International Archives of Photogrammetry and Remote Sensing, Vol. 32, Part 7-4-3 W6, Valladolid, Spain, 3-4 June, 1999
a measure of error. The inference requires the knowledge of the
prior model p(Q\M.) and the data prediction term or forward
model p(D\Q, M.) pursues Bayes’ rule and is applied
separately for each of the models {Af,}:
p(D\Q, M.)p(Q\M.)
=—jimr) <•>
The information extraction is a Maximum A Posteriori (MAP)
estimation:
0 = argmaxp(@\D, M.) (2)
The evidence termp(D\M.) in the denominator of (1) is
generally neglected in the model fitting, but is important in the
second level of inference, and here lies the novelty of the
Bayesian approach in data inversion.
Model comparison. The task of the second level of the Bayesian
inference is to find the most plausible model, given the data. The
inference applies in the space of models:
The objective of this work is to use the Bayesian inference in
order to obtain an estimate of a given physical parameter, using
observations acquired with different sensors. We assume a model
for the desired physical parameter and try to estimate it by fitting
this model to the data.
If we take into consideration two datasets D/ and D 2 separately
for observations, the information extraction can be splitted into
two separated problems with solutions given by the maximum of
the following a posteriori probabilities (MAP):
p(0|D 1 ,M 1 )
p(D^e,M { )-p(e\M x )
p(d v m 1 )
(4)
0, M 2 ) • p(0|M 2 )
p(£> 2 |m 2 ) (5)
where 0 is the desired physical parameter, and p(0IA/,)
encapsulates our a priori knowledge. The measure of fidelity to
the observed data is given in terms of the conditional
probabilities p(D,|0, Mf, i=l,2.
In Fig. 2 we introduce a first paradigm for data fusion. It refers
mainly to the extraction of image content information.
P(° 2
p(G\d 2 ,m 2 ) =
piM^ocpiDlMJpWt) (3)
The inference relies on the evidence of Af, carried by p(D\Mf and
the subjective prior over the assumed hypothesis space p(Mf.
p(M i ) shows how plausible we thought the alternative models
were, before the data arrived.
Inference of probability distributions from observation of
sensory data aims at finding the best stochastic models able to
consistently characterize classes of images (Datcu et al., 1998;
Minka and Picard, 1997). The Bayesian approach for data
modelling is used. The information contained in a dataset
(provided by a unique sensor) is extracted in different
assumptions. The assumptions are represented by different prior
models (Fig. 1). In the case of a multispectral sensor, the assumed
prior models can characterize either the spectral components or
the texture structures. The extracted information according to
these two models is not commensurable, it represents different
qualities.
Fig. 1. Information extraction using qualitatively different
models.
We call the process to extract information from sensory data
using different prior models, data fission.
Fig. 2. First paradigm for sensory information extraction and
data fusion.
Here, we can identify three cases. The first case, the case of a
unique source of information, was treated as data fission. A
second situation is the extraction of information from different
sources using the same prior model. The estimated parameters
will have identical representation, however their meaning can be
different. A simple example is texture parameter estimation from
data with different resolution. The scale plays the role of meta
information, thus the direct interpretation of the estimated
parameters is not consistent. The third case assumes information
extraction from different sources using different prior models.
The resulted information has incommensurable representations.
A full Bayesian approach for information fusion can be
formulated as maximizing the following a posteriori probability:
p(Q\d v d 2 ,m v m 2 ) =
p(D x |0, M^piD^Q, M 2 ) • P{p(G\M x )p{Q\M 2 )} (6 )
p(D v £> 2 | m i> M i)
where F is an operator representing the prior information in the
assumption of two different models. We observe that the problem
of fusion of information from two datasets is extended with
fusion of knowledge, in form of the specification of the a priori