Multimodal image registration involves aligning images derived from various modalities. Accurately obtaining a diffeomorphic mapping is challenging due to the distinct features these images present. In this work, we propose a bi-variant multimodal diffeomorphic image registration model with flexible deformation. We begin by applying the constraint of the Jacobian determinant, det(I-d + del u(x)) = v(x), where v(x) > 0 is a flexible function, to multimodal image registration. This approach ensures the accurate alignment of images from different modalities while preserving diffeomorphic mappings. A penalty-splitting method models the constraint as an unconstrained optimization problem, which we solve using a discretize-then-optimize technique and a hybrid optimization strategy. To optimize the variable u(x) for the mutual information similarity measure, where Hessian computation is computationally expensive, we employ the L-BFGS algorithm combined with the Armijo line search. For the variable v(x), where Hessian is easier to compute, we apply the Gauss-Newton method with the Armijo line search. Numerical results on 2D and 3D medical and synthetic images from different modalities demonstrate the flexibility and effectiveness of our approach.

The convergence of inexact Newton methods is studied for solving generalized equations on Riemannian manifolds by using the metric regularity property, which is also explored. Under appropriate conditions and without any additional geometric assumptions, local convergence results with linear and quadratic rates, as well as a semi-local convergence result, are obtained for the proposed method. Finally, the theory is applied to the problem of finding a singularity for the sum of two vector fields. In particular, the KKT system for the constrained Riemannian center of mass on the sphere is explored numerically.

Image segmentation plays a crucial role in extracting important objects of interest from images, enabling various applications. While existing methods have shown success in segmenting clean images, they often struggle to produce accurate segmentation results when dealing with degraded images, such as those containing noise or occlusions. To address this challenge, interactive segmentation has emerged as a promising approach, allowing users to provide meaningful input to guide the segmentation process. However, an important problem in interactive segmentation lies in determining how to incorporate minimal yet meaningful user guidance into the segmentation model. In this paper, we propose the quasi-conformal interactive segmentation (QIS) model, which incorporates user input in the form of positive and negative clicks. Users mark a few pixels belonging to the object region as positive clicks, indicating that the segmentation model should include a region around these clicks. Conversely, negative clicks are provided on pixels belonging to the background, instructing the model to exclude the region near these clicks from the segmentation mask. Additionally, the segmentation mask is obtained by deforming a template mask with the same topology as the object of interest using an orientation-preserving quasiconformal mapping. This approach helps to avoid topological errors in the segmentation results. We provide a thorough analysis of the proposed model, including theoretical support for the ability of QIS to include or exclude regions of interest or disinterest based on the user's indication. To evaluate the performance of QIS, we conduct experiments on synthesized images, medical images, natural images and noisy natural images. The results demonstrate the efficacy of our proposed method.

An inexact-Newton method with cubic regularization is designed for solving Riemannian unconstrained nonconvex optimization problems. The proposed algorithm is fully adaptive with at most 𝒪 (ϵ _g^-3/2) iterations to achieve the norm of the gradient smaller than ϵ _g for given ϵ _g>0 , and at most 𝒪(max{ϵ _g^-3/2 , ϵ _H^-3} ) iterations to reach a second-order stationary point respectively. Notably, the proposed algorithm remains applicable even in cases of the gradient and Hessian of the objective function are unknown. Numerical experiments are performed with gradient and Hessian being approximated by forward finite-differences to illustrate the theoretical results and numerical comparison.

We propose a manifold AdaGrad-Norm method (MAdaGrad), which extends the norm version of AdaGrad (AdaGrad-Norm) to Riemannian optimization. In contrast to line-search schemes, which may require several exponential map computations per iteration, MAdaGrad requires only one. Assuming the objective function f has Lipschitz continuous Riemannian gradient, we show that the method requires at most 𝒪(ε^-2) iterations to compute a point x such that grad f(x)≤ε. Under the additional assumptions that f is geodesically convex and the manifold has sectional curvature bounded from below, we show that the method takes at most 𝒪(ε^-1) to find x such that f(x)-f_low, where f_low is the optimal value. Moreover, if f satisfies the Polyak–Łojasiewicz condition globally on the manifold, we establish a complexity bound of 𝒪(log(ε^-1)), provided that the norm of the initial Riemannian gradient is sufficiently large. For the manifold of symmetric positive definite matrices, we construct a family of nonconvex functions satisfying the PL condition. Numerical experiments illustrate the remarkable performance of MAdaGrad in comparison with Riemannian Steepest Descent equipped with Armijo line-search.

Image segmentation plays a crucial role in extracting objects of interest and identifying their boundaries within an image. However, accurate segmentation becomes challenging when dealing with occlusions, obscurities, or noise in corrupted images. To tackle this challenge, prior information is often utilized, with recent attention on star-shape priors. In this paper, we propose a star-shape segmentation model based on the registration framework. By combining the level set representation with the registration framework and imposing constraints on the deformed level set function, our model enables both full and partial star-shape segmentation, accommodating single or multiple centers. Additionally, our approach allows for the enforcement of identified boundaries to pass through specified landmark locations. We tackle the proposed models using the alternating direction method of multipliers. Through numerical experiments conducted on synthetic and real images, we demonstrate the efficacy of our approach in achieving accurate star-shape segmentation.

Diffeomorphic image registration is crucial for various medical imaging applications because it can preserve the topology of the transformation. This study introduces DCCNN-LSTM-Reg, a learning framework that evolves dynamically and learns a symmetrical registration path by satisfying a specified control increment system. This framework aims to obtain symmetric diffeomorphic deformations between moving and fixed images. To achieve this, we combine deep learning networks with diffeomorphic mathematical mechanisms to create a continuous and dynamic registration architecture, which consists of multiple Symmetric Registration (SR) modules cascaded on five different scales. Specifically, our method first uses two U-nets with shared parameters to extract multiscale feature pyramids from the images. We then develop an SR-module comprising a sequential CNN-LSTM architecture to progressively correct the forward and reverse multiscale deformation fields using control increment learning and the homotopy continuation technique. Through extensive experiments on three 3D registration tasks, we demonstrate that our method outperforms existing approaches in both quantitative and qualitative evaluations.

Image registration matches the features of two images by minimizing the intensity difference, so that useful and complementary information can be extracted from the mapping. However, in real life problems, images may be affected by the imaging environment, such as varying illumination and noise during the process of imaging acquisition. This may lead to the local intensity distortion, which makes it meaningless to minimize the intensity difference in the traditional registration framework. To address this problem, we propose a variational model for joint image registration and intensity correction. Based on this model, a related greedy matching problem is solved by introducing a multiscale approach for joint image registration and intensity correction. An alternating direction method (ADM) is proposed to solve each multiscale step, and the convergence of the ADM method is proved. For the numerical implementation, a coarse-to-fine strategy is further proposed to accelerate the numerical algorithm, and the convergence of the proposed coarse-to-fine strategy is also established. Some numerical tests are performed to validate the efficiency of the proposed algorithm.

With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing n-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two- and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling.

Image segmentation is to extract meaningful objects from a given image. For degraded images due to occlusions, obscurities or noises, the accuracy of the segmentation result can be severely affected. To alleviate this problem, prior information about the target object is usually introduced. In Chan et al. (J Math Imaging Vis 60(3):401–421, 2018), a topology-preserving registration-based segmentation model was proposed, which is restricted to segment 2D images only. In this paper, we propose a novel 3D topology-preserving registration-based segmentation model with the hyperelastic regularization, which can handle both 2D and 3D images. The existence of the solution of the proposed model is established. We also propose a converging iterative scheme to solve the proposed model. Numerical experiments have been carried out on the synthetic and real images, which demonstrate the effectiveness of our proposed model.

Image segmentation aims to extract the target objects or to identify the corresponding boundaries. For corrupted images due to occlusions, obscurities or noises, to get an accu-rate segmentation result is very difficult. To overcome this issue, the prior information is often introduced and particularly, the convex prior attracts more and more attentions re-cently. In this paper, we propose a topology-and convexity-preserving registration-based segmentation model, which can be suitable for both 2D and 3D cases. By incorporating the level set representation and imposing the constraints on the suitable regions, we can explicitly force the fully convex segmentation results or the partially convex segmentation results. To solve the proposed model, we employ the alternating direction method of mul-tipliers and numerical experiments on 2/3D synthetic and real images demonstrate that the proposed model can indeed lead to the accurately topology-and convexity-preserving segmentation. (c) 2021 Elsevier Inc. All rights reserved.

In this Chapter we discuss multi-modality image registration models and efficient algorithms. We propose a simple method to enhance a variational model to generate a diffeomorphic transformation. The idea is illustrated by using a particular model based on reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. By adding a control term motivated by quasi-conformal maps and Beltrami coefficients, the model has the ability to guarantee a diffeomorphic transformation. Without this feature, the model may lead to visually pleasing but invalid results. To solve the model numerically, we present both a Gauss-Newton method and an augmented Lagrangian method to solve the resulting discrete optimization problem. A multilevel technique is employed to speed up the initialization and reduce the possibility of getting local minima of the underlying functional. Finally numerical experiments demonstrate that this new model can deliver good performances for multi-modal image registration and simultaneously generate an accurate diffeomorphic transformation.

The Beltrami coefficient from complex analysis has recently been found to provide a robust constraint for obtaining orientation-preserving and diffeomorphic transformations for registration of planar images. There exists no such concept of the Beltrami coefficient in three or higher dimensions, although a generalized theory of quasi-conformal maps in high dimensions exists. In this paper, we first propose a new algebraic measure in three dimensions (3D) that mimics the Beltrami concept in two dimensions (2D) and then propose a corresponding registration model based on it. We then establish the existence of solutions for the proposed model and further propose a converging generalized Gauss-Newton iterative method to solve the resulting nonlinear optimization problem. In addition, we also provide another two possible regularizers in 3D. Numerical experiments show that the new model can produce more accurate orientation-preserving transformations than competing state-of-the-art registration models.

In this paper, we propose new multilevel optimization methods for minimizing continuously differentiable functions obtained by discretizing models for image registration problems. These multilevel schemes rely on a novel two-step Gauss-Newton method, in which a second step is computed within each iteration by minimizing a quadratic approximation of the objective function over a certain two-dimensional subspace. Numerical results on image registration problems show that the proposed methods can outperform the standard multilevel Gauss-Newton method.

In this work, we investigate image registration by mapping one image to another in a variational framework and focus on both model robustness and solver efficiency. We first propose a new variational model with a special regularizer, based on the quasi-conformal theory, which can guarantee that the registration map is diffeomorphic. It is well known that when the deformation is large, many variational models including the popular diffusion model cannot ensure diffeomorphism. One common observation is that the fidelity error appears small while the obtained transform is incorrect by way of mesh folding. However, direct reformulation from the Beltrami framework does not lead to effective models; our new regularizer is constructed based on this framework and added to the diffusion model to get a new model, which can achieve diffeomorphism. However, the idea is applicable to a wide class of models. We then propose an iterative method to solve the resulting nonlinear optimization problem and prove the convergence of the method. Numerical experiments can demonstrate that the new model can not only get a diffeomorphic registration even when the deformation is large, but also possess the accuracy in comparing with the currently best models.