George Vosselman 
  
Pfeifer et al. [1998] describe a filter method based on an iterative linear least squares interpolation. Using a weight 
function which assigns low weights to relatively high points, a robust estimation of the ground surface is obtained. The 
weight function is described by four parameters. 
All methods described above use the assumption of a locally horizontal terrain or a uniform point distribution 10 
determine the ground points. In order to preserve the ground points in sloped terrain, the window sizes need to be 
restricted or distance thresholds need to be increased. The optimal values of the filter parameters clearly differ from one 
terrain type to another. Based on experience with these filters, a list of optimal parameter values can be made, but there 
is no direct relationship between these values and characteristics of the terrain. 
In this paper a new filter method is presented in which the height differences between ground points are used to 
determine the optimal filter function. The next section describes the principle of this method, the relationship to 
mathematical morphology, and some implementation aspects. In section three several methods for deriving a filter 
function are introduced. Different filter objectives will lead to different filter functions. In section four and five the set- 
up and results of a series of experiments are presented. Conclusions are given in the last section. 
2 SLOPE BASED FILTERING 
2.1 Filtering principle 
The basic idea, like in the methods mentioned above, is based on the observation that a large height difference between 
two nearby points is unlikely to be caused by a steep slope in the terrain. More likely, the higher point is not a ground 
point. Clearly, for some height difference, the probability that the higher point could be a ground point decreases if the 
distance between the two points decreases. Therefore, Kilian et al. [1996] introduce weights depending on the size of 
the morphological filter kernel and Pfeifer et al. [1998] implicitly weigh the heights by a covariance function which 
depends on the distance between two points. 
Instead of introducing weights, we explicitly define the acceptable height difference between two points as a function of 
the distance between the points: A/max(d). In general, this will be a non-decreasing function. In section three, several 
methods for deriving such a function will be described. 
The filter function can now be used to define the set of points that are classified as ground points. Let A be the set of all 
points and DEM be the set of ground points, then 
DEM - i», e AINp,eA:h, —h € Ah, Ad Cp; p;))] (1) 
In words: a point p; is classified as a terrain point if there is no other point p; such that the height difference between 
these points is larger than the allowed maximum height difference at the distance between these points. This filter 
definition is closely related to some concepts from (grey scale) mathematical morphology. 
2.2 Relation to mathematical morphology 
The erosion e(x,y) of a two-dimensional signal (x, y) with a kernel K(Ax, Ay) is defined as [Haralick and Shapiro, 1992]: 
e(x, y) = min min| A(x + Ax, y+ Ay) — k(Ax,Ay)] (2) 
Ax Ay 
For a point p; of a discrete set of points A, this corresponds to 
nis min[h, Shir, Tt Np, 7n J eo 
If we define the kernel function as 
k(Ax, y) 2 Mix (Vax? +Ay“ ) (4) 
(note the minus sign!), the eroded value at point p; can be written as 
  
936 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.