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Applications of Size Theory in Shape Comparison
by
Andrea Cerri
Università di Bologna
Size Theory was proposed in the early 90's as a
geometrical/topological approach to the problem of shape
comparison. The main idea is to translate the task of comparing two
objects in a database (e.g. images, 3D models or sounds) into the
one of comparing two suitable topological spaces M,
N (non-empty, compact and locally connected Hausdorff
spaces), endowed with two continuous functions
j:M→R, y:N→R
that are chosen according to the applications. These functions are
called measuring functions and can be seen as descriptors of
the features considered relevant for the comparison. The pairs
(M, j), (N, y) are said to be
size pairs and provide a representation of the considered shapes:
In Size Theory, such pairs can be compared by size functions,
whose role is to capture qualitative aspects of a shape and
represent them in a quantitative way. The idea is to study the
pairs (M〈j ≤ x〉, M〈j ≤ y〉), where
M〈j ≤ t〉 is defined by setting
M〈j ≤ t〉 = {P ∈ M:j(P) ≤ t} for t ∈ R: The size function
l(M, j):{(x, y) ∈ R2:x < y}→N
is then the function that takes each point (x, y) of the domain
into the number of connected components of
M〈j ≤ y〉 containing at least one
point of M〈j ≤ x〉 [1]. By means of
Size Theory, we can then model a shape by a size pair, and describe
it by considering the associated size function: As a consequence,
the comparison of two shapes can be translated into the simpler task
of comparing two functions from the half-plane
{(x, y) ∈ R2:x < y} to the natural numbers. However, a
common scenario in applications is to deal with multidimensional
information: Indeed, a shape can be more thoroughly characterized by
means of a set of real functions, each investigating specific
features of the shape under study. This problem can be faced by
observing that size functions are modular descriptors: In order to
study different properties of a shape, we only need to change the
measuring function. Since its introduction, Size Theory has been
studied and applied in quite a lot of applications: An example is
given by [2], where the authors propose an automatic retrieval
system for trademark images based on size functions, to support
human labor in guaranteeing copyright policy. Other examples on the
use of Size Theory in applications can be found in several fields,
ranging from leukocyte classification in medical context [3] to
image retrieval in the World Wide Web [4]: This work proposes to be
an overview on some meaningful experimental results, in order to
show the capability and the flexibility of this theoretical
framework in dealing with concrete applications.
References
[1] P. Frosini, and C. Landi, Size Theory as a
Topological Tool for Computer Vision, Pattern Recogn. Image Anal.
9(4) (1999),
596-603.
[2] A. Cerri, M. Ferri, and D. Giorgi, Retrieval of
trademark images by means of size functions, Graphical Models
68 (2006), 451-471.
[3] M. Ferri, S. Lombardini and
C. Pallotti, Leukocyte classification by size functions, In
Proc. of the 2nd IEEE Workshop on Applications of Computer
Vision, IEEE Computer Society Press, Los Alamitos, CA (1994),
223-229.
[4] A. Cerri, M. Ferri, P. Frosini, and D. Giorgi,
Keypics: free-hand drawn iconic keywords, International
Journal of Shape Modelling 13(2) (2007), in press.
Date received: May 21, 2008
Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caxd-20.