The ground-breaking performance obtained by deep convolutional neural networks (CNNs) for image processing tasks is inspiring research efforts attempting to extend it for 3D geometric tasks. One of the main challenge in applying CNNs to 3D shape analysis is how to define a natural convolution operator on non-euclidean surfaces. In this paper, we present a method for applying deep learning to 3D surfaces using their spherical descriptors and alt-az anisotropic convolution on 2-sphere. A cascade set of geodesic disk filters rotate on the 2-sphere and collect spherical patterns and so to extract geometric features for various 3D shape analysis tasks. We demonstrate theoretically and experimentally that our proposed method has the possibility to bridge the gap between 2D images and 3D shapes with the desired rotation equivariance/invariance, and its effectiveness is evaluated in applications of non-rigid/ rigid shape classification and shape retrieval. A recent research effort in computer vision and geometric processing communities is towards replicating the incredible success of deep convolutional neural networks (CNNs) from the image analysis to 3D shape analysis. A straightforward extension is to treat a 3D shape as a voxel grid BID38 ; BID16 ; BID30 ; BID36 ; BID22 . ) Alternative methods include encoding a 3D shape as a collection of 2D renderings from multiple cameras BID20 ; BID31 ; ,) or projecting a 3D object onto geometric entities which can be flattened as 2D images BID27 ; BID4 ; BID25 . ) All these methods convert a 3D shape into an Euclidean grid structure which supports shift (translational) equivariance/invariance, such that conventional CNNs can work out-of-the box.Although embedded in R 3 , 3D shapes are typically represented as manifold surfaces. Recent research has particularly focused on convolutional networks for non-Euclidean domains such as manifolds or graphs. One of the main difficulties of adopting CNNs and similar methods in these nonEuclidean domains is the lack of shift-invariance on surfaces or graphs BID15 . ) Our motivation comes from the representation of 3D shapes as functions on spheres. We transfer the problem of manifold surface convolution into spherical convolution with the primary benefit of rotation invariance. Although shift-invariance is hard to achieve on general surfaces, by replacing filter translations with filter rotations, rotation equivariance/invariance can be obtained on the 2-sphere. Furthermore, spherical descriptors of 3D shapes are compact and require a network of lower capacity, compared to voxel or multi-view representations. In this work, we are primarily interested in analyzing 3D geometric data using a specific type of spherical convolution either for classification or retrieval tasks. In this paper, we presented and analyzed a convolutional neural network based on alt-az anisotropic spherical convolution operator which is different from the existing types of networks. Numerically, we implemented an efficient algorithm for computing spherical convolution with locally-supported geodesic filters using icosahedron-sphere grid. We demonstrated the efficacy of our approach for non-rigid/ rigid shape classification and retrieval and showed that it compares favorably to competing methods. Furthermore, we have shown that the proposed method can effectively generalize across rotations, and achieve state-of-the-art results on competitive 3D shape recognition tasks, without excessive data augmentation, feature engineering and task-tuning.
A method for applying deep learning to 3D surfaces using their spherical descriptors and alt-az anisotropic convolution on 2-sphere. Presents a polar anisotropic convolution scheme on a unit sphere by replacing filter translation with filter rotation. This paper explores deep learning of 3D shapes using alt-az anisotropic 2-sphere convolution