nal mean value
(10)
(11)
(12)
; (13)
(14)
(15)
(16)
lity of recursive
(17)
> the parameter
| past necessary
nstructed. On
re known: we
in such a way
olved using the
ween close lines
(18)
estimator (11),
own contextual
e spectral band
5), (16), (17).
correlations on
lation of these
ION
and M» with
fi = B.)
ft sets I, ;, 1, ;
xis. According
rule for mini-
or chooses the
a model whose
he highest one.
ompleted as in
$e gi W^ p(Mi|YC=9) > p(Mo|Y 79)
DT equas otherwise
(19)
where Zi. are data vectors corresponding to 7; ;. Following
the Bayesian framework used in our paper and choosing uni-
form a priori model in the absence of contrary information,
»( M;|YC7D) » p(YU-U|M,;), the simultaneous conditional
probability density can be evaluated from
AY CC) =
[ [vo meatus o oan += (20)
Under the already assumed conditional pixel independence,
the analytical solution has the form
à "1 ating
p( MiYU79) s k [Vi ua3| 3 Aue oi wll)
where k is a common constant. To evaluate MM; ID),
we have to use a similar approximation (18) as for the pre-
dictor (10). All statistics related to a model Mi (15), (16),
(17), (21) are computed from data on one side of the recon-
structed line while symmetrical statistics of the model M»
are computed from the opposite side.
The solution of (21) uses the following notations:
WA OT, (22)
Aii m Vou) 7 Voute=1) Ven Vauts-1) (23)
The determinant |V(;)| as well as A; can be evaluated recur-
sively see [Haindl, 1992]:
Val = Ve Ze Via Ze), (24)
Ac m Aca + (Yi — PT
(Y - BE AZ)Yür ziv Any. (25)
In the case when some data necessary for the approximation
are missing the corresponding model probability is set to zero
»(MiY*?)zo. (26)
If the reconstructed line is located in a boundary image area
then the reconstruction algorithm uses only one model (one
of the model probabilities is permanently zero).
4 MULTI-SPECTRAL LINE REGRESSION MODEL
Let as assume that all spectral components of the multi-
spectral line are missing. Such a multi-spectral line can be
modelled using a multi-dimensional regression model:
Y, ) AY; + Fy; (27)
‘el;
where {= (m,n); Y; is v x1 reconstructed multi-spectral
pixel value, A; are unknown v x v model parameters
matrices, E; is the white noise vector.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
Parameter vector P7 (5) become now the » x fv matrix:
PTS Ag (28)
and
B* 2 vcardl, = vf (29)
Zi (T) isthe 8* data vector, Vo become a positive definite
(8* --v) x (8* --») matrix and 4(0) » 8* —2 . Equations
(7)-(26) remain unchanged and can be used for the multi-
dimensional prediction and optimal multi-dimensional model
selection (19),(21) as well.
If A; = diag[a1x,..., av] Vi then the multi-dimensional
model reconstruction is identical with separately applied
single-dimensional model reconstruction G on every spectral
line component.
5 DIRECTIONAL FORGETTING
The reconstruction model in sections 2 and 3 was devel-
oped under the assumption that model parameters are strictly
location-invariant. This assumption is not realistic for most
real image reconstruction problems. The Bayesian solution of
the case of location-variant parameters is given by equation
p( Pi Ord lY O?) = ] [0 oan acm
p( A, 0; * IY (OYd Pd! (30)
Unfortunately the required conditional distribution for this re-
cursion p(Pi41, Q7, |, Q7, Y(?) is seldom known. Usual
solution of this problem is the constant exponential forget-
ting. It increases the uncertainty of the old parameter esti-
mate by a constant factor equal for all data. This results in
modification of equations (17),(24),(25) see [Haindl, 1996]
for details.
The exponential forgetting permanently loses old informa-
tion even if there is lack of a new one (parameters re-
main unchanged in some directions). Some ideas to over-
come this insufficiency were suggested in [Hägglund, 1983],
[Kulhavy, 1993]. We propose another solution based on di-
rectional forgetting coefficients to control individually every
data item [Y;.; : i € L]^ forgetting depending on the
corresponding directional derivation change, i.e.
o min(| 22, |.)
6 ói
= ie : (31)
max {| 241,12)
If there is no change in the direction ? then o; — 1 (no
data forgetting), otherwise o; « 1 and the increase of
old information uncertainty is proportional to the directional
change of the derivation.
Let us denote the matrix of directional forgetting parameters
Q1 SEL 0
zl uoi. Mu :
aw s © ; (32)
0 sut At hu
where
ay z diag[a1, .:.,o,]
and
Os m diag[au ki teret] ie
