Recent attempts at introducing rotation invariance or equivariance in 3D deep learning approaches have shown promising results, but these methods still struggle to reach the performances of standard 3D neural networks. In this work we study the relation between equivariance and invariance in 3D point convolutions. We show that using rotation-equivariant alignments, it is possible to make any convolutional layer rotation-invariant. Furthermore, we improve this simple alignment procedure by using the alignment themselves as features in the convolution, and by combining multiple alignments together. With this core layer, we design rotation-invariant architectures which improve state-of-the-art results in both object classification and semantic segmentation and reduces the gap between rotation-invariant and standard 3D deep learning approaches.
Nanjing University of Science and Technology（南京理工大学）
This paper proposes a Rotation-equivariant Attention Feature Fusion Pyramid Networks for Aerial Object Detection named ReAFFPN. ReAFFPN aims at improving the effect of rotation-equivariant features fusion between adjacent layers which suffers from the semantic and scale discontinuity. Due to the particularity of rotational equivariant convolution, general methods are unable to achieve their original effect while ensuring rotation equivariance of the network. To solve this problem, we design a new Rotation-equivariant Channel Attention which has the ability to both generate channel attention an…
We propose a semantic segmentation model that exploits rotation and reflection symmetries. We demonstrate significant gains in sample efficiency due to increased weight sharing, as well as improvements in robustness to symmetry transformations. The group equivariant CNN framework is extended for segmentation by introducing a new equivariant (G->Z2)-convolution that transforms feature maps on a group to planar feature maps. Also, equivariant transposed convolution is formulated for up-sampling in an encoder-decoder network. To demonstrate improvements in sample efficiency we evaluate on multipl…
Objects in aerial images have greater variations in scale and orientation than in typical images, so detection is more difficult. Convolutional neural networks use a variety of frequency- and orientation-specific kernels to identify objects subject to different transformations; these require many parameters. Sampling equivariant networks can adjust sampling from input feature maps according to the transformation of the object, allowing a kernel to extract features of an object under different transformations. Doing so requires fewer parameters, and makes the network more suitable for represent…
University of Zurich（苏黎世大学）；Ecole des Ponts, Paris Tech（巴黎高科桥路学院）
In many computer vision tasks, we expect a particular behavior of the output with respect to rotations of the input image. If this relationship is explicitly encoded, instead of treated as any other variation, the complexity of the problem is decreased, leading to a reduction in the size of the required model. In this paper, we propose the Rotation Equivariant Vector Field Networks (RotEqNet), a Convolutional Neural Network (CNN) architecture encoding rotation equivariance, invariance and covariance. Each convolutional filter is applied at multiple orientations and returns a vector field repre…
PreCM: The Padding-based Rotation Equivariant Convolution Mode for Semantic Segmentation
Shanghai Key Laboratory of Intelligent Sensing and Recognition, School of Sensing Science and Engineering, Shanghai Jiao Tong University（上海智能感知与识别重点实验室，传感科学与工程学院，上海交通大学）；Intelligent Photoelectric Sensing Institute, School of Sensing Science and Engineering, Shanghai Jiao Tong University（智能光电传感研究院，传感科学与工程学院，上海交通大学）
Shanghai Jiao Tong University（上海交通大学）；GIPSA-lab, Université Grenoble Alpes, CNRS, Grenoble INP（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔国立理工学院联合GIPSA实验室）；University of Iceland（冰岛大学）；University of Leicester（莱斯特大学）
Detection of objects is extremely important in various aerial vision-based applications. Over the last few years, the methods based on convolution neural networks have made substantial progress. However, because of the large variety of object scales, densities, and arbitrary orientations, the current detectors struggle with the extraction of semantically strong features for small-scale objects by a predefined convolution kernel. To address this problem, we propose the rotation equivariant feature image pyramid network (REFIPN), an image pyramid network based on rotation equivariance convolutio…
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…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Purdue University（普渡大学）
We propose learnable polyphase sampling (LPS), a pair of learnable down/upsampling layers that enable truly shift-invariant and equivariant convolutional networks. LPS can be trained end-to-end from data and generalizes existing handcrafted downsampling layers. It is widely applicable as it can be integrated into any convolutional network by replacing down/upsampling layers. We evaluate LPS on image classification and semantic segmentation. Experiments show that LPS is on-par with or outperforms existing methods in both performance and shift consistency. For the first time, we achieve true shi…
Technical University of Munich（慕尼黑工业大学）；Munich Center for Machine Learning（慕尼黑机器学习中心）；Friedrich-Alexander University Erlangen-Nuremberg（埃尔朗根-纽伦堡弗里德里希-亚历山大大学）
Michigan State University（密歇根州立大学）；Meta AI；Ford Motor Company（福特汽车公司）
Modern neural networks use building blocks such as convolutions that are equivariant to arbitrary 2D translations. However, these vanilla blocks are not equivariant to arbitrary 3D translations in the projective manifold. Even then, all monocular 3D detectors use vanilla blocks to obtain the 3D coordinates, a task for which the vanilla blocks are not designed for. This paper takes the first step towards convolutions equivariant to arbitrary 3D translations in the projective manifold. Since the depth is the hardest to estimate for monocular detection, this paper proposes Depth EquiVarIAnt NeTwo…
Chalmers University of Technology（查尔姆斯理工大学）；Gothenburg University（哥德堡大学）；Umeå University（于默奥大学）；Zenseact
We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\mathcal{M}$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces $\mathcal{M}=G/K$, which are instead equivariant with respect to the global symmetry $G$ on $\mathcal{M}$. Group equivariant layers can be interpreted as intertwiners between ind…
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）
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…
Group Equivariant Convolutions (GConvs) enable convolutional neural networks to be equivariant to various transformation groups, but at an additional parameter and compute cost. We investigate the filter parameters learned by GConvs and find certain conditions under which they become highly redundant. We show that GConvs can be efficiently decomposed into depthwise separable convolutions while preserving equivariance properties and demonstrate improved performance and data efficiency on two datasets. All code is publicly available at github.com/Attila94/SepGrouPy.
It has been shown that equivariant convolution is very helpful for many types of computer vision tasks. Recently, the 2D filter parametrization technique plays an important role when designing equivariant convolutions. However, the current filter parametrization method still has its evident drawbacks, where the most critical one lies in the accuracy problem of filter representation. Against this issue, in this paper we modify the classical Fourier series expansion for 2D filters, and propose a new set of atomic basis functions for filter parametrization. The proposed filter parametrization met…
The convolutional layers of standard convolutional neural networks (CNNs) are equivariant to translation. However, the convolution and fully-connected layers are not equivariant or invariant to other affine geometric transformations. Recently, a new class of CNNs is proposed in which the conventional layers of CNNs are replaced with equivariant convolution, pooling, and batch-normalization layers. The final classification layer in equivariant neural networks is invariant to different affine geometric transformations such as rotation, reflection and translation, and the scalar value is obtained…
Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）；University of Amsterdam（阿姆斯特丹大学）
Although group convolutional networks are able to learn powerful representations based on symmetry patterns, they lack explicit means to learn meaningful relationships among them (e.g., relative positions and poses). In this paper, we present attentive group equivariant convolutions, a generalization of the group convolution, in which attention is applied during the course of convolution to accentuate meaningful symmetry combinations and suppress non-plausible, misleading ones. We indicate that prior work on visual attention can be described as special cases of our proposed framework and show …
South China University of Technology（华南理工大学）；DexForce Co. Ltd.（DexForce有限公司）；Peng Cheng Laboratory（鹏城实验室）；SmartMore Technology Co. Ltd.（思谋科技有限公司）；Pazhou Laboratory（琶洲实验室）
As a basic component of SE(3)-equivariant deep feature learning, steerable convolution has recently demonstrated its advantages for 3D semantic analysis. The advantages are, however, brought by expensive computations on dense, volumetric data, which prevent its practical use for efficient processing of 3D data that are inherently sparse. In this paper, we propose a novel design of Sparse Steerable Convolution (SS-Conv) to address the shortcoming; SS-Conv greatly accelerates steerable convolution with sparse tensors, while strictly preserving the property of SE(3)-equivariance. Based on SS-Conv…
Convolution is conventionally defined as a linear operation on functions of one or more variables which commutes with shifts. Group convolution generalizes the concept to linear operations on functions of group elements representing more general geometric transformations and which commute with those transformations. Since similarity transformation is the most general geometric transformation on images that preserves shape, the group convolution that is equivariant to similarity transformation is the most general shape preserving linear operator. Because similarity transformations have four fre…
University of Southern California（南加州大学）；USC Institute for Creative Technologies（南加州大学创意技术研究所）；Tencent Game AI Research Center（腾讯游戏AI研究中心）；Pinscreen
Features that are equivariant to a larger group of symmetries have been shown to be more discriminative and powerful in recent studies. However, higher-order equivariant features often come with an exponentially-growing computational cost. Furthermore, it remains relatively less explored how rotation-equivariant features can be leveraged to tackle 3D shape alignment tasks. While many past approaches have been based on either non-equivariant or invariant descriptors to align 3D shapes, we argue that such tasks may benefit greatly from an equivariant framework. In this paper, we propose an effec…
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…
A Turn Toward Better Alignment: Few-Shot Generative Adaptation with Equivariant Feature Rotation
Hohai university（河海大学）；School of Electronic Engineering, Xidian University（西安电子科技大学电子工程学院）；School of Computer Science and Technology, Xidian University（西安电子科技大学计算机科学与技术学院）；Institute for Infocomm Research (I 2 R), A*STAR（信息通信研究院（I2R），A*STAR）
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…
Rotation-invariance is a desired property of machine-learning models for medical image analysis and in particular for computational pathology applications. We propose a framework to encode the geometric structure of the special Euclidean motion group SE(2) in convolutional networks to yield translation and rotation equivariance via the introduction of SE(2)-group convolution layers. This structure enables models to learn feature representations with a discretized orientation dimension that guarantees that their outputs are invariant under a discrete set of rotations. Conventional approaches fo…
UvA-Bosch Delta Lab（阿姆斯特丹大学-博世三角洲实验室）；University of Amsterdam（阿姆斯特丹大学）
Scale is often seen as a given, disturbing factor in many vision tasks. When doing so it is one of the factors why we need more data during learning. In recent work scale equivariance was added to convolutional neural networks. It was shown to be effective for a range of tasks. We aim for accurate scale-equivariant convolutional neural networks (SE-CNNs) applicable for problems where high granularity of scale and small kernel sizes are required. Current SE-CNNs rely on weight sharing and kernel rescaling, the latter of which is accurate for integer scales only. To reach accurate scale equivari…
DEGMC: Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutions
NAGIP-Nonlinear Analysis and Geometric Information Processing Research Group（非线性分析与几何信息处理研究组）；Department of Mathematics, University Iba Der Thiam（数学系，Iba Der Thiam大学）；Institut Mines-Telecom Nord Europe, Centre for Digital Systems, CNRS Centrale Lille UMR 9189 CRIStAL, University of Lille（Institut Mines-Telecom Nord Europe数字系统中心，CNRS Centrale Lille UMR 9189 CRIStAL，Lille大学）
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…
We present group equivariant capsule networks, a framework to introduce guaranteed equivariance and invariance properties to the capsule network idea. Our work can be divided into two contributions. First, we present a generic routing by agreement algorithm defined on elements of a group and prove that equivariance of output pose vectors, as well as invariance of output activations, hold under certain conditions. Second, we connect the resulting equivariant capsule networks with work from the field of group convolutional networks. Through this connection, we provide intuitions of how both meth…
Recent research has shown that incorporating equivariance into neural network architectures is very helpful, and there have been some works investigating the equivariance of networks under group actions. However, as digital images and feature maps are on the discrete meshgrid, corresponding equivariance-preserving transformation groups are very limited. In this work, we deal with this issue from the connection between convolutions and partial differential operators (PDOs). In theory, assuming inputs to be smooth, we transform PDOs and propose a system which is equivariant to a much more genera…
Group equivariant convolutional networks (GCNNs) endow classical convolutional networks with additional symmetry priors, which can lead to a considerably improved performance. Recent advances in the theoretical description of GCNNs revealed that such models can generally be understood as performing convolutions with G-steerable kernels, that is, kernels that satisfy an equivariance constraint themselves. While the G-steerability constraint has been derived, it has to date only been solved for specific use cases - a general characterization of G-steerable kernel spaces is still missing. This wo…
Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision tasks. This paper proposes to decompose the convolutional filters over joint steerable bases across the space and the group geometry simultaneously, namely a rotation-equivariant CNN with decomposed convolutional filters (RotDCF). This decomposition facilitates computing the joint convolution, which is proved to be necessary for the group equivariance. It significantly reduces the model size and compu…
Learning Coordinate-based Convolutional Kernels for Continuous SE(3) Equivariant and Efficient Point Cloud Analysis
Interdisciplinary Program in Neuroscience, Seoul National University（首尔国立大学神经科学跨学科项目）；Département d’Informatique, École Normale Supérieure (ENS)（规范高等学校计算机系）；Department of Computing, Imperial College London（伦敦帝国学院计算系）；Department of Computer Science and Engineering, Seoul National University（首尔国立大学计算机科学与工程系）
The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of $E(2)$-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations…
Recent work in equivariant deep learning bears strong similarities to physics. Fields over a base space are fundamental entities in both subjects, as are equivariant maps between these fields. In deep learning, however, these maps are usually defined by convolutions with a kernel, whereas they are partial differential operators (PDOs) in physics. Developing the theory of equivariant PDOs in the context of deep learning could bring these subjects even closer together and lead to a stronger flow of ideas. In this work, we derive a $G$-steerability constraint that completely characterizes when a …
University of Washington（华盛顿大学）；Purdue University（普渡大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
Incorporating group symmetry directly into the learning process has proved to be an effective guideline for model design. By producing features that are guaranteed to transform covariantly to the group actions on the inputs, group-equivariant convolutional neural networks (G-CNNs) achieve significantly improved generalization performance in learning tasks with intrinsic symmetry. General theory and practical implementation of G-CNNs have been studied for planar images under either rotation or scaling transformation, but only individually. We present, in this paper, a roto-scale-translation equ…
Convolutional neural networks lack shift equivariance due to the presence of downsampling layers. In image classification, adaptive polyphase downsampling (APS-D) was recently proposed to make CNNs perfectly shift invariant. However, in networks used for image reconstruction tasks, it can not by itself restore shift equivariance. We address this problem by proposing adaptive polyphase upsampling (APS-U), a non-linear extension of conventional upsampling, which allows CNNs with symmetric encoder-decoder architecture (for example U-Net) to exhibit perfect shift equivariance. With MRI and CT reco…
Several popular approaches to 3D vision tasks process multiple views of the input independently with deep neural networks pre-trained on natural images, achieving view permutation invariance through a single round of pooling over all views. We argue that this operation discards important information and leads to subpar global descriptors. In this paper, we propose a group convolutional approach to multiple view aggregation where convolutions are performed over a discrete subgroup of the rotation group, enabling, thus, joint reasoning over all views in an equivariant (instead of invariant) fash…
Transmute AI Lab（Transmute AI实验室）；IIT Dhanbad（丹巴德印度理工学院）
Convolutional Neural Networks(CNN) are inherently equivariant under translations, however, they do not have an equivalent embedded mechanism to handle other transformations such as rotations and change in scale. Several approaches exist that make CNNs equivariant under other transformation groups by design. Among these, steerable CNNs have been especially effective. However, these approaches require redesigning standard networks with filters mapped from combinations of predefined basis involving complex analytical functions. We experimentally demonstrate that these restrictions in the choice o…
Recently, 3D face reconstruction and face alignment tasks are gradually combined into one task: 3D dense face alignment. Its goal is to reconstruct the 3D geometric structure of face with pose information. In this paper, we propose a graph convolution network to regress 3D face coordinates. Our method directly performs feature learning on the 3D face mesh, where the geometric structure and details are well preserved. Extensive experiments show that our approach gains superior performance over state-of-the-art methods on several challenging datasets.
University of Amsterdam（阿姆斯特丹大学）；Qualcomm AI Research（高通人工智能研究院）
Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's infer…
Recently, learning equivariant representations has attracted considerable research attention. Dieleman et al. introduce four operations which can be inserted into convolutional neural network to learn deep representations equivariant to rotation. However, feature maps should be copied and rotated four times in each layer in their approach, which causes much running time and memory overhead. In order to address this problem, we propose Deep Rotation Equivariant Network consisting of cycle layers, isotonic layers and decycle layers. Our proposed layers apply rotation transformation on filters ra…
Face Alignment is an active computer vision domain, that consists in localizing a number of facial landmarks that vary across datasets. State-of-the-art face alignment methods either consist in end-to-end regression, or in refining the shape in a cascaded manner, starting from an initial guess. In this paper, we introduce DeCaFA, an end-to-end deep convolutional cascade architecture for face alignment. DeCaFA uses fully-convolutional stages to keep full spatial resolution throughout the cascade. Between each cascade stage, DeCaFA uses multiple chained transfer layers with spatial softmax to pr…
Training a Convolutional Neural Network (CNN) to be robust against rotation has mostly been done with data augmentation. In this paper, another progressive vision of research direction is highlighted to encourage less dependence on data augmentation by achieving structural rotational invariance of a network. The deep equivariance-bridged SO(2) invariant network is proposed to echo such vision. First, Self-Weighted Nearest Neighbors Graph Convolutional Network (SWN-GCN) is proposed to implement Graph Convolutional Network (GCN) on the graph representation of an image to acquire rotationally equ…
UvA-Bosch Delta Lab（阿姆斯特丹大学-博世德尔塔实验室）；University of Amsterdam（阿姆斯特丹大学）
The effectiveness of Convolutional Neural Networks (CNNs) has been substantially attributed to their built-in property of translation equivariance. However, CNNs do not have embedded mechanisms to handle other types of transformations. In this work, we pay attention to scale changes, which regularly appear in various tasks due to the changing distances between the objects and the camera. First, we introduce the general theory for building scale-equivariant convolutional networks with steerable filters. We develop scale-convolution and generalize other common blocks to be scale-equivariant. We …
In this paper, we propose Deep Alignment Network (DAN), a robust face alignment method based on a deep neural network architecture. DAN consists of multiple stages, where each stage improves the locations of the facial landmarks estimated by the previous stage. Our method uses entire face images at all stages, contrary to the recently proposed face alignment methods that rely on local patches. This is possible thanks to the use of landmark heatmaps which provide visual information about landmark locations estimated at the previous stages of the algorithm. The use of entire face images rather t…
This paper is concerned with a fundamental problem in geometric deep learning that arises in the construction of convolutional neural networks on surfaces. Due to curvature, the transport of filter kernels on surfaces results in a rotational ambiguity, which prevents a uniform alignment of these kernels on the surface. We propose a network architecture for surfaces that consists of vector-valued, rotation-equivariant features. The equivariance property makes it possible to locally align features, which were computed in arbitrary coordinate systems, when aggregating features in a convolution la…
