Rotation Equivariant Vector Field Networks
29 Oct 2017-pp 5058-5067
TL;DR: RotEqNet as discussed by the authors is a convolutional neural network (CNN) architecture encoding rotation equivariance, invariance and covariance, instead of treating as any other variation, leading to a reduction in the size of the required model.
Abstract: 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 representing magnitude and angle of the highest scoring orientation at every spatial location. We develop a modified convolution operator relying on this representation to obtain deep architectures. We test RotEqNet on several problems requiring different responses with respect to the inputs’ rotation: image classification, biomedical image segmentation, orientation estimation and patch matching. In all cases, we show that RotEqNet offers extremely compact models in terms of number of parameters and provides results in line to those of networks orders of magnitude larger.
Citations
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Posted Content•
TL;DR: Tensor field neural networks are introduced, which are locally equivariant to 3D rotations, translations, and permutations of points at every layer, and demonstrate the capabilities of tensor field networks with tasks in geometry, physics, and chemistry.
Abstract: We introduce tensor field neural networks, which are locally equivariant to 3D rotations, translations, and permutations of points at every layer. 3D rotation equivariance removes the need for data augmentation to identify features in arbitrary orientations. Our network uses filters built from spherical harmonics; due to the mathematical consequences of this filter choice, each layer accepts as input (and guarantees as output) scalars, vectors, and higher-order tensors, in the geometric sense of these terms. We demonstrate the capabilities of tensor field networks with tasks in geometry, physics, and chemistry.
542 citations
TL;DR: This work builds up the existing state-of-the-art object detection systems and proposes a simple but effective method to train rotation-invariant and Fisher discriminative CNN models to further boost object detection performance.
Abstract: The performance of object detection has recently been significantly improved due to the powerful features learnt through convolutional neural networks (CNNs). Despite the remarkable success, there are still several major challenges in object detection, including object rotation, within-class diversity, and between-class similarity, which generally degenerate object detection performance. To address these issues, we build up the existing state-of-the-art object detection systems and propose a simple but effective method to train rotation-invariant and Fisher discriminative CNN models to further boost object detection performance. This is achieved by optimizing a new objective function that explicitly imposes a rotation-invariant regularizer and a Fisher discrimination regularizer on the CNN features. Specifically, the first regularizer enforces the CNN feature representations of the training samples before and after rotation to be mapped closely to each other in order to achieve rotation-invariance. The second regularizer constrains the CNN features to have small within-class scatter but large between-class separation. We implement our proposed method under four popular object detection frameworks, including region-CNN (R-CNN), Fast R- CNN, Faster R- CNN, and R- FCN. In the experiments, we comprehensively evaluate the proposed method on the PASCAL VOC 2007 and 2012 data sets and a publicly available aerial image data set. Our proposed methods outperform the existing baseline methods and achieve the state-of-the-art results.
367 citations
01 Jun 2018
TL;DR: In this article, steerable filter convolutional neural networks (SFCNNs) are proposed to achieve joint equivariance under translations and rotations by design, which achieves state-of-the-art performance on the rotated MNIST benchmark and on the ISBI 2012 2D EM segmentation challenge.
Abstract: 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 interpolation artifacts from filter rotation. We utilize group convolutions which guarantee an equivariant mapping. In addition, we generalize He's weight initialization scheme to filters which are defined as a linear combination of a system of atomic filters. Numerical experiments show a substantial enhancement of the sample complexity with a growing number of sampled filter orientations and confirm that the network generalizes learned patterns over orientations. The proposed approach achieves state-of-the-art on the rotated MNIST benchmark and on the ISBI 2012 2D EM segmentation challenge.
320 citations
Proceedings Article•
03 Dec 2018
TL;DR: The experimental results confirm the effectiveness of 3D Steerable CNNs for the problem of amino acid propensity prediction and protein structure classification, both of which have inherent SE(3) symmetry.
Abstract: We present a convolutional network that is equivariant to rigid body motions. The model uses scalar-, vector-, and tensor fields over 3D Euclidean space to represent data, and equivariant convolutions to map between such representations. These SE(3)-equivariant convolutions utilize kernels which are parameterized as a linear combination of a complete steerable kernel basis, which is derived analytically in this paper. We prove that equivariant convolutions are the most general equivariant linear maps between fields over ℝ3. Our experimental results confirm the effectiveness of 3D Steerable CNNs for the problem of amino acid propensity prediction and protein structure classification, both of which have inherent SE(3) symmetry.
292 citations
Posted Content•
TL;DR: In this paper, the authors give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not only translations, but also the action of any compact group.
Abstract: Convolutional neural networks have been extremely successful in the image recognition domain because they ensure equivariance to translations. There have been many recent attempts to generalize this framework to other domains, including graphs and data lying on manifolds. In this paper we give a rigorous, theoretical treatment of convolution and equivariance in neural networks with respect to not just translations, but the action of any compact group. Our main result is to prove that (given some natural constraints) convolutional structure is not just a sufficient, but also a necessary condition for equivariance to the action of a compact group. Our exposition makes use of concepts from representation theory and noncommutative harmonic analysis and derives new generalized convolution formulae.
282 citations
References
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Abstract: There is large consent that successful training of deep networks requires many thousand annotated training samples. In this paper, we present a network and training strategy that relies on the strong use of data augmentation to use the available annotated samples more efficiently. The architecture consists of a contracting path to capture context and a symmetric expanding path that enables precise localization. We show that such a network can be trained end-to-end from very few images and outperforms the prior best method (a sliding-window convolutional network) on the ISBI challenge for segmentation of neuronal structures in electron microscopic stacks. Using the same network trained on transmitted light microscopy images (phase contrast and DIC) we won the ISBI cell tracking challenge 2015 in these categories by a large margin. Moreover, the network is fast. Segmentation of a 512x512 image takes less than a second on a recent GPU. The full implementation (based on Caffe) and the trained networks are available at http://lmb.informatik.uni-freiburg.de/people/ronneber/u-net .
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TL;DR: It is shown that dropout improves the performance of neural networks on supervised learning tasks in vision, speech recognition, document classification and computational biology, obtaining state-of-the-art results on many benchmark data sets.
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30,843 citations
Posted Content•
TL;DR: It is shown that such a network can be trained end-to-end from very few images and outperforms the prior best method (a sliding-window convolutional network) on the ISBI challenge for segmentation of neuronal structures in electron microscopic stacks.
Abstract: There is large consent that successful training of deep networks requires many thousand annotated training samples. In this paper, we present a network and training strategy that relies on the strong use of data augmentation to use the available annotated samples more efficiently. The architecture consists of a contracting path to capture context and a symmetric expanding path that enables precise localization. We show that such a network can be trained end-to-end from very few images and outperforms the prior best method (a sliding-window convolutional network) on the ISBI challenge for segmentation of neuronal structures in electron microscopic stacks. Using the same network trained on transmitted light microscopy images (phase contrast and DIC) we won the ISBI cell tracking challenge 2015 in these categories by a large margin. Moreover, the network is fast. Segmentation of a 512x512 image takes less than a second on a recent GPU. The full implementation (based on Caffe) and the trained networks are available at this http URL .
19,534 citations
Posted Content•
TL;DR: It is shown that convolutional networks by themselves, trained end- to-end, pixels-to-pixels, improve on the previous best result in semantic segmentation.
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9,803 citations