Research Overview of Prof. Guojun Gordon Liao
Prof. Guojun Gordon Liao Joined the University of Texas at Arlington in the Fall semester of 1989. He received a PhD degree in mathematics from the University of California at Berkeley in 1985. His dissertation is on the local regularity of harmonic maps on Riemannian manifolds. Prof. Rick Schoen was his academic adviser at UC Berkeley. In 1986, Liao conducted research in nonlinear science as a postdoc under the supervision of Prof. S.T. Yau. After three years as a research instructor at the University of Utah, Prof. Liao joined the mathematics department as an associate professor and received the tenure status in 1992. In 2000, Prof. Liao became a full professor of mathematics.
Prof. Liao’s recent research is focused on medical image analysis, which is a significant change in his research direction. Switching to medical image analysis was a natural and interesting outgrowth of his previous work.
1. The Deformation Method
Liao continued his research in differential geometry and published several works in the regularity of harmonic maps by variational methods. In the joint paper with Ziyong Gao of Rice University: “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, a previous work on manifolds with Ricci curvature bounds by Professor Gao was significantly improved.
Around that time, Prof. Liao discovered a link between differential geometry and computational grid generation for field simulation. In the 90s, Liao was led to the grid generation problem by Prof. Steinberg and Prof. Costello who asked him to verify a couple of published papers on harmonic maps for construction of invertible transformation in two and three dimensions. Their arguments were naïve extensions of Rado’s theorem in two dimensions. Suggested by Professor Peter Li, Liao lectured on the work by Professor Hans Levi with a counterexample to a core argument in the paper under review. Later, Prof. Li sent me a reprint by Malas on a counterexample to the three dimensions version of Rado’s theorem. Master student Haisheng Liu and Prof. Liao were able to provide a stronger counterexample. So the harmonic maps have limitations in three dimensions. What shall we do? It was exciting when Liao discovered the seminal paper by Jurgen Moser: Volume elements of a Riemann Manifold, Trans AMS, 120, 1965. In that paper, Moser proposed the deformation method which was further developed by Dacorogna into a method which could generate mappings with prescribed Jacobian determinant in any dimensions in 1991. Prof. Liao introduced the method to the grid generation community and the deformation method for grid generation and adaptation was coined! The deformation method is used to generate an invertible transformation (visualized as an unfolded grid) with a prescribed Jacobian determinant, which models the grid cell size. A few years later, Prof. Liao was able to work out a dynamic version of the deformation method that generates time-dependent transformations with prescribed time-dependent Jacobian determinants. Prof. Liao’s research in grid generation was supported by NSF grants in Computational Mathematics Program.
2. Medical Image Analysis
Prof. Liao’s interest in medical image analysis was inspired by the image registration problem, which is the process of finding one-to-one correspondence between similar pixels of two or more images. This is a key technology in medical image analysis, which uses registration to compare the anatomies of different individuals to assist in diagnosing diseases and monitoring treatments. From a lecture by Prof. Miller, Liao learned that the registration is presented as an unfolded grid on the image domain, which is a rectangle in two dimensions or a cube in three dimensions. Over the past decades, the medical image research community has published hundreds of papers with different transformation models based on the B-splines, elastic or fluid mechanics, and other ideas. The aims of the existing methods are to find search space for minimization of dissimilarity measures that provide positive Jacobian determinants. For instance, a leading method by the name of large deformation diffeomorphic metric mapping (LDDMM) is based on sophisticated mathematics such as the Lie group of diffeomorphisms, developed in the early 2000s. The method minimizes a dissimilarity measure such as SSD (sum of squared distances) by controlling a velocity vector field which in turn satisfies a linear differential equation involving the Laplace operator. Despite well-sounded theoretical foundations, this, and other leading methods still produce folded registration grids when struggling to match images with complicated differences. After a long period of observing the problem, Prof. Liao proposed a novel methodology that is based on controlling both the Jacobian determinants (JD) and the curl vector (CV) which models the local gridline rotation. The curl vector (CV) is a fundamental geometric property of a grid. Intuitively, a grid is characterized by its cell size modeled by JD and the gridline rotation modeled by CV. Grids with the same cell size distribution may be deformed to have very different CVs. Therefore, in order to register (match) complicated images, we must directly control CV. Computational experiments by our recently developed Variational Principle (the VP) confirmed that general transformations in two or three dimensions can be accurately reconstructed by their JDs and CVs. Moreover, a novel construction of the average of a set of grids is developed based on their average of JDs and CVs. This method is applied to the construction of an unbiased template that serves as the target image, to which all images in a dataset are registered. This is the main technology proposed to the NIH and funded in May 2020 in R03MH120627. The reviewers were positive about the significance of adding the curl vector field to the Jacobian determinant to improve diffeomorphic brain registration. The review group on Emerging Imaging Technologies in Neuroscience Study Section gave the application a high Impact Score:20 and Percentile:5 +.
3. Recent Research Work
3.1 The Variational Principle (the VP)
We developed a variational method which constructs diffeomorphism with prescribed Jacobian determinant and curl vector. This method is a significant advancement of the deformation method which generates diffeomorphisms with the Jacobian determinant only. The ability of prescribing the curl vector is the foundation of our novel methodology for medical image analysis. Based on the VP, we developed a new approach to averaging a set of diffeomorphisms; which in turn, gives rise to a new approach to averaging a set of images. This is the method we proposed in the grant R03MH120627 for the construction of unbiased templates.
3.2 Uniqueness Conjecture
The VP supports the following conjecture: the Jacobian determinant and the curl vector uniquely determine a transformation.
This uniqueness conjecture is proved in a simple case: If a transformation has the three properties: (1) its Jacobian determinant equals 1, (2) its curl vector equals 0, and (3) it is close to the identity transformation Id in a norm, then it is the ID.
A numerical scheme of searching for counterexamples is developed and so far, no potential counterexamples are found.
3.3 Image registration controlled by the Jacobian determinant (JD) and the curl vector (CV)
Image registration is the process of establishing one-to-one correspondence between similar pixels (voxels) of two or more images. It is a key component of medical image analysis. The main challenge is to formulate transformation models that are sensitive and robust, Sensitivity means the models can reproduce grids associated with complicated variability among the images; robustness means the ability to guarantee one-to-one property.
Based on the researches mentioned above, using both the Jacobian determinant and curl vector as control function, we can provide the sensitivity and robustness that are required for accurate registration.
We have developed several versions of this approach and continue to develop the methods based on these two fundamental geometric properties JD and CV.
3.4 New Loss functions for Deep Learning Methods of Image Registration
Optimal control variational methods are formulated in the Lagrange Multiplier’s framework, with JD and CV. Optimality conditions are derived, which can be solved by traditional computational methods or by learning-based methods.
New loss functions for deep learning methods are constructed from the residuals of the optimality conditions. Research results are published in proceedings of leading conferences.
