According to this equation a surface can be treated as the 
superimposition of a finite number of sinusoidal components of 
the adequate amplitudes, frequencies and phases. 
The function that emerged from FFT does not have a simple 
physical interpretation. Certain conversion of spectrum is 
necessary in order to determine physical parameters. An 
amplitude spectrum it is the absolute value of the FFT 
spectrum: 
X n (v) = X(w) = ol (e) + Xn (@) (4) 
The phase spectrum is calculated using following equation: 
X imag (v) 
# & rol (c ) 
X, (0) 7 arctg (5) 
The frequency structure of tested surface is represented by the 
amplitude spectrum and phase spectrum. Using the spectrum 
one can unambiguously reconstruct a topographical surface. 
The spectrum is thus an alternative and equivalent way of a 
surface representation. 
The power spectrum is defined as the squared magnitude of the 
FFT of the data divided by the number of measured points. A 
periodogram it is an estimation of the spectral density function. 
It is important to know the characteristic of such curve in order 
to be able to distinguish between portion representing true 
elevation of surface and portion representing noise (so-called 
cut-off frequency @,). 
2.1.2 Digital filtering: The digital filters, which do not pass 
a given frequency components of topographical surface, can be 
applied when the cut-off frequency is known. 
According to definition digital filter is a discrete system that 
converts the input data in a certain way, by changes in the 
spectrum. 
In a numerical form the filter is described by an P/ / operator, 
which converts z(n) input data in z-(n) output data, called a 
filter response (Ionescu, 1996): 
z, (n) - Plz(n)] (6) 
Digital filter can be presented in the spatial domain as a discrete 
convolution sum of the measured data and impulse response 
h(n): 
zy, (n) 2 h(n)* z(n) (7) 
Digital filter may be also described in frequency domain as a 
so-called frequency response i.e. Fourier transform of impulse 
response: 
N-| — F-qeAx- mo 
H(p)- Y hgn.e "em 
n=0 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
The frequency response of a low-pass filter is presented in the 
Fig. 1. 
H(o) A 
  
  
  
  
  
> 
© 
  
Figure 1. The frequency response of low-pass filter (cut-off 
frequency o.) 
2.2 Filtering procedure 
The principal assumption is based on the idea, that the low 
frequencies are responsible for the run of topographical surface, 
while the high frequencies are connected to the field objects, 
located on the ground. It is necessary, therefore, to design a 
low-pass filter, which let lower frequencies pass, and blocks 
higher frequencies (Marmol, 2002). The designed method it is 
an iterative process, which consists of the following phases: 
Interpolation 
The theory of digital transformation of signals is based on the 
assumption, that a considered flow of the initial data it is a set 
of data distributed in equal distances. To fulfil that condition it 
is necessary to interpolate points located in regular matrices out 
of the cloud of points registered during the laser scanning. As it 
is well known, the laser data comprise not only ground points, 
but also the points located on various field objects. An 
interpolation process can contribute to the significant 
falsification of the topographical surface run, and can make the 
filtration process difficult. To reduce the errors the Nearest 
Neighbour interpolation was applied to the original data. 
The trend analysis 
The subsequent research phase was aimed on determination of 
the trend of analysed surface. The removal of trend must 
necessarily be executed before applying the FFT, to omit an 
influence of trend during estimation of spectrum. 
The trend analysis consist in a least square approximation of a 
set of points with the use of a polynomial function. The degree 
of fitting the data by the trend is evaluated by determination 
coefficient. (Kokesz, 1984). 
The significance of empirical data approximation was verified 
with the use of F-Snedecor test with the significance level 
a=0.01 (Marmol, 2003). 
The influence of the polynomial degree on a quality of fitting of 
empirical data by the trend was also tested. 
Designing the digital filters 
In the research a FIR filters (Finite Impulse Response) are used. 
The simple method to design a FIR filter is the window method. 
The procedure consists of following stages: 
    
  
    
  
    
    
  
   
  
   
    
  
    
  
   
   
   
   
   
   
    
   
   
    
     
    
   
  
  
   
  
   
  
    
   
   
    
  
  
  
  
  
  
  
     
Inter 
  
The 
coef 
In tl 
then 
dime 
were 
FFT 
Usir 
dete 
char 
dom 
pres 
45 
te 
Amplitude 
The 
Tha 
Det 
the 
The 
tre: 
tha 
me 
syr 
det