Page (PS/TeX): 1 / 1, COMPOSITE ✐ ✐ Two Extensions of Fractional Brownian Motion. Application to Trabecular Bone Radiographs 1 Chapter 1 T WO E XTENSIONS OF F RACTIONAL B ROWNIAN MOTION .A PPLICATION TO T RABECULAR B ONE R ADIOGRAPHS Rachid JENNANE (1) ∗ , Rachid HARBA (1) , Aline BONAMI (2) , Eric LESPESSAILLES (3) 1 Institut PRISME-Université d’Orléans 2 Laboratoire MAPMO-Université d’Orléans 3 Equipe Inserm U658-CHR d’Orléans Key Words: Fractals, fractional Brownian motion, Fractional Gaussian noises, bone, os- teoporosis Abstract In this chapter, starting from fractional Brownian motion of unique parameter H (0 < H < 1), we propose two variants which extend the classical fBm. First, the piece-wise fractional Brownian motion model of parameters H o , H i and γ presents two spectral regimes: it behaves like fBm of parameter H o for low frequencies | f | < γ and like fBm of parameter H i for high frequencies | f | > γ. Second, the nth-order fractional Brownian motion enables the H parameter to be within the range ]n − 1, n[, where n is any strictly positive integer. The properties of these two processes are investigated. ∗ E-mail address: Rachid.Jennane@univ-orleans.fr