No Arabic abstract
We present a novel method for estimation of the fiber orientation distribution (FOD) function based on diffusion-weighted Magnetic Resonance Imaging (D-MRI) data. We formulate the problem of FOD estimation as a regression problem through spherical deconvolution and a sparse representation of the FOD by a spherical needlets basis that form a multi-resolution tight frame for spherical functions. This sparse representation allows us to estimate FOD by an $l_1$-penalized regression under a non-negativity constraint. The resulting convex optimization problem is solved by an alternating direction method of multipliers (ADMM) algorithm. The proposed method leads to a reconstruction of the FODs that is accurate, has low variability and preserves sharp features. Through extensive experiments, we demonstrate the effectiveness and favorable performance of the proposed method compared with two existing methods. Particularly, we show the ability of the proposed method in successfully resolving fiber crossing at small angles and in automatically identifying isotropic diffusion. We also apply the proposed method to real 3T D-MRI data sets of healthy elderly individuals. The results show realistic descriptions of crossing fibers that are more accurate and less noisy than competing methods even with a relatively small number of gradient directions.
Due to recent technological advances, large brain imaging data sets can now be collected. Such data are highly complex so extraction of meaningful information from them remains challenging. Thus, there is an urgent need for statistical procedures that are computationally scalable and can provide accurate estimates that capture the neuronal structures and their functionalities. We propose a fast method for estimating the fiber orientation distribution(FOD) based on diffusion MRI data. This method models the observed dMRI signal at any voxel as a convolved and noisy version of the underlying FOD, and utilizes the spherical harmonics basis for representing the FOD, where the spherical harmonic coefficients are adaptively and nonlinearly shrunk by using a James-Stein type estimator. To further improve the estimation accuracy by enhancing the localized peaks of the FOD, as a second step a super-resolution sharpening process is then applied. The resulting estimated FODs can be fed to a fiber tracking algorithm to reconstruct the white matter fiber tracts. We illustrate the overall methodology using both synthetic data and data from the Human Connectome Project.
Diffusion tractography is routinely used to study white matter architecture and brain connectivity in vivo. A key step for successful tractography of neuronal tracts is the correct identification of tract directions in each voxel. Here we propose a fingerprinting-based methodology to identify these fiber directions in Orientation Distribution Functions, dubbed ODF-Fingerprinting (ODF-FP). In ODF-FP, fiber configurations are selected based on the similarity between measured ODFs and elements in a pre-computed library. In noisy ODFs, the library matching algorithm penalizes the more complex fiber configurations. ODF simulations and analysis of bootstrapped partial and whole-brain in vivo datasets show that the ODF-FP approach improves the detection of fiber pairs with small crossing angles while maintaining fiber direction precision, which leads to better tractography results. Rather than focusing on the ODF maxima, the ODF-FP approach uses the whole ODF shape to infer fiber directions to improve the detection of fiber bundles with small crossing angle. The resulting fiber directions aid tractography algorithms in accurately displaying neuronal tracts and calculating brain connectivity.
We discuss Spherical Needlets and their properties. Needlets are a form of spherical wavelets which do not rely on any kind of tangent plane approximation and enjoy good localization properties in both pixel and harmonic space; moreover needlets coefficients are asymptotically uncorrelated at any fixed angular distance, which makes their use in statistical procedures very promising. In view of these properties, we believe needlets may turn out to be especially useful in the analysis of Cosmic Microwave Background (CMB) data on the incomplete sky, as well as of other cosmological observations. As a final advantage, we stress that the implementation of needlets is computationally very convenient and may rely completely on standard data analysis packages such as HEALPix.
Accurate lighting estimation is challenging yet critical to many computer vision and computer graphics tasks such as high-dynamic-range (HDR) relighting. Existing approaches model lighting in either frequency domain or spatial domain which is insufficient to represent the complex lighting conditions in scenes and tends to produce inaccurate estimation. This paper presents NeedleLight, a new lighting estimation model that represents illumination with needlets and allows lighting estimation in both frequency domain and spatial domain jointly. An optimal thresholding function is designed to achieve sparse needlets which trims redundant lighting parameters and demonstrates superior localization properties for illumination representation. In addition, a novel spherical transport loss is designed based on optimal transport theory which guides to regress lighting representation parameters with consideration of the spatial information. Furthermore, we propose a new metric that is concise yet effective by directly evaluating the estimated illumination maps rather than rendered images. Extensive experiments show that NeedleLight achieves superior lighting estimation consistently across multiple evaluation metrics as compared with state-of-the-art methods.
This article describes mathematical methods for estimating the top-tail of the wealth distribution and therefrom the share of total wealth that the richest $p$ percent hold, which is an intuitive measure of inequality. As the data base for the top-tail of the wealth distribution is inevitably less complete than the data for lower wealth, the top-tail distribution is replaced by a parametric model based on a Pareto distribution. The different methods for estimating the parameters are compared and new simulations are presented which favor the maximum-likelihood estimator for the Pareto parameter $alpha$. New criteria for the choice of other parameters are presented which have not yet been discussed in the literature before. The methods are applied to the 2012 data from the ECB Household and Consumption Survey (HFCS) for Germany and the corresponding rich list from the Manager Magazin. In addition to a presentation of all formulas, R scripts implementing them are provided by the author.