Abstract
In this work, we wish to denoise HARDI (High Angular Resolution Diffusion Imaging) data arising in medical brain imaging. Diffusion imaging is a relatively new and powerful method to measure the three-dimensional profile of water diffusion at each point in the brain. These images can be used to reconstruct fiber directions and pathways in the living brain, providing detailed maps of fiber integrity and connectivity. HARDI data is a powerful new extension of diffusion imaging, which goes beyond the diffusion tensor imaging (DTI) model: mathematically, intensity data is given at every voxel and at any direction on the sphere. Unfortunately, HARDI data is usually highly contaminated with noise, depending on the b-value which is a tuning parameter pre-selected to collect the data. Larger b-values help to collect more accurate information in terms of measuring diffusivity, but more noise is generated by many factors as well. So large b-values are preferred, if we can satisfactorily reduce the noise without losing the data structure. Here we propose two variational methods to denoise HARDI data. The first one directly denoises the collected data S, while the second one denoises the so-called sADC (spherical Apparent Diffusion Coefficient), a field of radial functions derived from the data. These two quantities are related by an equation of the form S = S(S)exp (-b · sADC) (in the noise-free case). By applying these two different models, we will be able to determine which quantity will most accurately preserve data structure after denoising. The theoretical analysis of the proposed models is presented, together with experimental results and comparisons for denoising synthetic and real HARDI data.
Highlights
Introduction to the HARDI dataCurrently, HARDI data is used to map cerebral connectivity through fiber tractography in the brain
HARDI is a type of diffusion MRI, which was introduced in the mid-1980s by Le Bihan et al [27,28,29] and Merboldt et al [35]
It is based on the idea that the magnetic resonance signal, which forms the basis of MRI, is attenuated when water diffuses out of a voxel, and the degree of attenuation can be used to measure the rate of water diffusion in any arbitrary threedimensional direction, via the Stejskal-Tanner equation [42]
Summary
HARDI data is used to map cerebral connectivity through fiber tractography in the brain. Each voxel's signal intensity in the k-th image is decreased, by water diffusion, according to the Stejskal-Tanner equation [42]: where S0 is the non-diffusion weighted signal intensity, D is the 3×3 diffusion tensor, gk is the direction of the diffusion gradient and b is Le Bihan's factor with information on the pulse sequence, gradient strength, and physical constants It is widely used, the diffusion tensor model breaks down for voxels in which fiber pathways cross or mix together, and these are ubiquitous in the brain which is highly interconnected. A very active area of research has grown up in processing the HARDI signals, leading to methods for HARDI denoising, segmentation, and registration using metrics on spherical functions (Lenglet et al [30]) Most of these signal processing methods still model the diffusion signal as a tensor, rather than exploiting the full information in the spherical harmonic expansion. We would like to mention that a preliminary version of this work has been presented and published in IPMI 2009 [26]
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