# On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging

H. Akkari; I. Bhouri; P. Dubois; M. H. Bedoui

Mathematical Modelling of Natural Phenomena (2008)

- Volume: 3, Issue: 6, page 48-75
- ISSN: 0973-5348

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topAkkari, H., et al. "On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging." Mathematical Modelling of Natural Phenomena 3.6 (2008): 48-75. <http://eudml.org/doc/222329>.

@article{Akkari2008,

abstract = {
The inner knowledge of volumes from images
is an ancient problem. This question becomes complicated when it
concerns quantization, as the case of any measurement and in
particular the calculation of fractal dimensions. Trabecular bone
tissues have, like many natural elements, an architecture which
shows a fractal aspect. Many studies have already been developed
according to this approach. The question which arises however is to
know to which extent it is possible to get an exact determination of
the fractal dimension of a given volume only from the fractal
measurement made on the projections or slice images given by medical
imaging. This paper gives general results about the Minkowski
dimensions and contents of projections and sections of a set. We
also show with examples that they depend essentially on the
directions of the planes and so there is - in general case - no
relation between 3D and 2D fractal dimensions. This consideration is
then illustrated with examples from synthetic models and from CT
scan images of wrists. In conclusion, this study reveals that the
quantitative characterization of an organic volume (in particular
osseous) requires taking into account the whole volume, and not only
some of its slices or projections.
},

author = {Akkari, H., Bhouri, I., Dubois, P., Bedoui, M. H.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {minkowski dimensions and contents;
Hausdorff measure; projection; sections; osteoporosis; medical
imaging; Minkowski dimensions and contents; Hausdorff measure; medical imaging},

language = {eng},

month = {12},

number = {6},

pages = {48-75},

publisher = {EDP Sciences},

title = {On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging},

url = {http://eudml.org/doc/222329},

volume = {3},

year = {2008},

}

TY - JOUR

AU - Akkari, H.

AU - Bhouri, I.

AU - Dubois, P.

AU - Bedoui, M. H.

TI - On the Relations Between 2D and 3D Fractal Dimensions: Theoretical Approach and Clinical Application in Bone Imaging

JO - Mathematical Modelling of Natural Phenomena

DA - 2008/12//

PB - EDP Sciences

VL - 3

IS - 6

SP - 48

EP - 75

AB -
The inner knowledge of volumes from images
is an ancient problem. This question becomes complicated when it
concerns quantization, as the case of any measurement and in
particular the calculation of fractal dimensions. Trabecular bone
tissues have, like many natural elements, an architecture which
shows a fractal aspect. Many studies have already been developed
according to this approach. The question which arises however is to
know to which extent it is possible to get an exact determination of
the fractal dimension of a given volume only from the fractal
measurement made on the projections or slice images given by medical
imaging. This paper gives general results about the Minkowski
dimensions and contents of projections and sections of a set. We
also show with examples that they depend essentially on the
directions of the planes and so there is - in general case - no
relation between 3D and 2D fractal dimensions. This consideration is
then illustrated with examples from synthetic models and from CT
scan images of wrists. In conclusion, this study reveals that the
quantitative characterization of an organic volume (in particular
osseous) requires taking into account the whole volume, and not only
some of its slices or projections.

LA - eng

KW - minkowski dimensions and contents;
Hausdorff measure; projection; sections; osteoporosis; medical
imaging; Minkowski dimensions and contents; Hausdorff measure; medical imaging

UR - http://eudml.org/doc/222329

ER -

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