Incremental Meshfree Approximation of Real Geographic Data

The most frequent task for many engineering problems is the reconstruction of the given data. There have been developed several algorithms for interpolation or approximation of the given data. Nevertheless, they mostly expect some kind of data ordering, e.g. rectangular mesh, structured mesh, unstructured mesh, etc. This requirement is not necessary when the meshless techniques such as the Radial Basis Function (RBF) methods originally introduced in [1, 2] are used. RBF techniques can be used in many fields of the technical or non-technical problems, e.g. reconstruction of surfaces [3, 4, 5], visualization of data [6], solving partial differential equations [7, 8]. The RBF methods are based on the distance computation between two points and they are independent of the dimension of the space. When the RBF techniques are applied, the given data can be described using analytical formula. The significant compression of the given data is also achieved by using RBF approximation.

A significant role in terms of the quality of RBF approximation and the compression ratio plays the appropriate placement of the reference points for RBF approximation. In the case of geographic data, this requirement is met for placement along significant features such as ridges, peaks, valleys, etc. A new incremental approach for RBF approximation that puts the emphasis on good placement of reference points and significantly improves the compression ratio will be described in this paper.

In the following sections, the fundamental theoretical background needed for description of the proposed approach will be mentioned. The proposed incremental RBF approximation will be described in Sect. 3. In Sect. 4, the results of our proposed algorithm will be presented. Finally, a final discussion of results will be performed.

In this section, some theoretical aspects needed for description of the proposed incremental approach will be introduced.

In this section, the proposed incremental approach for approximation geographic data using radial basis functions is described.

In every following level \(k > 2\), the residues \(\varvec{r}_{k-1}\) are filtered by applying the Gaussian low-pass filter due to eliminating insignificant local maxima. Then, the set of stationary points for filtered residues are determined using algorithm in [10] and only local maxima are added to the set of reference points. Moreover, the uniqueness of the added reference points is checked. When the new set of reference points is obtained, the RBF approximation (described in Sect. 2.1) is again computed and residues \(\varvec{r}_k\) are calculated using Eq. (1). The whole process is repeated until the required accuracy of approximation is achieved or the maximum permissible compression ratio is exceeded.

Finally, it should be noted that the value of standard deviation \(\sigma _k\) of Gaussian low-pass filter in \(k^{th}\) level is set as:

$$\begin{aligned} \sigma _k = {\left\{ \begin{array}{ll} \displaystyle \sigma &{} \displaystyle k= 1, 2\\ \displaystyle \frac{\sigma _{k-1}}{2} &{} \displaystyle k= 3,\dots , L \end{array}\right. } \end{aligned}$$

(2)

where \(\sigma \) is initial value. The whole pseudocode is in Algorithm 1.

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In this section, the experimental results for our proposed approach will be presented. The implementation was performed in Matlab. The thin plate spline (TPS) function \(r^2\log (r^2)\) which is shape parameter free and divergent as radius increases has been used for RBF approximation.

For the purposes of below mentioned experiments, two geographic point clouds were used. The first dataset was obtained from GPS data of the mount Veľký Rozsutec in the Malá Fatra, Slovakia (Fig. 1a) and contains 24,190 points. The second dataset is GPS data of the part of Pennine Alps, Switzerland (Fig. 2a) and contains 131,044 points.

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Results for different levels of RBF approximation of the mount Veľký Rozsutec are shown in Figs. 1b–d. We can see that the quality of approximation in terms of error is improving with increasing level of the incremental RBF approximation. For \(8^{th}\) level (see Fig. 1d), the many details of the original terrain are already apparent.

In Fig. 2b–d, the results for different levels of incremental RBF approximation of the part of Pennine Alps are shown. It can be again seen that the quality of approximation is improving with increasing level of the incremental approach. For the first level (see Fig. 2b), it is evident, that the small number of reference points is defined for the ridge in the foreground, and therefore, this ridge is approximated by several peaks in the first level. This problem is eliminated with increasing level of the incremental RBF approximation. For \(7^{th}\) level (see Fig. 2d), the many details of the original terrain are again apparent.

The mean relative error in dependency on compression ratio is presented for both geographic datasets in Fig. 3. Moreover, the comparison of the proposed incremental approach with the classical RBF approximation [9] is performed. From the results, it can be seen that the proposed approach achieves the better quality of results in terms of error.

In this paper, a new incremental approach for RBF approximation for geographic data is presented. Selection of the set of reference points for proposed incremental approximation is based on the determination of stationary points of the input point cloud in the first level and the finding local maxima of residues at each hierarchical level. In addition, the Gaussian low-pass filter is used to smooth the trend of the input points, resp. the residues before finding significant points.

The proposed approach achieves the improvement of results in comparison with other existing methods because the features of the given dataset are respected.

In the future work, the proposed approach can be extended to higher dimensions, as the extension should be straightforward. Also, the improving the computational performance without loss of accuracy can be explored.