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

Abstract

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.

Image segmentation via convolution of a level-set function with a Rigaut Kernel

“…Let f(K) be its density function with respect to some carrier measure dK on n . (This model has been presented in the context of the diffusion weighted MR signal attenuation by Jian and Vemuri in [3] and later used in the context of image smoothing by Subakan et al in [22].) We propose to model the orientation distribution by a continuous mixture of Gaussian functions: (2) where ξ encodes the coordinates, G(ξ; g) is the response of the Gabor filter with an orientation determined by g a unit direction vector, G 0 denotes the maximal filter response.…”

“…(2) is a continuous mixture of Gaussian functions with f(K) being a mixing density. This integral can be recognized as the Laplace transform (matrix variable case) of [16]: (3) where f denotes the Laplace transform of a function f which takes its argument as symmetric positive definite matrices from n , and B = gg T . In this expression, we are faced with the problem of recovering a distribution defined on n that best explains the observed orientation data G(ξ g).…”

“…and normally distributed. Since the maximization of the likelihood function under a conditional Gaussian noise distribution for a linear model is equivalent to minimizing a sum-of-squares error function, we use a non-negative least squares (NNLS) minimization which achieves an accurate and sparse solution for (6) Jian and Vemuri have shown that this de-convolution method outperforms many other methods in achieving accuracy and sparsity [4,3]. After w is estimated for orientations, we locally convolve the signed distance function with the spatially varying Rigaut kernel formulated as (7) dF = f(K)dK is the mixing density described above, i.e.…”

Extracting Tractosemas from a Displacement Probability Field for Tractography in DW-MRI

“…Some of the models that have been proposed in literature include discrete [3] and continuous [4] mixture of Gaussians, higher-order tensors [5], and the spherical harmonic transformation [6]. After reconstruction of the signal, one has to compute its Fourier transform in order to obtain the displacement probability whose peaks correspond to distinct fiber orientations.…”

“…Multiple fiber orientations can also be estimated by reconstructing the orientation distribution function (ODF) [8] using the so called Q-ball imaging [9]. Most of the above techniques ( [1,3,8,4]) can be expressed as a special case of a more generalized method in which the DW-MR signal can be expressed as the convolution over the sphere of a fiber bundle response function with the ODF [10,11]. In this spherical deconvolution approach there is no limitation regarding the number of the distinct fiber populations in the estimated ODF.…”