No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
DETERMINISTIC TRACTOGRAPHY USING ORIENTATION DISTRIBUTION FUNCTIONS ESTIMATED WITH PROBABILITY DENSITY CONSTRAINTS AND SPATIAL REGULARITY
... Diffusion estimations can be refined by de-noising and smoothing methods that also incorporate diffusion information from a spatial neighborhoods [32] for example. Feature Analysis. ...
... However, ODFs are PDFs on the unit sphere, and therefore should be non-negative and sum to 1, conditions which are often violated due to noisy measurements. While ensuring the distribution sums to 1 is done by rescaling after estimation, enforcing non-negativity has been addressed by adding constraints on the SH coefficients in (2.28) as [32]: ...
... Incorporating spatial information into voxel-wise reconstruction is a well utilized technique for increasing the accuracy of reconstruction. The following is a general formulation for including spatial regularization into the angular sparse coding problem: spatial coherence (see [32] for example). Some have found incorporating both angular sparsity λ||A|| 1 and spatial coherence R(A) beneficial for applications such as de-noising [99,101,123,124] and tractography [125]. ...
... In addition, some researchers have combined their estimation methods with neighborhood correlation that smoothes the HARDI data and estimates fiber orientation simultaneously. [29][30][31] Inspired by these techniques, and considering both the neighborhood correlation and Rician distribution of HARDI data, we present a novel spherical deconvolution approach that is based on the Richardson-Lucy algorithm. Our new deconvolution method can reduce noise and correct Rician bias simultaneously. ...
... The RL-type algorithms were developed from the maximum-likelihood method. 29 The log-likelihood of Eq. (6) is ...
... In the first experiment, we wanted to determine which of the modifications (Rician bias correction or local neighborhood smoothness) are actually the most beneficial for a RL-type algorithm. We followed the schema for simulated HARDI data in Ref. 29. We first synthesized an anisotropic HARDI field containing differing numbers of crossing fibers [ Fig. 1(a)]. ...
Purpose:
Diffusion
tensor
imaging is widely used for studying neural fiber trajectories in white matter and for quantifying changes in tissue using diffusion properties at each voxel in the brain. To better model the nature of crossing fibers within complex architectures, rather than using a simplified tensor model that assumes only a single fiber direction at each image voxel, a model mixing multiple diffusion
tensors is used to profile diffusion signals from high angular resolution
diffusion
imaging (HARDI) data. Based on the HARDI signal and a multiple tensors model, spherical deconvolution methods have been developed to overcome the limitations of the diffusion
tensor model when resolving crossing fibers. The Richardson–Lucy algorithm is a popular spherical deconvolution method used in previous work. However, it is based on a Gaussian distribution, while HARDI data are always very noisy, and the distribution of HARDI data follows a Rician distribution. This current work aims to present a novel solution to address these issues.
... This is a problem in population studies, where one is interested in applying statistical methods to ODFs to differentiate between healthy and diseased populations, which cannot be accurately done without axiomatically correct distributions. To address this problem, [10] enforces non-negativity at finitely many directions on the sphere, but ODF interpolation and registration methods may require evaluating ODFs outside discrete grids. [11] uses a non-ODF constrained spherical deconvolution method that reduces the occurrence of negative values, but does not completely eliminate them. ...
... However, disregarding the nonnegativity constraints could result in negative values for p(ϕ, ϑ). To address this issue, [10] proposes to enforce the non-negativity constraints at finitely many 1 To solve this problem, [10] defines the discrete ODF p ∈ R M whose i-th entry is ...
... However, disregarding the nonnegativity constraints could result in negative values for p(ϕ, ϑ). To address this issue, [10] proposes to enforce the non-negativity constraints at finitely many 1 To solve this problem, [10] defines the discrete ODF p ∈ R M whose i-th entry is ...
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 from the spherical harmonic representation of the ODF. We study the distribution of the eigenvalues of these matrices and use it to derive an iterative semi-definite program that enforces non-negativity on the continuous domain. We illustrate the performance of our method and compare it to the state-of-the-art with experiments on synthetic and real data.
... Solving for an ODF subject to nonnegativity constraints requires solving an optimization problem with infinitely many constraints, one per each direction on the sphere. The work of [5] aims to address this issue by enforcing nonnegativity at a finite number of fixed directions. However, this does not guarantee the nonnegativity of the ODF in all directions. ...
... The approximate method of [5] enforces nonnegativity at a finite number of fixed grid points x ...
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.
... The phantom was from the Fiber Cup held in London in 2009 during the Medical Image Computing and Computer Assisted Intervention (MICCAI 2009) [1] and is publicly available on the following website: http://www.lnao.fr/spip.php?rubrique79. This phantom's associated advantages are: (1) this phantom was elaborately manufactured (the whole procedure can be referred to from [1] and [13]); (2) this phantom contained various extreme conditions of fiber tracking: fiber crossing, fiber kissing, as well as bundles of different curvatures; (3) many famous fiber tracking methods [14][15][16][17][18][19][20][21][22][23]have already conducted testing on this phantom and have been given a score based on an evaluation system, so the present method can be quantitatively compared with the previous tracking methods. The parameters for the phantom acquisition were as follows: spatial resolution was 3 mm, field of view FOV=19.2 ...
... This evaluation system relies upon the point-based Root Mean Square Error (RMSE) between the construction results and the ground truth [1]. The other ten methods and their associated scores are the following (listed in decreasing score order): Global tractography [5] 116 score, FOD-SH with constrained spherical deconvolution and streamline tractography [16] 87 score, Combined 2-DT model estimation and streamline tractography [19] 31 score, ODF-SH with positivity and regularity constraints and streamline tractography [23] 19 score, PAS-MRI and streamline tractography [20] 16 score, Adaptive 1 or 2-DT model and streamline tractography [15] 5 score, Single-DT and streamline tractography [21] 5 score, FOD-SH with streamline tractography [22] 5 score, Single-DT with tensor deflection [18] 4 score, and Single-DT with streamline tractography and RK4 integration [4] 0 score. The present method obtained a score of 78, placing it third in the previous list. ...
Diffusion tensor imaging (DTI) is a tractography algorithm that provides the only means of mapping white matter fibers. Furthermore, because of its wealth of applications, diffusion MRI tractography is gaining importance in clinical and neuroscience research. This paper presents a novel brain white matter fiber reconstruction method based on the snake model by minimizing the energy function, which is composed of both external energy and internal energy. Internal energy represents the assembly of the interaction potential between connected segments, whereas external energy represents the differences between predicted DTI signals and measured DTI signals. Through comparing the proposed method with other tractography algorithms in the Fiber Cup test, the present method was shown to perform superiorly to the majority of the other methods. In fact, the proposed test performed the third best out of the ten available methods, which demonstrates that present method can accurately formulate the brain white matter fiber reconstruction.
... The resulting estimates {α,γ} are restricted to be in [0, π) due to the antipodal symmetry of the ODFs. 2 This last step finalizes the estimation of the rotation asR = R(α,β,γ). 1 In reality, it is seldom that β > π 2 , hence we discard the root in ( π 2 , π]. 2 Notice that ifβ ≈ 0, we can take γ = 0 and solve (14) for α. ...
... Here, Tr denotes the r-th tensor and we set b = 3, 000 s/mm 2 . We simulate the noise-free signal at 81 gradient directions and reconstruct 100 ODFs using the method proposed in [14]. We rotate each ODF with known parameters α, γ ∈ {0, π 6 , . . . ...
We consider the problem of aligning high angular resolution diffusion images characterized by orientation distribution functions (ODFs). We cast this problem as an optimization problem where we seek the rotation that aligns the source and target ODFs. This rotation induces a linear transformation of the spherical harmonic coefficients of the ODFs, which can be parametrized by the rotation Euler angles. We propose an algebraic approach to estimate this transformation from a number of ODF correspondences. We evaluate the proposed method on synthetic ODFs as well as on a diffusion MR phantom dataset.
... New tractography methods that can account for fanning configurations in the tracking are being developed, but results are still preliminary, and the more complicated subvoxel configurations are not addressed (105,123,135). Others have started to include local geometry information from the neighboring voxels and spatial regularization (136)(137)(138). ...
This article covers the young history of high angular resolution diffusion imaging (HARDI), from basic diffusion principles and diffusion tensor imaging (DTI) to q-space imaging, advanced modeling, and high-order reconstruction techniques. HARDI has opened up new perspectives to noninvasively measure microstructural features and study white matter connectivity using HARDI-based fiber tractography. HARDI is thus at the heart of modern neuroscience research and several clinical applications.
... The dMRI literature has produced a wide array of dMRI reconstruction algorithms for different acquisition protocols, an artillery of q-space bases and varying models for estimating fiber tract PDFs. The vast majority of research reconstructs q-space signals in each voxel with a q-space basis while adding a set of constraints C(a v ) on the coefficients to enforce desirable properties such as non-negativity of PDFs [12,13], smoothing [14] or spatial coherence [15], solving: The constraint of particular interest in our paper is that of enforcing sparsity on the coefficients of the reconstruction, known as Sparse Coding, which has applications in CS as well as super-resolution [16] and de-noising [17]. ...
Diffusion MRI (dMRI) provides the ability to reconstruct neuronal fibers in the brain, in vivo, by measuring water diffusion along angular gradient directions in q-space. High angular resolution diffusion imaging (HARDI) can produce better estimates of fiber orientation than the popularly used diffusion tensor imaging, but the high number of samples needed to estimate diffusivity requires longer patient scan times. To accelerate dMRI, compressed sensing (CS) has been utilized by exploiting a sparse dictionary representation of the data, discovered through sparse coding. The sparser the representation, the fewer samples are needed to reconstruct a high resolution signal with limited information loss, and so an important area of research has focused on finding the sparsest possible representation of dMRI. Current reconstruction methods however, rely on an angular representation per voxel with added spatial regularization, and so, for non-zero signals, one is required to have at least one non-zero coefficient per voxel. This means that the global level of sparsity must be greater than the number of voxels. In contrast, we propose a joint spatial-angular representation of dMRI that will allow us to achieve levels of global sparsity that are below the number of voxels. A major challenge, however, is the computational complexity of solving a global sparse coding problem over large-scale dMRI. In this work, we present novel adaptations of popular sparse coding algorithms that become better suited for solving large-scale problems by exploiting spatial-angular separability. Our experiments show that our method achieves significantly sparser representations of HARDI than is possible by the state of the art.
... It is important to point out at this juncture that a key result of our analysis leading to the GO-ESP method (Galinsky & Frank, 2015) is that the generally accepted view that the diffusion PDF is the fundamental quantity in diffusion MRI methods is predicated on the assumption that voxel diffusion profiles are independent. Although there are currently methods that introduce bridging for the local and global scales, i.e. through spatially regularized ODF reconstruction (Goh et al., 2009;Reisert et al., 2011), or by augmenting streamline tractography with some pseudo global looking schemes (Kreher et al., 2008;Fillard et al., 2009;Reisert et al., 2014;Christiaens et al., 2015), the majority of these methods perform diffusion estimation and tractography independently. The fact that voxel diffusion profiles measured in diffusion weighted images of continuous underlying white matter structures are clearly not independent leads to the logically inconsistent procedures whereby local diffusion is estimated based upon this assumption of independence whereas tractography is constructed based upon the implicit assumption of dependence. ...
We present a quantitative statistical analysis of pairwise crossings for all fibers obtained from whole brain tractography that confirms with high confidence that the brain grid theory (Wedeen et al., 2012a) is not supported by the evidence. The overall fiber tracts structure appears to be more consistent with small angle treelike branching of tracts rather than with near-orthogonal gridlike crossing of fiber sheets. The analysis uses our new method for high-resolution whole brain tractography that is capable of resolving fibers crossing of less than 10 degrees and correctly following a continuous angular distribution of fibers even when the individual fiber directions are not resolved. This analysis also allows us to demonstrate that the whole brain fiber pathway system is very well approximated by a lamellar vector field, providing a concise and quantitative mathematical characterization of the structural connectivity of the human brain.
... Classical HARDI reconstruction methods model the q-space signals from each voxel separately and add a regularization term R to enforce desirable properties such as spatial coherence, ODF non-negativity, or sparsity. Some recent methods [1,5,12,13] have considered simultaneous voxel-based reconstruction over an entire volume by solving ...
High angular resolution diffusion imaging (HARDI) can produce better estimates of fiber orientation and richer sets of features for disease classification than diffusion tensor imaging. However, existing HARDI reconstruction algorithms require a large number of gradient directions, making the acquisition time too long to be clinically viable. State-of-the-art compressed sensing methods can reduce the number of measurements needed for accurate reconstruction by exploiting angular sparsity at each voxel, but the global sparsity level is therefore bounded below by the number of voxels. In this work, we aim to find a significantly sparser representation of HARDI by exploiting redundancies in both the spatial and angular domains jointly with a global HARDI basis. However, this leads to a massive global optimization problem over the whole brain which cannot be solved using existing sparse coding methods. We present a novel Kronecker extension to ADMM that exploits the separable spatial-angular structure of HARDI data to efficiently find a globally sparse reconstruction. We validate our method on phantom and real HARDI brain data by showing that we can achieve accurate reconstructions with a global sparsity level corresponding to less then one atom per voxel, surpassing the absolute limit of the state-of-the-art.
... To improve the data on which the tractography is performed, different regularization methods can be used. Methods exist that apply filtering for the reduction of noise directly on the DW-MRI data [8][9][10], other methods aim to regularize the DTI tensor fields [11][12][13][14][15]. On HARDI data the regularization can be performed on individual voxels [16][17][18] or in combination with the local spatial information [19][20][21][22][23]. ...
We propose two strategies to improve the quality of tractography results
computed from diffusion weighted magnetic resonance imaging (DW-MRI) data. Both
methods are based on the same PDE framework, defined in the coupled space of
positions and orientations, associated with a stochastic process describing the
enhancement of elongated structures while preserving crossing structures. In
the first method we use the enhancement PDE for contextual regularization of a
fiber orientation distribution (FOD) that is obtained on individual voxels from
high angular resolution diffusion imaging (HARDI) data via constrained
spherical deconvolution (CSD). Thereby we improve the FOD as input for
subsequent tractography. Secondly, we introduce the fiber to bundle coherence
(FBC), a measure for quantification of fiber alignment. The FBC is computed
from a tractography result using the same PDE framework and provides a
criterion for removing the spurious fibers. We validate the proposed
combination of CSD and enhancement on phantom data and on human data, acquired
with different scanning protocols. On the phantom data we find that PDE
enhancements improve both local metrics and global metrics of tractography
results, compared to CSD without enhancements. On the human data we show that
the enhancements allow for a better reconstruction of crossing fiber bundles
and they reduce the variability of the tractography output with respect to the
acquisition parameters. Finally, we show that both the enhancement of the FODs
and the use of the FBC measure on the tractography improve the stability with
respect to different stochastic realizations of probabilistic tractography.
This is shown in a clinical application: the reconstruction of the optic
radiation for epilepsy surgery planning.
... Lastly, the bounded variation model could be substituted by an alternative model, which could (possibly) provide a better account for the spatial regularity of HARDI signals. Among alternative regularization models are the diffusion-based method of [59], the LMMSE filtering of [61], the weighted least-square regularization approach of [62], and the recent nonlocal mean denoising of [63]. Exploring the above options constitutes essential part of our ongoing research. ...
Despite the relative recency of its inception, the theory of compressive sampling (aka compressed sensing) (CS) has already revolutionized multiple areas of applied sciences, a particularly important instance of which is medical imaging. Specifically, the theory has provided a different perspective on the important problem of optimal sampling in magnetic resonance imaging (MRI), with an ever-increasing body of works reporting stable and accurate reconstruction of MRI scans from the number of spectral measurements which would have been deemed unacceptably small as recently as five years ago. In this paper, the theory of CS is employed to palliate the problem of long acquisition times, which is known to be a major impediment to the clinical application of high angular resolution diffusion imaging (HARDI). Specifically, we demonstrate that a substantial reduction in data acquisition times is possible through minimization of the number of diffusion encoding gradients required for reliable reconstruction of HARDI scans. The success of such a minimization is primarily due to the availability of spherical ridgelet transformation, which excels in sparsifying HARDI signals. What makes the resulting reconstruction procedure even more accurate is a combination of the sparsity constraints in the diffusion domain with additional constraints imposed on the estimated diffusion field in the spatial domain. Accordingly, the present paper describes an original way to combine the diffusion- and spatial-domain constraints to achieve a maximal reduction in the number of diffusion measurements, while sacrificing little in terms of reconstruction accuracy. Finally, details are provided on an efficient numerical scheme which can be used to solve the aforementioned reconstruction problem by means of standard and readily available estimation tools. The paper is concluded with experimental results which support the practical value of the proposed reconstruction methodology.
... 5 (10)) and FOD-(ODF with spherical deconvolution) based (Fig. 5 (2) and (9)) methods qualitatively give a good match with the ground truth. However, method 10 (Fig. 5 (10)) (Goh et al., 2009) produced a very irregular and tortuous result. This is very likely to be caused by curve averaging as in a former submission competitors returned several fibers per seed which was not compliant with our requirements (a revised submission was then resent). ...
We consider the problem of improving the accuracy and reliability of probabilistic white matter tractography methods by improving the built-in sampling scheme, which randomly draws, from a diffusion model such as the orientation distribution function (ODF), a direction of propagation. Existing methods employing inverse transform sampling require an ad hoc thresholding step to prevent the less likely directions from being sampled. We herein propose to perform importance sampling of spherical harmonics, which redistributes an input point set on the sphere to match the ODF using hierarchical sample warping. This produces a point set that is more concentrated around the modes, allowing the subsequent inverse transform sampling to generate orientations that are in better accordance with the local fiber configuration. Integrated into a Kalman filter-based framework, our approach is evaluated through experiments on synthetic, phantom, and real datasets.
A novel method for estimating a field of fiber orientation distribution (FOD) based on signal de-convolution from a given set of diffusion weighted magnetic resonance (DW-MR) images is presented. We model the FOD by higher order Cartesian tensor basis using a parametrization that explicitly enforces the positive semi-definite property to the computed FOD. The computed Cartesian tensors, dubbed Cartesian Tensor-FOD (CT-FOD), are symmetric positive semi-definite tensors whose coefficients can be efficiently estimated by solving a linear system with non-negative constraints. Next, we show how to use our method for converting higher-order diffusion tensors to CT-FODs, which is an essential task since the maxima of higher-order tensors do not correspond to the underlying fiber orientations. Finally, we propose a diffusion anisotropy index computed directly from CT-FODs using higher order tensor distance measures thus consolidating the whole analysis pipeline of diffusion imaging solely using CT-FODs. We evaluate our method qualitatively and quantitatively using simulated DW-MR images, phantom images, and human brain real dataset. The results conclusively demonstrate the superiority of the proposed technique over several existing multi-fiber reconstruction methods.
High angular resolution diffusion imaging (HARDI) is a magnetic resonance imaging (MRI) technique, determining the diffusion
of water molecules in tissue in vivo. HARDI is advantageous over the well-known diffusion tensor imaging (DTI), since it is
able to extract more than one fiber orientation within a voxel and can therefore resolve crossing, kissing or fanning fiber
tracts. However, multiple orientations per voxel require more sophisticated tractography approaches. In this paper we introduce
a new deterministic fiber tracking method using the complete orientation distribution function (ODF) reconstructed from Q-ball
imaging to enable tractography in challenging regions. Anisotropy classifiers are used to differentiate intra-voxel fiber
populations and adjust a curvature threshold for one and multiple fiber configurations, respectively. In addition, we determine
the most appropriate propagation direction in complex white matter regions, using the course of the current tract. To ensure
tractography running within fiber bundles, a distance-based approach is integrated, which aims to maintain the initial distance
of the seed point to the white matter boundary through the whole tracking. We evaluated our method using a phantom dataset
featuring crossing, kissing and fanning fiber configurations and a human brain dataset, reconstructing the fanning of the
corpus callosum and considering the region of the centrum semiovale.
A key question in diffusion imaging is how many diffusion-weighted images suffice to provide adequate signal-to-noise ratio (SNR) for studies of fiber integrity. Motion, physiological effects, and scan duration all affect the achievable SNR in real brain images, making theoretical studies and simulations only partially useful. We therefore scanned 50 healthy adults with 105-gradient high-angular resolution diffusion imaging (HARDI) at 4T. From gradient image subsets of varying size (6<or=N<or=94) that optimized a spherical angular distribution energy, we created SNR plots (versus gradient numbers) for seven common diffusion anisotropy indices: fractional and relative anisotropy (FA, RA), mean diffusivity (MD), volume ratio (VR), geodesic anisotropy (GA), its hyperbolic tangent (tGA), and generalized fractional anisotropy (GFA). SNR, defined in a region of interest in the corpus callosum, was near-maximal with 58, 66, and 62 gradients for MD, FA, and RA, respectively, and with about 55 gradients for GA and tGA. For VR and GFA, SNR increased rapidly with more gradients. SNR was optimized when the ratio of diffusion-sensitized to non-sensitized images was 9.13 for GA and tGA, 10.57 for FA, 9.17 for RA, and 26 for MD and VR. In orientation density functions modeling the HARDI signal as a continuous mixture of tensors, the diffusion profile reconstruction accuracy rose rapidly with additional gradients. These plots may help in making trade-off decisions when designing diffusion imaging protocols.
Q-ball imaging (QBI) is a high angular resolution diffusion imaging (HARDI) technique which has been proven very successful in resolving multiple intravoxel fiber orientations in MR images. The standard computation of the orientation distribution function (ODF, the probability of diffusion in a given direction) from q-ball uses linear radial projection, neglecting the change in the volume element along the ray, thereby resulting in distributions different from the true ODFs. For instance, they are not normalized or as sharp as expected, and generally require post-processing, such as sharpening or spherical deconvolution. In this paper, we consider the mathematically correct definition of the ODF and derive a closed-form expression for it in QBI. The derived ODF is dimensionless and normalized, and can be efficiently computed from q-ball acquisition protocols. We describe our proposed method and demonstrate its significantly improved performance on artificial data and real HARDI volumes.
We propose an integral concept for tractography to describe crossing and splitting fibre bundles based on the fibre orientation distribution function (ODF) estimated from high angular resolution diffusion imaging (HARDI). We show that in order to perform accurate probabilistic tractography, one needs to use a fibre ODF estimation and not the diffusion ODF. We use a new fibre ODF estimation obtained from a sharpening deconvolution transform (SDT) of the diffusion ODF reconstructed from q-ball imaging (QBI). This SDT provides new insight into the relationship between the HARDI signal, the diffusion ODF, and the fibre ODF. We demonstrate that the SDT agrees with classical spherical deconvolution and improves the angular resolution of QBI. Another important contribution of this paper is the development of new deterministic and new probabilistic tractography algorithms using the full multidirectional information obtained through use of the fibre ODF. An extensive comparison study is performed on human brain datasets comparing our new deterministic and probabilistic tracking algorithms in complex fibre crossing regions. Finally, as an application of our new probabilistic tracking, we quantify the reconstruction of transcallosal fibres intersecting with the corona radiata and the superior longitudinal fasciculus in a group of eight subjects. Most current diffusion tensor imaging (DTI)-based methods neglect these fibres, which might lead to incorrect interpretations of brain functions.
We present new diffusion phantoms dedicated to the study and validation of high-angular-resolution diffusion imaging (HARDI) models. The phantom design permits the application of imaging parameters that are typically employed in studies of the human brain. The phantoms were made of small-diameter acrylic fibers, chosen for their high hydrophobicity and flexibility that ensured good control of the phantom geometry. The polyurethane medium was filled under vacuum with an aqueous solution that was previously degassed, doped with gadolinium-tetraazacyclododecanetetraacetic acid (Gd-DOTA), and treated by ultrasonic waves. Two versions of such phantoms were manufactured and tested. The phantom's applicability was demonstrated on an analytical Q-ball model. Numerical simulations were performed to assess the accuracy of the phantom. The phantom data will be made accessible to the community with the objective of analyzing various HARDI models.
We propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace-Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk-Radon transform which approximates the ODF. We prove a new corollary of the Funk-Hecke theorem to obtain this simplification. Then, we show that the Laplace-Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace-Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method.