# A Study of Well-Composedness in n-D

## Abstract

Digitization of the real world using real sensors has many drawbacks; in particular, we loose “well-composedness” in the sense that two digitized objects can be connected or not depending on the connectivity we choose in the digital image, leading then to ambiguities. Furthermore, digitized images are arrays of numerical values, and then do not own any topology by nature, contrary to our usual modeling of the real world in mathematics and in physics. Loosing all these properties makes difficult the development of algorithms which are “topologically correct” in image processing: e.g., the computation of the tree of shapes needs the representation of a given image to be continuous and well-composed; in the contrary case, we can obtain abnormalities in the final result. Some well-composed continuous representations already exist, but they are not in the same time ${\displaystyle n}$-dimensional and self-dual. In fact${\displaystyle n}$-dimensionality is crucial since usual signals are more and more 3-dimensional (like 2D videos) or 4-dimensional (like 4D Computerized Tomography-scans), and self-duality is necessary when a same image can contain different objects with different contrasts. We developed then a new way to make images well-composed by interpolation in a self-dual way and in ${\displaystyle n}$-D; followed with a span-based immersion, this interpolation becomes a self-dual continuous well-composed representation of the initial ${\displaystyle n}$-D signal. This representation benefits from many strong topological properties: it verifies the intermediate value theorem, the boundaries of any threshold set of the representation are disjoint union of discrete surfaces, and so on.

## Bibtex (lrde.bib)

@PhDThesis{	  boutry.16.phd,
author	= {Nicolas Boutry},
title		= {A Study of Well-Composedness in \$n\$-D},
school	= {Universit\'e Paris-Est},
year		= 2016,
month		= dec,
abstract	= {Digitization of the real world using real sensors has many
drawbacks; in particular, we loose ``well-composedness'' in
the sense that two digitized objects can be connected or
not depending on the connectivity we choose in the digital
image, leading then to ambiguities. Furthermore, digitized
images are arrays of numerical values, and then do not own
any topology by nature, contrary to our usual modeling of
the real world in mathematics and in physics. Loosing all
these properties makes difficult the development of
algorithms which are ``topologically correct'' in image
processing: e.g., the computation of the tree of shapes
needs the representation of a given image to be continuous
and well-composed; in the contrary case, we can obtain
abnormalities in the final result. Some well-composed
continuous representations already exist, but they are not
in the same time \$n\$-dimensional and self-dual. In fact,
\$n\$-dimensionality is crucial since usual signals are more
and more 3-dimensional (like 2D videos) or 4-dimensional
(like 4D Computerized Tomography-scans), and self-duality
is necessary when a same image can contain different
objects with different contrasts. We developed then a new
way to make images well-composed by interpolation in a
self-dual way and in \$n\$-D; followed with a span-based
immersion, this interpolation becomes a self-dual
continuous well-composed representation of the initial
\$n\$-D signal. This representation benefits from many strong
topological properties: it verifies the intermediate value
theorem, the boundaries of any threshold set of the
representation are disjoint union of discrete surfaces, and
so on.}
}