Communications in Mathematical Sciences

Volume 17 (2019)

Number 8

$3D$ crystal image analysis based on fast synchro-squeezed transforms

Pages: 2113 – 2140



Tao Zhang (Department of Mathematics, National University of Singapore)

Ling Li (Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, U.S.A.)

Haizhao Yang (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


We propose an efficient algorithm to analyze $3D$ crystal structure at the individual particle level based on a fast $3D$ synchro-squeezed wave packet transform. The proposed algorithm can automatically extract microscopic information from $3D$ atomic/particle resolution crystal images, e.g., crystal orientation, defects, and deformation, which are important information for characterizing material properties as well as potentially understanding the underlying formation processes. The effectiveness of our algorithms is illustrated by experiments of synthetic datasets and real $3D$ microscopic colloidal images.


crystal defect, elastic deformation, crystal rotation, $3D$ general wave shape, $3D$ band-limited synchro-squeezed transforms

2010 Mathematics Subject Classification

65T99, 74B20, 74E15, 74E25

H. Yang thanks the support of the start-up grant of the Department of Mathematics at the National University of Singapore and the Ministry of Education in Singapore under the grant MOE2018-T2- 2-147. L. Li gratefully acknowledges the Department of Mechanical Engineering at Virginia Tech for the start-up support.

Received 30 January 2019

Accepted 4 July 2019

Published 3 February 2020