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

Abstract

The inner knowledge of volumes from imagesis an ancient problem. This question becomes complicated when itconcerns quantization, as the case of any measurement and inparticular the calculation of fractal dimensions. Trabecular bonetissues have, like many natural elements, an architecture whichshows a fractal aspect. Many studies have already been developedaccording to this approach. The question which arises however is toknow to which extent it is possible to get an exact determination ofthe fractal dimension of a given volume only from the fractalmeasurement made on the projections or slice images given by medicalimaging. This paper gives general results about the Minkowskidimensions and contents of projections and sections of a set. Wealso show with examples that they depend essentially on thedirections of the planes and so there is - in general case - norelation between 3D and 2D fractal dimensions. This consideration isthen illustrated with examples from synthetic models and from CTscan images of wrists. In conclusion, this study reveals that thequantitative characterization of an organic volume (in particularosseous) requires taking into account the whole volume, and not onlysome of its slices or projections.