program

The lectures will be held at the sixth floor of the Department of Mathematics, room "Aula Tonelli". Coffee breaks and lunches will be served at the seventh floor.

Friday, May 25, 2012
09:00 - 10:30
Alessandro Bevilacqua (ARCES Università di Bologna)
Computer Vision view abstract Alessandro Bevilacqua
Computer Vision
Computer vision is the science whose purpose is studying and developing theoretical, algorithmic and computational methods to automatically extract and analyze useful information from one or a set of images, by employing common PCs or high performance computers. In general, an image is formed when a sensor registers radiations (from the whole electromagnetic spectrum – that is, besides the visible wavelengths) that has interacted with physical objects. Simply, the goal of computer vision is to make computers understand and interpret information that can be imaged. The complex human activity that stands behind scene understanding or a medical image-based diagnosis relies on a very fast analysis of a huge amount of data that requires pattern recognition capabilities. That is, decisions are taken depending on whether a pattern belongs to a given category. In computer vision, we need to teach a machine to recognize and classify patterns, either in a supervised or unsupervised manner. Advanced pattern recognition methods can be employed at different image processing stages, to recognize objects, detect motion from an image sequence, infer people behaviour and comprehend object’s interactions. Many application fields can take great advantages of computer vision, ranging from industrial inspection to surveillance and security, from remote sensing to computer aided diagnosis.
10:30 - 11:00
Coffee Break
11:00 - 12:30
Herbert Edelsbrunner (Institute of Science and Technology of Austria)
Persistent Topology view abstract Herbert Edelsbrunner
Persistent Topology
In these two lectures, I will introduce the concept of persistent homology, discuss algorithms, and present applications. A. Persistent Homology (Algebra and Algorithms) Homology groups belong to the oldest topics in Algebraic Topology, dating back to the founding years. Persistence is a recent addition to the theory, motivated by practical needs that clash with the instability of homology computations for noisy data. In contrast, persistence is stable. It can be computed with a variant of the classic Smith normal form algorithm. B. Persistent Homology (Applications and Algorithms) Persistent homology quantifies scale through a measure of the features in a discrete or continuous structure. It has been used to prove new mathematical theorems, to analyze scientific datasets, and to generally shed light on multiscale phenomena in nature. Some of these applications pose new computational challenges, such as 3-dimensional images whose sheer size requires an essentially hierarchical approach.
12:30 - 14:00
Lunch
14:00 - 15:30
Alessandro Bevilacqua (ARCES Università di Bologna)
Computer Vision view abstract Alessandro Bevilacqua
Computer Vision
Computer vision is the science whose purpose is studying and developing theoretical, algorithmic and computational methods to automatically extract and analyze useful information from one or a set of images, by employing common PCs or high performance computers. In general, an image is formed when a sensor registers radiations (from the whole electromagnetic spectrum – that is, besides the visible wavelengths) that has interacted with physical objects. Simply, the goal of computer vision is to make computers understand and interpret information that can be imaged. The complex human activity that stands behind scene understanding or a medical image-based diagnosis relies on a very fast analysis of a huge amount of data that requires pattern recognition capabilities. That is, decisions are taken depending on whether a pattern belongs to a given category. In computer vision, we need to teach a machine to recognize and classify patterns, either in a supervised or unsupervised manner. Advanced pattern recognition methods can be employed at different image processing stages, to recognize objects, detect motion from an image sequence, infer people behaviour and comprehend object’s interactions. Many application fields can take great advantages of computer vision, ranging from industrial inspection to surveillance and security, from remote sensing to computer aided diagnosis.
15:30 - 16:00
Coffee Break
16:00 - 17:30
Herbert Edelsbrunner (Institute of Science and Technology of Austria)
Persistent Topology view abstract Herbert Edelsbrunner
Persistent Topology
In these two lectures, I will introduce the concept of persistent homology, discuss algorithms, and present applications. A. Persistent Homology (Algebra and Algorithms) Homology groups belong to the oldest topics in Algebraic Topology, dating back to the founding years. Persistence is a recent addition to the theory, motivated by practical needs that clash with the instability of homology computations for noisy data. In contrast, persistence is stable. It can be computed with a variant of the classic Smith normal form algorithm. B. Persistent Homology (Applications and Algorithms) Persistent homology quantifies scale through a measure of the features in a discrete or continuous structure. It has been used to prove new mathematical theorems, to analyze scientific datasets, and to generally shed light on multiscale phenomena in nature. Some of these applications pose new computational challenges, such as 3-dimensional images whose sheer size requires an essentially hierarchical approach.
17:40 - 18:30
Presentation of the ACAT Project
Saturday, May 26, 2012
09:00 - 10:30
Marian Mrozek (Institute of Computer Science Jagiellonian University)
Computational Homology view abstract Marian Mrozek
Computational Homology
Motivation Homology groups have been invented in early XX century as a topological tool motivated by some problems in differential equations. In fact, certain difficult existence problems in the theory of differential equations, in particular the existence of chaotic dynamics, can be reduced to questions about the homology of some sets in R^n and continuous maps acting on them. The sets and some information about the maps may be derived algorithmically via rigorous numerical enclosures of the trajectories of the system based on interval arithmetic. The problem is that although the topology of the sets is in general simple, the sets are often huge. For many problems the classical homology algorithms are not fast enough for the computer assisted proofs to succeed. Also, there are no standard algorithms which can find the homology maps on the basis of the information available from the rigorous numerics of differential equations. However, some non standard techniques based on geometric reduction algorithms and cubical homology of multivalued maps make such computations possible. The algorithms invented for this purpose turn out to be useful also in other areas. Outline 1) Discretization of topological spaces: simplicial complexes, cubical sets, CW complexes 2) Homology groups and homology functor 3) The classical algorithm for homology groups 4) Algebraic and geometric reductions 4) Reductions via Discrete Morse Theory 5) Multivalued representations of continuous maps 6) Computing homology of maps via chain selectors and graph projections 7) Applications
10:30 - 11:00
Coffee Break
11:00 - 12:30
Neža Mramor Kosta (University of Ljubljana)
Discrete Morse Theory view abstract Neža Mramor Kosta
Discrete Morse Theory
The topic of the short course will be discrete Morse functions, a combinatorial extension of smooth Morse functions to the PL category proposed by Robin Forman in the 1990's. In discrete Morse theory, the classical concept of the vector field of a smooth Morse function on a manifold is substituted by a pairing on the set of cells of a regular cell complex. This enables purely combinatorial extensions of classical smooth concepts like critical points, their ascending and descending discs, gradient paths, and the Morse-Smale decomposition to be carried over to the case of general regular cell complexes. In the short course we will introduce these basic constructions and give a number of applications of discrete Morse theory to combinatorial problems and to data analysis and reconstruction, in particular to image analysis.
12:30 - 14:00
Lunch
14:00 - 15:30
Marian Mrozek (Institute of Computer Science Jagiellonian University)
Computational Homology view abstract Marian Mrozek
Computational Homology
Motivation Homology groups have been invented in early XX century as a topological tool motivated by some problems in differential equations. In fact, certain difficult existence problems in the theory of differential equations, in particular the existence of chaotic dynamics, can be reduced to questions about the homology of some sets in R^n and continuous maps acting on them. The sets and some information about the maps may be derived algorithmically via rigorous numerical enclosures of the trajectories of the system based on interval arithmetic. The problem is that although the topology of the sets is in general simple, the sets are often huge. For many problems the classical homology algorithms are not fast enough for the computer assisted proofs to succeed. Also, there are no standard algorithms which can find the homology maps on the basis of the information available from the rigorous numerics of differential equations. However, some non standard techniques based on geometric reduction algorithms and cubical homology of multivalued maps make such computations possible. The algorithms invented for this purpose turn out to be useful also in other areas. Outline 1) Discretization of topological spaces: simplicial complexes, cubical sets, CW complexes 2) Homology groups and homology functor 3) The classical algorithm for homology groups 4) Algebraic and geometric reductions 4) Reductions via Discrete Morse Theory 5) Multivalued representations of continuous maps 6) Computing homology of maps via chain selectors and graph projections 7) Applications
15:30 - 16:00
Coffee Break
16:00 - 17:30
Neža Mramor Kosta (University of Ljubljana)
Discrete Morse Theory view abstract Neža Mramor Kosta
Discrete Morse Theory
The topic of the short course will be discrete Morse functions, a combinatorial extension of smooth Morse functions to the PL category proposed by Robin Forman in the 1990's. In discrete Morse theory, the classical concept of the vector field of a smooth Morse function on a manifold is substituted by a pairing on the set of cells of a regular cell complex. This enables purely combinatorial extensions of classical smooth concepts like critical points, their ascending and descending discs, gradient paths, and the Morse-Smale decomposition to be carried over to the case of general regular cell complexes. In the short course we will introduce these basic constructions and give a number of applications of discrete Morse theory to combinatorial problems and to data analysis and reconstruction, in particular to image analysis.