Incorporating Cyclic Group Equivariance into Deep Learning for Reliable Reconstruction of Rotationally Symmetric Tomography Systems

Rotational symmetry is a defining feature of many tomography systems, including computed tomography (CT) and emission computed tomography (ECT), where detectors are arranged in a circular or periodically rotating configuration. This study revisits the image reconstruction process from the perspective of hardware-induced rotational symmetry and introduces a cyclic group equivariance framework for deep learning-based reconstruction. Specifically, we derive a mathematical correspondence that couples cyclic rotations in the projection domain to discrete rotations in the image domain, both arising from the same cyclic group inherent in the hardware design. This insight also reveals the uniformly distributed circular structure of the projection space. Building on this principle, we provide a cyclic rotation equivariant convolution design method to preserve projection domain symmetry and a cyclic group equivariance regularization approach that enforces consistent rotational transformations across the entire network. We further integrate these modules into a domain transform reconstruction framework and validate them using digital brain phantoms, training on discrete models and testing on more complex and realistic fuzzy variants. Results indicate markedly improved generalization and stability, with fewer artifacts and better detail preservation, especially under data distribution deviation. These findings highlight the potential of cyclic group equivariance as a unifying principle for tomographic reconstruction in rotationally symmetric systems, offering a flexible and interpretable solution for scenarios with limited data.