A Fast and Robust ODF Estimation Algorithm in Q-Ball Imaging

Descoteaux, Maxime; Angelino, Elaine; Fitzgibbons, Shaun; Deriche, Rachid

doi:10.1109/isbi.2006.1624857

Cited by 25 publications

(48 citation statements)

References 9 publications

(14 reference statements)

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“…and normally distributed, the maximum likelihood estimate of w naturally leads to the L 2 norm as a measure of goodness of the fit. Without inequality constraints, the corresponding quadratic programming (QP) problem minimizing the residual sum of squares (9) can be efficiently solved by solving a linear system using for instance, direct methods when the size of the linear system in Eq. (9) is not large as in our application.…”

Section: Stable Sparse and Positive Deconvolutionmentioning

confidence: 99%

“…The q-ball imaging (QBI) method approximates the radial integral of the displacement probability distribution function (PDF) by the spherical Funk-Radon transform [6]. More recent studies have expressed QBI's Funk-Radon transform in a spherical harmonic basis [7][8][9]. Diffusion spectrum imaging (DSI) can measure the microscopic diffusion function directly based on the Fourier relation between the diffusion signal and the diffusion function, but is limited by the time-intensive q-space sampling burden [10].…”

Section: Introductionmentioning

confidence: 99%

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Multi-fiber Reconstruction from Diffusion MRI Using Mixture of Wisharts and Sparse Deconvolution

Jian

Vemuri

2007

Lecture Notes in Computer Science

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In this paper, we present a novel continuous mixture of diffusion tensors model for the diffusionweighted MR signal attenuation. The relationship between the mixing distribution and the MR signal attenuation is shown to be given by the Laplace transform defined on the space of positive definite diffusion tensors. The mixing distribution when parameterized by a mixture of Wishart distributions (MOW) is shown to possess a closed form expression for its Laplace transform, called the Rigauttype function, which provides an alternative to the Stejskal-Tanner model for the MR signal decay. Our model naturally leads to a deconvolution formulation for multi-fiber reconstruction. This deconvolution formulation requires the solution to an ill-conditioned linear system. We present several deconvolution methods and show that the nonnegative least squares method outperforms all others in achieving accurate and sparse solutions in the presence of noise. The performance of our multi-fiber reconstruction method using the MOW model is demonstrated on both synthetic and real data along with comparisons with state-of-the-art techniques.

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Section: Stable Sparse and Positive Deconvolutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Multi-fiber Reconstruction from Diffusion MRI Using Mixture of Wisharts and Sparse Deconvolution

Jian

Vemuri

2007

Lecture Notes in Computer Science

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“…Among these, Tuch et al (2003) proposed the so called q-ball imaging (QBI) method, in which the radial integral of the displacement PDF is approximated by the spherical Funk-Radon transform (Tuch, 2004). Recent studies have expressed QBI in a spherical harmonic basis (Anderson, 2005;Hess et al, 2006;Descoteaux et al, 2006). Another reconstruction algorithm referred to as persistent angular structure (PAS) MRI was proposed by Jansons and Alexander (2003).…”

Section: Introductionmentioning

confidence: 99%

A novel tensor distribution model for the diffusion-weighted MR signal

et al. 2007

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Diffusion MRI is a non-invasive imaging technique that allows the measurement of water molecule diffusion through tissue in vivo. The directional features of water diffusion allow one to infer the connectivity patterns prevalent in tissue and possibly track changes in this connectivity over time for various clinical applications. In this paper, we present a novel statistical model for diffusion-weighted MR signal attenuation which postulates that the water molecule diffusion can be characterized by a continuous mixture of diffusion tensors. An interesting observation is that this continuous mixture and the MR signal attenuation are related through the Laplace transform of a probability distribution over symmetric positive definite matrices. We then show that when the mixing distribution is a Wishart distribution, the resulting closed form of the Laplace transform leads to a Rigaut-type asymptotic fractal expression, which has been phenomenologically used in the past to explain the MR signal decay but never with a rigorous mathematical justification until now. Our model not only includes the traditional diffusion tensor model as a special instance in the limiting case, but also can be adjusted to describe complex tissue structure involving multiple fiber populations. Using this new model in conjunction with a spherical deconvolution approach, we present an efficient scheme for estimating the water molecule displacement probability functions on a voxel-by-voxel basis. Experimental results on both simulations and real data are presented to demonstrate the robustness and accuracy of the proposed algorithms.

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“…[6]. The integral is evaluated as a discrete sum (Simpson numerical integration) over the available range ʦ [0,100].The approach used to compute QBI ODFs is similar to those previously described (8,29,30). The ODFs were computed through Eq.…”

mentioning

confidence: 99%

“…Energy of the truncated plane wave, defined as the square of the magnitude of the wave computed using Eq. [29]. The direction of r was assumed along the z-axis.…”

mentioning

confidence: 99%

Mathematical description of q‐space in spherical coordinates: Exact q‐ball imaging

Canales-Rodríguez

Melie‐García

Iturria-Medina

2009

Magnetic Resonance in Med

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Novel methodologies have been recently developed to characterize the microgeometry of neural tissues and porous structures via diffusion MRI data. In line with these previous works, this article provides a detailed mathematical description of q-space in spherical coordinates that helps to highlight the differences and similarities between various related q-space methodologies proposed to date such as q-ball imaging (QBI), diffusion spectrum imaging (DSI), and diffusion orientation transform imaging (DOT). This formulation provides a direct relationship between the orientation distribution function (ODF) and the diffusion data without using any approximation. Under this relationship, the exact ODF can be computed by means of the Radon transform of the radial projection (in q-space) of the diffusion MRI signal. This new methodology, termed exact q-ball imaging (EQBI), was put into practice using an analytical ODF estimation in terms of spherical harmonics that allows obtaining model-free and model-based reconstructions. This work provides a new framework for combining information coming from diffusion data recorded on multiple spherical shells in q-space (hybrid diffusion imaging encoding scheme), which is capable of mapping ODF to a high accuracy. This represents a step toward a more efficient development of diffusion MRI experiments for obtaining better ODF estimates. Diffusion MRI (dMRI) data are routinely used for studying molecular diffusion in fluids (1). When the diffusion is constrained by the presence of obstacles, dMRI experiments yield information about the confining geometry (2). Recently, there has been a large amount of interest in using dMRI techniques to obtain information about the microgeometry of porous structures and biological cells, which is supported by the noninvasive nature of these measurements. Also, dMRI techniques are able to probe the pore geometry over a range of length scales not accessible to standard techniques like x-ray and small-angle neutron scattering.Diffusion tensor imaging (DTI) is the first dMRI technique that has been proposed to quantify in vivo the anisotropy of the water self-diffusion process in fibrous tissues (3). This approach allows estimating a second-order self-diffusion tensor D from a series of dMRI data without requiring exogenous contrast agents. DTI provides relevant information that is not available from other imaging modalities; however, this technique can only resolve a single fiber direction within each voxel.In the last few years a number of novel methodologies have been developed to characterize more accurately the microgeometry of neural tissues (for a partial list of such publications, see . Among these advanced methods and algorithms, q-ball imaging (QBI) (6) and diffusion spectrum imaging (DSI) (11) have generated considerable interest due to a number of benefits such as model independence, theoretical soundness, and the ability to resolve intravoxel orientational heterogeneity.In DSI the molecular displacement probability density function (also ...

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A Fast and Robust ODF Estimation Algorithm in Q-Ball Imaging

Cited by 25 publications

References 9 publications

Multi-fiber Reconstruction from Diffusion MRI Using Mixture of Wisharts and Sparse Deconvolution

Multi-fiber Reconstruction from Diffusion MRI Using Mixture of Wisharts and Sparse Deconvolution

A novel tensor distribution model for the diffusion-weighted MR signal

Mathematical description of q‐space in spherical coordinates: Exact q‐ball imaging

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