pradition: 
(3) 
À s= (N-1) 
in which @ 
and f=1/L, 
N/2 relation 
) condition, 
md @. Thus 
um limes do 
possible. 
estimations 
| by errors 
pidation at a 
this case the 
n the space 
ated with a 
cen Z(s) real 
ss, through 
‘vene for a 
1 spectrum, 
is called im 
, Root. W.L, 
h Menning 
wN) I m= 
m order to 
components 
presion: 
(4) 
-0,@-1 ) 
using the 
) (6) 
sema, out of 
»ughmess is 
(fie) cut-off 
ough which 
arated from 
in signals 
h retaining 
    
certain component parts of a signal, when it passed 
through the filter or window. In their numerical variant 
the filters are described mathematically by am PI 1 
operator, which converts Z( nAs ) input signal in z( nAs) 
output signal; 
z(nAs)=P IZ(mAs) 1 (7) 
achieving fillerig functions of lowpass type, highpass, 
bandpass and bandstop. According to PI 1 operator 
properties different filter classes can be achieved. 
Invariant in space ( or time ), linear systems have the 
largest use, because they allow a easier mathematical 
treatement. 
The condition of linearity as well as that of invariance are 
imposed to the operator in order do get these systems 
(Hamming. RB. M, 1977 ). According to the first condition, 
if z1( mAs ) and z2( nAs ) represent the filter respons to 
Z1( mAs) and Z2( nAs ) input samples, the filter is linear 
only when: 
P [ aZi(mAs) * bZ2( nAs) 1 7 
=aP i Zi(nAs)] * bP I Z2(nAs)? 1-7 (8) 
= azl( mAs ) + bz2( mAs) 
where a and b are dwo arbitrary constants. The second 
condition require that the filtering effect be the same 
irrespectively of the filtered sample position. Thus if the 
respons (0 Z( nAs) input sample is z( nAs ), then respons 
to Z((n-k) As ) sample will be z((n-k) As ). 
hi») is considered the result of the transformation 
obtained through the application of PI 1 operator to 
ö (m) impulse signal: 
h(n) = PLo(m 1 (9) 
Also, PI ] operator is applied to the signal expressed by 
the relation: 
N 
PI Zín) 1» PI X Z(i)d(m-i) ] (10) 
i=1 
By using the linearity condition, ( 10 ) becomes: 
N 
z(nm) = 3X Z(i) PLO (m-i) 1 (11) 
i=1 
The following relation results taking into accoumt of 
invariance condition: 
(12) 
N N 
z(n) D ZCi)h(n-i)= Z h(i)Z(n-D=h(n)"ZCn) 
i=1 i=l 
where the second equality is obtained from the ( n-i ) => i 
change of variable and third represents the shortening 
forma. 
The relation ( 12), called also the convolution sum, proves 
that a space invariant and discrete linear filter is entirely 
characterised by its respons to signal impulse. Thus 
knowing h( n ) kernel, by computing the convolution sum 
with input data Z( n), will result z( n ) filters respons. 
The z( n ) response values can be obtained by directly 
calculating the convolution sum. But this method means 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
an important operation volume because N multiplications 
are mecessary for each Zi height sample. The cutting of 
operations number is achievsd by using fast Fourier 
transformation algoritms. Thus ,if F1z(k) 1, 5 [ Z(k 
) 1 and FI h(k) } are Fourier transforms corresponding 
to z( m), Z( n ) and h( n ), can be write: 
Fluk)l= F1 h(k) 1* F1 ZAKk) 1 = 
-HC R)* F(K) (13) 
and respectively: 
z(k)7 Z'(ZIh(k)1* 7IZ(k)1)-7 
- 2 [HO * FK) 1 (14) 
( 13 ) and ( 14 ) relations associated to a fast Fourier 
transformation ( FFT ) algorithm implemented in a 
subroutine lead to the following operating procedure: 
31 h(k) ] and 71 Z( kK) 1 discrete Fourier transforms 
are computed at a begining, then F1 h(k) 1° F1 Z(k) 1 
product is dome and finally z( k ) respoms filter is 
determined through inverse Fourier transformation 
( Stolojanu. G, Podaru. V, Cetina. F, 1984 ). 
In this approach context, the roughness is determinated 
through a filtering process, frequentially achieved, im 
which 271 h( Kk) 1 transform or H( k ) is reffered as 
process transfer function, respectively as of the applied 
filter. 
Butterworth filter was chosen for filtering execution, 
starting from idea to use fr frequency level corresponding 
io roughness, that main factor within terrain data 
processing. The reason of this choise is that it poses a 
transfer function which offers possibility of being directly 
conditioned to operate taking into account this parameter. 
It is used as low frequency filter, having fc cutt-off 
frequency equal to fir for the separation of components 
which represents terrain roughness. 
The transfer function for Butterworth lowpass filter 
(BLPF ) of n order is given by the relation: 
H(ji)-1/L1-7(£/íc)?^ ] (15) 
(fe = cutt-off frequency ) 
Grafically represented for n = 1, has the form show in 
figure 1: 
Hljf) 
1 
1 i A Jmm. 
gies 2 3 wp, 
Fig. 1 The transfer function of the Butterworth’ s filter 
(m=1) 
The analyse of function graphic reveals that it cannot 
strictly define the separation between filtered frequency 
and unfiltered ones. That is why the place of fc cutt- off 
frequency is established in a position for which H( jf ) is 
smaller with a certain fraction than its maximum. A 
common value used is ( 1/V2) H(jf max. Changed in this 
sense, relation (15) will have the following form 
( Gonzalez. R. C, Wintz. P, 1987): 
  
  
  
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