Estimation of Non-negative ODFs Using the Eigenvalue Distribution of Spherical Functions

Schwab, Evan; Afsari, Bijan; Vidal, René

doi:10.1007/978-3-642-33418-4_40

Cited by 6 publications

(6 citation statements)

References 17 publications

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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%

“…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%

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Nonnegative ODF estimation via optimal constraint selection

Wolfers

Schwab

Vidal

2014

2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI)

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We consider the problem of estimating a nonnegative orientation distribution function (ODF) from high angular resolution diffusion images. Since enforcing nonnegativity of the ODF for all directions on the sphere leads to an optimization problem with infinitely many constraints, prior work cannot guarantee the nonnegativity of the estimated ODF. The first contribution of this paper is to show that, under certain conditions, a single constraint is sufficient to guarantee the nonnegativity of the estimated ODF in all directions. Otherwise, when these conditions are violated, we propose an iterative algorithm that enforces one constraint at a time and is guaranteed to converge to the optimal nonnegative ODF. Experiments on synthetic and real data show that our methods produce more accurate solutions than prior work at a reduced runtime.Index Terms-Diffusion MRI, HARDI, estimation of nonnegative ODFs, semi-infinite optimization.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonnegative ODF estimation via optimal constraint selection

Wolfers

Schwab

Vidal

2014

2014 IEEE 11th International Symposium on Biomedical Imaging (ISBI)

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“…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%

Rotation Invariant Features for HARDI

Schwab

Çetingül

Afsari

et al. 2013

Lecture Notes in Computer Science

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Reducing the amount of information stored in diffusion MRI (dMRI) data to a set of meaningful and representative scalar values is a goal of much interest in medical imaging. Such features can have far reaching applications in segmentation, registration, and statistical characterization of regions of interest in the brain, as in comparing features between control and diseased patients. Currently, however, the number of biologically relevant features in dMRI is very limited. Moreover, existing features discard much of the information inherent in dMRI and embody several theoretical shortcomings. This paper proposes a new family of rotation invariant scalar features for dMRI based on the spherical harmonic (SH) representation of high angular resolution diffusion images (HARDI). These features describe the shape of the orientation distribution function extracted from HARDI data and are applicable to any reconstruction method that represents HARDI signals in terms of an SH basis. We further illustrate their significance in white matter characterization of synthetic, phantom and real HARDI brain datasets.

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“…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%

Non-Negative Spherical Deconvolution (NNSD) for estimation of fiber Orientation Distribution Function in single-/multi-shell diffusion MRI

et al. 2014

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Spherical Deconvolution (SD) is commonly used for estimating fiber Orientation Distribution Functions (fODFs) from diffusion-weighted signals. Existing SD methods can be classified into two categories: 1) Continuous Representation based SD (CR-SD), where typically Spherical Harmonic (SH) representation is used for convenient analytical solutions, and 2) Discrete Representation based SD (DR-SD), where the signal profile is represented by a discrete set of basis functions uniformly oriented on the unit sphere. A feasible fODF should be non-negative and should integrate to unity throughout the unit sphere S(2). However, to our knowledge, most existing SH-based SD methods enforce non-negativity only on discretized points and not the whole continuum of S(2). Maximum Entropy SD (MESD) and Cartesian Tensor Fiber Orientation Distributions (CT-FOD) are the only SD methods that ensure non-negativity throughout the unit sphere. They are however computational intensive and are susceptible to errors caused by numerical spherical integration. Existing SD methods are also known to overestimate the number of fiber directions, especially in regions with low anisotropy. DR-SD introduces additional error in peak detection owing to the angular discretization of the unit sphere. This paper proposes a SD framework, called Non-Negative SD (NNSD), to overcome all the limitations above. NNSD is significantly less susceptible to the false-positive peaks, uses SH representation for efficient analytical spherical deconvolution, and allows accurate peak detection throughout the whole unit sphere. We further show that NNSD and most existing SD methods can be extended to work on multi-shell data by introducing a three-dimensional fiber response function. We evaluated NNSD in comparison with Constrained SD (CSD), a quadratic programming variant of CSD, MESD, and an L1-norm regularized non-negative least-squares DR-SD. Experiments on synthetic and real single-/multi-shell data indicate that NNSD improves estimation performance in terms of mean difference of angles, peak detection consistency, and anisotropy contrast between isotropic and anisotropic regions.

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Estimation of Non-negative ODFs Using the Eigenvalue Distribution of Spherical Functions

Cited by 6 publications

References 17 publications

Nonnegative ODF estimation via optimal constraint selection

Nonnegative ODF estimation via optimal constraint selection

Rotation Invariant Features for HARDI

Non-Negative Spherical Deconvolution (NNSD) for estimation of fiber Orientation Distribution Function in single-/multi-shell diffusion MRI

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