The model equations (17),(24),(25) become (33)-(35), re- 
spectively. 
Paz Ta RE d 
z(t—1) 
Va ufu e e, PT. 2:37] (33) 
v4 8* 
Vol = ()_ ail Veen) + ZT SZ) (34) 
J=v 
At ee ayAi—10y c (Y: — dudit ardt) 
(Ye — au PLZ) (1+ ZV 2)™ (85) 
Z. = es Dr (36) 
The filtering matrices ay, a, are diagonal so the increase of 
computational complexity for the algorithm | is very moder- 
ate. 
6 NUMERICAL REALIZATION 
The predictors in (19) can be evaluated using updating of 
matrices Vi, (12) and their following inversion. Another 
possibility is the direct updating of P;, (17), (33). To ensure 
the numerical stability of the solution, it is advantageous to 
calculate P;, (17), (33) using a square-root filter, which 
guarantees the positivity of matrix (12). The filter updates 
directly the Cholesky square root of matrices Vi : 
Alternatively it is possible to use the UDU filter (a factoriza- 
tion into two triangular and one diagonal matrices) for this 
purpose. Note that the same square-root filters can be used 
also for the updating of statistics of the directional forgetting 
algorithm version. They only difference is in input filter val- 
ues. Initialization of recursive (17), (24) and (25) must keep 
the condition of positive definiteness of matrices Vio (8). 
We implemented in our algorithm the uniform a priori start : 
Y oc. (37) 
This solution not only conforms with the initial lack of in- 
formation at the start of algorithm, but also simplifies the 
calculation of the integral (20). Another possibility could be 
for example a local condition start, which ensures a quicker 
adaptation . 
7 RESULTS 
In this section we present simulation results of the pro- 
posed reconstruction method and compare them with meth- 
ods briefly surveyed in the introductory section. The per- 
formance of the methods is compared on artificially created 
bad lines (removed from the unspoiled parts of the images 
so that the original data are known) using the criterion of 
mean absolute difference between original and replaced pixel 
values 
lo xx ; 
MAD = — pk | gir 
ny 2.2 Wo Yinl (38) 
JT i= 
where v = 1 for the mono-spectral model. 
812 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
  
  
  
  
monospectral multispectral 
method | radar | SPOT | TM | SPOT 
MAD | MAD | MAD | MAD 
A 701 2.8 24 1.25 
B 412 1.3 20 0.7 
C 394 1.8 19 0.9 
D - - 23 2.6 
E - - 21 1.1 
F - - 19 1.5 
G 135 1.23 16 0.7 
| 132 1.1 14 0.5 
| 130 0.72 10.1 0.34 
  
  
  
  
  
  
  
Table 1: Single spectral band reconstruction. 
Pixels corresponding to the l;; are denoted ^ and the 
reconstructed pixel o , respectively. 
The first example is the defective radiospectrograph image 
shown in Fig.1 from the Ondfejov Observatory 1000 - 2000 
MHz radiospectrograph observation of the solar radio emis- 
sion. The frequency band is divided into 256 channels (the 
frequency resolution of about 4 MHz) and the grey level ra nge 
of pixels is 0-2800. 
The optimal reconstruction models M; for the radiospectro- 
graph were found to be: 
* xk * + 
AS * + 9 + x Mo x * O * * 
The second tested image Fig.2 was the agricultural type of 
the Thematic Mapper seven spectral band sub-scene from 
North Moravia. The failed line was located in the TM1 band 
and the selected models are: 
* 
Mi X ok q9iow M3 * O0 o + 
* * 
The last two examples are SPOT multi-spectral image Fig.3 
(agricultural scene from Moravia, failed line located in the 
green visible band) 
Mit M; ^ 
* 
Q UR + *^ x O + 
and SPOT panchromatic image (agricultural scene from 
vicinity of La Rochelle, both models are + * o * x 
y 
Table 1 contains monospectral line reconstruction results. 
Method | (regression method with the directional forgetting) 
demonstrates improvement in comparison with the regres- 
sion method using a constant exponential " forgetting factor" 
a = 0.99. These results show the superiority of our method 
over the classical ones. The last table row demonstrates iso- 
late pixels reconstruction. In this case there is no need for 
approximation, because the predictor is used with complete 
knowledge of all past data. 
The radar example demonstrates properly found better es- 
timation data side (approximation line for model parame- 
ters estimation) in the case of one superior side, on the 
   
rei 
rat