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1. Quaternion Equivariant Capsule Networks for 3D Point Clouds

   Zhao, Yongheng; Birdal, Tolga; Lenssen, Jan Eric; Menegatti, Emanuele; Guibas, Leonidas; Tombari, Federico (November 2020, European Conference on Computer Vision)

   null (Ed.)

   We present a 3D capsule module for processing point clouds that is equivariant to 3D rotations and translations, as well as invariant to permutations of the input points. The operator receives a sparse set of local reference frames, computed from an input point cloud and establishes end-to-end transformation equivariance through a novel dynamic routing procedure on quaternions. Further, we theoretically connect dynamic routing between capsules to the well-known Weiszfeld algorithm, a scheme for solving iterative re-weighted least squares (IRLS) problems with provable convergence properties. It is shown that such group dynamic routing can be interpreted as robust IRLS rotation averaging on capsule votes, where information is routed based on the final inlier scores. Based on our operator, we build a capsule network that disentangles geometry from pose, paving the way for more informative descriptors and a structured latent space. Our architecture allows joint object classification and orientation estimation without explicit supervision of rotations. We validate our algorithm empirically on common benchmark datasets. 

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2. Quaternion Equivariant Capsule Networks for 3D Point Clouds

   https://doi.org/10.1007/978-3-030-58452-8_1  

   Zhao, Yongheng; Birdal, Tolga; Lenssen, Jan Eric; Menegatti, Emanuele; Guibas, Leonidas; Tombari, Federico (November 2020, European Conference on Computer Vision)

   null (Ed.)

   We present a 3D capsule module for processing point clouds that is equivariant to 3D rotations and translations, as well as invariant to permutations of the input points. The operator receives a sparse set of local reference frames, computed from an input point cloud and establishes end-to-end transformation equivariance through a novel dynamic routing procedure on quaternions. Further, we theoretically connect dynamic routing between capsules to the well-known Weiszfeld algorithm, a scheme for solving iterative re-weighted least squares (IRLS) problems with provable convergence properties. It is shown that such group dynamic routing can be interpreted as robust IRLS rotation averaging on capsule votes, where information is routed based on the final inlier scores. Based on our operator, we build a capsule network that disentangles geometry from pose, paving the way for more informative descriptors and a structured latent space. Our architecture allows joint object classification and orientation estimation without explicit supervision of rotations. We validate our algorithm empirically on common benchmark datasets. 

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3. Image to Sphere: Learning Equivariant Features for Efficient Pose Prediction

   Klee, David M; Biza, Ondrej; Platt, Robert; Robin Walters (January 2023, The Eleventh International Conference on Learning Representations)

   Predicting the pose of objects from a single image is an important but difficult computer vision problem. Methods that predict a single point estimate do not predict the pose of objects with symmetries well and cannot represent uncertainty. Alternatively, some works predict a distribution over orientations in SO(3). However, training such models can be computation- and sample-inefficient. Instead, we propose a novel mapping of features from the image domain to the 3D rotation manifold. Our method then leverages SO(3) equivariant layers, which are more sample efficient, and outputs a distribution over rotations that can be sampled at arbitrary resolution. We demonstrate the effectiveness of our method at object orientation prediction, and achieve state-of-the-art performance on the popular PASCAL3D+ dataset. Moreover, we show that our method can model complex object symmetries, without any modifications to the parameters or loss function. Code is available at https://dmklee.github.io/image2sphere/ 

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4. Single-Stage Keypoint- Based Category-Level Object Pose Estimation from an RGB Image

   https://doi.org/10.1109/ICRA46639.2022.9812299  

   Lin, Yunzhi; Tremblay, Jonathan; Tyree, Stephen; Vela, Patricio A.; Birchfield, Stan (May 2022, International Conference on Robotics and Automation)

   Prior work on 6-DoF object pose estimation has largely focused on instance-level processing, in which a textured CAD model is available for each object being detected. Category-level 6- DoF pose estimation represents an important step toward developing robotic vision systems that operate in unstructured, real-world scenarios. In this work, we propose a single-stage, keypoint-based approach for category-level object pose estimation that operates on unknown object instances within a known category using a single RGB image as input. The proposed network performs 2D object detection, detects 2D keypoints, estimates 6- DoF pose, and regresses relative bounding cuboid dimensions. These quantities are estimated in a sequential fashion, leveraging the recent idea of convGRU for propagating information from easier tasks to those that are more difficult. We favor simplicity in our design choices: generic cuboid vertex coordinates, single-stage network, and monocular RGB input. We conduct extensive experiments on the challenging Objectron benchmark, outperforming state-of-the-art methods on the 3D IoU metric (27.6% higher than the MobilePose single-stage approach and 7.1 % higher than the related two-stage approach). 

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5. A Functional Approach to Rotation Equivariant Non-Linearities for Tensor Field Networks

   Poulenard, Adrien; Guibas, Leonidas (January 2021, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition)

   null (Ed.)

   Learning pose invariant representation is a fundamental problem in shape analysis. Most existing deep learning algorithms for 3D shape analysis are not robust to rotations and are often trained on synthetic datasets consisting of pre-aligned shapes, yielding poor generalization to unseen poses. This observation motivates a growing interest in rotation invariant and equivariant methods. The field of rotation equivariant deep learning is developing in recent years thanks to a well established theory of Lie group representations and convolutions. A fundamental problem in equivariant deep learning is to design activation functions which are both informative and preserve equivariance. The recently introduced Tensor Field Network (TFN) framework provides a rotation equivariant network design for point cloud analysis. TFN features undergo a rotation in feature space given a rotation of the input pointcloud. TFN and similar designs consider nonlinearities which operate only over rotation invariant features such as the norm of equivariant features to preserve equivariance, making them unable to capture the directional information. In a recent work entitled "Gauge Equivariant Mesh CNNs: Anisotropic Convolutions on Geometric Graphs" Hann et al. interpret 2D rotation equivariant features as Fourier coefficients of functions on the circle. In this work we transpose the idea of Hann et al. to 3D by interpreting TFN features as spherical harmonics coefficients of functions on the sphere. We introduce a new equivariant nonlinearity and pooling for TFN. We show improvments over the original TFN design and other equivariant nonlinearities in classification and segmentation tasks. Furthermore our method is competitive with state of the art rotation invariant methods in some instances. 

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