Free form deformation and symmetry constraint‐based multimodal brain image registration using generative adversarial nets

Multi‐modal brain image registration has been widely applied to functional localisation,neurosurgery and computational anatomy.The existing registration methods based on the dense deformation fields involve too many parameters,which is not conducive to the exploration of correct spatial correspondence between the float and reference images.Meanwhile,the unidirectional registration may involve the deformation folding,which will result in the change of topology during registration.To address these issues,this work has presented an unsupervised image registration method using the free form deformation(FFD)and the symmetry constraint‐based generative adversarial networks(FSGAN).The FSGAN utilises the principle component analysis network‐based structural representations of the reference and float images as the inputs and uses the generator to learn the FFD model parameters,thereby producing two deformation fields.Meanwhile,the FSGAN uses two discriminators to decide whether the bilateral registration have been realised simultaneously.Besides,the symmetry constraint is utilised to construct the loss function,thereby avoiding the deformation folding.Experiments on BrainWeb,high grade gliomas,IXI and LPBA40 show that compared with state‐of‐the‐art methods,the FSGAN provides superior performance in terms of visual comparisons and such quantitative indexes as dice value,target registration error and computational efficiency.

High-Order Binary Symmetry Constraints of a Liouville Integrable Hierarchy and Its Integrable Couplings

A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra A^-2, its integrable couplings are established.

An implicit symmetry constraint of the modified Korteweg-de Vries (mKdV) equation

In this paper, an implicit symmetry constraint is calculated and its associated binary nonlinearization of the Lax pairs and the adjoint Lax pairs is carried out for the modified Korteweg-de Vries (mKdV) equation. After introducing two new inde-pendent variables, we find that under the implicit symmetry constraint, the spatial part and the temporal part of the mKdV equation are decomposed into two finite-dimensional systems. Furthermore we prove that the obtained finite-dimensional systems are Hamiltonian systems and completely integrable in the Liouville sense.

ADJOINT SYMMETRY CONSTRAINTS OF MULTICOMPONENT AKNS EQUATIONS

A soliton hierarchy of multicomponent AKNS equations is generated from an arbitraryorder matrix spectral problem, along with its bi-Hamiltonian formulation. Adjoint symmetry constraints are presented to manipulate binary nonlinearization for the associated arbitrary order matrix spectral problem. The resulting spatial and temporal constrained flows are shown to provide integrable decompositions of the multicomponent AKNS equations.

The Bargmann Symmetry Constraint and Binary Nonlinearization of the Super Dirac Systems

An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super Dirac systems. Under the obtained symmetry constraint, the n-th flow of the super Dirac hierarchy is decomposed into two super finite-diinensional integrable Hamiltonian systems, defined over the super- symmetry manifold R^4N{2N with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.

Binary nonlinearization of the super classical-Boussinesq hierarchy

The symmetry constraint and binary nonlinearization of Lax pairs for the super classical-Boussinesq hierarchy is obtained. Under the obtained symmetry constraint, the n-th flow of the super classical-Boussinesq hierarchy is decomposed into two super finite-dimensional integrable Hamiltonian systems, defined over the super-symmetry manifold with the corresponding dynamical variables x and tn. The integrals of motion required for Liouville integrability are explicitly given.

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems （FDHS） in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint, we obtain a finite dimensional system from the Lax pair of the nonlinear Schrodinger equation. We show that this system is a completely integrable Hamiltonian system.