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

Issue		Math. Model. Nat. Phenom. Volume 3, Number 6, 2008 Medical imaging


Page(s)		48 - 75
DOI		10.1051/mmnp:2008081
Published online		24 December 2008

Math. Model. Nat. Phenom. Vol. 3, No. 6, 2008, pp. 48-75
DOI: 10.1051/mmnp:2008081

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

H. Akkari¹, I. Bhouri², P. Dubois³ and M.H. Bedoui¹

¹ Laboratoire de Biophysique, TIM, Faculté de Médecine de Monastir, Tunisia
² Unité de recherche Multifractals et Ondelettes, Faculté des Sciences de Monastir, Tunisia
³ INSERM, U 703, ITM, CHRU Lille, France

bhouri_imen@yahoo.fr

Published online: 24 December 2008

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.

Mathematics Subject Classification. 28A78, 28A80, 28A75

Key words: minkowski dimensions and contents -- Hausdorff measure -- projection -- sections -- osteoporosis -- medical imaging

© EDP Sciences 2008

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