Contributions and Publications

(1)  Contributions in Differential Geometry

Dr. Liao’s dissertation is about isolated singularities of harmonic maps in three dimensional Riemannian manifolds. It was proved that isolated singularities are removable if the harmonic map energy is sufficiently small. This result generalized a well known two dimensional result. Dr. Liao continued research in analysis on manifolds and made significant contribution in Ricci curvature flow study in a joint paper.

  • [Liao 1985] Dissertation: Variational Problems for Maps of Riemannian Manifolds, 1985, under the supervision of Professor Richard Schoen, UC Berkeley.
  • [Gao and Liao 1994] On the Compactness of a Class of Riemannian Manifolds, Z. Gao and G. Liao, The Pacific J. of Mathematics, Vol. 166, No. 1, p. 23-41, 1994

(2) Contributions in Computational Fluid Dynamics and Mesh Generation

Dr. Liao discovered a link between differential geometry and adaptive mesh generation for computational fluid dynamics. He received two NSF grants to develop the deformation method for grid generation and adaptation. The deformation method is validated in publications co-authored with experts in computational fluid dynamics.

  • [Liu et al. 1998] An Adaptive Grid Method and its Application to Steady Euler Flow Calculations, F. Liu, S. Ji, and G. Liao, SIAM (Society for Industrial and Applied Mathematics) Journal for Scientific Computing, Vol. 20, No. 3, p. 811-825, 1998
  • [Liao et al. 2000] Level Set Based Deformation Methods for Adaptive Grids, G. Liao, F. Liu, G. de la Pena, D. Pang, and S. Osher, Journal of Computational Physics, vol. 159, p. 103-122, 2000
  • [Liao et al. 2002] Adaptive Grids for Resolution Enhancement, G. Liao, Z. Lei and G. de la Pena, vol. 12, p.153-156, Shock Waves (2002) https://doi.org/10.1007/s00193-002-0149-y
  • [Liao and Xue 2006] Moving Meshes by the Deformation Method, Liao, G. and J. Xue, Special issue: The International Symposium on Computing and Information, Journal of Computational and Applied Mathematics, Volume 195, Issue 1, pp. 83-92, 10/2006.

(3)  Contributions in Image Processing by Classical and Learned-based Approaches

Dr. Liao proposed the optimal control approach to nonrigid image registration, which was funded by a grant from the Division of Mathematics of the National Science Foundation. He was also awarded a NIH R03 grant to develop a novel approach to image averaging based on Jacobian determinant and curl vector.

Dr. Liao is the PI of R03MH120627 from the National Institute of Mental Health of the National Institute of Health, 05/2020 to 05/2022. Project Title: “Novel Construction of Unbiased Templates for Brain Morphometry,” Total grant $152,328. More detail of the project will be provided in a separate page.

Dr. Liao was the PI of the NSF Grant, DMS 0612998, 2006 to 2010, from the Computational Mathematics Program, Division of Mathematical Science, NSF. The project title is: ” Collaborative Research: A Geometric Method for Image Registration.” In this project, an optimal control approach to image registration was developed. This method does not contain the commonly used Tikhonov regularizing terms. Instead, the optimal control method minimizes a dis-similarity measure under the constraints of a differential system consisting of the divergence and the curl operators. This method does not have any trade-off parameters. This is in contrast with other mainstream methods that add regularizing terms to the dis-similarity term that has an unknown trade-off parameter. Dr. Liao and his team at UT Arlington worked on the algorithm developments and their implementation in Matlab. In summary, this method minimizes a dis-similarity method with respect to the control functions, which are the values of divergence and curl of the evolving registration deformations.

This method will be further enhanced to guarantee positive Jacobian determinant. We will also explore the use of the Jacobian determinant and the curl vector in the constraints. This large deformation diffeomorphic formulation will ensure that the dis-similarity measure is reduced on diffeomorphisms with lower bound for Jacobian determinant.

In addition to classical approach, Dr. Liao and collaborators have proposed new loss functions for image registration in unsupervised deep learning based on the Jacobian determinant (JD) and curl vector (CV). It is shown that the additional information provided by JD and CV enhanced performance in image segmentation and super-resolution. One paper is published in IEEE Access, and several conference papers have been accepted.

Classical Approach

  • [Liao et al. 2008] Optimal Control Approach to Data Alignment, G. Liao, X. Cai, D. Fleitas, X. Luo, J. Wang, J. Xue, Volume 21, Issue 9, September 2008, Pages 898-905, Applied Mathematics Letters
  • [Chu et al. 2009] Adaptive Grid Generation Based Non-Rigid Image Registration Using Mutual Information for Breast MRI, Mei-Yi Chu, Hua-Mei Chen, Chih-Yao Hsieh, Ting-Hung Lin, Hsi-Yue Hsiao, Guojun Liao, and Qi Peng, Journal of Signal Process System (2009) 54:45–63
  • [Liao et al. 2009] Construction of differentiable transformations, Guojun Liao, Xianxing Cai , Jie Liu, Xiaonan Luo, Jianmin Wang, Jiaxing Xue , 22 (2009) 1543-1548, Applied Math. Letters
  • [Hsiao et al. 2014] Hsiao, H-Y., Hsieh, C-Y., Chen, X., Gong, Y., Luo, X. and Liao, G., New Development of Nonrigid Registration. ANZIAM Journal, Vol. 55, pp. 289-297, 2014. doi:10.1017/S1446181114000091
  • [Zicong Zhou and Guojun Liao 2021], A Novel Approach to form Normal Distribution of Medical Image Segmentation based on Multiple Doctors’ Annotations, SPIE Image Processing, February 2021, San Diego, California (https://spie.org/medical-imaging/presentation/A-novel-approach-to-form-normal-distribution-of-medical-image/12032-112)
  • [Zicong Zhou and Guojun Liao 2022a] Construction of Diffeomorphisms with Prescribed Jacobian Determinant and Curl, International Congress of Geometry and Graph (ICGG 2022)
  • [Zicong Zhou and Guojun Liao 2022b] Recent Developments of an Optimal Control Approach to Nonrigid Image Registration, the 10th Workshop on Biological Image Registration, (WBIR 2022)

Learned-based Approach

  • [Zhu et al. 2019] Effects of Differential Geometry Parameters on Grid Generation and Segmentation of MRI Brain Image, Zhu YP, Zhou ZC, Liao GJ, Yang QX, Yuan KH, IEEE Access. 7(1), 68529-8539 (2019). https://ieeexplore.ieee.org/document/8719974.
  • [Zhu et al. 2020a] New Loss Functions for Medical Image Registration Based on VoxelMorph, Yongpei Zhu, Zicong Zhou, Guojun Liao, Kehong Yuan,  Proc. SPIE 11313, Medical Imaging 2020: Image Processing, 113132E (10 March 2020).
  • [Zhu et al. 2020b]  Y. Zhu, Z. Zhou, G. Liao, and K. Yuan, “Csrgan: Medical Image Super-Resolution Using A Generative Adversarial Network,” 2020 IEEE 17th International Symposium on Biomedical Imaging Workshops (ISBI Workshops), 2020, pp. 1-4, doi: 10.1109/ISBIWorkshops50223.2020.9153436.
  • [Zhu et al. 2021a]  Yongpei Zhu, Zicong Zhou, Guojun Liao, Kehong Yuan, “A novel unsupervised learning model for diffeomorphic image registration,” Proc. SPIE 11596, Medical Imaging 2021: Image Processing, 115960M (15 February 2021) https://doi.org/10.1117/12.2580815
  • [Zhu et al. 2021b]  Y. Zhu, Z. Zhou, G. Liao, and K. Yuan, “A New Unsupervised Learning Method for 3D Deformable Medical Image Registration,” 2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI), 2021, pp. 908-912, doi: 10.1109/ISBI48211.2021.9433835.

(4)  Other Manuscripts

  • [Liao et al. 2015]  A New Method for Triangular Mesh Generation, Jul 2015, Guojun G Liao, Xi Chen, Xianxin Cai, Ben Hildebrand, Dion Fleitas, arXiv:1507.03699
  • [Chen and Liao 2015]  New Variational Method of Grid Generation with Prescribed Jacobian determinant and Prescribed Curl, Chen, X. and Liao, G. https://arxiv.org/pdf/1507.03715.pdf, 2015.
  • [Chen and Liao 2016]  New method of averaging diffeomorphisms based on Jacobian determinant and curl vector, Chen, X. and Liao, G. https://arxiv.org/pdf/1611.03946.pdf, 2016
  • [Zhou et al. 2017]  Uniqueness of Transformation Based on Jacobian Determinant and Curl Vector, Zicong Zhou, Xi Chen, Xian Xin Cai, Guojun Liao, arXiv: 1712.03443: 11 pages, eight figures Subjects: Computational Geometry (cs.CG)
  • [Zhou et al. 2017]  A Novel Deformation Method for Higher Order Mesh Generation, Zicong Zhou, Xi Chen, Guojun Liao, arXiv:1710.00291, Subjects: Computational Geometry (cs.CG)
  • [Zhou et al. 2018] Computational Technologies for Brain Morphometry, Zicong Zhou, Ben Hildebrand, Xi Chen, and Guojun Liao, http://arxiv.org/pdf/1810.04833.pdf, 2018
  • [Zhu et al. 2018]  The method of multimodal MRI brain image segmentation based on differential geometric features, Y. Zhu, Z. Zhou, G. Liao, K. Yuen, arXiv: 1811.04281, 2018
  • [Liu and Liao 2018]  Multi-Block Grid Deformation Method in 3D, Jie Liu (liu@dixie.edu) Department of Mathematics Dixie State College St George, UT 84770 USA, and Guojun Liao (Corresponding Author: liao@uta.edu) Department of Mathematics University of Texas, Arlington, TX 76019-0408 USA, [1811.08974] https://arxiv.org/abs/1811.08974