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.
It is worth mentioning that standard data augmentation strategies belong to this first family. They rely on random rotations and flips of the training samples @cite_4 : given abundant training samples and enough model capacity, a CNN might learn that different orientations should score the same by learning equivalent filters at different orientations @cite_9 . Unlike this, RotEqNet is well suited for problems with limited training samples that can profit from reduced model sizes, since the behavior with respect to rotations is hardcoded and it does not need to be learned.
@cite_9 @cite_4
2949360407 2569680626 2801243570 2775143585
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. 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. Despite breakthroughs in image classification due to the evolution of deep learning and, in particular, convolutional neural networks (CNNs), state-of-the-art models only possess a very limited amount of rotational invariance. Known workarounds include artificial rotations of the training data or ensemble approaches, where several models are evaluated. These approaches either increase the workload of the training or inflate the number of parameters. Further approaches add rotational invariance by globally pooling over rotationally equivariant features. Instead, we propose to incorporate rotational invariance into the feature-extraction part of the CNN directly. This allows to train on unrotated data and perform well on a rotated test set. We use rotational convolutions and introduce a rotational pooling layer that performs a pooling over the back-rotated output feature maps. We show that when training on the original, unrotated MNIST training dataset, but evaluating on rotations of the MNIST test dataset, the error rate can be reduced substantially from 58.20 to 12.20 . Similar results are shown for the CIFAR-10 and CIFAR-100 datasets. In recent years, convolutional neural networks (CNN) have played an important role in the field of deep learning. Variants of CNN's have proven to be very successful in classification tasks across different domains. However, there are two big drawbacks to CNN's: their failure to take into account of important spatial hierarchies between features, and their lack of rotational invariance. As long as certain key features of an object are present in the test data, CNN's classify the test data as the object, disregarding features' relative spatial orientation to each other. This causes false positives. The lack of rotational invariance in CNN's would cause the network to incorrectly assign the object another label, causing false negatives. To address this concern, propose a novel type of neural network using the concept of capsules in a recent paper. With the use of dynamic routing and reconstruction regularization, the capsule network model would be both rotation invariant and spatially aware. The capsule network has shown its potential by achieving a state-of-the-art result of 0.25 test error on MNIST without data augmentation such as rotation and scaling, better than the previous baseline of 0.39 . To further test out the application of capsule networks on data with higher dimensionality, we attempt to find the best set of configurations that yield the optimal test error on CIFAR10 dataset.