We present a spherical harmonics-based convolutional neural network (CNN) for cortical parcellation, which we call SPHARM-Net. Recent advances in CNNs offer cortical parcellation on a fine-grained triangle mesh of the cortex. Yet, most CNNs designed for cortical parcellation employ spatial convolution that depends on extensive data augmentation and allows only predefined neighborhoods of specific spherical tessellation. On the other hand, a rotation-equivariant convolutional filter avoids data augmentation, and rotational equivariance can be achieved in spectral convolution independent of a neighborhood definition. Nevertheless, the limited resources of a modern machine enable only a finite set of spectral components that might lose geometric details. In this paper, we propose (1) a constrained spherical convolutional filter that supports an infinite set of spectral components and (2) an end-to-end framework without data augmentation. The proposed filter encodes all the spectral components without the full expansion of spherical harmonics. We show that rotational equivariance drastically reduces the training time while achieving accurate cortical parcellation. Furthermore, the proposed convolution is fully composed of matrix transformations, which offers efficient and fast spectral processing. In the experiments, we validate SPHARM-Net on two public datasets with manual labels: Mindboggle-101 (N=101) and NAMIC (N=39). The experimental results show that the proposed method outperforms the state-of-the-art methods on both datasets even with fewer learnable parameters without rigid alignment and data augmentation. Our code is publicly available at https://github.com/Shape-Lab/SPHARM-Net .

中文翻译：

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SPHARM-Net：用于皮质分割的基于球面谐波的卷积

我们提出了一种用于皮质分割的基于球谐函数的卷积神经网络 (CNN)，我们称之为 SPHARM-Net。CNN 的最新进展在皮质的细粒度三角形网格上提供了皮质分割。然而，大多数为皮质分割设计的 CNN 都采用空间卷积，这依赖于广泛的数据增强，并且只允许特定球形镶嵌的预定义邻域。另一方面，旋转等变卷积滤波器避免了数据增强，并且可以在独立于邻域定义的谱卷积中实现旋转等变。然而，现代机器的有限资源只能启用可能丢失几何细节的有限光谱分量集。在本文中，我们提出（1）一个支持无限组光谱分量的约束球形卷积滤波器和（2）一个没有数据增强的端到端框架。所提出的滤波器对所有光谱分量进行编码，而无需完全扩展球谐函数。我们表明，旋转等效性极大地减少了训练时间，同时实现了准确的皮质分割。此外，所提出的卷积完全由矩阵变换组成，提供了高效和快速的频谱处理。在实验中，我们在两个带有手动标签的公共数据集上验证 SPHARM-Net：Mindboggle-101 (N=101) 和 NAMIC (N=39)。实验结果表明，所提出的方法在两个数据集上都优于最先进的方法，即使在没有严格对齐和数据增强的情况下使用较少的可学习参数也是如此。我们的代码可在以下位置公开获得https://github.com/Shape-Lab/SPHARM-Net .