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首页> 外文会议>IEEE Conference on Computer Vision and Pattern Recognition >Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images
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Multivariate General Linear Models (MGLM) on Riemannian Manifolds with Applications to Statistical Analysis of Diffusion Weighted Images

机译：黎曼流形上的多元通用线性模型（MGLM）及其在扩散加权图像统计分析中的应用

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摘要

Linear regression is a parametric model which is ubiquitous in scientific analysis. The classical setup where the observations and responses, i.e., (xi, yi) pairs, are Euclidean is well studied. The setting where yi is manifold valued is a topic of much interest, motivated by applications in shape analysis, topic modeling, and medical imaging. Recent work gives strategies for max-margin classifiers, principal components analysis, and dictionary learning on certain types of manifolds. For parametric regression specifically, results within the last year provide mechanisms to regress one real-valued parameter, xi ε R, against a manifold-valued variable, yi ε M. We seek to substantially extend the operating range of such methods by deriving schemes for multivariate multiple linear regression -- a manifold-valued dependent variable against multiple independent variables, i.e., f: Rn → M. Our variational algorithm efficiently solves for multiple geodesic bases on the manifold concurrently via gradient updates. This allows us to answer questions such as: what is the relationship of the measurement at voxel y to disease when conditioned on age and gender. We show applications to statistical analysis of diffusion weighted images, which give rise to regression tasks on the manifold GL(n)/O(n) for diffusion tensor images (DTI) and the Hilbert unit sphere for orientation distribution functions (ODF) from high angular resolution acquisition. The companion open-source code is available on nitrc.org/projects/riem_mglm.
机译：线性回归是在科学分析中普遍存在的参数模型。对观察和响应（即（xi，yi）对）为欧几里得的经典设置进行了很好的研究。 yi在多个方面的设置是一个非常受关注的主题，这是由形状分析，主题建模和医学成像中的应用所激发。最近的工作提供了针对最大类型的分类器，主成分分析和某些类型的流形词典学习的策略。具体来说，对于参数回归，去年的结果提供了一种回归一个实数值参数xi＆epsi;的机制。 R，针对流形值变量yi＆epsi; M.我们试图通过推导多元多元线性回归的方案（一种针对多个自变量的流形值因变量，即f：Rn＆rarr;）来扩展这种方法的操作范围。 M.我们的变分算法通过梯度更新有效地同时解决了流形上的多个测地线。这使我们能够回答以下问题：以年龄和性别为条件的体素y测量值与疾病之间的关系是什么。我们展示了在扩散加权图像的统计分析中的应用，这引起了扩散张量图像（DTI）的流形GL（n）/ O（n）和方向分布函数（ODF）的希尔伯特单位球面从高到高的回归任务。角分辨率采集。随附的开放源代码可在nitroc.org/projects/riem_mglm上找到。

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