Massimo Ferri

A graph-theoretical representation of PL-manifolds — A survey on crystallizations

This is a survey of the techniques and results developed by M. Pezzana and his group, which includes, besides the authors, A. Cavicchioli, P. Bandieri and A Donati.The original concept is that of “contracted triangulation”, which was introduced with the main goal of finding a “minimal” atlas for topological manifolds ([P1 1968], [P2 1974], [P3 1974], [FG2 1979]). Only later did the possibility of deducing a graph-theoretical tool — the crystallization — for representing P.L. manifolds occur as a major aspect of the theory ([P4 1975], [F1 1976]). This leads to an application of graph theory to P.L. topology, which seems not to have been explored before. Recently, other authors outside Italy have independently become interested in this subject.For the sake of conciseness, definitions and statements often appear in a form other than that of the quoted references.

On the use of size functions for shape analysis

According to a recent mathematical theory a shape can be represented by size functions, which convey information on both the topological and metric properties of the viewed shape. In this paper the relevance of the theory of size functions to visual perception is investigated. An algorithm for the computation of the size functions is presented, and many theoretical properties of the theory are demonstrated on real images. It is shown that the representation of shape in terms of size functions (1) can be tailored to suit the invariance of the problem at hand and (2) is stable against small qualitative and quantitative changes of the viewed shape. A distance between size functions is used as a measure of similarity between the representations of two different shapes. The results obtained indicate that size functions are likely to be very useful for object recognition. In particular, they seem to be well suited for the recognition of natural and articulated objects.

Retrieval of trademark images by means of size functions

We propose a new, effective system for content-based retrieval of figurative images, which is based on size functions, a geometrical-topological tool for shape description and matching. Three different classes of shape descriptors are introduced and integrated, for a total amount of 25 measuring functions. The evaluation of our fully automatic retrieval system has been performed on a benchmark database of 10,745 real trademark images, supplied by the United Kingdom Patent Office. Comparative results show that our method actually outperforms other existing whole-image matching techniques, comprising features incorporated in the MPEG-7 standard.

One-dimensional reduction of multidimensional persistent homology

A recent result on size functions is extended to higher homology modules: the persistent homology based on a multidimensional measuring function is reduced to a 1-dimensional one. This leads to a stable distance for multidimensional persistent homology. Some reflections on i-essentiality of homological critical values conclude the paper.

Molecular Basis of Learning and Memory: Modelling Based on Receptor Mosaics

The intercellular communication in the Central Nervous System (CNS) occurs via two main modes: the Wiring Transmission (WT) and the Volume Transmission (VT) (see Fig. 9.1). WT is a point to point transmission (one source one target) whose best known realisation is the synapse; VT is a diffuse type of extracellular transmission (one source many targets) whose best known examples are the recently discovered gaseous transmitters like nitric oxide (NO), but data are also available for classical neurotransmitters (such as dopamine) and neuropeptides (such as β-endorphin) [Agnati and Fuxe, 2000; Agnati et al., 2000].

Qualitative asymmetry measure for melanoma detection

Size functions and support vector machines are used to implement a new automatic classifier of melanocytic lesions. This is mainly based on a qualitative assessment of asymmetry. The program is used, at clinical level, with two thresholds, so that comparison of the two outputs produces a report of low-middle-high risk. Experimental results with cross-validation are reported.

Leukocyte classifications by size functions

Leukocytes are divided into classes. Their automatic classification is accomplished by means of site functions, based on two measuring functions defined expressly for taking into account the specific morphological features of the cell classes. A successful experimentation on 45 cells is reported. The original contribution resides in the use of this new geometrical-topological technique, size theory, so confirming its suitableness for recognition of natural objects.<<ETX>>

The asymptotic number of acyclic diagraphs, II

Abstract Let A n , q be the number of labeled digraphs with n labeled vertices, q edges, and no directed cycles. Let C n , q be the corresponding number of weakly connected ones, and let a n , q and c n , q be the corresponding numbers of unlabeled ones. We show that A n , q ∼ C n , q ∼ n ! a n , q ∼ n ! c n , q when eN q e ) N , where N = ( 2 n ). An asymptotic formula for A n , q was obtained in an earlier paper.

Size Functions from a Categorical Viewpoint

A new categorical approach to size functions is given. Using this point of view, it is shown that size functions of a Morse map, f: M→ℜ can be computed through the 0-dimensional homology. This result is extended to the homology of arbitrary degree in order to obtain new invariants of the shape of the graph of the given map.

Size functions for the morphological analysis of melanocytic lesions

Size Functions and Support Vector Machines are used to implement a new automatic classifier of melanocytic lesions. This is mainly based on a qualitative assessment of asymmetry, performed by halving images by several lines through the center of mass, and comparing the two halves in terms of color, mass distribution, and boundary. The program is used, at clinical level, with two thresholds, so that comparison of the two outputs produces a report of low-middle-high risk. Experimental results on 977 images, with cross-validation, are reported.