The practice of binning point data to form a type of 2D histogram, density plot, or what is sometimes called a heatmap, is quite useful as an alternative for the cartographic display of of very dense points sets. This is particularly the case when the points experience significant overlap at the displayed scale. This is the type of operation that is used (in feature space) in Whitebox‘s Feature Space Plot tool. Normally when we perform binning, however, it is using a regular square or perhaps rectangular grid. The other day I read an interesting blog on hexbinning (see also the excellent post by Zachary Forest Johnson also on the topic of hexbinning), or the use of a hexagonal-shaped grid for the creation of density plots.
A Hex-binned heatmap (click to enlarge)
These blogs got me excited, in part because I have used hex-grids for various spatial operations in the past and am aware of several advantages to this tessellation structure. The linked hexbinning blogs point out that hexagonal grids are useful because the hexagon is the most circular-shaped polygon that tessellates. A consequence of this circularity is that hex grids tend to represent curves more naturally than square grids, which includes better representation of polygon boundaries during rasterization. But there are several other advantages that make hexbinning a worthwhile practice. For example, one consequence of the nearly circular shape of hexagons is that hex-grids are very closely packed compared with square grids. That is, the average spacing between hexagon centre points in a hex-grid is smaller than the equivalent average spacing in a square grid. One way of thinking about this characteristic is that it means that hexagonal grids require about 13% fewer data points then the square grid to represent a distribution at a comparable level of detail. Also, unlike a square grid, each cell in a hex-grid shares an equal-length boundary with its six neighbouring grid cells. With square grids, four of the eight neighbouring cells are connected through a point only. This causes all sorts of problems for spatial analysis, not the least of which is the characteristic orientation sensitivity of square grids; and don’t get me started on the effects of this characteristics for surface flow-path modelling on raster digital elevation models. Hex-grid cells are also equally distant to each of their six neighbours. I’ve even heard it argued before that given the shape of those cone and rod cells in our eyes, hex-grids are more naturally suited to the human visual system, although I’m not sure how true this is.
Hexagonal grids are certainly worthwhile data structures and hex-binning is a great way to make a heatmap. So, that said, I decided to write a tool to perform hex-binning in Whitebox GAT. The tool will be publicly available in the 3.2.1 release of the software, which will hopefully be out soon but here is a preview:
Whitebox’s Hexbinning tool (click to enlarge)
The tool will take either an input shapefile of POINT or MULTIPOINT ShapeType, or a LiDAR LAS file and it will be housed in both the Vector Tools and LiDAR Tools toolboxes. Here is an example of a hex-binned density map (seen on right) derived using the tool applied to a distribution of 7.7 million points (seen on left) contained in a LAS file and derived using a terrestrial LiDAR system:
Hexbinned LiDAR points (Click to enlarge)
Notice that the points in the image on the left are so incredibly dense in places that you cannot effectively see individual features; they overlap completely to form blanket areas of points. It wouldn’t matter how small I rendered the points, at the scale of the map, they would always coalesce into areas. The hex-binned heatmap is a far more effective way of visualizing the distribution of LiDAR points in this case.
The hexbinning tool is also very efficient. It took about two seconds to perform the binning on the 7.7 million LiDAR points above using my laptop. In the CUNY blog linked above, the author (I think it was Lee Hachadoorian) describes several problems that they ran into when they performed the hexbinning operation on their 285,000 points using QGIS. I suspect that their problems were at least partly the result of the fact that they performed the binning using point-in-polygon operations to identify the hexagon in which each point plotted. Point-in-polygon operations are computationally expensive and I think there is a better way here. A hex-grid is essentially the Voronoi diagram for the set of hexagon centre points, meaning that every location within a hexagon grid cell is nearest the hexagon’s centre than another other neighbouring cell centre point. As such, a more efficient way of performing hexbinning would be to enter each hexagon’s centre point into a KD-tree structure and perform a highly efficient nearest-neighbour operation on each of the data points to identify their enclosing hexagon. This is a fast operation in Whitebox and so the hexbinning tool works well even with massive point sets, as you commonly encounter when working with LiDAR data and other types of spatial data.
So I hope you have fun with this new tool and make some beautiful maps. Leave your comments below and, as always, best wishes and happy geoprocessing.