Non-rigid Medical Image Registration Models And Algorithms Based On Functions Of Bounded Deformation

Image registration is a widely used technology in the fields of computer vision and medical image processing and analysis.Generally speaking,the task of registration is to find the correspondence of some or all points in two or more images so that each aligned pair of points corresponds to the same point of the imaging object.Therefore,the essence of image registration is to find a spatial geometric transformation between two or more images.Image registration involves geometric transformation such as rigid transforma-tion,affine transformation,projection transformation and non-rigid transformation.As to the former three types of transformations,the image is warpped as a whole,that is,the transformation parameters of each point in the image are consistent.As to non-rigid transformations,the image is warpped locally,that is,the transformation parameters of each point in the image can be inconsistent.Note that the focusing objects in medical im-ages are often soft tissues or soft organs,thus non-rigid deformations occur when they are affected by respiratory,external pressure or tumors,and therefore,non-rigid registration methods are more suitable for medical image registration.In fact,non-rigid registration methods have already become a research hotspot in the field of medical image regis-tration.Medical imaging objects usually have complex peripheral structures and may contain lesions.At present,the accuracy of medical imaging equipment is not enough,and usually medical images are affected by noises,with different distributions and to different degrees.In addition,images obtained by different imaging devices may have very different resolutions.All of these bring great difficulties to non-rigid registration of medical images.The basic task of non-rigid image registration is to find the displacement field u between the pixel coordinates of the images to be registered,and then use a certain interpolation technology to warp the moving image with this specific displacement field,finally to obtain the registered image.Many existing non-rigid image registration methods formulate the non-rigid registration task as a variational model in a functional analysis framework or directly obtain the corresponding partial differential equation system by formulating the non-rigid registration task as a classic physical model.In these methods,the displacement field between images is usually regarded as a function in a specific function space,which has a certain degree of continuity and smoothness.However,for real medical images,there are often intensity inhomogeneity,diseased tissues and noises.All of these make the displacement field function not necessarily continuous,let alone smooth.In the classical theory of elastoplastic mechanics,the space of functions of bounded deformation,denoted by BD(Ω),is often used to describe the discontinuous displacement field.Inspired by this,we consider the displacement field u between the images to be registered as a function of bounded deformation,and propose a novel non-rigid medical image registration method based on bounded deformation function.The model requires less regularity of the displacement field and is more consistent with the real situation of medical images,that is,displacement fields may be discontinuous and thus not smooth.Then we generalize BD(Ω)space to the space of functions of bounded generalized deformation,denoted as BGDk(Ω),and construct new non-rigid registration models using the second-order generalized deformation of the displacement field as a regular term.Numerical results show the effectiveness of the proposed models in this paper.The main results of this paper are as follows:1.For two medical images with the same modality and high consistency of intensity distributions,a novel non-rigid image registration model,denoted as BDSSD model,is proposed in this paper.The classical SSD(sum of square difference)term is used as the data term,and the semi-norm in bounded deformation function space is used as the reg-ularization term.We prove the existence of the solution of the model and the uniqueness of the solution under certain conditions.In 2D numerical experiments,BDSSD model surpasses the classical Demons model,Diffeomorphic Demons model and the non-rigid registration model based on vector total variation in four commonly used evaluation in-dexes.In numerical experiments on two open three-dimensional datasets(4D-CT datasets and COPDgene datasets),BDSSD model achieves relatively better results compared with the other fourteen models.2.For medical images to be registered with the same modal but different intensity distributions and local intensity biases,a novel non-rigid registration model,denoted as BDLCC model,is proposed by using LCC(local correlation coefficient)term as the data term and combining semi-norm in bounded deformable function space as the regulariza-tion term.We prove the existence of solutions to the model.Two-dimensional numerical experiments show that,BDLCC model effectively registers medical images,and achieves better registration results than BDSSD model.On 4D-CT and COPDgene datasets,BDLCC model achieves better registration results than BDSSD model,and also achieves better registration results than the other fourteen models.3.This paper presents a primal-dual algorithm for solving the variational model of non-rigid image registration,and uses this algorithm to solve the above-mentioned BDSSD model and BDLCC model.Numerical results show that the primal-dual fast algorithm proposed in this paper can solve the corresponding model effectively,and its computing time is significantly less than the classical gradient descent algorithm.4.In this paper,we introduce the space of functions of bounded generalized defor-mation,and propose two non-rigid image registration models,BGDSSD and BGDLCC,both using the second-order generalized deformation of displacement field as the regular-ization term and using SSD and LCC as data terms,respectively.We prove the existence of solutions of these models,and use an adaptive primal-dual fast algorithm to solve these two non-rigid registration models.The results of two-dimensional and three-dimensional numerical experiments show that given data terms,the second-order generalized defor-mation regularization term can be obtain better registration results than usual bounded deformation regularization term,despite it takes longer to solve the model.