
Methodologies
The project uses two broad classes of techniques to attack its problems :
In addition to applying these techniques to specific cases, the project advances these techniques more generally, through innovative modeling and theoretical analysis, and a comparative study of the two classes. An important recent theme, for example, is the incorporation of geometric information into both classes of techniques, in the probabilistic case via the use of stochastic geometry, and in the variational case via the use of higherorder active contours.
The project also concerns itself with a number of important, related problems, in particular the development of the parameter estimation procedures necessary to render the above methods automatic or semiautomatic, and the study of the optimization algorithms used to solve the problems (for example, reversible jump Markov chain Monte Carlo (RJMCMC)).
Following a Bayesian methodology as far as possible,
probabilistic models are used within the Ariana project, as elsewhere,
for two purposes : to describe the class of images to be expected
from any given scene, and to describe prior knowledge about the
scene in the absence of the current data.
Markov random fields
Markov random fields were introduced to image
processing in the Eighties, and were quickly applied to the
full range of inverse problems in computer vision. They owe
their popularity to their flexible and intuitive nature, which
makes them an ideal modeling tool, and to the existence of
standard and easytoimplement algorithms for their solution.
In the Ariana project, attention is focused
on their use in image modeling, in particular of textures,
on the development of improved prior models for segmentation,
and on the lightening of the heavy computational load traditionally
associated with these techniques, in particular via the study
of varieties of hierarchical random field.

Wavelets
The development of wavelets as an alternative
to the pixel and Fourier bases has had a big impact on image
processing due to their spatial and frequency localization,
and the sparse nature of many types of image data when expressed
in these bases. In particular, wavelet bases have opened up
many possibilities for probabilistic modeling due to the existence
of not one but two natural correlation structures, intra
and interscale, leading to adaptive wavelet packet models
and tree models respectively.
In Ariana, attention is focused on the use of
tree models for denoising and deconvolution, adaptive wavelet
packet models for texture description, and on the use of complex
wavelets for their improved translation invariance and directional
selectivity.

Stochastic geometry
One of the grand challenges of computer vision
and image processing is the expression and use of prior geometric
information. For satellite and aerial imagery, this problem
has become increasingly important as the increasing resolution
of the data results in the necessity to model geometric structures
hitherto invisible. One of the most promising approaches to
the inclusion of this type of information is stochastic geometry,
which is a new and important line of research in the Ariana
project.
Instead of defining probabilities for different
types of image, probabilities are defined for configurations
of an indeterminate number of interacting, parameterized objects
located in the image. Such probability distribution are called
"marked point processes". For instance, two examples
that have been developed in Ariana use interacting cuboids
of varying length, width, height and orientation for modeling
buildings; and interacting line segments of varying length
and orientation for modeling road and other networks.

Regularization and functional analysis
The use of variational models for the regularization
of inverse problems in image processing is longestablished.
Attention in Ariana is focused on the theoretical study of
these models and their associated algorithms, and in particular
on the convergence of sequences of functionals and on projection
algorithms. Recent research concerns the definition and computation
of a function space containing oscillatory patterns, a sort
of dual space to the BV space that captures the geometry of
the image. Variational methods are also applied to a variety
of problems, including phase unwrapping and image decomposition.

Contours and regions
In addition to the regularization of inverse
problems, variational methods are much used in the modeling
of boundaries in images using contours. In Ariana, attention
is focused on the use of such models for image segmentation,
in particular texture segmentation; on the theoretical study
of the models and their associated algorithms, in particular
level set methods; and on the incorporation of prior geometric
information concerning the regions sought using higherorder
active contour energies.

Wavelets
Wavelets are important to variational approaches
in two ways. They enter theoretically, through the study of
Besov spaces, and they enter practically, in models of texture
for segmentation, and in the denoising of the oscillatory
parts of images.

One of the most important problems studied in the
Ariana project is how to estimate the parameters that appear in
the models. For probabilistic models, the problem is easily framed,
but is not necessarily easy to solve, particularly in the case when
it is necessary to extract simultaneously from the data both the
information of interest and the parameters. For variational models,
there are few methods available, and the problem is consequently
more difficult.
