Defining and reliably finding a canonical orientation for 3D surfaces is key to many Computer Vision and Robotics applications. This task is commonly addressed by handcrafted algorithms exploiting geometric cues deemed as distinctive and robust by the designer. Yet, one might conjecture that humans learn the notion of the inherent orientation of 3D objects from experience and that machines may do so alike. In this work, we show the feasibility of learning a robust canonical orientation for surfaces represented as point clouds. Based on the observation that the quintessential property of a canonical orientation is equivariance to 3D rotations, we propose to employ Spherical CNNs, a recently introduced machinery that can learn equivariant representations defined on the Special Orthogonal group SO(3). Specifically, spherical correlations compute feature maps whose elements define 3D rotations. Our method learns such feature maps from raw data by a self-supervised training procedure and robustly selects a rotation to transform the input point cloud into a learned canonical orientation. Thereby, we realize the first end-to-end learning approach to define and extract the canonical orientation of 3D shapes, which we aptly dub Compass. Experiments on several public datasets prove its effectiveness at orienting local surface patches as well as whole objects.
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
Detecting robust keypoints from an image is an integral part of many computer vision problems, and the characteristic orientation and scale of keypoints play an important role for keypoint description and matching. Existing learning-based methods for keypoint detection rely on standard translation-equivariant CNNs but often fail to detect reliable keypoints against geometric variations. To learn to detect robust oriented keypoints, we introduce a self-supervised learning framework using rotation-equivariant CNNs. We propose a dense orientation alignment loss by an image pair generated by synth…
Beijing Jiaotong University（北京交通大学）；Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）
Existing panoramic depth estimation methods based on convolutional neural networks (CNNs) focus on removing panoramic distortions, failing to perceive panoramic structures efficiently due to the fixed receptive field in CNNs. This paper proposes the panorama transformer (named PanoFormer) to estimate the depth in panorama images, with tangent patches from spherical domain, learnable token flows, and panorama specific metrics. In particular, we divide patches on the spherical tangent domain into tokens to reduce the negative effect of panoramic distortions. Since the geometric structures are es…
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
Extracting discriminative local features that are invariant to imaging variations is an integral part of establishing correspondences between images. In this work, we introduce a self-supervised learning framework to extract discriminative rotation-invariant descriptors using group-equivariant CNNs. Thanks to employing group-equivariant CNNs, our method effectively learns to obtain rotation-equivariant features and their orientations explicitly, without having to perform sophisticated data augmentations. The resultant features and their orientations are further processed by group aligning, a n…
Convolutional neural networks (CNNs) have been widely used in various vision tasks, e.g. image classification, semantic segmentation, etc. Unfortunately, standard 2D CNNs are not well suited for spherical signals such as panorama images or spherical projections, as the sphere is an unstructured grid. In this paper, we present Spherical Transformer which can transform spherical signals into vectors that can be directly processed by standard CNNs such that many well-designed CNNs architectures can be reused across tasks and datasets by pretraining. To this end, the proposed method first uses loc…
We propose a spherical kernel for efficient graph convolution of 3D point clouds. Our metric-based kernels systematically quantize the local 3D space to identify distinctive geometric relationships in the data. Similar to the regular grid CNN kernels, the spherical kernel maintains translation-invariance and asymmetry properties, where the former guarantees weight sharing among similar local structures in the data and the latter facilitates fine geometric learning. The proposed kernel is applied to graph neural networks without edge-dependent filter generation, making it computationally attrac…
Columbia University（哥伦比亚大学）；Imperial College（帝国理工学院）
In neuroimaging studies, the human cortex is commonly modeled as a sphere to preserve the topological structure of the cortical surface. Cortical neuroimaging measures hence can be modeled in spherical representation. In this work, we explore analyzing the human cortex using spherical CNNs in an Alzheimer's disease (AD) classification task using cortical morphometric measures derived from structural MRI. Our results show superior performance in classifying AD versus cognitively normal and in predicting MCI progression within two years, using structural MRI information only. This work demonstra…
We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations…
Peking University（北京大学）；The Chinese University of Hong Kong（香港中文大学）；Pazhou Lab（琶洲实验室）
Spherical signals exist in many applications, e.g., planetary data, LiDAR scans and digitalization of 3D objects, calling for models that can process spherical data effectively. It does not perform well when simply projecting spherical data into the 2D plane and then using planar convolution neural networks (CNNs), because of the distortion from projection and ineffective translation equivariance. Actually, good principles of designing spherical CNNs are avoiding distortions and converting the shift equivariance property in planar CNNs to rotation equivariance in the spherical domain. In this …
Omni-directional cameras have many advantages overconventional cameras in that they have a much wider field-of-view (FOV). Accordingly, several approaches have beenproposed recently to apply convolutional neural networks(CNNs) to omni-directional images for various visual tasks.However, most of them use image representations defined inthe Euclidean space after transforming the omni-directionalviews originally formed in the non-Euclidean space. Thistransformation leads to shape distortion due to nonuniformspatial resolving power and the loss of continuity. Theseeffects make existing convolution…
Convolutional neural networks (CNNs) constructed natively on the sphere have been developed recently and shown to be highly effective for the analysis of spherical data. While an efficient framework has been formulated, spherical CNNs are nevertheless highly computationally demanding; typically they cannot scale beyond spherical signals of thousands of pixels. We develop scattering networks constructed natively on the sphere that provide a powerful representational space for spherical data. Spherical scattering networks are computationally scalable and exhibit rotational equivariance, while th…
Information Technologies Institute (ITI), Centre for Research and Technology Hellas (CERTH)（希腊研究与技术中心（CERTH）信息技术研究所（ITI））；Universidad Politécnica de Madrid (UPM)（马德里理工大学（UPM））
Learning based approaches for depth perception are limited by the availability of clean training data. This has led to the utilization of view synthesis as an indirect objective for learning depth estimation using efficient data acquisition procedures. Nonetheless, most research focuses on pinhole based monocular vision, with scarce works presenting results for omnidirectional input. In this work, we explore spherical view synthesis for learning monocular 360 depth in a self-supervised manner and demonstrate its feasibility. Under a purely geometrically derived formulation we present results f…
GRASP Laboratory（GRASP实验室）；University of Pennsylvania（宾夕法尼亚大学）；Google Research（谷歌研究院）
Learning equivariant representations is a promising way to reduce sample and model complexity and improve the generalization performance of deep neural networks. The spherical CNNs are successful examples, producing SO(3)-equivariant representations of spherical inputs. There are two main types of spherical CNNs. The first type lifts the inputs to functions on the rotation group SO(3) and applies convolutions on the group, which are computationally expensive since SO(3) has one extra dimension. The second type applies convolutions directly on the sphere, which are limited to zonal (isotropic) …
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Facebook AI Research（Facebook人工智能研究院）
Ideally, 360° imagery could inherit the deep convolutional neural networks (CNNs) already trained with great success on perspective projection images. However, existing methods to transfer CNNs from perspective to spherical images introduce significant computational costs and/or degradations in accuracy. In this work, we present the Kernel Transformer Network (KTN). KTNs efficiently transfer convolution kernels from perspective images to the equirectangular projection of 360° images. Given a source CNN for perspective images as input, the KTN produces a function parameterized by a polar angle …
University of Applied Sciences Western Switzerland (HES-SO)（瑞士西部应用科学大学）；CHUV（洛桑大学医院）；Harvard School of Engineering and Applied Sciences（哈佛工程与应用科学学院）
Locally Rotation Invariant (LRI) operators have shown great potential in biomedical texture analysis where patterns appear at random positions and orientations. LRI operators can be obtained by computing the responses to the discrete rotation of local descriptors, such as Local Binary Patterns (LBP) or the Scale Invariant Feature Transform (SIFT). Other strategies achieve this invariance using Laplacian of Gaussian or steerable wavelets for instance, preventing the introduction of sampling errors during the discretization of the rotations. In this work, we obtain LRI operators via the local pr…
Möbius transformations play an important role in both geometry and spherical image processing - they are the group of conformal automorphisms of 2D surfaces and the spherical equivalent of homographies. Here we present a novel, Möbius-equivariant spherical convolution operator which we call Möbius convolution, and with it, develop the foundations for Möbius-equivariant spherical CNNs. Our approach is based on a simple observation: to achieve equivariance, we only need to consider the lower-dimensional subgroup which transforms the positions of points as seen in the frames of their neighbors. T…
Many problems across computer vision and the natural sciences require the analysis of spherical data, for which representations may be learned efficiently by encoding equivariance to rotational symmetries. We present a generalized spherical CNN framework that encompasses various existing approaches and allows them to be leveraged alongside each other. The only existing non-linear spherical CNN layer that is strictly equivariant has complexity $\mathcal{O}(C^2L^5)$, where $C$ is a measure of representational capacity and $L$ the spherical harmonic bandlimit. Such a high computational cost often…
Efficient 3D affinely equivariant CNNs with adaptive fusion of augmented spherical Fourier-Bessel bases
School of Computer and Artificial Intelligence, Nanjing University of Finance and Economics（南京金融学院计算机与人工智能学院）；Interdisciplinary Center for Scientific Computing, Mannheim Institute for Intelligent Systems in Medicine, Medical Faculty Mannheim, Heidelberg University（科学计算交叉中心、医学智能系统研究所、曼海姆医学院、海德堡大学）；Computer Assisted Clinical Medicine, Mannheim Institute for Intelligent Systems in Medicine, Medical Faculty Mannheim, Heidelberg University（临床医学辅助计算机、医学智能系统研究所、曼海姆医学院、海德堡大学）；Department of Diagnostic and Interventional Radiology, University Hospital Bonn（诊断与介入放射科、波恩大学医院）；Interdisciplinary Center for Scientific Computing, Central Institute for Computer Engineering, CSZ Heidelberg Center for Model-Based AI, Data Analysis and Modeling in Medicine, Mannheim Institute for Intelligent Systems in Medicine, Medical Faculty Mannheim, Heidelberg University（科学计算交叉中心、计算机工程中央研究所、海德堡模型驱动AI中心、医学数据与建模中心、曼海姆医学智能系统研究所、曼海姆医学院、海德堡大学）
Shanghai Jiao Tong University（上海交通大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）
Pooling and unpooling are two essential operations in constructing hierarchical spherical convolutional neural networks (HS-CNNs) for comprehensive feature learning in the spherical domain. Most existing models employ downsampling-based pooling, which will inevitably incur information loss and cannot adapt to different spherical signals and tasks. Besides, the preserved information after pooling cannot be well restored by the subsequent unpooling to characterize the desirable features for a task. In this paper, we propose a novel framework of HS-CNNs with a lifting structure to learn adaptive …
Convolutional neural networks have many hyperparameters such as the filter size, number of filters, and pooling size, which require manual tuning. Though deep stacked structures are able to create multi-scale and hierarchical representations, manually fixed filter sizes limit the scale of representations that can be learned in a single convolutional layer. This paper introduces a new adaptive filter model that allows variable scale and orientation. The scale and orientation parameters of filters can be learned using back propagation. Therefore, in a single convolution layer, we can create filt…
Lausanne University Hospital（洛桑大学医院）；University of Applied Sciences Western Switzerland (HES-SO)（瑞士西部应用科学大学（HES-SO））
Convolutional Neural Networks (CNNs) traditionally encode translation equivariance via the convolution operation. Generalization to other transformations has recently received attraction to encode the knowledge of the data geometry in group convolution operations. Equivariance to rotation is particularly important for 3D image analysis due to the large diversity of possible pattern orientations. 3D texture is a particularly important cue for the analysis of medical images such as CT and MRI scans as it describes different types of tissues and lesions. In this paper, we evaluate the use of 3D g…
MultiModal Remote Sensing, University of Zurich（苏黎世大学多模态遥感）；Laboratory of GeoInformation Science and Remote Sensing, Wageningen University and Research（瓦赫宁根大学与研究中心地理信息科学与遥感实验室）
In remote sensing images, the absolute orientation of objects is arbitrary. Depending on an object's orientation and on a sensor's flight path, objects of the same semantic class can be observed in different orientations in the same image. Equivariance to rotation, in this context understood as responding with a rotated semantic label map when subject to a rotation of the input image, is therefore a very desirable feature, in particular for high capacity models, such as Convolutional Neural Networks (CNNs). If rotation equivariance is encoded in the network, the model is confronted with a simp…
University College London（伦敦大学学院）；Kagenova Limited（Kagenova有限公司）
This paper demonstrates spherical convolutional neural networks (S-CNN) offer distinct advantages over conventional fully-connected networks (FCN) at estimating scalar parameters of tissue microstructure from diffusion MRI (dMRI). Such microstructure parameters are valuable for identifying pathology and quantifying its extent. However, current clinical practice commonly acquires dMRI data consisting of only 6 diffusion weighted images (DWIs), limiting the accuracy and precision of estimated microstructure indices. Machine learning (ML) has been proposed to address this challenge. However, exis…
UC Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）；Lawrence Berkeley Nat’l Lab（劳伦斯伯克利国家实验室）；Technical University of Munich（慕尼黑工业大学）
We present an efficient convolution kernel for Convolutional Neural Networks (CNNs) on unstructured grids using parameterized differential operators while focusing on spherical signals such as panorama images or planetary signals. To this end, we replace conventional convolution kernels with linear combinations of differential operators that are weighted by learnable parameters. Differential operators can be efficiently estimated on unstructured grids using one-ring neighbors, and learnable parameters can be optimized through standard back-propagation. As a result, we obtain extremely efficien…
University of Amsterdam（阿姆斯特丹大学）；University of Heidelberg（海德堡大学）
In many machine learning tasks it is desirable that a model's prediction transforms in an equivariant way under transformations of its input. Convolutional neural networks (CNNs) implement translational equivariance by construction; for other transformations, however, they are compelled to learn the proper mapping. In this work, we develop Steerable Filter CNNs (SFCNNs) which achieve joint equivariance under translations and rotations by design. The proposed architecture employs steerable filters to efficiently compute orientation dependent responses for many orientations without suffering int…
Kagenova Limited（Kagenova有限公司）；University College London (UCL)（伦敦大学学院）
No existing spherical convolutional neural network (CNN) framework is both computationally scalable and rotationally equivariant. Continuous approaches capture rotational equivariance but are often prohibitively computationally demanding. Discrete approaches offer more favorable computational performance but at the cost of equivariance. We develop a hybrid discrete-continuous (DISCO) group convolution that is simultaneously equivariant and computationally scalable to high-resolution. While our framework can be applied to any compact group, we specialize to the sphere. Our DISCO spherical convo…
Key Laboratory of Biomedical Engineering of Ministry of Education, Zhejiang University（浙江大学教育部生物医学工程重点实验室）；Department of Radiology and BRIC, University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校放射科和BRIC（生物医学研究信息中心））；Department of Psychiatry, University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校精神病学系）
Convolutional Neural Networks (CNNs) have been providing the state-of-the-art performance for learning-related problems involving 2D/3D images in Euclidean space. However, unlike in the Euclidean space, the shapes of many structures in medical imaging have a spherical topology in a manifold space, e.g., brain cortical or subcortical surfaces represented by triangular meshes, with large inter-subject and intrasubject variations in vertex number and local connectivity. Hence, there is no consistent neighborhood definition and thus no straightforward convolution/transposed convolution operations …
Comsats University（科姆萨特大学）；University of Leicester（莱斯特大学）；University of Calabria（卡拉布里亚大学）；Free University of Bozen-Bolzano（博岑-博尔扎诺自由大学）；Edinburgh Napier University（爱丁堡龙比亚大学）
Object recognition from live video streams comes with numerous challenges such as the variation in illumination conditions and poses. Convolutional neural networks (CNNs) have been widely used to perform intelligent visual object recognition. Yet, CNNs still suffer from severe accuracy degradation, particularly on illumination-variant datasets. To address this problem, we propose a new CNN method based on orientation fusion for visual object recognition. The proposed cloud-based video analytics system pioneers the use of bi-dimensional empirical mode decomposition to split a video frame into i…
State-of-the-art deep learning systems often require large amounts of data and computation. For this reason, leveraging known or unknown structure of the data is paramount. Convolutional neural networks (CNNs) are successful examples of this principle, their defining characteristic being the shift-equivariance. By sliding a filter over the input, when the input shifts, the response shifts by the same amount, exploiting the structure of natural images where semantic content is independent of absolute pixel positions. This property is essential to the success of CNNs in audio, image and video re…
University of Applied Sciences Western Switzerland (HES-SO)（瑞士西部应用科学大学）；Harvard School of Engineering and Applied Sciences（哈佛大学工程与应用科学学院）；Centre Hospitalier Universitaire Vaudois (CHUV)（沃州大学中心医院）；Hopitaux Universitaires de Genève (HUG)（日内瓦大学医院）
Locally Rotation Invariant (LRI) image analysis was shown to be fundamental in many applications and in particular in medical imaging where local structures of tissues occur at arbitrary rotations. LRI constituted the cornerstone of several breakthroughs in texture analysis, including Local Binary Patterns (LBP), Maximum Response 8 (MR8) and steerable filterbanks. Whereas globally rotation invariant Convolutional Neural Networks (CNN) were recently proposed, LRI was very little investigated in the context of deep learning. LRI designs allow learning filters accounting for all orientations, whi…
Histology images are inherently symmetric under rotation, where each orientation is equally as likely to appear. However, this rotational symmetry is not widely utilised as prior knowledge in modern Convolutional Neural Networks (CNNs), resulting in data hungry models that learn independent features at each orientation. Allowing CNNs to be rotation-equivariant removes the necessity to learn this set of transformations from the data and instead frees up model capacity, allowing more discriminative features to be learned. This reduction in the number of required parameters also reduces the risk …
Qualcomm AI Research（高通AI研究院）；Qualcomm Technologies, Inc.（高通技术公司）；University of Amsterdam（阿姆斯特丹大学）
A common approach to define convolutions on meshes is to interpret them as a graph and apply graph convolutional networks (GCNs). Such GCNs utilize isotropic kernels and are therefore insensitive to the relative orientation of vertices and thus to the geometry of the mesh as a whole. We propose Gauge Equivariant Mesh CNNs which generalize GCNs to apply anisotropic gauge equivariant kernels. Since the resulting features carry orientation information, we introduce a geometric message passing scheme defined by parallel transporting features over mesh edges. Our experiments validate the significan…
Chalmers University of Technology（查尔姆斯理工大学）；Berlin Institute of Technology（柏林工业大学）；Berlin Institute for the Foundations of Learning and Data (BIFOLD)（柏林学习与数据基础研究所（BIFOLD））；University of Gothenburg（哥德堡大学）；Umeå University（于默奥大学）；Zenseact（Zenseact（智己汽车科技有限公司））
We analyze the role of rotational equivariance in convolutional neural networks (CNNs) applied to spherical images. We compare the performance of the group equivariant networks known as S2CNNs and standard non-equivariant CNNs trained with an increasing amount of data augmentation. The chosen architectures can be considered baseline references for the respective design paradigms. Our models are trained and evaluated on single or multiple items from the MNIST or FashionMNIST dataset projected onto the sphere. For the task of image classification, which is inherently rotationally invariant, we f…
We present a new and general framework for convolutional neural network operations on spherical (or omnidirectional) images. Our approach represents the surface as a graph of connected points that doesn't rely on a particular sampling strategy. Additionally, by using an interpolated version of SelectionConv, we can operate on the sphere while using existing 2D CNNs and their weights. Since our method leverages existing graph implementations, it is also fast and can be fine-tuned efficiently. Our method is also general enough to be applied to any surface type, even those that are topologically …
Otto von Guericke University Magdeburg（马格德堡奥托·冯·格里克大学）
External localization is an essential part for the indoor operation of small or cost-efficient robots, as they are used, for example, in swarm robotics. We introduce a two-stage localization and instance identification framework for arbitrary robots based on convolutional neural networks. Object detection is performed on an external camera image of the operation zone providing robot bounding boxes for an identification and orientation estimation convolutional neural network. Additionally, we propose a process to generate the necessary training data. The framework was evaluated with 3 different…
Confocal histology provides an opportunity to establish intra-voxel fiber orientation distributions that can be used to quantitatively assess the biological relevance of diffusion weighted MRI models, e.g., constrained spherical deconvolution (CSD). Here, we apply deep learning to investigate the potential of single shell diffusion weighted MRI to explain histologically observed fiber orientation distributions (FOD) and compare the derived deep learning model with a leading CSD approach. This study (1) demonstrates that there exists additional information in the diffusion signal that is not cu…
University of Pennsylvania（宾夕法尼亚大学）；Google Research（谷歌研究院）
Spherical convolutional networks have been introduced recently as tools to learn powerful feature representations of 3D shapes. Spherical CNNs are equivariant to 3D rotations making them ideally suited to applications where 3D data may be observed in arbitrary orientations. In this paper we learn 2D image embeddings with a similar equivariant structure: embedding the image of a 3D object should commute with rotations of the object. We introduce a cross-domain embedding from 2D images into a spherical CNN latent space. This embedding encodes images with 3D shape properties and is equivariant to…
Swansea University（斯旺西大学）；Université de Toulon（土伦大学）；Aix Marseille Univ（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）；Université de Strasbourg（斯特拉斯堡大学）
Robustness to transformation is desirable in many computer vision tasks, given that input data often exhibits pose variance. While translation invariance and equivariance is a documented phenomenon of CNNs, sensitivity to other transformations is typically encouraged through data augmentation. We investigate the modulation of complex valued convolutional weights with learned Gabor filters to enable orientation robustness. The resulting network can generate orientation dependent features free of interpolation with a single set of learnable rotation-governing parameters. By choosing to either re…
The Hong Kong University of Science and Technology（香港科技大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）
In this paper, we propose PointSeg, a real-time end-to-end semantic segmentation method for road-objects based on spherical images. We take the spherical image, which is transformed from the 3D LiDAR point clouds, as input of the convolutional neural networks (CNNs) to predict the point-wise semantic map. To make PointSeg applicable on a mobile system, we build the model based on the light-weight network, SqueezeNet, with several improvements. It maintains a good balance between memory cost and prediction performance. Our model is trained on spherical images and label masks projected from the …
We present a rotation-equivariant unsupervised learning framework for the sparse deconvolution of non-negative scalar fields defined on the unit sphere. Spherical signals with multiple peaks naturally arise in Diffusion MRI (dMRI), where each voxel consists of one or more signal sources corresponding to anisotropic tissue structure such as white matter. Due to spatial and spectral partial voluming, clinically-feasible dMRI struggles to resolve crossing-fiber white matter configurations, leading to extensive development in spherical deconvolution methodology to recover underlying fiber directio…
Quaternion Approximation Networks for Enhanced Image Classification and Oriented Object Detection
Department of Electrical, Systems and Computer Engineering（电气系统与计算机工程系；凯斯西储大学）；Case Western Reserve University（机械与航空航天工程系；凯斯西储大学）；Department of Mechanical and Aerospace Engineering, Case Western Reserve University
University College London（伦敦大学学院）；Massachusetts General Hospital（麻省总医院）；Harvard Medical School（哈佛医学院）；Massachusetts Institute of Technology（麻省理工学院）
Retrospective analysis of brain MRI scans acquired in the clinic has the potential to enable neuroimaging studies with sample sizes much larger than those found in research datasets. However, analysing such clinical images "in the wild" is challenging, since subjects are scanned with highly variable protocols (MR contrast, resolution, orientation, etc.). Nevertheless, recent advances in convolutional neural networks (CNNs) and domain randomisation for image segmentation, best represented by the publicly available method SynthSeg, may enable morphometry of clinical MRI at scale. In this work, w…
Mitsubishi Electric Research Laboratories（三菱电机研究实验室）；Massachusetts Institute of Technology（麻省理工学院）
Omnidirectional images and spherical representations of $3D$ shapes cannot be processed with conventional 2D convolutional neural networks (CNNs) as the unwrapping leads to large distortion. Using fast implementations of spherical and $SO(3)$ convolutions, researchers have recently developed deep learning methods better suited for classifying spherical images. These newly proposed convolutional layers naturally extend the notion of convolution to functions on the unit sphere $S^2$ and the group of rotations $SO(3)$ and these layers are equivariant to 3D rotations. In this paper, we consider th…
