We propose a framework for rotation and translation covariant deep learning using SE(2) group convolutions. The group product of the special Euclidean motion group SE(2) describes how a concatenation of two roto-translations results in a net roto-translation. We encode this geometric structure into convolutional neural networks (CNNs) via SE(2) group convolutional layers, which fit into the standard 2D CNN framework, and which allow to generically deal with rotated input samples without the need for data augmentation. We introduce three layers: a lifting layer which lifts a 2D (vector valued) image to an SE(2)-image, i.e., 3D (vector valued) data whose domain is SE(2); a group convolution layer from and to an SE(2)-image; and a projection layer from an SE(2)-image to a 2D image. The lifting and group convolution layers are SE(2) covariant (the output roto-translates with the input). The final projection layer, a maximum intensity projection over rotations, makes the full CNN rotation invariant. We show with three different problems in histopathology, retinal imaging, and electron microscopy that with the proposed group CNNs, state-of-the-art performance can be achieved, without the need for data augmentation by rotation and with increased performance compared to standard CNNs that do rely on augmentation.
In work by @cite_1 , an effective template matching method was proposed using group correlations in orientation scores, which are @math images obtained from a 2D image via lifting convolutions with a specific choice of kernel @cite_2 . The @math templates were put in a B-spline basis (allowing for exact kernel rotations) and optimized via logistic regression. Their architecture fits within our framework as a single channel G-CNN of depth 2 with a fixed lifting kernel.
@cite_1 @cite_2
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We propose a template matching method for the detection of 2D image objects that are characterized by orientation patterns. Our method is based on data representations via orientation scores, which are functions on the space of positions and orientations, and which are obtained via a wavelet-type transform. This new representation allows us to detect orientation patterns in an intuitive and direct way, namely via cross-correlations. Additionally, we propose a generalized linear regression framework for the construction of suitable templates using smoothing splines. Here, it is important to recognize a curved geometry on the position-orientation domain, which we identify with the Lie group SE(2): the roto-translation group. Templates are then optimized in a B-spline basis, and smoothness is defined with respect to the curved geometry. We achieve state-of-the-art results on three different applications: detection of the optic nerve head in the retina (99.83 percent success rate on 1,737 images), of the fovea in the retina (99.32 percent success rate on 1,616 images), and of the pupil in regular camera images (95.86 percent on 1,521 images). The high performance is due to inclusion of both intensity and orientation features with effective geometric priors in the template matching. Moreover, our method is fast due to a cross-correlation based matching approach. This paper proposes a new template matching method that is robust to outliers and fast enough for real-time operation. The template and image are densely transformed in binary code form by projecting and quantizing histograms of oriented gradients. The binary codes are matched by a generic method of robust similarity applicable to additive match measures, such as L p - and Hamming distances. The robust similarity map is computed efficiently via a proposed Inverted Location Index structure that stores pixel locations indexed by their values. The method is experimentally justified in large image patch datasets. Challenging applications, such as intra-category object detection, object tracking, and multimodal image matching are demonstrated. In this paper we aim to determine the location and orientation of a ground-level query image by matching to a reference database of overhead (e.g. satellite) images. For this task we collect a new dataset with one million pairs of street view and overhead images sampled from eleven U.S. cities. We explore several deep CNN architectures for cross-domain matching – Classification, Hybrid, Siamese, and Triplet networks. Classification and Hybrid architectures are accurate but slow since they allow only partial feature precomputation. We propose a new loss function which significantly improves the accuracy of Siamese and Triplet embedding networks while maintaining their applicability to large-scale retrieval tasks like image geolocalization. This image matching task is challenging not just because of the dramatic viewpoint difference between ground-level and overhead imagery but because the orientation (i.e. azimuth) of the street views is unknown making correspondence even more difficult. We examine several mechanisms to match in spite of this – training for rotation invariance, sampling possible rotations at query time, and explicitly predicting relative rotation of ground and overhead images with our deep networks. It turns out that explicit orientation supervision also improves location prediction accuracy. Our best performing architectures are roughly 2.5 times as accurate as the commonly used Siamese network baseline. We propose a combinatorial solution to determine the optimal elastic matching of a deformable template to an image. The central idea is to cast the optimal matching of each template point to a corresponding image pixel as a problem of finding a minimum cost cyclic path in the three-dimensional product space spanned by the template and the input image. We introduce a cost functional associated with each cycle, which consists of three terms: a data fidelity term favoring strong intensity gradients, a shape consistency term favoring similarity of tangent angles of corresponding points, and an elastic penalty for stretching or shrinking. The functional is normalized with respect to the total length to avoid a bias toward shorter curves. Optimization is performed by Lawler's Minimum Ratio Cycle algorithm parallelized on state-of-the-art graphics cards. The algorithm provides the optimal segmentation and point correspondence between template and segmented curve in computation times that are essentially linear in the number of pixels. To the best of our knowledge, this is the only existing globally optimal algorithm for real-time tracking of deformable shapes.