Generalized 3D fragmentation index derived from lidar point clouds
© The Author(s) 2017
Received: 7 December 2016
Accepted: 22 March 2017
Published: 20 April 2017
Point clouds with increased point densities create new opportunities for analyzing landscape structure in 3D space. Taking advantage of these dense point clouds we have extended a 2D forest fragmentation index developed for regional scale analyses into a 3D index for analyzing vegetation structure at a much finer scale.
Based on the presence or absence of points in a 3D raster (voxel model) the 3D fragmentation index is used to evaluate the configuration of a cell’s 3D neighborhood resulting in fragmentation classes such as interior, edge, or patch. In order to incorporate 3D fragmentation into subsequent conventional 2D analyses, we developed a transformation of this 3D fragmentation index into a series of 2D rasters based on index classes.
We applied this method to a point cloud obtained by airborne lidar capturing a suburban area with mixed forest cover. All processing and visualization was done in GRASS GIS, an open source, geospatial processing and remote sensing tool. The newly developed code is also publicly available and open source. The entire processing chain is available and executable through Docker for maximum reproducibility.
We demonstrated that this proposed index can be used to describe different types of vegetation structure making it a promising tool for remote sensing and landscape ecology. Finally, we suggest that processing point clouds using 3D raster methods including 3D raster algebra is as straightforward as using well-established 2D raster and image processing methods.
Keywords3D raster Voxel model Spatial pattern Lidar Raster algebra Spatial indices
Data acquired by airborne lidar have transformed how the Earth’s surface and vegetation structure are mapped and analyzed leading to many applications, for example, in terrain modeling and ecosystem studies .
Lidar point clouds have been used not only to map the spatial distribution of vegetation [2–4], but also to analyze the vertical structure of forested and savanna ecosystems [5–8]. With the increasing density of points obtained by the new types of lidar technologies, such as single-photon lidar, which produce orders of magnitude more points , there is a need for new techniques that would take advantage of high point densities and provide analyses to support improved ecosystem management.
Many existing methods for 3D point cloud analyses are limited to 2D or 2.5D [7, 10, 11], have been implemented in a specialized lidar-processing software [5, 12, 13], or use custom low-level code . To make advanced analysis of point clouds more general and accessible, we use 3D rasters and associated 3D raster algebra as the basis for developing new methods for lidar data analysis. 3D rasters, also referred to as voxels, voxel models, voxel-based space, or 3D grids, are used in many fields such as soil science , geology , atmospheric sciences , human anatomy , and 3D printing . In the fields of remote sensing and geographic information systems, 3D rasters have been used with airborne lidar data to characterize fine-scale bird habitat  and with terrestrial lidar to characterize forest canopy fuel properties  and detailed tree models [21, 22]. In ecology, spatio-temporal data in 3D rasters have been used to quantify the complexity of simulated population dynamics  and 3D rasters representing trees have been used to assess lighting conditions . Remotely-sensed hyperspectral data have been represented and processed as 3D rasters to extract textures .
In order to describe vertical vegetation structure, we define a 3D version of a 2D forest fragmentation index introduced by Riitters et al. . Different spatial indices have been used to describe land cover structure [26–31]. These indices were implemented in various software packages including SPAN software , the r.le software package coupled with GRASS GIS  and later replaced by different set of modules for GRASS GIS called r.li , the well-known FRAGSTATS software package for computing spatial indices , the GuidosToolbox software package for the assessment of pattern, connectivity, and fragmentation , and the SDMTools R package for species distribution modeling . Jjumba and Dragicevic  presented a set of indices for the basic analysis of data represented as 3D rasters. Parrot et al.  defined 3D metrics for the analysis of spatio-temporal data in ecology.
The original 2D forest fragmentation index by Riitters et al.  was created to characterize the spatial configuration and structure of a forest at a global scale. The presented 3D fragmentation index can be used in applications describing 3D vegetation structure, classifying vegetation types, characterizing fine-scale bird habitat in three dimensions, or describing overall landscape characteristics. We present this new 3D fragmentation index as an example how a 2D index or a 2D filter can be extended into 3D and implemented in a similar way as its 2D version. To use the 3D rasters with established 2D raster processing methods and tools, we also present several methods for converting a 3D raster into a series of 2D rasters.
We provide source code for all the presented methods, which we implemented as modules for GRASS GIS, so that they can be used together with other open source geospatial processing tools . We also provide a repository with all the materials needed to fully reproduce the research presented here using Docker .
Vegetation structure reconstruction
A 3D raster can be created from a lidar point cloud using a process called binning, rasterization, or voxelization. A value of a 3D cell (voxel) is determined by presence, count, or properties of the points which fall into a 3D space occupied by a given cell [12, 21, 38, 42]. We use point heights relative to the ground surface. Generally, binning produces outputs such as number of points per cell or, if points have values associated with them, a mean of these values or other statistics. We use binning where each cell which contains one or more points is assigned value 1, while empty cells have value 0. Alternatively, we could use a threshold for point count, mean intensity, or a percentage of points within a vertical column [8, 42].
2D forest fragmentation index
The operator ∧ is logical AND, which we define so that it yields 1 when both cells have value 1 and 0 otherwise.
The operator ∨ is logical OR, which we define so that it yields 1 when at least one of the cells has value 1 and 0 otherwise.
The final classification used to create the index is described in the next section.
3D fragmentation index
Most raster and image processing algorithms operate on 2D rasters, not 3D rasters. 2D rasters are easier to combine with other 2D data and more suitable for creating printed or on-line 2D maps. Therefore, it is necessary to convert a 3D raster into 2D representations so that different approaches can be adopted based on the information one wishes to preserve or highlight.
The basic conversion involves splitting the 3D raster into horizontal slices, which will be represented as a series of 2D rasters. Each 2D raster represents a slice at a certain depth in the original 3D raster. This approach preserves the information about the relative height based on the order of the 2D raster in the series. The resulting series of 2D rasters can then be processed as any other series or used as image bands in subsequent analysis.
Number of cells per vertical column with a given class
The value which is the result for the central cell is then assigned to the corresponding cell of the 2D raster. This also creates a continuous raster from classified (i.e. categorical) data such as the fragmentation index. Given the fragmentation index, we can, for example, measure the number of cells within the patch class and define a measure of patchiness based on that.
For the purpose of this study we replace b i,j with 0 when we apply this method to the fragmentation index because we take into account all vegetation above the ground level and we use heights relative to the ground. An example at Fig. 4 shows the differences in between the absolute and relative counts and count without the surface constraint.
Once we have relative count for each of the classes, we can also determine the most common (i.e. dominant) class for each vertical column by finding a class with maximum q.
For this study we used an airborne lidar point cloud. The study site, depicted in Fig. 5, is a 16 hectare (38 acre) area on North Carolina State University’s campus. The data were collected during leaf-off conditions in January 2015 by the North Carolina Floodplain Mapping Program. The point cloud was classified by the data provider. We used only points classified as ground (class 2) and vegetation (classes 3, 4, and 5).
Reconstructed vegetation structure
The computations were performed with cubical cells with approximately 0.9 m (3 feet) edges. With this data it was necessary to reconstruct the vegetation structure as some parts did not have enough cells with at least one point. The average point density is 2.0 points per 2D cell (0.9 m × 0.9 m) and the point density in 3D is 0.044 points per 3D cell (0.9 m × 0.9 m × 0.9 m). However, as a profile (vertical slice) in Fig. 2 shows, the structure of the vegetation emerges after the reconstruction step with a 3×3×3 neighborhood.
A profile of the 3D raster representing the fragmentation index in Fig. 2 shows areas with different structures and distributions of the index classes using 3×3×3 neighborhood. The low vegetation contains only a few exterior cells under the top of the canopy, while the higher vegetation has a lot of exterior cells in some areas and higher numbers of transitional and edge cells in other areas (likely indicating different vegetation type, not only height). The middle part of the profile in between the low and high vegetation is very dense, resulting in a lot of interior and perforated cells. Figure 6 shows the profile from Fig. 2 placed into the 3D raster together with an orthophoto.
Although we see generally the same behavior for all the zones in Fig. 10, such as high number of exterior cells, we can also observe more unique behavior for some of the zones. The zones number 2 and 6 have the lowest ratio of exterior cells under the vegetation surface, and the zones 1, 4, and 5 seem to follow the same pattern for both absolute and relative counts.
The 3D (forest) fragmentation index is implemented in a new GRASS GIS module r3.forestfrag, which shares most of its code with the 2D version, the r.forestfrag module. This was possible thanks to the extensive use of raster algebra in both modules. The r.forestfrag module received a major code update as part of this work.
Furthermore, we implemented the counting of cells with a given class (i.e. category counting) in a vertical column of a 3D raster in a new module called r3.count.categories. To create profiles of 3D rasters (Fig. 2), we implemented the r3.profile module, which slices 3D raster vertically between two given points. Once again we based the code of the r3.profile module on its 2D equivalent, the r.profile module, which creates a profile from a surface map.
Finally, we prepared a publicly available Git repository hosted on GitHub. The repository contains the data for the study area, scripts to perform the analyses presented here, and details about the dependencies. Using Docker, this repository can be turned into a complete runtime environment to produce the figures (except for Figs. 1, 3 and 6), plots, and all underlying data for this manuscript. We also connected the repository with a continuous integration service, Travis CI, which will show if the basic functionality was broken by any future changes.
Metrics for data represented as 3D rasters were presented in the past for plot and artificial data [5, 38] and specific applications [12, 20, 43]. We present a general fragmentation index based on its 2D version  and apply it to a sample study area. Although the initial testing of the methods and the software was done on 13 million points, the study area we selected for the manuscript is much smaller and contains only about 900 thousand points to make all the figures quickly reproducible and the data easily distributable as detailed at the end of this section.
The individual classes of the 3D fragmentation index can have different meanings. A specific application can assign ranks or weights to individual classes based on their importance for a particular use case. The interior class can be important for forest structure study, but certain bird species habitat may be associated with perforated or edge classes. Alternatively, P f and P f f variables or their ratio can be used directly, leaving out the classification completely.
We applied the 3D fragmentation index on a 3D raster containing zeros and ones which was based on presence or absence of lidar points in a cell or surrounding cells in a window 3×3×3 at resolution 0.9 m. Alternatively, the 3D raster with zeros and ones can be derived from cells falling above or below some threshold for point count, total or mean intensity per cell, or their combination. The window size may be altered as well depending on how many cells contain at least one point. However, larger window size could cause creation of artificial interior cells. The 3×3×3 window creates an envelope around the cells with points which is subsequently classified as edge. The overall structure is not influenced as visible from the profiles in Fig. 2, but we must be careful when using the results in applications where the exact position of the edge cell matters. With dense enough point clouds, for example from waveform or single-photon lidars, there isn’t any need for a moving window and structure reconstruction step.
The choice of resolution is influenced by the application and point density. We used small cubical cells to capture details in tree canopy or understory, but for studying individual trees or vegetation patches, it may be appropriate to choose different vertical and horizontal resolutions. Using a coarser resolution may also remove the need for the vegetation structure reconstruction step because when the point density is high enough, the cells with points will start to touch each other and there won’t be any gaps for structure reconstruction to fill. On the other hand, no window at all in combination with sparse point cloud would lead to potentially incomplete model of the tree structure, no interior cells, and many transitional and patch cells. Again, the significance of this depends on the context in which the resulting fragmentation index is used.
To compute the fragmentation index, we again used 3×3×3 window. The size of the window together with the vegetation structure reconstruction causes only insignificant number of cells to be classified as patch as visible from Fig. 10. Larger window size would yield larger number of patch cells and it would change the overall fragmentation result as well [26, 29, 44]. The horizontal proportions of the window should be kept the same, i.e. the 2D projection of the window should be square. However, the size in the vertical direction can be different because, depending on the choice of resolution, the vertical relations may be different than the horizontal ones. In other words, we expect the horizontal relations to be isotropic, but in the vertical direction we may encounter anisotropy depending on what the vertical dimension in the 3D raster represents.
The choice of the window size for the fragmentation index depends on the application and using multiple window sizes may be appropriate for regressions and classifications.
The 3D fragmentation index depends on lidar pulses penetrating through the top of the canopy. When the canopy is dense, lidar pulses may not penetrate it resulting in no points under the canopy and limited applicability of 3D raster methods. The penetration also depends on the sensors used. If the point cloud comes from processing unmanned aerial system (UAS) imagery, it typically captures only the top of the canopy, while the fragmentation index works on a full 3D raster (as opposed to surface represented by 3D cells). However, UAS equipped with lidar may provide a more complete representation of the vegetation.
We applied the index strictly on vegetation, specifically different types of forest and groups of trees using lidar point cloud where the points were already classified. However, using the fragmentation index together with the technique of counting cells of a given class per vertical column under a given surface, we could support classification of the point cloud because, for example, the buildings would be characterized by high number of exterior cells under the surface and only one patch or transitional cell right below the surface.
Processing of 3D rasters
In the context of point cloud and land cover analysis, we can now take advantage of combining the 3D raster data and processing with the 2D data so that the techniques currently in use can be enhanced by the explicit information about the 3D structure. We described and demonstrated the 3D and subsequent 2D analyses on the presented 3D fragmentation index which can be combined with the commonly used 2D (or 2.5D) metrics such as canopy height, point density, mean intensity, some of the more specific point cloud measures such as canopy-relief ratio , or spectral and hyperspectral data. However, alternative approaches exist, for example, 2D rasters can be extruded into 3D rasters (with the same value for all depths) and further used in 3D computations fully preserving the 3D relationships captured in the other 3D rasters.
The 3D version of the 2D forest fragmentation index, similarly to the 3D Moran’s I presented by Jjumba and Dragicevic , shows that 2D indices can be transfered to 3D while keeping similar semantics as the 2D version. Additionally, we can observe that 3D raster algebra and 3D raster tools in general can be used for analysis in the same manner as the 2D equivalents. For example, the raster algebra expressions to determine fragmentation class based on P f and P f f values is the same for 2D and 3D and also the P f value is computed in a similar way; only the expression for P f f is different in 2D and 3D due to additional members and third value for referencing the neighboring cells.
The processing times and disk space will be typically higher for 3D rasters simply because there is one more dimension to take care of in comparison to 2D rasters, however the parallelization of the computations can be as straightforward as in 2D. In general, 3D rasters can be used for development of new remotely sensed data processing techniques and subsequent landscape ecology methodologies and measures such as landscape indices or connectivity metrics. Specifically 3D rasters in GRASS GIS were used in past for modeling of evaporation processes in 3D .
We not only provide newly written source code and use open source software, but we also prepared a repository for a full reproducibility which is considered equal with replicability, repeatability, and recomputability  for the purpose of this manuscript. The source code is a necessary part of the method description [48, 49] and, together with documentation, a step towards re-usability. In addition to it, the use of open source software for the dependencies, most notably GRASS GIS, makes reproducibility possible  and opens the whole underlying computational environment for review.
However, the full and easy reproducibility is possible only when the whole processing chain and environment is shared. Thanks to Docker and the way we prepared the repository with data and code, our processing environment and the results can be reproduced on any computer . The results cannot be reproduced within 10 minutes on a standard workstation as required by Schwab et al.  for easily reproducible result because of the environment building and processing time. However, obtaining the result requires up to 10 minutes of preparation, depending on whether Docker is installed on the computer or not, and the total time including building and processing is within an hour.
In the fields of remote sensing, GIS, and landscape ecology, scientists often process remotely sensed data as 2D rasters (images). Comparing to 2D rasters, 3D rasters are less common, despite the fact they explicitly preserve the 3D relations in the data such as lidar point clouds.
We show how 3D rasters can be utilized in a general remote sensing and GIS environment to process lidar point clouds. We used a point cloud, which is dense enough for reconstructing the structure of vegetation. The 3D raster we obtained represents the spatial information about the structure of vegetation in all three dimensions.
Well-known concepts such as moving windows and indices used in 2D processing in remote sensing or landscape ecology can be also applied in 3D. The process of doing so is very straightforward. We redefined an existing 2D forest fragmentation index as a general 3D fragmentation index, which we used to describe 3D vegetation structure. This index is represented as a 3D raster, however we show how to use it in 2D to leverage common 2D processing techniques.
The newly developed code is open source and is implemented in a well-established GIS and remote sensing software GRASS GIS. Standard GRASS GIS tools, most notably 3D raster support, were used for rest of the analysis showing how GRASS GIS can be leveraged for processing lidar point clouds in three dimensions as well as in two dimensions.
Availability of data and materials
We used a suite of existing and newly developed software to implement the new methods and perform the analyses. We used publicly available data and aim for full reproducibility of the results presented in this study.
Newly developed software
We implemented the 3D version of the forest fragmentation index as a GRASS GIS module called r3.forestfrag. We developed the r3.count.categories module to count cells with a given class (i.e. category) and we also developed the r3.profile module to create profiles (i.e. vertical slices) from 3D rasters. The r3.profile module is written in C programming language, while r3.forestfrag and r3.count.categories modules are in Python. All three modules are distributed under the terms of the GNU General Public License 2  or higher (GNU GPL). The modules are now available in the GRASS GIS Add-ons repository. The documentation for the modules is available online .
The rest of the study used GRASS GIS , its graphical user interface, and modules which can run in command line. The most important modules used in this study were r3.in.lidar  and r3.to.rast. GRASS GIS is platform independent, is supported on all common desktop operating systems and can run on servers and clusters. For our study we used a desktop computer with Ubuntu . GRASS GIS is licensed under GNU GPL and, therefore, does not impose any special restrictions for use by non-academics. GRASS GIS is written in C and Python. The r3.in.lidar module requires libLAS library for reading lidar point clouds in LAS format.
The lidar point cloud data used in the study are from the North Carolina Floodplain Mapping Program and are available through North Carolina’s Spatial Data Download website . We used tile LA_37_20079301_20160228 from phase 3 from January 2015.
Code for full reproducibility
The Git  repository with data for the study area and scripts in Bash for running the analyses is hosted on GitHub and publicly available  under GNU GPL. The repository contains a Dockerfile so that Docker  can be used to create the exact environment used to run the analyses for this study. The repository’s web page contains instructions how to download it and the two commands needed to reproduce all of the data and figures presented in this study. The status of the code can be reviewed at Travis CI , a continuous integration service. A snapshot of the repository is also available as Additional file 1.
Two dimensional, two dimensions
Two and half dimensional, two and half dimensions
Three dimensional, three dimensions
Geographic information system
GNU’s Not Unix!
- GNU GPL:
GNU General Public License
Unmanned aerial system
We are grateful to the GRASS GIS developer and user community for developing and maintaining GRASS GIS software package. We acknowledge Emmanuel Sambale, Stefan Sylla, and Paulo van Breugel who originally implemented the r.forestfrag module in GRASS GIS. We would like to also acknowledge the creators of the open source software used for this study, including, but not limited to, Python, GNU Compiler Collection (GCC), Linux, Ubuntu, ImageMagick, Kile, LaTeX, Docker, Travis CI, and Git. We acknowledge GitHub, Travis CI, and Overleaf services. We also acknowledge Paul Tol for the color-blind safe and printer-friendly color scheme for qualitative data  used in Figs. 8, 10 and 12. We acknowledge the authors of the viscm tool , which we used to design the color table for the fragmentation index (Fig. 3). We also acknowledge the authors of Matplotlib color tables viridis and plasma which we used in Fig. 5 and modified for Fig. 9. Finally we also acknowledge Brendan A. Harmon and Anna Petrasova for providing feedback on the text of the manuscript.
VP developed the methods, case study, processed data and prepared the manuscript. HM and DJN provided critical revisions to the manuscript. All authors read and approved the final manuscript.
The authors declare that they have no competing interests.
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