Visualizing Shape Deformations with Variation of Geometric Spectrum
Visualizing Shape Deformations with Variation of Geometric Spectrum
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DOI：
10.1109/tvcg.2016.2598790
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发表时间：
2017
影响因子：
5.2
通讯作者：
Jiaxi Hu;Hajar Hamidian;Z. Zhong;Jing Hua
中科院分区：
文献类型：
--
作者：
Jiaxi Hu;Hajar Hamidian;Z. Zhong;Jing Hua
本文提出了一种基于谱几何的新颖方法，通过映射两个流形来量化和可视化 3D 表面的非等距变形。该方法可以通过两种形状的Laplace-Beltrami谱的变化来确定多尺度、非等距变形。给定两个三角形网格，光谱可以通过在每个顶点上定义的比例函数而从一个到另一个变化。该变化表示为两个形状的特征值的线性插值。在每个迭代步骤中，基于我们导出的频谱变化定理和平滑能量约束构建二次规划问题来计算频谱变化。尺度函数的推导就是此类问题的解决。因此，最终的尺度函数可以通过对每一步的推导进行积分来求解，进而定量地描述两个形状之间的非等距变形。为了评估该方法，我们对合成数据和真实数据进行了广泛的实验。我们采用真实的癫痫患者成像数据来量化癫痫大脑中左右海马体的形状变化。此外，我们使用纵向阿尔茨海默病数据来比较患病和健康海马体的形状变形。为了证明所提出方法的准确性和有效性，我们还将其与基于空间配准的方法进行了比较，例如非刚性迭代最近点（ICP）和基于体素的方法。这些实验证明了我们方法的优点。
This paper presents a novel approach based on spectral geometry to quantify and visualize non-isometric deformations of 3D surfaces by mapping two manifolds. The proposed method can determine multi-scale, non-isometric deformations through the variation of Laplace-Beltrami spectrum of two shapes. Given two triangle meshes, the spectra can be varied from one to another with a scale function defined on each vertex. The variation is expressed as a linear interpolation of eigenvalues of the two shapes. In each iteration step, a quadratic programming problem is constructed, based on our derived spectrum variation theorem and smoothness energy constraint, to compute the spectrum variation. The derivation of the scale function is the solution of such a problem. Therefore, the final scale function can be solved by integral of the derivation from each step, which, in turn, quantitatively describes non-isometric deformations between two shapes. To evaluate the method, we conduct extensive experiments on synthetic and real data. We employ real epilepsy patient imaging data to quantify the shape variation between the left and right hippocampi in epileptic brains. In addition, we use longitudinal Alzheimer data to compare the shape deformation of diseased and healthy hippocampus. In order to show the accuracy and effectiveness of the proposed method, we also compare it with spatial registration-based methods, e.g., non-rigid Iterative Closest Point (ICP) and voxel-based method. These experiments demonstrate the advantages of our method.