Processing Data in Lie Groups : An Algebraic Approach. Application to Non-Linear Registration and Diffusion Tensor MRI. (Traitement de données dans les groupes de Lie : une approche algébrique. Application au recalage non-linéaire et à l'imagerie du tenseur de diffusion)
Processing Data in Lie Groups : An Algebraic Approach. Application to Non-Linear Registration and Diffusion Tensor MRI. (Traitement de données dans les groupes de Lie : une approche algébrique. Application au recalage non-linéaire et à l'imagerie du tenseur de diffusion)
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发表时间：
2006-11
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通讯作者：
Vincent Arsigny
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作者：
Vincent Arsigny
最近，在医学成像中，对非线性数据处理的严格框架的需求已经大大增加。在本文中，我们提出了几种通用框架来处理各种类型的非线性数据，这些数据都属于李群。为此，我们依靠这些空间的代数性质。因此，我们提出了对称正定矩阵的一般处理框架，称为对数-欧几里得，非常简单易用，具有良好的理论性质。它特别适合于MRI扩散张量的处理。我们还提出了几个框架，称为多仿射，以一种保证其可逆性的方式参数化局部刚性或仿射变换。它们的使用说明了在局部刚性注册的组织学切片和局部仿射三维注册的mri的人类大脑的情况下。这导致我们提出了在有限维李群中计算统计的两种一般框架：第一个是对数欧几里得框架，它推广了我们在张量上的工作，第二个是基于双不变均值的新概念的框架，它的性质推广到李群的算术均值。最后，我们将对数-欧几里得框架推广到微分同构几何变换，这为计算解剖学中统计学的一般和一致框架开辟了道路。
Recently, the need for rigorous frameworks for the processing of non-linear data has grown considerably in medical imaging. In this thesis, we propose several general frameworks to process various types of non-linear data, which all belong to Lie groups. To this end, we rely on the algebraic properties of these spaces. Thus, we propose a general processing framework for symmetric and positive-definite matrices, named Log-Euclidean, very simple to use and which has excellent theoretical properties. It is particularly well-adapted to the processing of diffusion tensor MRI. We also propose several frameworks, called polyaffine, to parameterize locally rigid or affine transformations, in a way that guarantees their invertibility. Their use is illustrated in the case of the locally rigid registration of histological slices and of the locally affine 3D registration of MRIs of the human brain. This led us to propose two general frameworks for computing statistics in finite-dimensional Lie groups: first the Log-Euclidean one, which generalizes our work on tensors, and second a framework based on the novel notion of bi-invariant mean, whose properties generalize to Lie groups those of the arithmetic mean. Finally, we generalize our Log-Euclidean framework to diffeomorphic geometrical transformations, which opens the way to a general and consistent framework for statistics in computational anatomy.