Deconvolution of High Dimensional Mixtures via Boosting, with Application to Diffusion-Weighted MRI of Human Brain
Diffusion-weighted magnetic resonance imaging (DWI) and fiber tractography are the only methods to measure the structure of the white matter in the living human brain. The diffusion signal has been modelled as the combined contribution from many individual fascicles of nerve fibers passing through each location in the white matter. Typically, this is done via basis pursuit, but estimation of the exact directions is limited due to discretization [1,2]. The difficulties inherent in modeling DWI data are shared by many other problems involving fitting non-parametric mixture models. Ekanadaham et al. [3]  proposed an approach, continuous basis pursuit, to overcome discretization error in the 1-dimensional case (e.g., spikesorting). Here, we propose a more general algorithm that fits mixture models of any dimensionality without discretization. Our algorithm uses the principles of L2-boost [4], together with refitting of the weights and pruning of the parameters. The addition of these steps to L2-boost both accelerates the algorithm and assures its accuracy. We refer to the resulting algorithm as elastic basis pursuit, or EBP, since it expands and contracts the active set of kernels as needed. We show that in contrast to existing approaches to fitting mixtures, our boosting framework (1) enables the selection of the optimal bias-variance tradeoff along the solution path, and (2) scales with high-dimensional problems. In simulations of DWI, we find that EBP yields better parameter estimates than a non-negative least squares (NNLS) approach, or the standard model used in DWI, the tensor model, which serves as the basis for diffusion tensor imaging (DTI) [5]. We demonstrate the utility of the method in DWI data acquired in parts of the brain containing crossings of multiple fascicles of nerve fibers.
This paper proposes elastic basis pursuit, which is an algorithm for fitting sparse mixture models via convex optimisation.  The method combines boosting with an active set method for continuous basis pursuit.  The authors provide proofs of convergence of the algorithm and test it experimentally within the application of estimating fibre directions from DWI data.The algorithm is well motivated, and the authors clearly understand well the wider space of solutions to this family of problem and their pros and cons. It is also well constructed and combines the benefits of different existing approaches retaining computational efficiency, convexity, and accuracy of solution. I am not intimately familiar with the latest literature on this problem, but as far as I am aware, the algorithm is novel and represents a fairly minor but not insignificant advance over the current state of the art.  I have not checked the details of the proofs of convergence, but, assuming they are correct, their presence adds to the value of the paper.The chosen application to demonstrate the technique is timely and appropriate, as diffusion MRI is used widely for estimating microstructural features of brain tissue, including fibre orientations, which then leads to algorithms that reconstruct global brain connectivity and are widely used in neuroscience and neurology. Resolving the orientations of crossing fibres is a long standing problem, which saw a lot of attention from 2000-2010, and still has outstanding issues so improved solutions have significant value.  However, the experimental results presented here are very limited.  They show one simulated example, for which the proposed algorithm will clearly perform better than the others tested, as well as a largely qualitative result comparing fitting error maps over a small brain region.  The simulation results would be much stronger if they included a quantitative evaluation of how much better and how often the new algorithm does in comparison to NNLS and other algorithms used commonly in the diffusion MRI literature.  The brain-data results show no obvious difference between the NNLS and EBP output, although presumably there are significant differences. Maps of number of fibre directions or subsequent tractography would add a great deal to convince the reader that the new fitting algorithm really makes a difference worth thinking about. Even a simple test of the significance of the difference in error scores between NNLS and EBP would add something.Minor points:page 6.  The diffusion tensor in 3D is rank 3 in general.page 7. The form of the oracle function is not clear.page 7.  Should w1, w2, w3 sum to 1?  Or does the value of c ensure the sum of compartment fractions sums to 1? Either way, where is the constraint imposed?  Also, it is unclear how c represents an isotropic diffusion component.page 8. Figures 4 and 5 not referred to from text; fig 4 erroneously referred to as fig 3, but fig 5 never mentioned.