Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of the wavelet formalism developed by Antoine & Vandergheynst (1999) and Wiaux et al. (2005). The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation is firstly identified. A scale discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background (CMB) data, such as the imprint of topological defects, in particular cosmic strings, and for their reconstruction after separation from the other signal components.Comment: 22 pages, 2 figures. Version 2 matches version accepted for publication in MNRAS. Version 3 (identical to version 2) posted for code release announcement - "Steerable scale discretised wavelets on the sphere" - S2DW code available for download at http://www.mrao.cam.ac.uk/~jdm57/software.htm

Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the M\"obius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions is publicly available at https://github.com/mauriceweiler/MobiusCNN

Exact reconstruction with directional wavelets on the sphere

A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of a previously developed wavelet formalism developed by Antoine & Vandergheynst and Wiaux et al. The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation are first identified. A scale-discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background data, such as the imprint of topological defects, in particular, cosmic strings, and for their reconstruction after separation from the other signal component

Roto-Translation Equivariant Convolutional Networks: Application to Histopathology Image Analysis

Rotation-invariance is a desired property of machine-learning models for medical image analysis and in particular for computational pathology applications. We propose a framework to encode the geometric structure of the special Euclidean motion group SE(2) in convolutional networks to yield translation and rotation equivariance via the introduction of SE(2)-group convolution layers. This structure enables models to learn feature representations with a discretized orientation dimension that guarantees that their outputs are invariant under a discrete set of rotations. Conventional approaches for rotation invariance rely mostly on data augmentation, but this does not guarantee the robustness of the output when the input is rotated. At that, trained conventional CNNs may require test-time rotation augmentation to reach their full capability. This study is focused on histopathology image analysis applications for which it is desirable that the arbitrary global orientation information of the imaged tissues is not captured by the machine learning models. The proposed framework is evaluated on three different histopathology image analysis tasks (mitosis detection, nuclei segmentation and tumor classification). We present a comparative analysis for each problem and show that consistent increase of performances can be achieved when using the proposed framework