed by selecting the best match with the minimum Euclidean distance to the target feature.

[0004] While spatially-oriented feature tracking methods offer high understanding, ease of implementation, and parallelism, most lack a global view of the tracked features, focusing only on the feature similarity between two adjacent time steps, making them susceptible to noise interference. To address these issues, inspiration can be drawn from image segmentation, a key focus of computer vision research. Image segmentation typically involves training with a set of images and pixel-level semantic labels to classify image pixels in semantically supervised scenes. In recent years, deep learning has been successfully applied to image segmentation. Various deep learning networks show great potential in extracting detailed features from image pixels. Theoretically, deep learning networks may possess similar potential in extracting features from scalar field volumetric data, which is essential for the feasibility of segmenting time-varying volumetric data based on unsupervised neural networks. Summary of the Invention

[0005] This invention addresses the problems existing in the prior art and provides an optimized global feature tracking method for time-varying data based on unsupervised segmentation. It utilizes deep learning and directed acyclic graphs to globally track time-varying data for feature extraction and tracking.

[0006] The technical solution adopted in this invention is a global feature tracking method for temporal volume data based on unsupervised segmentation. The method constructs a volume data segmentation network to segment the input volume data and achieve global tracking.

[0007] Preferably, the preprocessing of the input volume data includes the following steps:

[0008] Step 1.1: Obtain the original time variant data Enhance contrast and normalize the data, where N is the number of volume data points and n is a positive integer from 1 to N;

[0009] Step 1.2: Divide the processed raw time-varying data into hypervoxels according to the SLIC algorithm. , which is the volume data to be input into the segmentation network, and K is the number of supervoxels to be segmented, where k is a positive integer from 1 to K.

[0010] Preferably, the constructed volume data segmentation network includes a feature extraction module; a p-dimensional mapping is calculated from the feature extraction module. The constructed feature extraction module includes a 3D convolutional filter, a batch normalization function, an h-swish activation function, and M bottleneck components arranged sequentially. The bottleneck components include a 1x1x1 unfolded convolution, a 3x3x3 dilated convolution, and a 1x1x1 projection layer. The input and output of the bottleneck components are connected through the residual module.

[0011] The feature extraction module is followed by a 3D convolutional filter, a batch normalization function, and a linear classifier in sequence; the linear classifier satisfies... and Where q is the number of clusters, c is the number of convolution channels, p is the feature dimension, and V is the volume data. and These are the parameters of the classifier;

[0012] Map the response Normalization ,in, , , The mean and unit variance are zero, by selecting in The cluster label is obtained from the dimension with the maximum value. Improve clustering labels, making yes The middle belongs to the first The number of voxels in each cluster, replacing the cluster label with For all In other words, That is, calculate the category that appears most frequently in each cluster, and then record all voxels in that cluster as that category.

[0013] The loss between the response and the improved clustering labels is calculated using the softmax loss function, and the parameters of the convolutional filter are updated using the backpropagation algorithm. and the parameters of the classifier M is the number of layers in the 3D convolutional filter;

[0014] The 3D connected component algorithm assigns different index values ​​to two spatially disconnected clusters. Voxels with the same index value belong to the same feature. The volume data is divided into domains containing different index values. Each domain may be a feature needed by the expert or a meaningless background, thus obtaining the domain corresponding to the background or feature.

[0015] Preferably, The i-th cluster in is,

[0016]

[0017] in, and They represent The i-th and j-th elements.

[0018] Preferably, the global tracking includes the following steps:

[0019] Step 2.1: Construct a graph for recording tracking;

[0020] Step 2.2: Locate the tracking path for the features in the graph;

[0021] Step 2.3: Implement feature tracking under time and space constraints.

[0022] Preferably, step 2.1 includes the following steps:

[0023] Step 2.1.1: Construct a directed acyclic graph (DAG) to record tracking information. The nodes of the DAG are the partitioned domains. Each node stores the label of its time step and the histogram information of the corresponding domain.

[0024] Step 2.1.2: For each feature domain at time step t and the feature domain at time step t+1 Calculate the centroid and center of mass Distance between ,

[0025]

[0026] In the segmented temporal variant data, any two regions Chi-square histogram distance between for,

[0027] ;

[0028] in, and These are domains , The i-th bin value of the histogram,

[0029] by As a domain Heyu The weight of the edge corresponding to the node.

[0030]

[0031] in, [0,1];

[0032] Step 2.1.3: When the centroid distance between two domains < or chi-square distance < At that time, establish an edge between them. , , and The parameters can be adjusted appropriately according to the different characteristics.

[0033] Preferably, step 2.2 includes the following steps:

[0034] Step 2.2.1: Select the node containing the target domain in the DAG as the target node, and set the label in the segmented volume data to be the same as the index value of the node in the DAG. The domain corresponding to the target node is called the target feature.

[0035] Step 2.2.2: Calculate the shortest path from each node to the target node at all time steps using Dijkstra's algorithm, and save the shortest path information to the new directed acyclic graph SPG;

[0036] Step 2.2.3: Track the features forward and backward from the current time step of the target node, and record the forward and backward tracking paths P+ and P-; traverse the next node in turn until the endpoint or start node is found; when the current node has the smallest distance to the target node among all nodes in the current time step and has no outgoing edges, the current node is considered to be the endpoint or start node.

[0037] Step 2.2.4: Find the path from the endpoint to the target node in SPG as the forward path. Find the path from the starting node to the target node in SPG as the forward path. , path and Merge to obtain the complete tracking path of the target domain. .

[0038] Preferably, step 2.3 includes the following steps:

[0039] Step 2.3.1: Obtain all paths for features similar to the target features corresponding to the target node at the same time step;

[0040] Step 2.3.2: Filter candidate features similar to the target feature at time step ti using chi-square histogram distance. , where n is the number of candidate features; when the target feature and candidate features Chi-square histogram distance hour, The value is the same as in step 2.1.3. Same, with Candidate features;

[0041] Step 2.3.3: Calculate each candidate matching region The trajectory P' is obtained, and it is compared with the trajectory of the target feature using the longest common subsequence (LCSS). Compare; assuming the input subpath A = B= Then the length of LCSS is,

[0042] ,

[0043] Here In spatial constraints The values ​​are the same;

[0044] The formula for the similarity of the longest common subsequence is:

[0045]

[0046] The candidate features obtained in step 2.3.2 are further filtered, which means that the candidate paths of the candidate nodes are constrained in terms of time.

[0047] Preferably, the segmented volume data is visualized;

[0048] Set the opacity of the passed function.

[0049]

[0050] in, Each voxel representing a foreground feature, and These represent the normalized intensity and gradient, respectively. log() represents the logarithmic function, and γ represents the parameter controlling the opacity of the boundary. The value of γ can be adjusted according to the visualization effect.

[0051] This invention relates to an optimized method for global feature tracking of temporal volume data based on unsupervised segmentation, which constructs a volume data segmentation network to segment the input volume data and achieve global tracking.

[0052] The beneficial effects of this invention are as follows:

[0053] (1) Deep learning was introduced to segment volume data, and the feature domain was separated from the background. It can automatically and accurately track specific features in complex data without any manual annotation, which reduces the complexity of tracking and improves the accuracy of tracking.

[0054] (2) A global feature tracking method is introduced. As long as the target feature is selected, the trajectory of features similar to the target feature can be tracked in all time steps. Users can select any feature at any time step and then track the feature of interest in time-varying data. It can track the extracted features from a global perspective, thereby avoiding the tracking errors caused by local tracking methods and avoiding the defects caused by local tracking.

[0055] (3) Based on the tracking of target features, the tracking of spatiotemporal similar features has been added, which can simultaneously track the complete path of spatial similar features of target features, thereby improving the accuracy of feature tracking. Attached Figure Description

[0056] Figure 1 This is an overall flowchart of the present invention;

[0057] Figure 2 This is a segmentation diagram in which volume data is divided into supervoxels in this invention;

[0058] Figure 3 This is a diagram showing the result of segmenting the volume data in this invention;

[0059] Figure 4 This refers to the results of Halfcylinder data tracking of individual secondary features and individual primary features in this invention;

[0060] Figure 5 This refers to the tracking results of the target features and candidate features of the secondary and primary features of the Halfcylinder data in this invention. Detailed Implementation

[0061] The present invention will be further described in detail below with reference to embodiments, but the scope of protection of the present invention is not limited thereto.

[0062] This invention relates to a global feature tracking method for time-varying data based on unsupervised segmentation, mainly addressing the two challenging problems of feature extraction and tracking for time-varying data. Figure 1 This shows the entire process from feature extraction to feature tracking.

[0063] A global feature tracking method based on unsupervised segmentation of time-varying data includes the following three main steps: constructing a network for volume data segmentation, globally tracking volume data, and visualization.

[0064] Reference Figure 1 A global tracking method for temporal variable data based on unsupervised segmentation can be divided into three steps: constructing a volumetric data segmentation network, global feature tracking, and visualization.

[0065] 1) Generate the segmented volume data, the process is as follows:

[0066] 1.1, For the original time-varying data The contrast of these structures is enhanced by applying an automatic contrast enhancement method based on histograms, and they are normalized to their maximum and minimum values ​​using global maximum and minimum values. [0, 1] Within the specified range, the input data is used as the input to the segmentation network;

[0067] 1.2 The processed raw data is divided into hypervoxels according to the SLIC algorithm. The data is then input into the network. Each segmented supervoxel block is a cluster, where K is the number of voxel blocks to be divided, set to K=8000. The SLIC algorithm first segments the original volume data into cubes of uniform size. For each segmented voxel block, a point is randomly sampled as the cluster center, and the distance between each voxel and the cluster center is calculated. The original SLIC algorithm only calculates the distance between pixels within the radius of each cluster center and that cluster center, considering both spatial and color distances. Since the input is volume data with only grayscale values, it only considers the distance between voxels surrounding the cluster center and the cluster center, considering both 3D spatial distance and grayscale distance. Voxels are clustered into different clusters based on spatial and grayscale values, thus dividing the volume into supervoxels.

[0068] Figure 2 The images show supervoxel slices from Salt Dissolving in Water, Square Cylinder, and Halfcylinder data, corresponding to (a), (b), and (c) in the figure, respectively. Different gray levels represent different supervoxels.

[0069] Next, we construct a network for the segmented data. The network architecture is referenced from [reference needed]. Figure 1 The first part. Calculate a p-dimensional mapping from the feature extraction module. The system consists of a 3D convolutional filter, a batch normalization function, an h-swish activation function, and M bottleneck components arranged sequentially. Each bottleneck component includes a 1x1x1 unfolded convolution, a 3x3x3 dilated convolution, and a 1x1x1 projection layer. The input and output of the bottleneck components are connected via residual modules. Following this, a 3D convolutional filter, a batch normalization function, and a linear classifier are arranged sequentially. The linear classifier satisfies... and Where q is the number of clusters, c is the number of convolution channels, p is the feature dimension, and V is the volume data. and These are the parameters for the classifier; next, a response map is obtained. Map the response Normalization , , , The mean and unit variance are zero, by selecting in The cluster label is obtained from the dimension with the maximum value. The final response The i-th cluster can be written as:

[0070] (1)

[0071] in, and They represent The i-th and j-th elements are used to assign each voxel to the nearest point among the q representative points.

[0072] Next, we need to improve the clustering labels so that... yes The middle belongs to the first The number of voxels in each cluster, replacing the cluster label with For all In other words, ;set up yes The middle belongs to the first The number of voxels in each cluster, selection (For all) In other words, Replace the cluster label with .

[0073] The loss between the response and the improved clustering labels is calculated using the softmax loss function, and the parameters of the convolutional filter are updated using the backpropagation algorithm. and the parameters of the classifier M represents the number of layers in the 3D convolutional filter, and the parameters are updated using stochastic gradient descent with momentum.

[0074] The network finally outputs clustered volumetric data. Since the data only contains grayscale values, similar features may be assigned the same label. To facilitate the next step of tracking, the 3D Connected Component Analysis (3D Component Analysis) algorithm is used to relabel spatially disconnected clusters. Specifically, the 3D Component Analysis algorithm assigns different index values ​​to two spatially disconnected clusters, and voxels with the same index value belong to the same feature. The clustered volumetric data is then spatially divided into different domains, each of which could be either background or feature data. Figure 3 The results are shown after 3D connected component processing, where (a), (b), and (c) are compared with... Figure 2 correspond.

[0075] 2) Global feature tracking, the process is as follows:

[0076] 2.1 Constructing a graph for recording tracking:

[0077] 2.1.1 A Directed Acyclic Graph (DAG) is constructed to record tracking information. The nodes of the DAG are the segmented domains given above. Each node needs to store the label of its current time step and the histogram information of its corresponding domain. Edges exist only between nodes corresponding to domains at two consecutive time steps; there are no edges between domains at the same time step. The edges of the DAG record the similarity between the domains corresponding to two nodes. This paper selects two criteria to measure the similarity between two domains: centroid distance and histogram similarity. For each feature domain at time step t... and the feature domain at time step t+1 Calculate the centroid and center of mass Distance between as follows:

[0078] (2)

[0079] Calculate the chi-square histogram distance between two regions. as follows:

[0080] , (3)

[0081] 2.1.2, Centroid Distance Distance between the two sides Both are important for the tracking region. If there are two regions... and If both the centroid distance (dc) and chi-square distance are small, then the region is considered... It is very likely that the area will be tracked. Combinations using a linear combination To represent, and will As a region and region The weight of the edge corresponding to the node:

[0082] , (4)

[0083] in The value is in the range [0,1]. Experiments show that the value set in this paper is... When the value is 0.03, it is easier to produce correct results.

[0084] 2.1.3 For large and complex datasets, a single entity is often divided into hundreds or thousands of domains. Recording all information between each domain in a Directed Acyclic Graph (DAG) would generate millions of edges, which is impractical. To reduce the number of edges, two conditions are set to decrease the number of edges in the DAG:

[0085] (1) When the centroid distance between two regions < Only then should an edge be established between them;

[0086] (2) When the chi-square distance between the two regions < Only then is an edge established between them.

[0087] These two conditions greatly reduce unnecessary edges and are key to constructing a sparse DAG.

[0088] 2.2, Find the tracking path for the features in the graph:

[0089] 2.2.1 Given a DAG for recording tracking information, the user selects the target domain. The node is used as the target feature. To facilitate the display of tracking information, the label in the segmented volume data is set to be the same as the index value of the node in the DAG.

[0090] 2.2.2 Use Dijkstra's algorithm to calculate the shortest path from each node to the target node at all time steps, and save the shortest path information into a new directed acyclic graph SPG;

[0091] 2.2.3, then from the target node At the current time step tuser, features are tracked forward and backward, and the forward and backward tracking paths P+ and P- are recorded. The process iterates through the next node until an endpoint or start node is found. If the current node has the smallest distance to the target node among all nodes at the current time step and has no outgoing edges, then the current node is considered the endpoint or start node. Finding the endpoint involves... (The sentence is incomplete and requires further context to translate accurately.) The path is used as the forward path. Find the starting node in SPG The path is used as the backward path. Finally, the path and By combining them, we obtain the complete tracking path for the target domain. . Figure 4 The results show the Halfcylinder data tracking for a single secondary feature and a single primary feature.

[0092] 2.3 Feature tracking under temporal and spatial constraints:

[0093] Section 2.3.1 describes the tracing of the complete path of a single feature. For scientific simulation data, it is common to have multiple similar features at the same time step. To globally track the target feature across all time steps, we must first find similar candidate features within the same time step. Then, by applying the method for tracking a single feature from the previous section, we can obtain all paths of all features similar to the target feature at the same time step.

[0094] 2.3.2, Selecting the chi-square histogram distance to filter target features at time step ti Similar candidate features Here, n is the number of candidate features. When the target feature and candidate features Chi-square histogram distance hour, The value is the same as in step 2.1.3. same, Only those features can be used as candidate features. However, not all paths traced based on candidate features can be used as the final tracing path. This is because these features not only exhibit similarity at a given time step, but also show similar changes over time. This allows for the quantification of their periodic structural changes over time.

[0095] 2.3.3 Calculate each candidate matching region The trajectory P' is obtained and compared with the target feature using the longest common subsequence (LCSS). trajectory The comparison is then performed. The LCS algorithm is typically used to calculate the longest common subsequence (LCS) between two time series. The LCS length can be used to determine the differences between the two time series, and it allows for both time contraction and expansion, covering different time spans. The current goal is to match paths. and Consider the input subpath A= and B= The length of such an LCSS can be written as:

[0096] (5)

[0097] Here In spatial constraints If the values ​​are the same, based on the above formula, the similarity formula for the longest common subsequence is:

[0098] , (6)

[0099] This not only constrains candidate nodes spatially and further filters the candidate features obtained in step 2.3.2, but also further constrains candidate paths of candidate nodes temporally, resulting in more accurate tracking information. Figure 5 The results show the tracking of target features and candidate features of the secondary and primary features of the Halfcylinder data.

[0100] 3) Animation visualization, the process is as follows:

[0101] Volume rendering, more specifically, a volume data ray casting algorithm, is used to display the tracked results in an animated form. The opacity of its transfer function is set using Equation 7, where... Each voxel represents a foreground feature; Indicates its opacity; and These represent the normalization strength and gradient, respectively; log() represents the logarithmic function; γ represents the user-controlled parameter for the control boundary degree.

[0102] , (7)

[0103] The embodiments described in this specification are merely examples of implementations of the inventive concept and are for illustrative purposes only. The scope of protection of this invention should not be considered limited to the specific forms stated in these embodiments; rather, it extends to equivalent technical means conceived by those skilled in the art based on the inventive concept.

Claims

1. A method for global feature tracking of time-varying data based on unsupervised segmentation, characterized in that: The method constructs a volume data segmentation network to segment the input volume data and achieve global tracking. The global tracking includes the following steps: Step 2.1: Construct a graph for recording tracking, including the following steps: Step 2.1.1: Construct a directed acyclic graph (DAG) to record tracking information. The nodes of the DAG are the partitioned domains. Each node stores the label of its time step and the histogram information of the corresponding domain. Step 2.1.2: For each feature domain at time step t and the feature domain at time step t+1 Calculate the centroid and center of mass Distance between , ， In the segmented temporal variant data, any two regions Chi-square histogram distance between for, ； in, and These are domains , The i-th bin value of the histogram, by As a domain Heyu The weight of the edge corresponding to the node. ， in, [0,1]; Step 2.1.3: When the centroid distance between two domains < or chi-square distance < At that time, establish an edge between them. , ; Step 2.2: Locate the feature tracking path in the graph, including the following steps: Step 2.2.1: Select the node containing the target domain in the DAG as the target node, and set the label in the segmented volume data to be the same as the index value of the node in the DAG. The domain corresponding to the target node is called the target feature. Step 2.2.2: Calculate the shortest path from each node to the target node at all time steps using Dijkstra's algorithm, and save the shortest path information to the new directed acyclic graph SPG; Step 2.2.3: Track the features forward and backward from the current time step of the target node, and record the forward and backward tracking paths P+ and P-; traverse the next node in turn until the end node or the start node is found; when the distance from the current node to the target node is the smallest among all nodes in the current time step and there are no outgoing edges, the current node is considered to be the end node or the start node. Step 2.2.4: Find the path from the endpoint to the target node in SPG as the forward path. Find the path from the start node to the target node in SPG as the forward path. , path and Merge to obtain the complete tracking path of the target domain. ; Step 2.3: Implement feature tracking under time and space constraints, including the following steps: Step 2.3.1: Obtain all paths for features similar to the target features corresponding to the target node at the same time step; Step 2.3.2: Filter candidate features similar to the target feature at time step ti using chi-square histogram distance. , where n is the number of candidate features; when the target feature and candidate features Chi-square histogram distance hour, The value is the same as in step 2.1.

3. Same, with Candidate features; Step 2.3.3: Calculate each candidate matching region The trajectory P' is obtained, and it is compared with the trajectory of the target feature using the longest common subsequence (LCSS). Compare; assuming the input subpath A = B= Then the length of LCSS is, ， Here In spatial constraints The values ​​are the same; The formula for the similarity of the longest common subsequence is: ， The candidate features obtained in step 2.3.2 are further filtered, which means that the candidate paths of the candidate nodes are constrained in terms of time.

2. The method for global feature tracking of time-varying data based on unsupervised segmentation according to claim 1, characterized in that: Preprocessing of the input volume data includes the following steps: Step 1.1: Obtain the original time variant data Enhance contrast and normalize the data, where N is the number of volume data points and n is a positive integer from 1 to N; Step 1.2: Divide the processed raw time-varying data into hypervoxels according to the SLIC algorithm. , which is the volume data to be input into the segmentation network, and K is the number of supervoxels to be segmented, where k is a positive integer from 1 to K.

3. The method for global feature tracking of time-varying data based on unsupervised segmentation according to claim 2, characterized in that: The constructed volume data segmentation network includes a feature extraction module; Compute a p-dimensional mapping from the feature extraction module. The constructed feature extraction module includes a 3D convolutional filter, a batch normalization function, an h-swish activation function, and M bottleneck components arranged sequentially. The bottleneck components include a 1x1x1 unfolded convolution, a 3x3x3 dilated convolution, and a 1x1x1 projection layer. The input and output of the bottleneck components are connected through the residual module. The feature extraction module is followed by a 3D convolutional filter, a batch normalization function, and a linear classifier in sequence; the linear classifier satisfies... and Where q is the number of clusters, c is the number of convolution channels, p is the feature dimension, and V is the volume data. and These are the parameters of the classifier; Map the response Normalization ,in, , , The mean and unit variance are zero, by selecting in The cluster label is obtained from the dimension with the maximum value. Improve clustering labels, making yes The middle belongs to the first The number of voxels in each cluster, replacing the cluster label with For all In other words, ; The loss between the response and the improved clustering labels is calculated using the softmax loss function, and the parameters of the convolutional filter are updated using the backpropagation algorithm. and the parameters of the classifier M is the number of layers in the 3D convolutional filter; The 3D connected component algorithm assigns different index values ​​to two spatially disconnected clusters. Voxels with the same index value belong to the same feature. The volume data is divided into domains containing different index values, thus obtaining the corresponding background or feature domains.

4. The method for global feature tracking of time-varying data based on unsupervised segmentation according to claim 3, characterized in that: The i-th cluster in is, ， in, and They represent The i-th and j-th elements.

5. The method for global feature tracking of time-varying data based on unsupervised segmentation according to claim 1, characterized in that: Visualize the segmented volume data; Set the opacity of the passed function. ， in, Each voxel representing a foreground feature, normInten( )and These represent the normalized intensity and gradient, respectively; log() represents the logarithmic function; and γ represents the parameter controlling the boundary opacity.

Images