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