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4 a) b) c) d) FIGURE 1. Diffusion weighted data Sg. a) Slice of diffusion weighted image for a single (arbitrarily selected) diffusion weighting gradient direction. b) Same slice shown after averaging over four subsequent measurements. (A larger version of these images can be found in Figure 7.) c) Diffusion weighted data for all diffusion gradients in a single voxel in corpus callosum. The image is created from a 3D plot of the data, where the diffusion weighted values are shown in their corresponding gradient direction. d) Same after averaging over four subsequent measurements. an additional parameter κ which allows for a balance between distances on the sphere and in space. Finally, we get the following formula 3 k 2 6 i + k 2 i /κ 2 1/2 , ∆κ(g1, g2) := R −1 u2 (v1 − v2), R −1 u2 Ru1 R,κ ≈ where gi = (vi, ui) ∈ R 3 ⋊ S 2 , i = 1, 2, and Rui ∈ SO(3) is any rotation with Rui ez = ui as introduced above. 2.2. Position-orientation adaptive smoothing (POAS) the diffusion weighted images. De- note by Sg the measured diffusion weighted data at g = (v, b) ∈ R 3 ⋊ S 2 , i.e., for some voxel at position v and for an applied diffusion weighting gradient in direction b. The noise distribution of Sg follows a Rician distribution, which can be approximated by a Gaussian distribution for large values of the data [Johansen-Berg and Behrens, 2009]. In order to construct a structural adaptive smoothing algorithm for diffusion weighted data we first formulate a structural assumption on the data. This assumption represents the observation that the parameters of the distribution for the data show similarities in the vicinity of any voxel separated by sharp discontinuities, e.g., at tissue borders, see Figure 1. We therefore assume the existence of a non-trivial partition U of the space R 3 ⋊ S 2 into maximal homogeneity compartments. As a consequence for each g1 ∈ R 3 ⋊ S 2 there is a vicinity U(g1) such that i=1 ESg2 = ESg1 ∀ g2 ∈ U(g1), where E denotes the expectation with respect to the Rician distribution. This assumption can be extended to more complicated local models, e.g., local linear or local quadratic parameter models. Here, we only consider local constant models for the sake of computational simplicity! See Section 5 for the effects of a misspecified structural assumption. Our method POAS to infer on expected values ESg, g ∈ R 3 ⋊ S 2 , and their homogeneity regions U(g) is based on the Propagation-Separation approach [Polzehl and Spokoiny, 2006] and yields an iterative procedure with a prespecified number of steps k ∗ . At each iteration step i=4
the aggregated information on the homogeneity regions is used to obtain improved estimates for the image values ESg, using kernel estimation. These estimates are then again used to infer on the homogeneity regions. The estimate for the image value ESg1 at g1 ∈ R 3 ⋊ S 2 and iteration step k is (2) ˆ S (k) g1 := g2∈R 3 ⋊S 2 w (k) (k) g1g2Sg2/N g1 with N (k) g1 := g2 w(k) g1g2 and local weighting schemes w (k) g1g2 for each pair of measurements at g1, g2 ∈ R 3 ⋊ S 2 : (3) w (k) g1g2 := Kloc ∆κ( b1,k) (g1, g2) /h( (k) b1, k) Kst s g1g2 /λ . We call the weights w (k) g1g2 adaptive. We use the term non-adaptive if the factor Kst(s (k) g1g2/λ) is omitted in Eq. (3). In Figure 2 we show the non-adaptive weighting schemes for several bandwidths and values of κ in order to demonstrate effects of the discrepancy function ∆κ. Now, we shortly describe the terms used in Eq. (3): Kloc and Kst are kernel functions. ∆κ (g1, g2) is the discrepancy introduced above. {h( b, k)}k=0,··· ,k∗ is an increasing sequence of bandwidths depending on the gradient directionb ∈ S2 . The sequence {κ( b, k)}k=0,··· ,k∗ describes for each gradient directionb ∈ S2 the relation between distances on the sphere and in voxel space. The statistical penalty s (k) g1g2, defined by ˆS (k−1) s (k) (k−1) g1g2 := N g1 · K g1 ˆσ , ˆ S (k−1) g2 ˆσ depends on the Kullback-Leibler distance between the two standardized Rician distribu- tions in g1 and g2 with parameters ˆ S (k−1) g1 , /ˆσ and ˆ S (k−1) g1 /ˆσ, where ˆσ denotes an estimate for the scale parameter of the Rician distribution. Evaluating the distance between the estimated image values the statistical penalty tests for our structural assumption. The Kullback-Leibler distance is approximated numerically. N (k) g variance reduction at step k. λ is the adaptation parameter of the algorithm. 5 is an approximation for the Note that both, the sequence of bandwidths {h( b, k)}k=0,··· ,k∗ and the sequence of balancing parameters {κ( b, k)}k=0,··· ,k∗ possibly depend on the diffusion gradient direction b ∈ S2 , as most gradient schemes are not fully homogeneous on the sphere. Starting at very local initial estimates that coincide with non-adaptive-smoothing on the sphere, i.e. setting s (0) g1g2 := 1 and choosing h( bl, 0) = 1, and using the discrepancy derived in the previous section we iterate computation of weights (3) and estimation of ESg by ˆ S (k) g (2), increasing k with each iteration. This defines a position-orientation adaptive smoothing algorithm for diffusion weighted data in the space R3 ⋊ S2 .
Page 1 and 2: Weierstraß-Institut für Angewandt
Page 3 and 4: ABSTRACT. We introduce an algorithm
Page 5: R(α,β,γ) of SO(3) additionally d
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 and 22: For G := diag{1, 1, 1, 1, 1, 1} it
Page 23 and 24: APPENDIX B. CODE SNIPPETS Computati
Page 25: J. Polzehl and K. Tabelow. Beyond t

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