Visualising Scattered 3D Data

Scattered 3D data can be difficult to visualise, particularly when the data do not lie on a regular grid and contain regions of sparse measurements. This often occurs in geophysical or biophysical data. Fitting a functional model to the data can help visualise otherwise hidden structure in the data.

The following example illustrates the use of the FastRBFTM Matlab toolbox to visualise scattered data. The data consist of 471031 geophysical measurements made in 3D. In Figure (a) the data is shown in plan view, as a function of longitude and latitude (the axes are in degrees). Figure (b) is a histogram of data values. In Figure (c) the data are shown in 3D. The grey-level depicts the data values at a point.


(a) Plan view (scale in lattitude and longitude)
(b) Histogram of data values


(c) 3D view (scale in kilometres)

Fitting

It is difficult to see structure in the point-cloud of measurements shown in figures (a & c). Fitting an RBF to the data means that we can evaluate the data on a regular 3D grid. A 3D grid is useful for multi-planar reslicing as shown in (d) or for extracting iso-surfaces with algorithms such as Marching Cubes (e). Reslicing and iso-surfacing can be combined as in Figure (f).

Half a million points is a lot of points to fit a function to, fortunately the FastRBFTM software can fit to data sets of this size on a standard PC with modest RAM. In this example we have used an RBF to interpolate the data. New approximation and filtering techniques mean that RBFs can also be used to model noisy 3D data as well.


(d) Values on planes

(e) An iso-surface
(f) A combined view

Advanced Visualisation

Multiple surfaces can be visualised at once with appropriate choices of colour and transparency for the different surfaces (i). In Figure (g) several parallel slices through an iso-surface are shown. Only regions inside the iso-surface (i.e. data values greater the iso-surface threshold) are depicted. In Figure (h) multiple iso-surface thresholds are displayed in consecutive thick slices through the data. In Figure (j) the iso-surface has been shaded by the magnitude of the gradient at each point (which can easily be calculated by the FastRBF engine). This shows how fast the density is changing on the iso-surface. In this figure we can see that the density changes the fastest in the blue regions where the surface is more detailed.

All of the techniques shown here were performed in Matlab using FarField Technology's FastRBF Matlab toolbox.


(g) Slices with transparency threshold


(i) Transparent iso-surfaces
(h) Iso-surface slices


(j) Iso-surface shaded by gradient norm

Data provided courtesy of the Northern California Earthquake Data Center (NCEDC) and the Northern California Seismic Network, U.S. Geological Survey, Menlo Park.