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