7.3 Inversion
According to Table 4, the MAP criterion would choose
Z = Zo + AZ and Q = Qo + AQ in (20) such that
the conditional probability P(Z, Q | L) is maximized,
which is equivalent to maximize P(L | Z, Q)P(Z,Q),
if P(L) is constant.
As mentioned above, the conditional probability
P(L | Z, Q) can be simply assumed as the probability
that the observational residuals were produced by a
normally distributed random variable (cf. (4)). The
problem is, now, how to exploit our priori knowledge
about Z and @ to constitute their probabilities, i.e.
P(Z,Q). If Z and Q are statistically independent, we
have P(Z, Q) = P(Z) - P(Q).
To construct P(Z) and P(Q), one has to know the
meaning of Z and Q. The vector Z, for instance,
represents some elevations of discrete surface profile
points. So, our priori assumption about Z is the spa-
tial coherence of its elements. This suggests that the
local potential of the element Z; € Z, à € Z can be
written as
dee ur ar
2) - Y la (21)
JEN;
where N; denotes the set of totally connected sub-
graphs (cliques) with respect to the element Z;, l;; €
[0,1] is the connection strength between Z; and Z;,
and oz is a normalizing constant. According to the
Hammersley-Clifford theorem, the energy of Z can be
computed with
Zo Z4? i
£2)» 3 Y [uA] - epus rs on
cn oz
Z;€Z i#j
JEN;
and
Ezz$z2-7. (23)
where
1i.—1 0 0 Z1
0.1... —4 0 Z2
$; — ; , ZZ ,
0 0 ] - ZK
2 0 0
0 ge
0 0 TZ 0
Vp=|. dem]...
2
72
(K-1)K (
©
e .-
e
24)
Similarly, since we priori know that p should be
around 1 and q should be around 0, so the energy
of Q is
£(Q) - EQ Eg Eq, (25)
494
with
EQ — 99 Q — Va, (26)
where
] ^0 0 p
0 1 0 T2
oo — s : 3 Q = ,
0 0 1 PI
qJ
1 92 0 0...0
0 0 c2 0 0
Ya Bossy or c —- …
1 0-40: (4. 02 0
0 0-0. o9 al
(27)
and c, and c, are constants encoding the reliability
of oura priori knowledge about p and q.
Considering (3), (4), (5), (22) and (25), the MAP ba-
sed surface reconstruction is to solve the optimizing
problem
iyrzciy + HET: X2! Eg luf Eg. Eg — min,
(28)
with subject to (20), (23) and (26). Surely, the sur-
face Z which is inferred in this way is dependent on
T — (Z,07,V7,X7,09, VQ, Da) and they should be
determined using a priori knowledge, before the in-
version process takes place. The quality of Z is so-
metimes not satisfying if our knowledge is not good
enough to ensure an appropriate determination of T.
50, a very interesting question is how to enlarge our
knowledge and how to adapt I' during the inversion
in order to improve the quality of Z. Information pro-
cessing systems that improve their performance or en-
large their knowledge bases are said to “ learn”. This
ability would clearly have value in digital image inver-
sion. Using parameter estimation technique, we can,
for instance, adapt X, Xz, and XQ in (28) iterati-
vely during the inversion so that the result is robust
against image noises with different properties, against
surface discontinuities, and against different reliabili-
ties of our a priori knowledge (cf. Zheng and Hahn,
1990).
8 Examples
To demonstrate the feasibility of the methods for
the purpose of surface reconstruction an algorithm
has been developed based on the MAP criterion (cf.
Zheng, 1990). It was tested on a variety of data sets
including synthetic and real image data.
Z( M)
Soo æ
Figu
accu
Let
Imag
steep
then
the |
rithr
selec
Figu
and
tice
heigl
also
devic
The
