Novel 3D Binary Indexed Tree for Volume Computation of 3D Reconstructed Models from Volumetric Data

TL;DR

The paper addresses the need for fast, accurate intrinsic-volume computation from volumetric medical data (CT/MR) during interactive reconstruction workflows. It combines surface-integral concepts, inclusion-exclusion, triple-integrals, and marching cubes with a 3D Fenwick (Binary Indexed) Tree to enable rapid per-region volume queries and updates. A key contribution is the 30-volume-configuration scheme derived from Lorensen’s marching cubes table, stored in a VolumeLookup, coupled with a 3D BIT to support efficient updates and region sums, achieving near-constant query times and accuracy within $\pm 0.004$ cm$^3$. The approach enables real-time volume analysis during slicing or editing, with practical impact for cardiovascular imaging workflows and potential plugin-based integration into visualization tools.

Abstract

In the burgeoning field of medical imaging, precise computation of 3D volume holds a significant importance for subsequent qualitative analysis of 3D reconstructed objects. Combining multivariate calculus, marching cube algorithm, and binary indexed tree data structure, we developed an algorithm for efficient computation of intrinsic volume of any volumetric data recovered from computed tomography (CT) or magnetic resonance (MR). We proposed the 30 configurations of volume values based on the polygonal mesh generation method. Our algorithm processes the data in scan-line order simultaneously with reconstruction algorithm to create a Fenwick tree, ensuring query time much faster and assisting users' edition of slicing or transforming model. We tested the algorithm's accuracy on simple 3D objects (e.g., sphere, cylinder) to complicated structures (e.g., lungs, cardiac chambers). The result deviated within $\pm 0.004 \text{cm}^3$ and there is still room for further improvement.