Patrizio Frosini's abstract

Visual Attention in Robotics, Combinatorial Structures and Homology

In this talk we present a mathematical approach to the concept of shape. However shape can be defined, it is based on stable perceptions made by observers, at least in an empirical setting. This dependence on the observers follows from the large subjectivity we experience in shape comparison, while stability is requested by the fact that human judgements focus on persistent properties of the real world, while non-persistent properties are usually interpreted as noise.

We say that a property is persistent when it is robust with respect to changes in the perceptual setting, both as concerns space and time. As an example, let us think of an (almost) spherical stone. If we look at it, we see approximately a circle from any point of view, even if the stone has wrinkles and small bumps. We describe this perception by saying that we are looking at a sphere, since we disregard details that are not stable enough, during our observation and under small changes of our viewpoint.

Persistent Topology is devoted to give a mathematical formalization of this approach, endowing topological properties with the concept of persistence.

In order to express stability in a mathematical setting we need to model the set of observations as a topological space T, while the observer's perception can often be seen as a function ' taking each observation t 2 T to a vector in IRn. This function ' describes t from the point of view of the observer. In the spherical stone example the topological space can be a rectangle representing the image we see, while our perception is the color '(t) = (r(t); g(t); b(t)) for each point t in our rectangle.

When two pairs (T1; phi1), (T2; phi2) are chosen for "comparable perceptions" (ob- viously, we cannot compare the volume of an object to the colour of another), it is natural to try to match them in the best possible way. Therefore, we consider the functional O taking each homeomorphism h : T1 -> T2 (i.e. a bicontinuous mapping between T1 and T2) to the sup-norm of the function phi1 - phi2 o h. This functional represents the "cost" of the matching between perceptions that is induced by h. The lower this cost, the better the matching between the two perceptions is.

The natural pseudodistance d between the pairs (T1; phi1), (T2; phi2) is the infimum of this cost O(h), varying h. This pseudometric represents the concept of shape in Persistent Topology. However, other functionals can be used in place of O and some restrictions about the homeomorphisms can be applied.

Unfortunately, this pseudodistance is difficult to calculate, but we can get lower bounds for d by computing the size functions or the ranks of the persistent homology groups for the pairs (T1; phi1), (T2; phi2). This fact motivates the second part of this talk, where these concepts are brie y illustrated, together with some results recently obtained about multidimensional Persistent Topology and its algorithmic computation.

Universidad de Sevilla - ETS Ingeniería Informática - Avenida Reina Mercedes s/n - 41012 Sevilla - Teléfono: +34-954556921 - Fax: +34-954557878