Computational Optimization of the 3D Least-Squares Matching Algorithm by Direct Calculation of Normal Equations

Research output: Contribution to journalResearch articleContributedpeer-review



3D least-squares matching is an algorithm that allows to measure subvoxel-precise displacements between two data sets of computed tomography voxel data. The determination of precise displacement vector fields is an important tool for deformation analyses in in-situ X-ray micro-tomography time series. The goal of the work presented in this publication is the development and validation of an optimized algorithm for 3D least-squares matching saving computation time and memory. 3D least-squares matching is a gradient-based method to determine geometric (and optionally also radiometric) transformation parameters between consecutive cuboids in voxel data. These parameters are obtained by an iterative Gauss-Markov process. Herein, the most crucial point concerning computation time is the calculation of the normal equations using matrix multiplications. In the paper at hand, a direct normal equation computation approach is proposed, minimizing the number of computation steps. A theoretical comparison shows, that the number of multiplications is reduced by 28% and the number of additions by 17%. In a practical test, the computation time of the 3D least-squares matching algorithm was proven to be reduced by 27%.


Original languageEnglish
Pages (from-to)760-777
Number of pages18
JournalTomography : a journal of imaging research
Issue number2
Publication statusPublished - 14 Mar 2022

External IDs

PubMed 35314640
Scopus 85126836620


Research priority areas of TU Dresden

DFG Classification of Subject Areas according to Review Boards

Subject groups, research areas, subject areas according to Destatis

ASJC Scopus subject areas


  • 3D least-squares matching; cuboid tracking; 3D displacement field; voxel data; microtomography data

Library keywords