Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator
Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator
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DOI：
10.1109/tmi.2006.882143
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发表时间：
2006-10-01
影响因子：
10.6
通讯作者：
Miller, Michael I.
中科院分区：
文献类型：
--
作者：
Qiu, Anqi;Bitouk, Dmitri;Miller, Michael I.
在神经精神病学研究中，功能和结构图，如曲率、皮质厚度和功能磁共振成像(MRI)图，在皮质流形的局部坐标上索引起着重要的作用。由于大脑皮层高度复杂的性质和图像质量，如果没有适当的关联和平滑到局部坐标系的方法，这些函数通常是无法解释的。本文将(Wahba，1990)的样条光滑问题从球面推广到任意带边界的二维流形。首先求出二维流形AA上具有Neumann边界条件的Laplace-Beltrami(LB)算子的正交基函数的数值解，然后在再生核Hilbert空间(r.k.h.s)中求解样条光顺问题.由基函数构造核的流形M上的实值函数。通过直接在流形坐标上计算的有限元方法，得到了离散的LB表示，使得求解离散的LB正交基函数等价于求解一个代数特征值问题。然后，r.k.h.s中的平滑函数可以表示为基函数的线性组合。我们证明了单位球面上的球谐函数和时间流形上的脑正交基函数的数值解。然后利用合成数据来量化光滑度与地面真实情况的优度，并讨论了在光滑化中应该包含多少基函数。我们介绍了我们的方法在新大脑皮层的子流形上平滑沟平均曲率、皮质厚度和功能统计地图的应用。
Functional and structural maps, such as a curvature, cortical thickness, and functional magnetic resonance imaging (MRI) maps, indexed over the local coordinates of the cortical manifold play an important role in neuropsychiatric studies. Due to the highly convoluted nature of the cerebral cortex and image quality, these functions are generally uninterpretable without proper methods of association and smoothness onto the local coordinate system. In this paper, we generalized the spline smoothing problem (Wahba, 1990) from a sphere to any arbitrary two-dimensional (2-D) manifold with boundaries. We first seek a numerical solution to orthonormal basis functions of the Laplace-Beltrami (LB) operator with Neumann boundary conditions for a 2-D manifold AA then solve the spline smoothing problem in a reproducing kernel Hilbert space (r.k.h.s.) of real-valued functions on manifold M with kernel constructed from the basis functions. The explicit discrete LB representation is derived using the finite element method calculated directly on the manifold coordinates so that finding discrete LB orthonormal basis functions is equivalent to solving an algebraic eigenvalue problem. And then smoothed functions in r.k.h.s can be represented as a linear combination of the basis functions. We demonstrate numerical solutions of spherical harmonics on a unit sphere and brain orthonormal basis functions on a planum temporale manifold. Then synthetic data is used to quantify the goodness of the smoothness compared with the ground truth and discuss how many basis functions should be incorporated in the smoothing. We present applications of our approach to smoothing sulcal mean curvature, cortical thickness, and functional statistical maps on submanifolds of the neocortex.