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The research of the group main focus on different aspects of cognitive science:
Problems of
neuroscience, comunication sciences, and cognitive psicology.
Problems of
visual perception from the point of view
of the brain functionality, and of the psicology of the Gesthalt
Problems of
Lie analysis for the description of objects and their functionality.
- Instruments of neurosciences:
- the instruments of neuroimaging and integrate neuroscience allow to
mathematically model neural mechanisms of Human cognition
and to
build up computational algorithms of image processing
inspired to brain functionality.
An essential aspect of this problem concerns the relationship between the analysis of the Sensorial
Signal and the Geometric Structures of perceptive representations.
In particular this is the problem of of optimal compact representation of signals by
means of time-frequency or space-frequency packets. It goes from
classical Fourier Analysis methods to more innovative multi-scale
methods based on Wavelets and particularly to Gabor Transforms.
- algorithms of visual perception based on the psicology of the Gesthalt:
It has been proved that there is a clear distinction
between the physical images we look at and the
images we perceive.
Gaetano Kanizsa in [grammatica del vedere ] provided a taxonomy
of perceptual completion phenomena and outlined that they are
interesting test to understand how the visual system interpolates
existing information and builds the perceived units.
A tipical example is the famous triangle, where we clearly
perceive lines which are not really present in the image.
These boundaries are called subjective boudaries
and they have been classically modeled (Mumford Nithzberg and Shiota)
as minima of the elastica functional, which depeds on curvature.
These models have been widely extended with methods of calculus
of variations or evolution differential equations, in order to simultaneously segment the given image and
reconstruct the spatial ordering (depending on depth) of the objects contained in the image.
- algorithms of visual perception based on the functionality of the brain:
From the neurophysiological point of view, there is a
considerable evidence that perceptual phenomena are accomplished by
the first layer of the visual cortex by actively integrating local information.
The principal level of organization of the visual cortex V1 is the columnar structure. The integration mechanism
propagates columnar information in an orientation specific modality by means of long-range
horizontal connections. In this setting the formation of contours is explained as the meeting
of two neural activation flows shooted by local boundary inducers and integrating missing
information. The specificity of this information propagation is described by the "association
fields" that indicate boundary collinear directions as privileged diffusion directions to the
detriment of orthogonal ones. The columnar organization has been modeled as a structure of contact
on the Lie group of rototraslations. The notion of derivative is replaced in this setting by the
Lie derivative, which is performed along suitable horizontal vector fields, similar to the
association field recently proposed in vision neuroscience by Field, Hayes and Hess to explain
modal boundary completion.
- Lie group analysis for the description of objects and their functionality:
-The functionality of an object is usually described through its derivatives of
first and hygher order. Alternately it is possible to lift original object to a hyger
dimensional Lie group, whose elements are the original points together with their derivatives.
The differential relations between the derivatives induce in the lifted Lie group differential
costraints: not all the displacements are allowed, but only in the direction of integral curves
of suitable vector fields. These constraints completely describe the geometrical and functional
structure of the space: the distance, differential calculus and functional spaces.
Hence properties of distances, curves and surfaces (in particular of geodesics and minimal surfaces)
are object of studies.
This can be done with real analysis methods, calculus of variation instruments
and non linear partial differential equations
of curvature type
in Lie groups.
In particular the structure of
the visual cortex can be modeled with these instruments as the Lie group of rototraslations.
The subjective contours are modeled as geodesics, and the objects as surfaces of the group.
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