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 efficient neural networks that match or outperform state-of-the-art network architectures in terms of performance but with a significantly lower number of network parameters. We evaluate our algorithm in an extensive series of experiments on a variety of computer vision and climate science tasks, including shape classification, climate pattern segmentation, and omnidirectional image semantic segmentation. Overall, we present (1) a novel CNN approach on unstructured grids using parameterized differential operators for spherical signals, and (2) we show that our unique kernel parameterization allows our model to achieve the same or higher accuracy with significantly fewer network parameters.
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) …
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…
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…
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…
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…
Vegetation Indices based on paired images of the visible color spectrum (VIS) and near infrared spectrum (NIR) have been widely used in remote sensing applications. These vegetation indices are extended for their application in autonomous driving in unstructured outdoor environments. In this domain we can combine traditional vegetation indices like the Normalized Difference Vegetation Index (NDVI) and Enhanced Vegetation Index (EVI) with Convolutional Neural Networks (CNNs) pre-trained on available VIS datasets. By laying a focus on learning calibrated CNN outputs, we can provide an approach t…
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…
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）
Convolutional Neural Networks (CNNs) have been successful in solving tasks in computer vision including medical image segmentation due to their ability to automatically extract features from unstructured data. However, CNNs are sensitive to rotation and affine transformation and their success relies on huge-scale labeled datasets capturing various input variations. This network paradigm has posed challenges at scale because acquiring annotated data for medical segmentation is expensive, and strict privacy regulations. Furthermore, visual representation learning with CNNs has its own flaws, e.g…
Robert Bosch GmbH（罗伯特·博世有限公司）；University of Lübeck（吕贝克大学）
Unstructured pruning is well suited to reduce the memory footprint of convolutional neural networks (CNNs), both at training and inference time. CNNs contain parameters arranged in $K \times K$ filters. Standard unstructured pruning (SP) reduces the memory footprint of CNNs by setting filter elements to zero, thereby specifying a fixed subspace that constrains the filter. Especially if pruning is applied before or during training, this induces a strong bias. To overcome this, we introduce interspace pruning (IP), a general tool to improve existing pruning methods. It uses filters represented i…
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 …
Graph Convolutional Networks (GCNs) are widely used in a variety of applications, and can be seen as an unstructured version of standard Convolutional Neural Networks (CNNs). As in CNNs, the computational cost of GCNs for large input graphs (such as large point clouds or meshes) can be high and inhibit the use of these networks, especially in environments with low computational resources. To ease these costs, quantization can be applied to GCNs. However, aggressive quantization of the feature maps can lead to a significant degradation in performance. On a different note, Haar wavelet transform…
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…
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…
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…
Shanghai Jiao Tong University（上海交通大学）；Beijing Advanced Innovation Center for Intelligent Robots and Systems（智能机器人与系统北京先进创新中心）；Beijing Institute of Technology（北京理工大学）；University of Cambridge（剑桥大学）
The semantic segmentation of point clouds is an important part of the environment perception for robots. However, it is difficult to directly adopt the traditional 3D convolution kernel to extract features from raw 3D point clouds because of the unstructured property of point clouds. In this paper, a spherical interpolated convolution operator is proposed to replace the traditional grid-shaped 3D convolution operator. This newly proposed feature extraction operator improves the accuracy of the network and reduces the parameters of the network. In addition, this paper analyzes the defect of poi…
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 …
While 360° cameras offer tremendous new possibilities in vision, graphics, and augmented reality, the spherical images they produce make core feature extraction non-trivial. Convolutional neural networks (CNNs) trained on images from perspective cameras yield "flat" filters, yet 360° images cannot be projected to a single plane without significant distortion. A naive solution that repeatedly projects the viewing sphere to all tangent planes is accurate, but much too computationally intensive for real problems. We propose to learn a spherical convolutional network that translates a planar CNN t…
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 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…
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…
Convolutional Neural Networks (CNNs) have recently led to incredible breakthroughs on a variety of pattern recognition problems. Banks of finite impulse response filters are learned on a hierarchy of layers, each contributing more abstract information than the previous layer. The simplicity and elegance of the convolutional filtering process makes them perfect for structured problems such as image, video, or voice, where vertices are homogeneous in the sense of number, location, and strength of neighbors. The vast majority of classification problems, for example in the pharmaceutical, homeland…
University of Bologna（博洛尼亚大学）；Federal University of Technology - Paraná（巴拉那联邦理工大学）；Federal University of Paraná（巴拉那联邦大学）
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 cano…
Durham University（杜伦大学）；Institute for Infocomm Research（资讯通信研究所）
Scene understanding for autonomous vehicles is a challenging computer vision task, with recent advances in convolutional neural networks (CNNs) achieving results that notably surpass prior traditional feature driven approaches. However, limited work investigates the application of such methods either within the highly unstructured off-road environment or to RGBD input data. In this work, we take an existing CNN architecture designed to perform semantic segmentation of RGB images of urban road scenes, then adapt and retrain it to perform the same task with multichannel RGBD images obtained unde…
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…
Leveraging SO(3)-steerable convolutions for pose-robust semantic segmentation in 3D medical data
Support Center for Advanced Neuroimaging (SCAN), University Institute of Diagnostic and Interventional Neuroradiology, University of Bern, Inselspital, Bern University Hospital（高级神经影像支持中心（SCAN）、伯尔尼大学诊断与介入神经放射学研究所、伯尔尼大学、因塞尔斯皮塔尔医院、伯尔尼大学医院）；NVIDIA（英伟达）
University of Amsterdam（阿姆斯特丹大学）；TU Delft（代尔夫特理工大学）
Learning powerful feature representations with CNNs is hard when training data are limited. Pre-training is one way to overcome this, but it requires large datasets sufficiently similar to the target domain. Another option is to design priors into the model, which can range from tuned hyperparameters to fully engineered representations like Scattering Networks. We combine these ideas into structured receptive field networks, a model which has a fixed filter basis and yet retains the flexibility of CNNs. This flexibility is achieved by expressing receptive fields in CNNs as a weighted sum over …
Point clouds are unstructured and unordered data, as opposed to images. Thus, most machine learning approach developed for image cannot be directly transferred to point clouds. In this paper, we propose a generalization of discrete convolutional neural networks (CNNs) in order to deal with point clouds by replacing discrete kernels by continuous ones. This formulation is simple, allows arbitrary point cloud sizes and can easily be used for designing neural networks similarly to 2D CNNs. We present experimental results with various architectures, highlighting the flexibility of the proposed app…
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 …
Swiss Data Science Center (SDSC)（瑞士数据科学中心）；EPFL（洛桑联邦理工学院）；ETH Zurich（苏黎世联邦理工学院）
Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning toolbox and have led to many breakthroughs in Artificial Intelligence. These networks have mostly been developed for regular Euclidean domains such as those supporting images, audio, or video. Because of their success, CNN-based methods are becoming increasingly popular in Cosmology. Cosmological data often comes as spherical maps, which make the use of the traditional CNNs more complicated. The commonly used pixelization scheme for spherical maps is the Hierarchical Equal Area isoLatitude Pixelisation (HEALPix). We pr…
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…
Central South University（中南大学）；Chinese Academy of Geological Sciences（中国地质科学院）
The three-dimensional (3D) geological models are the typical and key data source in the 3D mineral prospecitivity modeling. Identifying prospectivity-informative predictor variables from the 3D geological models is a challenging and tedious task. Motivated by the ability of convolutional neural networks (CNNs) to learn the intrinsic features, in this paper, we present a novel method that leverages CNNs to learn 3D mineral prospectivity from the 3D geological models. By exploiting the learning ability of CNNs, the presented method allows for disentangling complex correlation to the mineralizati…
A Comparison and Evaluation of Fine-tuned Convolutional Neural Networks to Large Language Models for Image Classification and Segmentation of Brain Tumors on MRI
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…
Structured Pruning for Deep Convolutional Neural Networks: A survey
Centre for Frontier AI Research (CFAR), Agency for Science Technology and Research (A*STAR)（新加坡科技研究局前沿人工智能研究中心）；Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局高性能计算研究所）
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…
Convolutional Neural Networks have revolutionized vision applications. There are image domains and representations, however, that cannot be handled by standard CNNs (e.g., spherical images, superpixels). Such data are usually processed using networks and algorithms specialized for each type. In this work, we show that it may not always be necessary to use specialized neural networks to operate on such spaces. Instead, we introduce a new structured graph convolution operator that can copy 2D convolution weights, transferring the capabilities of already trained traditional CNNs to our new graph …
Convolutional neural networks are ubiquitous in Machine Learning applications for solving a variety of problems. They however can not be used in their native form when the domain of the data is commonly encountered manifolds such as the sphere, the special orthogonal group, the Grassmanian, the manifold of symmetric positive definite matrices and others. Most recently, generalization of CNNs to data domains such as the 2-sphere has been reported by some research groups, which is referred to as the spherical CNNs (SCNNs). The key property of SCNNs distinct from CNNs is that they exhibit the rot…
