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Variational Principle (VP) forms diffeomorphisms with prescribed Jacobian determinant (JD) and curl. Examples demonstrate that, (i) JD alone can not uniquely determine a diffeomorphism without curl; and (ii) the solutions by VP seem to satisfy properties of a Lie group. Hence, it is conjectured that a unique diffeomorphism can be assured by its JD and curl (Uniqueness Conjecture). In this paper, (1) an observation based on VP is derived that a counter example to the Conjecture, if exists, should satisfy a particular property; (2) from the observation, an experimental strategy is formulated to numerically test whether a given diffeomorphism is a valid counter example to the conjecture; (3) a proof of an intermediate step to the conjecture is provided and referred to as the semi-general case, which argues that, given two diffeomorphisms, $\pmb{\phi}$ and $\pmb{\psi}$, if they are close to the identity map, $\pmb{id}$, then $\pmb{\phi}$ is identical $\pmb{\psi}$.

The variational principle (VP) is designed to generate non-folding grids (diffeomorphisms) with prescribed Jacobian determinant (JD) and curl. Its solution pool of the original VP is based on an additive formulation and, consequently, is not invariant in the diffeomorphic Lie algebra. The original VP works well when the prescribed pair of JD and curl is calculated from a diffeomorphism, but not necessarily when the prescribed JD and curl are not known to come from a diffeomorphism. This issue is referred as the mismatched pair problem. In spite of that, the original VP works effectively in 2D grid generations. To resolve these issues, in this paper, we describe a new version of VP (revised VP), which is based on composition of transformations and, therefore, is invariant in the Lie algebra. The revised VP seems have overcome the inaccuracy of original VP in 3D grid generations. In the following sections, the mathematical derivations are presented. It is shown that the revised VP can calculate the inverse transformation of a known diffeomorphism. Its inverse consistency and transitivity of transformations are also demonstrated numerically. Moreover, a computational strategy is formulated based on the new version of VP to handle the mismatch issue and is demonstrated with preliminary result.

Accurate registration of medical images is vital for doctor's diagnosis and quantitative analysis. In this paper, we propose a new deformable medical image registration method based on average geometric transformations and VoxelMorph CNN architecture. We compute the differential geometric information including Jacobian determinant(JD) and the curl vector(CV) of diffeomorphic registration field and use them as multi-channel of VoxelMorph CNN for second train. In addition, we use the average transformation to construct a standard brain MRI atlas which can be used as fixed image. We verify our method on two datasets including ADNI dataset and MRBrainS18 Challenge dataset, and obtain excellent improvement on MR image registration with average Dice scores and non-negative Jacobian locations compared with MIT's original method. The experimental results show the method can achieve better performance in brain MRI diagnosis., Comment: 9 pages,9 figures

Multi-block grids provide the computational efficiency of structured grids and the flexibility for complex geometry. Thus, Multi-block structured grids are widely used for field simulation on complex domains. In this paper we propose a method which adapts multi-block grids according to a monitor function, which specifies cell volume distribution. The method is an extension of the deformation method on a single block to multi-blocks. Key words: multi-block grids, deformation method, adaptive grids

Accurate segmentation of brain tissue in magnetic resonance images (MRI) is a diffcult task due to different types of brain abnormalities. Using information and features from multimodal MRI including T1, T1-weighted inversion recovery (T1-IR) and T2-FLAIR and differential geometric features including the Jacobian determinant(JD) and the curl vector(CV) derived from T1 modality can result in a more accurate analysis of brain images. In this paper, we use the differential geometric information including JD and CV as image characteristics to measure the differences between different MRI images, which represent local size changes and local rotations of the brain image, and we can use them as one CNN channel with other three modalities (T1-weighted, T1-IR and T2-FLAIR) to get more accurate results of brain segmentation. We test this method on two datasets including IBSR dataset and MRBrainS datasets based on the deep voxelwise residual network, namely VoxResNet, and obtain excellent improvement over single modality or three modalities and increases average DSC(Cerebrospinal Fluid (CSF), Gray Matter (GM) and White Matter (WM)) by about 1.5% on the well-known MRBrainS18 dataset and about 2.5% on the IBSR dataset. Moreover, we discuss that one modality combined with its JD or CV information can replace the segmentation effect of three modalities, which can provide medical conveniences for doctor to diagnose because only to extract T1-modality MRI image of patients. Finally, we also compare the segmentation performance of our method in two networks, VoxResNet and U-Net network. The results show VoxResNet has a better performance than U-Net network with our method in brain MRI segmentation. We believe the proposed method can advance the performance in brain segmentation and clinical diagnosis., Comment: 18 pages,7 figures

In this paper, we described a set of computational technologies for image analysis with applications in Brain Morphometry. The proposed technologies are based on a new Variational Principle which constructs a transformation with prescribed Jacobian determinant (which models local size changes) and prescribed curl-vector (which models local rotations). The goal of this research is to convince the image research community that Jacobian determinant as well as curl-vector should be used in all steps of image analysis. Specifically, we develop an optimal control method for non-rigid registration; a new concept and construction of average transformation; and a general robust method for construction of unbiased template from a set of images. Computational examples are presented to show the effects of curl-vector and the effectiveness of optimal control methods for non-rigid registration and our method for construction of unbiased template.

Numerical examples demonstrated that a prescribed positive Jacobian determinant alone can not uniquely determine a diffeomorphism. It is conjectured that the uniqueness of a transformation can be assured by its Jacobian determinant and the curl-vector. In this work, we study the uniqueness problem analytically and propose an approach to the proof of the uniqueness of a transformation with prescribed Jacobian determinant and curl-vector., Comment: 11 pages, 8 figures

The development of higher order finite elements methods has become an active research area. The deformation method for mesh generation has achieved a prescribed positive Jacobian determinant constraint and it has been a useful method for mesh adaptation. In this work, the deformation method is combined with local refinement to generate higher order meshes. 2D and 3D numerical examples of element order p=3 are shown to demonstrate the effectiveness of our new algorithm.

Averaging diffeomorphisms is a challenging problem, and it has great applications in areas like medical image atlases. The simple Euclidean average can neither guarantee the averaged transformation is a diffeomorphism, nor get reasonable result when there is a local rotation. The goal of this paper is to propose a new approach to averaging diffeomorphisms based on the Jacobian determinant and the curl vector of the diffeomorphisms. Instead of averaging the diffeomorphisms directly, we average the Jacobian determinants and the curl vectors, and then construct a diffeomorphism based on the averaged Jacobian determinant and averaged curl vector as the average of diffeomorphisms. Numerical examples with convincible results are presented to demonstrate the method., Comment: 12 pages, 6 figures

Adaptive grid generation is an active research topic for numer- ical solution of differential equations. In this paper, we propose a variational method which generates transformations with prescribed Jacobian determinant and curl. Then we use this transformation to achieve adaptive grid generation task, and show the importance of curl in a transformation.

Computational mathematics plays an increasingly important role in computational fluid dynamics (CFD). The aeronautics and aerospace re- search community is working on next generation of CFD capacity that is accurate, automatic, and fast. A key component of the next generation of CFD is a greatly enhanced capacity for mesh generation and adaptivity of the mesh according to solution and geometry. In this paper, we propose a new method that generates triangular meshes on domains of curved boundary. The method deforms a Cartesian mesh that covers the domain to generate a mesh with prescribed boundary nodes. The deformation fields are generated by a system of divergence and curl equations which are solved effectively by the least square finite element method.