# The tree of shapes of an image

Coloma Ballester; Vicent Caselles; P. Monasse

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 9, page 1-18
- ISSN: 1292-8119

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topBallester, Coloma, Caselles, Vicent, and Monasse, P.. "The tree of shapes of an image." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 1-18. <http://eudml.org/doc/90691>.

@article{Ballester2010,

abstract = {
In [CITE], Kronrod proves that the connected components of isolevel
sets of a continuous function can be endowed with a tree
structure. Obviously, the connected components of upper level sets are an
inclusion tree, and the same is true for connected components of lower level
sets. We prove that in the case of semicontinuous functions, those trees can
be merged into a single one, which, following its use in image processing, we
call “tree of shapes”. This permits us to solve a classical representation
problem in mathematical morphology: to represent an image in such a way that
maxima and minima can be computationally dealt with simultaneously. We prove
the finiteness of the tree when the image is the result of applying any
extrema killer (a classical denoising filter in image processing). The shape
tree also yields an easy mathematical definition of adaptive image
quantization.
},

author = {Ballester, Coloma, Caselles, Vicent, Monasse, P.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Image representation; mathematical morphology; tree structure;
level sets.; image representation; level sets},

language = {eng},

month = {3},

pages = {1-18},

publisher = {EDP Sciences},

title = {The tree of shapes of an image},

url = {http://eudml.org/doc/90691},

volume = {9},

year = {2010},

}

TY - JOUR

AU - Ballester, Coloma

AU - Caselles, Vicent

AU - Monasse, P.

TI - The tree of shapes of an image

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 9

SP - 1

EP - 18

AB -
In [CITE], Kronrod proves that the connected components of isolevel
sets of a continuous function can be endowed with a tree
structure. Obviously, the connected components of upper level sets are an
inclusion tree, and the same is true for connected components of lower level
sets. We prove that in the case of semicontinuous functions, those trees can
be merged into a single one, which, following its use in image processing, we
call “tree of shapes”. This permits us to solve a classical representation
problem in mathematical morphology: to represent an image in such a way that
maxima and minima can be computationally dealt with simultaneously. We prove
the finiteness of the tree when the image is the result of applying any
extrema killer (a classical denoising filter in image processing). The shape
tree also yields an easy mathematical definition of adaptive image
quantization.

LA - eng

KW - Image representation; mathematical morphology; tree structure;
level sets.; image representation; level sets

UR - http://eudml.org/doc/90691

ER -

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