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20 A.4. Computing the discrepancy ∆κ(g1, g2). Let ˆg := R −1 u2 (v1 − v2), R −1 u2 Ru1 where gi = (vi, ui) ∈ R3 ⋊ S2 , i = 1, 2, and Rui ∈ SO(3) is any rotation with Rui ez = ui. We want to calculate the coordinates {ki}i=1,...,6 satisfying the equation ⎛ ⎞ ˆx 6 ⎜ exp(kiAi|ˆg) = ⎜ R (0, β,ˆγ) ˆ ˆy ⎟ ⎝ ˆz ⎠ i=1 0 0 0 1 = Mˆg ≡ ˆg as introduced in Section 2.1. In other words, we calculate the coordinates of the element ˆg w.r.t. the left-invariant basis matrices {Ai|ˆg}i=1,...,6 which have been defined in Eq. (7), see for comparison Eq. (6). Note that the coordinates are independent of rotation angle α since the Riemannian 2-norm is well-defined w.r.t. the embedding of R 3 ⋊ S 2 into SE(3). Further, the rotation matrices Rui ∈ SO(3) are defined by Rui ez = ui such that αi ∈ [0, 2π) can be freely chosen. So, we set α1 = α2 = ˆα = 0 yielding an easier form than (7) for the basis matrices {Ai|g}i=1,...,6. Then, it holds and ⎛ 1 0 0 k1 cos 3 ⎜ exp(kiAi|ˆg) = ⎜ ⎝ i=1 ˆ β + k3 sin ˆ β 0 1 0 k1 sin ˆ β sin ˆγ + k2 cos ˆγ − k3 cos ˆ β sin ˆγ 0 0 1 −k1 sin ˆ β cos ˆγ + k2 sin ˆγ + k3 cos ˆ ⎟ β cos ˆγ ⎠ 0 0 0 1 i=4 ⎛ 6 ⎜ exp(kiAi|ˆg) = ⎜ ⎝ N (k4,k5,k6) 0 0 0 0 0 0 1 where N(k4,k5,k6) denotes an appropriate matrix depending only on Further, it holds and ⎞ ⎟ ⎠ , A4|ˆg = (cos ˆ β) −1 X4 − tan ˆ β X6, A5|ˆg = X5, A6|ˆg = X6. ˆv = R −1 (0,β2,γ2) (v1 − v2), ˆβ = arcsin[sin β1 cos β2 − cos β1 sin β2 cos(γ1 − γ2)], ˆγ = arcsin[cos β1 sin(γ1 − γ2)(cos ˆ β) −1 ] R −1 (0,β,γ) = So, we can deduce directly that ⎛ ⎝ cos β sin β sin γ − sin β cos γ 0 cos γ sin γ sin β − cos β sin γ cos β cos γ (k1, k2, k3) T = R −1 (0, ˆ β,ˆγ) ˆv. The equation N(k4,k5,k6) = R (0, ˆ β,ˆγ) is solved numerically yielding the coordinates k4, k5 and k6. ⎞ ⎠ . ⎞
APPENDIX B. CODE SNIPPETS Computations are performed with our R-package dti [Tabelow and Polzehl, 2011] (version 1.0) on an Intel(R) Xeon CPU, Six-Core 2933MHz, 24 GB RAM. The operating system was OpenSuse 11.3 with R version 2.12.1. The R-package is freely available on CRAN (http: //cran.r-project.org) and NITRC (http://www.nitrc.org) and has been described in detail in Polzehl and Tabelow [2011]. Our current implementation uses R [R Development Core Team, 2010] with FORTRAN. Computation time for 3 Million voxels (approx. 180 million points in R 3 ⋊ S 2 ) in the experimental data using k ∗ = 12 has been approximately 2 h. In the following example the data, here in DICOM format, are supposed to reside in a directory s0002. The number of slices in the scan is 91, the file b-directions.txt contains the gradient directions including the non-diffusion weighted gradients in the order in which they where scanned. The following code reads the data, performs the position-orientation adaptive smoothing described in this paper, estimates the diffusion tensors and fractional anisotropy and displays the color-coded FA for slice number 57. > library(dti) > gradient dwiobj dwiobj.s dtiobj.s dtiind.s plot(dtiind.s, slice = 57) For more information on the usage of the package see the help files of the package dti and Polzehl and Tabelow [2011]. REFERENCES A. L. Alexander, K. Hasan, M. Lazar, J. S. Tsuruda, and D. L. Parker. Analysis of partial volume effects in diffusion-tensor MRI:. Magn Reson Med, 45:770–780, 2001. H.-E. Assemlal, D. Tschumperlé, L. Brun, and K. Siddiqi. Recent advances in diffusion MRI modeling: Angular and radial reconstruction. Med Image Anal, 15(4):369–396, 2011. P. J. Basser and S. Pajevic. Statistical artefacts in diffusion tensor MRI (DT-MRI) caused by background noise. Magn Reson Med, 44(1):41–50, 2000. C.A. Clark, G.J. Barker, and P.S. Tofts. Magnetic resonance diffusion imaging of the human cervical spinal cord in vivo. Magn Reson Med, 41:1269–1273, 1999. M. Descoteaux, E. Angelino, S. Fitzgibbons, and R. Deriche. Regularized, fast and robust analytical q-ball imaging. Magn Reson Med, 58:497–510, 2007. Z. Ding, J. C. Gore, and A. W. Anderson. Reduction of noise in diffusion tensor images using anisotropic smoothing. Magn Reson Med, 53(2):485–490, 2005. R. Duits. Perceptual Organization in Image Analysis. PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 2005. 21
Page 1 and 2: Weierstraß-Institut für Angewandt
Page 3 and 4: ABSTRACT. We introduce an algorithm
Page 5 and 6: R(α,β,γ) of SO(3) additionally d
Page 7 and 8: the aggregated information on the h
Page 9 and 10: weighted images: ˜w (k) := Kloc |
Page 11 and 12: a) k ⋆ = 0 b) k ⋆ = 4 c) k ⋆
Page 13 and 14: a) b) c) d) FIGURE 4. Color coded F
Page 15 and 16: a) b) c) d) FIGURE 7. a) Slice of d
Page 17 and 18: esulting reconstruction into a step
Page 19 and 20: denote the rotations around the x-,
Page 21: For G := diag{1, 1, 1, 1, 1, 1} it
Page 25: J. Polzehl and K. Tabelow. Beyond t

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