2012
DOI: 10.1007/978-3-642-33418-4_40
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Estimation of Non-negative ODFs Using the Eigenvalue Distribution of Spherical Functions

Abstract: Abstract. Current methods in high angular resolution diffusion imaging (HARDI) estimate the probability density function of water diffusion as a continuous-valued orientation distribution function (ODF) on the sphere. However, such methods could produce an ODF with negative values, because they enforce non-negativity only at finitely many directions. In this paper, we propose to enforce non-negativity on the continuous domain by enforcing the positive semi-definiteness of Toeplitz-like matrices constructed fro… Show more

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Cited by 6 publications
(6 citation statements)
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References 17 publications
(28 reference statements)
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“…We compare our methods OCS and ICS with LS, DC, and [9], which we call Eigenvalue Constraint (EC), on a synthetic field of 375 1-, 2-, and 3-fiber ODFs distorted with SNR of 4, 8, and 12 dB as well as a real HARDI brain data set. We evaluate the overall performance of each method using three metrics: 1) the average Riemannian distance to the optimal solution p * of (P ∞ ), i.e., the optimal nonnegative ODF computed by the DC method with 1 million constraints obtained from 1 million discrete grid points on the sphere, 2) the average percentage of negative values remaining in the estimated ODF, evaluated over 1 million discrete grid points, and 3) the average runtime per ODF.…”
Section: Methodsmentioning
confidence: 99%
See 1 more Smart Citation
“…We compare our methods OCS and ICS with LS, DC, and [9], which we call Eigenvalue Constraint (EC), on a synthetic field of 375 1-, 2-, and 3-fiber ODFs distorted with SNR of 4, 8, and 12 dB as well as a real HARDI brain data set. We evaluate the overall performance of each method using three metrics: 1) the average Riemannian distance to the optimal solution p * of (P ∞ ), i.e., the optimal nonnegative ODF computed by the DC method with 1 million constraints obtained from 1 million discrete grid points on the sphere, 2) the average percentage of negative values remaining in the estimated ODF, evaluated over 1 million discrete grid points, and 3) the average runtime per ODF.…”
Section: Methodsmentioning
confidence: 99%
“…However, this requires a full rank assumption, which may not guarantee nonnegativity everywhere. The work of [9] shows that enforcing nonnegativity in all directions is equivalent to enforcing the positive semi-definiteness of an infinite matrix built from the SH coefficients. However, we will see that their solution does not coincide with the optimal solution of the original optimization problem with infinite constraints.…”
Section: Introductionmentioning
confidence: 99%
“…For different experiments (Section 4) we use the eigenvalues associated to both the HARDI signal, s and the ODF p . Furthermore, we can even extend L by zeropadding the coefficient vector of our spherical function using the method in [23] to extract a larger number of features, of which the values such as minimum, maximum, range and variance of eigenvalues will better approximate the distribution of the function values.…”
Section: Invariant Features For Hardi Signals and Odfsmentioning
confidence: 99%
“…Ad-hoc normalization is also employed in these methods to obtain fODFs with unit integral. Some methods estimate continuously non-negative dODFs (Schwab et al, 2012;Cheng et al, 2012;Krajsek and Scharr, 2012) and EAPs (Cheng et al, 2012) using eigenvalue distribution of spherical functions and square root representation. But to our knowledge, none of these methods has been proposed to estimate continuously non-negative fODF in a SD framework.…”
Section: Introductionmentioning
confidence: 99%