A Lie group approach to steerable filters

Recently Freeman and Adelson (1991) published an approach to steer filters in their orientation by Fourier decompositions with respect to the angular coordinate of a polar representation. Simoncelli et al. (1992) generalized this method to steer other parameters than the orientation. In this paper we formulate the problem of steerability using the Lie group that performs the deformation of the filters. Within the presented theoretical framework we especially discuss the following points: (1) The possible scope and (2) the optimality of steerability by Fourier decompositions, (3) approximate steerability using a limited number of basis functions, (4) the nature of the singularity that occurs when steering the scale

Dynamic Steerable Blocks in Deep Residual Networks

Filters in convolutional networks are typically parameterized in a pixel basis, that does not take prior knowledge about the visual world into account. We investigate the generalized notion of frames designed with image properties in mind, as alternatives to this parametrization. We show that frame-based ResNets and Densenets can improve performance on Cifar-10+ consistently, while having additional pleasant properties like steerability. By exploiting these transformation properties explicitly, we arrive at dynamic steerable blocks. They are an extension of residual blocks, that are able to seamlessly transform filters under pre-defined transformations, conditioned on the input at training and inference time. Dynamic steerable blocks learn the degree of invariance from data and locally adapt filters, allowing them to apply a different geometrical variant of the same filter to each location of the feature map. When evaluated on the Berkeley Segmentation contour detection dataset, our approach outperforms all competing approaches that do not utilize pre-training. Our results highlight the benefits of image-based regularization to deep networks

On The Continuous Steering of the Scale of Tight Wavelet Frames

In analogy with steerable wavelets, we present a general construction of adaptable tight wavelet frames, with an emphasis on scaling operations. In particular, the derived wavelets can be "dilated" by a procedure comparable to the operation of steering steerable wavelets. The fundamental aspects of the construction are the same: an admissible collection of Fourier multipliers is used to extend a tight wavelet frame, and the "scale" of the wavelets is adapted by scaling the multipliers. As an application, the proposed wavelets can be used to improve the frequency localization. Importantly, the localized frequency bands specified by this construction can be scaled efficiently using matrix multiplication

Probing dark energy with steerable wavelets through correlation of WMAP and NVSS local morphological measures

Using local morphological measures on the sphere defined through a steerable wavelet analysis, we examine the three-year WMAP and the NVSS data for correlation induced by the integrated Sachs-Wolfe (ISW) effect. The steerable wavelet constructed from the second derivative of a Gaussian allows one to define three local morphological measures, namely the signed-intensity, orientation and elongation of local features. Detections of correlation between the WMAP and NVSS data are made with each of these morphological measures. The most significant detection is obtained in the correlation of the signed-intensity of local features at a significance of 99.9%. By inspecting signed-intensity sky maps, it is possible for the first time to see the correlation between the WMAP and NVSS data by eye. Foreground contamination and instrumental systematics in the WMAP data are ruled out as the source of all significant detections of correlation. Our results provide new insight on the ISW effect by probing the morphological nature of the correlation induced between the cosmic microwave background and large scale structure of the Universe. Given the current constraints on the flatness of the Universe, our detection of the ISW effect again provides direct and independent evidence for dark energy. Moreover, this new morphological analysis may be used in future to help us to better understand the nature of dark energy.Comment: 12 pages, 10 figures, replaced to match version accepted by MNRA

Boosting Deep Neural Networks with Geometrical Prior Knowledge: A Survey

While Deep Neural Networks (DNNs) achieve state-of-the-art results in many different problem settings, they are affected by some crucial weaknesses. On the one hand, DNNs depend on exploiting a vast amount of training data, whose labeling process is time-consuming and expensive. On the other hand, DNNs are often treated as black box systems, which complicates their evaluation and validation. Both problems can be mitigated by incorporating prior knowledge into the DNN. One promising field, inspired by the success of convolutional neural networks (CNNs) in computer vision tasks, is to incorporate knowledge about symmetric geometrical transformations of the problem to solve. This promises an increased data-efficiency and filter responses that are interpretable more easily. In this survey, we try to give a concise overview about different approaches to incorporate geometrical prior knowledge into DNNs. Additionally, we try to connect those methods to the field of 3D object detection for autonomous driving, where we expect promising results applying those methods.Comment: Survey Pape