Workshop on Computational Topology in Image Context
Poitiers - France - 16,17 June 2008
CTIC 2008 will offer three invited lectures:
from ENST, Paris, France
A few topological aspects in medical imaging
In this talk, an overview of various models taking topology into account
will be presented, for applications in medical imaging.
Preserving topology is often an important constraint in segmentation and
this will be addressed in the first part of the talk. We will show that
such constraints can be modeled on different digital 3D representations
(voxels, cellular complexes, tetrahedral meshes). This will be
illustrated with examples in brain segmentation from 3D images.
In the second part of the talk, the modeling of topological spatial
relations will be addressed, in the framework of fuzzy sets. The
interest of these models will be illustrated on model-based recognition
problems, in particular for anatomical structures.
from DAM, Sevilla, Spain
Digital Volume Homological Processing
In this talk, digital volumes are seen and analysed from an
algebraic-topological point of view . A polyhedral continuous analogous
$K(V)$ is uniquely associated to a digital volume $V$ (with $26$-adjacency
relation for neighbors voxels) and a "gradient" cellular field $\phi$ over
this complex can be
obtained using different classical homological techniques (incremental,
matrix, local contractibility, Morse-based-method,....) In fact, the map
$\phi$ determines an especial homotopy equivalence algebraically connecting
$K(V)$ and its homology, working on a field as a ground ring. From this
datum and a property of local simplicialization for $K(V)$, it is possible
to deduce advanced homological information for $V$: representative cycles of
homology generators, homological classification of cycles, homological
operations, ....). This idea of describing homology in terms of chain
homotopies is not new and comes back to Eilenberg-MacLane work on Algebraic
Topology and it has been developed later in algebraic-topological methods
like Effective Homology and Homological Perturbation Theory and in discrete
settings as Discrete Morse and AT-model theories.
from UJF, Grenoble, France
Constructive Algebraic Topology
"Standard" Algebraic Topology is not appropriately designed for
computations, for example the usual spectral sequences are most often
*not* computable from the data usually available. From a theoretical
point of view, standard Algebraic Topology is *not* constructive. To
make it constructive, according to the methods of the so-called
*Effective_Homology*, two main ingredients are to be used:
1) Functional programming allows to partially implement infinite
objects as "locally effective" objects on a theoretical or concrete
2) The Basic Perturbation Lemma becomes the main mathematical tool
to process the necessary connections between ordinary homology groups
and the corresponding locally effective objects.
1) Description of the main ideas of Effective Homology.
2) Results so obtained.
3) Open problems.