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.

中文翻译：

---

使用矢量总变异和对数势垒的 Hardi 数据去噪。

在这项工作中，我们希望对医学脑成像中出现的 HARDI（高角分辨率扩散成像）数据进行去噪。扩散成像是一种相对较新且功能强大的方法，用于测量大脑中每个点的水扩散的三维剖面。这些图像可用于重建活体大脑中的纤维方向和通路，提供纤维完整性和连接性的详细地图。HARDI 数据是扩散成像的强大新扩展，它超越了扩散张量成像 (DTI) 模型：在数学上，强度数据在球体上的每个体素和任何方向上给出。不幸的是，HARDI 数据通常受到噪声的严重污染，这取决于 b 值，该值是为收集数据而预先选择的调整参数。较大的 b 值有助于在测量扩散率方面收集更准确的信息，但许多因素也会产生更多噪声。如果我们可以在不丢失数据结构的情况下令人满意地降低噪声，那么大 b 值是首选。在这里，我们提出了两种对 HARDI 数据去噪的变分方法。第一个直接对收集的数据 S 进行去噪，而第二个对所谓的 sADC（球面表观扩散系数）进行去噪，这是一个从数据导出的径向函数场。这两个量通过 S = S(S)exp (-b · sADC) 形式的方程相关（在无噪声情况下）。通过应用这两种不同的模型，我们将能够确定去噪后哪个数量最能准确地保留数据结构。提出了所提出模型的理论分析，