Moment kernels: a simple and scalable approach for equivariance to rotations and reflections in deep convolutional networks

The principle of translation equivariance (if an input image is translated an output image should be translated by the same amount), led to the development of convolutional neural networks that revolutionized machine vision. Other symmetries, like rotations and reflections, play a similarly critical role, especially in biomedical image analysis, but exploiting these symmetries has not seen wide adoption. We hypothesize that this is partially due to the mathematical complexity of methods used to exploit these symmetries, which often rely on representation theory, a bespoke concept in differential geometry and group theory. In this work, we show that the same equivariance can be achieved using a simple form of convolution kernels that we call ``moment kernels,'' and prove that all equivariant kernels must take this form. These are a set of radially symmetric functions of a spatial position $x$, multiplied by powers of the components of $x$ or the identity matrix. We implement equivariant neural networks using standard convolution modules, and provide architectures to execute several biomedical image analysis tasks that depend on equivariance principles: classification (outputs are invariant under orthogonal transforms), 3D image registration (outputs transform like a vector), and cell segmentation (quadratic forms defining ellipses transform like a matrix).

Skeletonization of neuronal processes using Discrete Morse techniques from computational topology

To understand biological intelligence we need to map neuronal networks in vertebrate brains. Mapping mesoscale neural circuitry is done using injections of tracers that label groups of neurons whose axons project to different brain regions. Since many neurons are labeled, it is difficult to follow individual axons. Previous approaches have instead quantified the regional projections using the total label intensity within a region. However, such a quantification is not biologically meaningful. We propose a new approach better connected to the underlying neurons by skeletonizing labeled axon fragments and then estimating a volumetric length density. Our approach uses a combination of deep nets and the Discrete Morse (DM) technique from computational topology. This technique takes into account nonlocal connectivity information and therefore provides noise-robustness. We demonstrate the utility and scalability of the approach on whole-brain tracer injected data. We also define and illustrate an information theoretic measure that quantifies the additional information obtained, compared to the skeletonized tracer injection fragments, when individual axon morphologies are available. Our approach is the first application of the DM technique to computational neuroanatomy. It can help bridge between single-axon skeletons and tracer injections, two important data types in mapping neural networks in vertebrates.