Dmitrii Zhemchuzhnikov

Abstract:
In the dynamically evolving field of deep learning for spatial data analysis, the exploration of three-dimensional (3D) data structures presents unique challenges and opportunities. This thesis centers on developing and evaluating methods for detecting arbitrary-shaped patterns in both regular and irregular volumetric data, focusing on Fourier domain-based approaches. The work addresses challenges of global and local rotation invariance, enabling effective learning of hierarchical spatial patterns and providing accurate interpretation of neural network operations.

A range of methods is presented, each targeting different aspects of 3D data analysis. Starting with a Fourier-based approach, the research evaluates the applicability of Fourier representations for both regular and irregular data. One technique, inspired by X-ray and cosmology domains, demonstrated strong performance in protein structure data. The thesis further introduces methods like ILPONet, which is invariant to local pattern orientations, and EquiLoPONet, a network providing analytical equivariance concerning the continuous rotational space. Additionally, a method based on point cloud convolutions explores the challenges and potential of processing irregular data without relying on Fourier transformations.

This research underscores both the strengths and limitations of Fourier domain-based approaches in 3D spatial data analysis, particularly their utility in handling regular data. The findings contribute valuable insights for future applications in fields like medical imaging and molecular data analysis, paving the way for more robust and versatile deep learning models.