Solid Spherical Energy (SSE) CNNs for Efficient 3D Medical Image Analysis

Invariance to local rotation, to differentiate from the global rotation of images and objects, is required in various texture analysis problems. It has led to several breakthrough methods such as local binary patterns, maximum response and steerable filterbanks. In particular, textures in medical images often exhibit local structures at arbitrary orientations. Locally Rotation Invariant (LRI) Convolutional Neural Networks (CNN) were recently proposed using 3D steerable filters to combine LRI with Directional Sensitivity (DS). The steerability avoids the expensive cost of convolutions with rotated kernels and comes with a parametric representation that results in a drastic reduction of the number of trainable parameters. Yet, the potential bottleneck (memory and computation) of this approach lies in the necessity to recombine responses for a set of predefined discretized orientations. In this paper, we propose to calculate invariants from the responses to the set of spherical harmonics projected onto 3D kernels in the form of a lightweight Solid Spherical Energy (SSE) CNN. It offers a compromise between the high kernel specificity of the LRI-CNN and a low memory/operations requirement. The computational gain is evaluated on 3D synthetic and pulmonary nodule classification experiments. The performance of the proposed approach is compared with steerable LRI-CNNs and standard 3D CNNs, showing competitive results with the state of the art

A General Theory of Equivariant CNNs on Homogeneous Spaces

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also consider a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? Following Mackey, we show that such maps correspond one-to-one with convolutions using equivariant kernels, and characterize the space of such kernels