RDA-INR: Riemannian Diffeomorphic Autoencoding via Implicit Neural Representations

Diffeomorphic registration frameworks such as large deformation diffeomorphic metric mapping(LDDMM) are used in computer graphics and the medical domain for atlas building, statisticallatent modeling, and pairwise and groupwise registration. In recent years, researchers have devel-oped neural network--based approaches regarding diffeomorphic registration to improve the accuracyand computational efficiency of traditional methods. In this work, we focus on a limitation of neu-ral network--based atlas building and statistical latent modeling methods, namely that they either(i) are resolution dependent or (ii) disregard any data- or problem-specific geometry needed forproper mean-variance analysis. In particular, we overcome this limitation by designing a novel en-coder based on resolution-independent implicit neural representations. The encoder achieves res-olution invariance for LDDMM-based statistical latent modeling. Additionally, the encoder addsLDDMM Riemannian geometry to resolution-independent deep learning models for statistical la-tent modeling. We investigate how the Riemannian geometry improves latent modeling and isrequired for a proper mean-variance analysis. To highlight the benefit of resolution independence forLDDMM-based data variability modeling, we show that our approach outperforms current neuralnetwork--based LDDMM latent code models. Our work paves the way for more research into howRiemannian geometry; shape, respectively, image analysis; and deep learning can be combined.