Tensorial template matching for fast cross-correlation with rotations and its application for tomography

TL;DR

This work implements a new algorithm called tensorial template matching, based on a mathematical framework that represents all rotations of a template with a tensor field, which is much faster than template matching and has the potential to improve its accuracy.

Abstract

Object detection is a main task in computer vision. Template matching is the reference method for detecting objects with arbitrary templates. However, template matching computational complexity depends on the rotation accuracy, being a limiting factor for large 3D images (tomograms). Here, we implement a new algorithm called tensorial template matching, based on a mathematical framework that represents all rotations of a template with a tensor field. Contrary to standard template matching, the computational complexity of the presented algorithm is independent of the rotation accuracy. Using both, synthetic and real data from tomography, we demonstrate that tensorial template matching is much faster than template matching and has the potential to improve its accuracy

Figures (7)

• Figure 1: TM and its computational complexity. (A) TM consist in detecting positions, center localization, of the instances of an arbitrary input model, the template, in an image, and determining their rotations, or poses. (B) Computing time versus angular accuracy for TM in 2D images (blue) TM in 3D images (red) and the proposed TTM (green).
• Figure 2: TTM implementation scheme. Images represents 2D slices of tomograms, the input tomogram was taken from SHREC dataset Gubins2021, and the template was generated from https://www.rcsb.org/structure/5MRC atomic model. Black scale bars $200$ Å, white bars $2000$ Å.
• Figure 3: Synthetic data. (A) Isosurfaces of the 3D templates with 10 Å voxel size, from left to right: cylinder, L-shape, 3J9I, 3CF3, 4V4R, 1QvR, 4CR2 and 5MRC. The cylinder shows radial symmetry along the axis, 3J9I D7 and 3CF3 C6. 5MRC isosurface has transparency and also shows the atomic model used to generated the template density. Scale bar 100 Å. (B) 2D slice on XY-plane of the tomogram with SNR=0.1 containing instances of 4CR2 at random rotations. Scale bar 500 Å. (C) Isosurface of a noise free tomogram with 4CR2 instances. (D) Deviation in the rotations for the 4CR2 template at different $SNR$ levels. Horizontal axis is in log scale.
• Figure 4: Object detection in a real dataset. (A) A 2D slice on XY-plane of a real tomogram containing a region of a S. pombe cell. (B) Overlaid on the 2D slice the true positives (yellow), false positives (red) and false negative (blue) detected by TTM. (C) TTM and PyTOM global F1-scores for the dataset https://www.ebi.ac.uk/empiar/EMPIAR-10988/. (A-B) Scale bars 2000 Å.
• Figure 5: Running times. Comparison between our implementation of TTM and a well-known GPU implementation of TM (PyTOM) for processing a single https://www.ebi.ac.uk/empiar/EMPIAR-10988/ tomogram. Axes are in log scale.