Proc. XNPIG 2017 Zürich, Switzerland, 12-15 Sept. 2017, p. 49-50, https://www.psi.ch/xnpig2017
Nested Tomography: Application to Direct Ellipsoid Reconstruction in Anisotropic Darkfield Tomography J. DITTMANN1, S. ZABLER1,2, R. HANKE1,2,3 1
Lehrstuhl f. Röntgenmikroskopie, Universität Würzburg, Germany,
[email protected] 2
Fraunhofer IIS, Erlangen, Germany
3
Fraunhofer IZFP, Saarbrücken, Germany
Summary: Tomographic reconstruction of the orientation of anisotropic scatterers based on the fringe visibility (darkfield) contrast of Talbot interferometric X-ray imaging setups has recently been demonstrated. Three customized approaches to the modeling and reconstruction of this non-scalar information have been proposed so far. In order to improve the understanding of the fundamental problem and the limitations of tomographic reconstruction of anisotropic signals, we give a more abstract formulation as nested tomography problem and show the special case of ellipsoid (tensor) reconstruction. The relation to previous literature is discussed. Introduction. Demonstrations of the applicability of a Lambert-Beer relation to darkfield (fringe visibility) contrast by several authors have layed the foundation for tomographic applications. The anisotropy properties of the darkfield signal have initially been discussed for thin sheets of fibrous materials [1]. For tomographic reconstruction of anisotropic scattering distributions at each voxel location, Malecki and Vogel et al. [2, 3] employed an expansion into a non-orthogonal (pseudo-)basis of principal orientations with subsequent fitting to an ellispoid model (for each voxel), Wieczorek et al. [4] proposed an expansion into spherical harmonics and Bayer et al. [5] used a sinosoidal model describing the anisotropy of each voxel only within the tomographic plane. Respective reconstructions have been demonstrated using a modified SART algorithm [2] or generic optimization techniques based on an explicit system matrix [5]. [3, 4] attempt to disentangle the non-scalar tomographic task into seperate scalar tomography problems by approximate self consistent decomposition of the radiographies into independent contributions by each component of the anisotropic scattering model. We give a generic “nested tomography” interpretation of the problem and specifically show the application to direct tensor reconstruction without the necessity of a higher-dimensional intermediate representation as used by [2, 3]. Methods. The forward problem of anisotropic tomography based on scalar projections may simply be formulated in terms of the classic scalar forward problem and a problem specific transform describing the observable scalar projection of the non-scalar voxels. Given a 3D volume with generalized non-scalar voxels fj enumerated by j and a scalar transform t(i): fj → fj, the observable (scalar) projections gi may be modeled by the classic system matrix A: gi = Σj Aij fj(i) = Σj Aij t(i)(fj) (1) where i enumerates individual line integrals (i.e. source–detectorbin locations) encoded in A and the superscript (i) indicates some potential dependence of the scalar projection t on i. The inversion of Eq. (1) may be accomplished with any (iterative) reconstruction technique for scalar tomography with additional provision of a suiting pseudo-inverse
Jonas Graetz (Dittmann),
[email protected]
Proc. XNPIG 2017 Zürich, Switzerland, 12-15 Sept. 2017, p. 49-50, https://www.psi.ch/xnpig2017
t(i)T: fj → fj such that t(i)(t(i)T(s)) = s holds (approximately) for any scalar s. (2) (i)T The pseudo inverse t allows to transparently update multiple intrinsic parameters within the non-scalar object fj based on a single scalar update value determined by the employed classic reconstruction procedure. For an ellipsoid voxel model, fj is a symmetric rank 2 tensor (3x3 matrix) fjmn and the required transformations are t(i)(fj) = Σmn e(i)m fjmn e(i)n and t(i)T(fj) = e(i)m fj e(i)n (3) (i) with normalized sensing orientation ê (possibly) dependent on the current view (i). It can be easily verified that t(i)(t(i)T(s)) = Σmn e(i)m (e(i)m s e(i)n) e(i)n = s with Σn e(i)n2=1. Relation to previous work. The algorithm proposed by Malecki [3] prior to subsequent tensor fitting is equivalent to classic SART in combination with the following transforms: tM(i)(fj) = Σα v(i)α fjα and tM(i)T(fj) = fj v(i)α /(Σα v(i)α)2 (4) (i) where α enumerates principal scattering orientations and v α denote their respective contributions to the i’th projection. It can be easily verfied that Eq. (2) is not fulfilled. Vogel et al. [3] uses the same model with different reconstruction and tensor fitting techniques (without explicit t(i)T) . Wiezcorek et al. [4] alter the defintion of v(i)α based on spherical harmonics without fitting to an ellispoid. The in-plane sinosoidal model by Bayer et al. [5] is equivalent to the general tensor model (3) with the sensing orientation ê parallel to detector rows and perpendicular to the rotation axis (e.g. ê(i) = (cos φi, sin φi, 0)). The explicit definition of its ovservable scalar projections (Eq. 3) makes obvious that projections 90° apart on the circular trajectory image independent volumes (independent tensor components), indicating a considerable underdetermination of the problem in the case of only one scan trajectory for both problems (volume and tensor reconstruction). Discussion and Conclusion. The transformations t and tT are equivalent to projection and backprojection within a tomographic reconstruction algorithm. The non-scalar tomography problem thus effectively is a nested tomography problem. While the scalar problem is solvable by means of different projections of the same object, a view-dependent transformation t(i) may cause different projections to show another object (another component of f), effectively creating an entangled limited angle reconstruction problem for multiple components of f. Both problems may be decoupled by appropriate choice of the experimental setup, ideally rendering t(i) view (i) independent within circular scan trajectories. For the presented tensor tomography model this is the case for ê perpendicular to the tomographic plane and parallel to the rotational axis. If decoupling is not possible, the resulting problem is similar to limited angle tomography and both known limitations as well as known regularization techniques for this class of problems should apply and be applicable respectively. The special case of tensor (ellispoid) voxels can be directly formulated in the presented framework without the need for an intermediate representation. References [1] – V. Revol, C. Kottler, R. Kaufmann, A. Neels, A. Dommann. J. Appl. Phys 112, 114903, (2012) [2] – A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, F. Pfeiffer. EPL 105, 38002, (2014) [3] – J. Vogel, F. Schaff, A. Fehringer, C. Jud, M. Wieczorek, F. Pfeiffer, T. Lasser. Opt. Express 23, 15134, (2015) [4] – M. Wiezcorek, F. Schaff, F. Pfeiffer, T. Lasser. Phys. Rev. Lett. 117, 158101 (2016) [5] – F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, C. P. Riess, PNAS 111, 12699, (2014)
Jonas Graetz (Dittmann),
[email protected] 
