After a formal definition of segmentation as the largest partition
of the space according to a criterion σ and a function f, the notion
of a morphological connection is reminded. It is used as an input
to a central theorem of the paper (Theorem 8), that identifies segmentation
with the connections that are based on connective criteria. Just
as connections, the segmentations can then be regrouped by suprema
and infima. The generality of the theorem makes it valid for functions
from any space to any other one. Two propositions make precise the
AND and OR combinations of connective criteria. The soundness of
the approach is demonstrated by listing a series of segmentation
techniques. One considers first the cases when the segmentation under
study does not involve initial seeds. Various modes of regularity
are discussed, which all derive from Lipschitz functions. A second
category of examples involves the presence of seeds around which
the partition of the space is organized. An overall proposition shows
that these examples are a matter for the central theorem. Watershed
and jump connection based segmentations illustrate this type of situation.
The third and last category of examples deals with cases when the
segmentation occurs in an indirect space, such as an histogram, and
is then projected back on the actual space under study. The relationships
between filtering and segmentation are then investigated. A theoretical
chapter introduces and studies the two notions of a pulse opening
and of a connected operator. The conditions under which a family
of pulse openings can yield a connected filter are clarified. The
ability of segmentations to generate pyramids, or hierarchies, is
analyzed. A distinction is made between weak hierarchies where the
partitions increase when going up in the pyramid, and the strong
hierarchies where the various levels are structured as semi-groups,
and particularly as granulometric semi-groups. The last section is
based on one example, and goes back over the controversy about 'lattice'
versus 'functional' optimization. The problem is now tackled via
a case of colour segmentation, where the saturation serves as a cursor
between luminance and hue. The emphasis is put on the difficulty
of grouping the various necessary optimizations into a single one.
%0 Journal Article
%1 Sera2006
%A Sera, Jean
%D 2006
%J Journal of Mathematical Imaging and Vision
%K , 3-D Lipschitz colour connected connection connective crieteria jump levelling measurements methods operators processing quasiflat seed segmentation set smooth variational watersheds zones {$\omega$}-continuity
%N 1
%P 83--130
%R 10.1007/s10851-005-3616-0
%T A Lattice Approach to Image Segmentation
%U http://www.springerlink.com/content/pg819063895l7377/?p=2838d86992ca49f585f3201505e4b07d&pi=2
%V 24
%X After a formal definition of segmentation as the largest partition
of the space according to a criterion σ and a function f, the notion
of a morphological connection is reminded. It is used as an input
to a central theorem of the paper (Theorem 8), that identifies segmentation
with the connections that are based on connective criteria. Just
as connections, the segmentations can then be regrouped by suprema
and infima. The generality of the theorem makes it valid for functions
from any space to any other one. Two propositions make precise the
AND and OR combinations of connective criteria. The soundness of
the approach is demonstrated by listing a series of segmentation
techniques. One considers first the cases when the segmentation under
study does not involve initial seeds. Various modes of regularity
are discussed, which all derive from Lipschitz functions. A second
category of examples involves the presence of seeds around which
the partition of the space is organized. An overall proposition shows
that these examples are a matter for the central theorem. Watershed
and jump connection based segmentations illustrate this type of situation.
The third and last category of examples deals with cases when the
segmentation occurs in an indirect space, such as an histogram, and
is then projected back on the actual space under study. The relationships
between filtering and segmentation are then investigated. A theoretical
chapter introduces and studies the two notions of a pulse opening
and of a connected operator. The conditions under which a family
of pulse openings can yield a connected filter are clarified. The
ability of segmentations to generate pyramids, or hierarchies, is
analyzed. A distinction is made between weak hierarchies where the
partitions increase when going up in the pyramid, and the strong
hierarchies where the various levels are structured as semi-groups,
and particularly as granulometric semi-groups. The last section is
based on one example, and goes back over the controversy about 'lattice'
versus 'functional' optimization. The problem is now tackled via
a case of colour segmentation, where the saturation serves as a cursor
between luminance and hue. The emphasis is put on the difficulty
of grouping the various necessary optimizations into a single one.
@article{Sera2006,
abstract = {After a formal definition of segmentation as the largest partition
of the space according to a criterion σ and a function f, the notion
of a morphological connection is reminded. It is used as an input
to a central theorem of the paper (Theorem 8), that identifies segmentation
with the connections that are based on connective criteria. Just
as connections, the segmentations can then be regrouped by suprema
and infima. The generality of the theorem makes it valid for functions
from any space to any other one. Two propositions make precise the
AND and OR combinations of connective criteria. The soundness of
the approach is demonstrated by listing a series of segmentation
techniques. One considers first the cases when the segmentation under
study does not involve initial seeds. Various modes of regularity
are discussed, which all derive from Lipschitz functions. A second
category of examples involves the presence of seeds around which
the partition of the space is organized. An overall proposition shows
that these examples are a matter for the central theorem. Watershed
and jump connection based segmentations illustrate this type of situation.
The third and last category of examples deals with cases when the
segmentation occurs in an indirect space, such as an histogram, and
is then projected back on the actual space under study. The relationships
between filtering and segmentation are then investigated. A theoretical
chapter introduces and studies the two notions of a pulse opening
and of a connected operator. The conditions under which a family
of pulse openings can yield a connected filter are clarified. The
ability of segmentations to generate pyramids, or hierarchies, is
analyzed. A distinction is made between weak hierarchies where the
partitions increase when going up in the pyramid, and the strong
hierarchies where the various levels are structured as semi-groups,
and particularly as granulometric semi-groups. The last section is
based on one example, and goes back over the controversy about 'lattice'
versus 'functional' optimization. The problem is now tackled via
a case of colour segmentation, where the saturation serves as a cursor
between luminance and hue. The emphasis is put on the difficulty
of grouping the various necessary optimizations into a single one.},
added-at = {2011-03-27T19:47:06.000+0200},
author = {Sera, Jean},
biburl = {https://www.bibsonomy.org/bibtex/283f11dd1cc558fe2e7a6d07cc43870c5/cocus},
doi = {10.1007/s10851-005-3616-0},
file = {:./seralatticeimage.pdf:PDF},
interhash = {bcef84c2c7c9cbd28a23083012c96ab5},
intrahash = {83f11dd1cc558fe2e7a6d07cc43870c5},
issn = {0924-9907 (Print) 1573-7683 (Online)},
journal = {Journal of Mathematical Imaging and Vision},
keywords = {, 3-D Lipschitz colour connected connection connective crieteria jump levelling measurements methods operators processing quasiflat seed segmentation set smooth variational watersheds zones {$\omega$}-continuity},
month = jan,
number = 1,
owner = {CK},
pages = {83--130},
timestamp = {2011-03-27T19:47:09.000+0200},
title = {A Lattice Approach to Image Segmentation},
url = {http://www.springerlink.com/content/pg819063895l7377/?p=2838d86992ca49f585f3201505e4b07d&pi=2},
volume = 24,
year = 2006
}