We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured and geometric input, e.g., graphs or meshes. Our main contribution is a novel convolution operator based on B-splines, that makes the computation time independent from the kernel size due to the local support property of the B-spline basis functions. As a result, we obtain a generalization of the traditional CNN convolution operator by using continuous kernel functions parametrized by a fixed number of trainable weights. In contrast to related approaches that filter in the spectral domain, the proposed method aggregates features purely in the spatial domain. In addition, SplineCNN allows entire end-to-end training of deep architectures, using only the geometric structure as input, instead of handcrafted feature descriptors. For validation, we apply our method on tasks from the fields of image graph classification, shape correspondence and graph node classification, and show that it outperforms or pars state-of-the-art approaches while being significantly faster and having favorable properties like domainindependence. Our source code is available on GitHub 1 . Recently, a set of methods brought together under the term geometric deep learning [3] emerged, which aim to achieve this transfer by defining convolution operations for deep neural networks that can handle irregular input data. Existing work in this field can loosely be divided into two different subsets: the spectral and the spatial filtering approaches. The former is based on spectral graph theory [5] , where eigenvalues of a graph's Laplacian matrix are interpreted as frequencies of node signals [22] . They are filtered in the spectral domain, analogously to Fourier domain filtering of traditional signals. The latter subset, the spatial approaches, perform convolution in local Euclidean neighborhoods w.r.t. local positional relations between points, represented for example as polar, spherical or Cartesian coordinates, as shown as examples in Figure 1 . Contribution. We present Spline-based Convolutional Neural Networks (SplineCNNs), a variant of deep neural networks for irregular structured data. The main contribution is a trainable, spatial, continuous convolution kernel that leverages properties of B-spline bases to efficiently filter geometric input of arbitrary dimensionality. We show
To process graph inputs, Spline-CNN REF extends convolution by using continuous B-spline basis, which is parametrized by a constant number of trainable control values.
SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels
2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition
2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition
Computer Science