A New Tensorial Framework for Single-Shell High Angular Resolution Diffusion Imaging

Abstract

Single-shell high angular resolution diffusion imaging data (HARDI) may be decomposed into a sum of eigenpolynomials of the Laplace-Beltrami operator on the unit sphere. The resulting representation combines the strengths hitherto offered by higher order tensor decomposition in a tensorial framework and spherical harmonic expansion in an analytical framework, but removes some of the conceptual weaknesses of either. In particular it admits analytically closed form expressions for Tikhonov regularization schemes and estimation of an orientation distribution function via the Funk-Radon Transform in tensorial form, which previously required recourse to spherical harmonic decomposition. As such it provides a natural point of departure for a Riemann-Finsler extension of the geometric approach towards tractography and connectivity analysis as has been stipulated in the context of diffusion tensor imaging (DTI), while at the same time retaining the natural coarse-L. Florack ( )

Linear transforms for Fourier data on the sphere: Application to high angular resolution diffusion MRI of the brain

“…Many alternatives to the FRT have been proposed in recent years for estimating orientation information from the same kind of HARDI data, including multi-tensor models (Alexander et al, 2001; Tuch et al, 2002; Hosey et al, 2005; Kreher et al, 2005; Peled et al, 2006; Behrens et al, 2007; Jian et al, 2007; Ramirez-Manzanares et al, 2007; Pasternak et al, 2008; Melie-García et al, 2008; Leow et al, 2009; Tabelow et al, 2012), higher-order generalizations of tensor models (Alexander et al, 2002; Frank, 2002; Özarslan and Mareci, 2003; Liu et al, 2004; Schultz and Seidel, 2008; Barmpoutis et al, 2009; Liu et al, 2010; Florack et al, 2010), directional function modeling (Kaden et al, 2007; Rathi et al, 2009, 2010), spherical polar Fourier expansion (Assemlal et al, 2009, 2011), independent component analysis (Singh and Wong, 2010), sparse spherical ridgelet modeling (Michailovich and Rathi, 2010), diffusion circular spectrum mapping (Zhan et al, 2004), deconvolution (Tournier et al, 2007; Anderson, 2005; Descoteaux et al, 2009; Patel et al, 2010; Yeh et al, 2011; Reisert and Kiselev, 2011), the diffusion orientation transform (Özarslan et al, 2006; Canales-Rodríguez et al, 2010), estimation of persistent angular structure (Jansons and Alexander, 2003), generalized q -sampling imaging (Yeh et al, 2010), and orientation estimation with solid angle considerations (Tristán-Vega et al, 2009; Aganj et al, 2010; Tristán-Vega et al, 2010). However, the FRT has a unique combination of useful characteristics: it does not require a strict parametric model of the diffusion signal, it is linear and its theoretical characteristics can be explored analytically, and it can be computed very quickly using efficient algorithms (Anderson, 2005; Hess et al, 2006; Descoteaux et al, 2007; Kaden and Kruggel, 2011).…”

Higher-Order Tensors in Diffusion Imaging

“…Florack et al used the same Laplace-Beltrami regularization on the sphere, but for tensors instead of SHs [34]. This was based on an infinite inhomogeneous tensor basis representation, much like the SHs, with the diffusion function modified toD(u) = ∑ ∞ k=0 D (k) • k u.…”

Mathematical Methods for Signal and Image Analysis and Representation

“…This concept is frequently used in [114,157]. Results from differential geometry are then used to process this surface.…”