Research Article: High-speed tensor tomography: iterative reconstruction tensor tomography (IRTT) algorithm

Date Published: March 01, 2019

Publisher: International Union of Crystallography

Author(s): Zirui Gao, Manuel Guizar-Sicairos, Viviane Lutz-Bueno, Aileen Schröter, Marianne Liebi, Markus Rudin, Marios Georgiadis.


A fast and robust reconstruction algorithm for small-angle scattering tensor tomography, named iterative reconstruction tensor tomography, is presented. It employs a second-rank tensor model and an iterative error backpropagation to simplify and accelerate tensor tomography reconstruction.

Partial Text

Information about micro- or nanoscopic structural organization within a macroscopic sample is often of great importance. For example, in material science, alignment of carbon nanotubes strongly influences the resistivity of nanotube films (Behnam et al., 2007 ▸; Shekhar et al., 2011 ▸) and molecular anisotropy in an additive manufacturing process is shown to be crucial to controlling structure and morphology (Ivanova et al., 2013 ▸). In biology, structure is often optimized for its function (Fratzl & Weinkamer, 2007 ▸), and significant nano­structural alignment is found in many biological materials and tissues (Fratzl, 2012 ▸; Lichtenegger et al., 1999 ▸; Meek & Boote, 2009 ▸; Masic et al., 2015 ▸; Deymier-Black et al., 2014 ▸). For instance, the orientation of mineralized collagen fibers in bone tissue determines its local mechanical properties (Martin & Ishida, 1989 ▸; Granke et al., 2013 ▸), and is abnormal in different bone pathologies (Giannini et al., 2012 ▸). Similarly, in the brain, the direction of the neuronal axons is used to infer structural connectivity (Johansen-Berg & Rushworth, 2009 ▸), and aberration in structural and functional networks is associated with neuropathologies (Sundgren et al., 2004 ▸; Xie & He, 2011 ▸; Bakshi et al., 2008 ▸; Grefkes & Fink, 2014 ▸; Cao et al., 2015 ▸). A variety of techniques to investigate 3D tissue organization have been developed over the past few years (Georgiadis, Müller et al., 2016 ▸). However, most of them are restricted either to the analysis of tissue sections such as polarized light imaging (Axer et al., 2011 ▸; Bromage et al., 2003 ▸) and 3D scanning small-angle X-ray scattering (Georgiadis et al., 2015 ▸), or to very small sample volumes such as in volume light and electron microscopy (Helmstaedter et al., 2008 ▸; Briggman & Bock, 2012 ▸; Reznikov et al., 2013 ▸).

IRTT is introduced here as a novel, fast and robust method for tomographic reconstruction of the anisotropic nano­structure organization inside materials and tissues. IRTT uses experimental 2D anisotropy information in projections measured for multiple sample orientations (). The reconstruction is based on a tensor model for describing the ODF within each voxel. Model parameters are optimized by iterative backpropagation of the difference between experimental and reconstructed data for all voxels for a randomly chosen projection at each iteration step. IRTT has been shown to be more than an order of magnitude faster compared with previously described reconstruction algorithms (Liebi et al., 2015 ▸, 2018 ▸). This is due to (i) the use of a simpler physical model characterizing the structural anisotropy, i.e. a second-rank tensor versus spherical harmonics used in SASTT, and (ii) an optimization algorithm that employs linearization and error backpropagation to update the model based on a single projection for each iteration cycle. IRTT might be used as a robust tensor tomography reconstruction method, for examining anisotropic nanostructure in materials and tissues. Additionally, its speed makes it suitable for use as a quick first-line reconstruction method for identifying the main nanostructure orientation within each voxel, which can then be used as a starting point for a more refined or general reconstruction such as SASTT. This would significantly reduce the overall reconstruction time by eliminating the multiple steps needed for SASTT and by significantly reducing the number of iterations required to refine the solution.




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