Abstract. We introduce a novel formulation for curvature regularization by penalizing normal curvatures from multiple directions. This total normal curvature regularization is capable of producing solutions with sharp edges and precise isotropic properties. To tackle the resulting high-order nonlinear optimization problem, we reformulate it as the task of finding the steady-state solution of a time-dependent partial differential equation (PDE) system. Time discretization is achieved through operator splitting, where each subproblem at the fractional steps either has a closed-form solution or can be efficiently solved using advanced algorithms. Our method circumvents the need for complex parameter tuning and demonstrates robustness to parameter choices. The efficiency and effectiveness of our approach have been rigorously validated in the context of surface and image smoothing problems.

Traditional variational models often fail to segment images in the presence of inhomogeneity or weak boundaries, partly due to their reliance on unreliable region metrics that quantify inhomogeneity based on a single mean value or a smoothed image serving as a mean function. The former, such as variance-based methods, are highly sensitive to image inhomogeneity, whereas the latter, such as local convolution-based approaches, lack a global receptive field. To address these issues, we employ an optimal transport-based data fidelity term in our segmentation objective functional. This term accounts for global differences between regions, resolving problems arising from local convolutions. It can also adaptively seek an optimized match between two probability density functions, proving more robust than relying solely on their mean values. Our proposed functional is minimized by gradually performing region merging. Experimental results demonstrate that our model outperforms state-of-the-art variational and deep learning models.

Existing level set models employ regularization based only on gradient information, 1D curvature or 2D curvature. For 3D image segmentation, however, an appropriate curvature-based regularization should involve a well-defined 3D curvature energy. This is the first paper to introduce a regularization energy that incorporates 3D scalar curvature for 3D image segmentation, inspired by the Einstein-Hilbert functional. To derive its Euler-Lagrange equation, we employ a two-step gradient descent strategy, alternately updating the level set function and its gradient. The paper also establishes the existence and uniqueness of the viscosity solution for the proposed model. Experimental results demonstrate that our proposed model outperforms other state-of-the-art models in 3D image segmentation.

Image registration is a key technique in image processing and analysis. Due to its high complexity, the traditional registration frameworks often fail to meet real-time demands in practice. To address the real-time demand, several deep learning networks for registration have been proposed, including the supervised and the unsupervised networks. Unsupervised networks rely on large amounts of training data to minimize specific loss functions, but the lack of physical information constraints results in the lower accuracy compared with the supervised networks. However, the supervised networks in medical image registration face two major challenges: physical mesh folding and the scarcity of labeled training data. To address these two challenges, we propose a novel few-shot learning framework for image registration. The framework contains two parts: random diffeomorphism generator (RDG) and a supervised few-shot learning network for image registration. By randomly generating a complex vector field, the RDG produces a series of diffeomorphism. With the help of diffeomorphism generated by RDG, one can use only a few image data (theoretically, one image data is enough) to generate a series of labels for training the supervised few-shot learning network. Concerning the elimination of the physical mesh folding phenomenon, in the proposed network, the loss function is only required to ensure the smoothness of deformation (no other control for mesh folding elimination is necessary). The experimental results indicate that the proposed method demonstrates superior performance in eliminating physical mesh folding when compared to other existing learning-based methods. Our code is available at this link https://github.com/weijunping111/RDG-TMI.git.

Abstract. Intensity inhomogeneity is a challenging task in image registration. Few past works have addressed the case of intensity inhomogeneity due to texture noise. To address this difficulty, we propose a novel three-stage approach for 2D/3D diffeomorphic multimodality image registration. The proposed approach contains three stages: (1) [Formula: see text] decomposition which decomposes the image pairs into texture, noise, and smooth component; (2) Blake–Zisserman homogenization which transforms the geometric features from different modalities into approximately the same modality in terms of the first-order and second-order edge information; (3) image registration which combines the homogenized geometric features and mutual information. Based on the proposed approach, the greedy matching for multimodality image registration is discussed and a coarse-to-fine algorithm is also proposed. Furthermore, several numerical tests are performed to validate the efficiency of the proposed approach.

Image segmentation is a fundamental task in the field of imaging and vision. Supervised deep learning for segmentation has achieved unparalleled success when sufficient training data with annotated labels are available. However, annotation is known to be expensive to obtain, especially for histopathology images where the target regions are usually with high morphology variations and irregular shapes. Thus, weakly supervised learning with sparse annotations of points is promising to reduce the annotation workload. In this work, we propose a contrast-based variational model to generate segmentation results, which serve as reliable complementary supervision to train a deep segmentation model for histopathology images. The proposed method considers the common characteristics of target regions in histopathology images and can be trained in an end-to-end manner. It can generate more regionally consistent and smoother boundary segmentation, and is more robust to unlabeled ‘novel’ regions. Experiments on two different histology datasets demonstrate its effectiveness and efficiency in comparison to previous models. Code is available at: https://github.com/hrzhang1123/CVM_WS_Segmentation.

The geometric high-order regularization methods such as mean curvature and Gaussian curvature, have been intensively studied during the last decades due to their abilities in preserving geometric properties including image edges, corners, and contrast. However, the dilemma between restoration quality and computational efficiency is an essential roadblock for high-order methods. In this paper, we propose fast multi-grid algorithms for minimizing both mean curvature and Gaussian curvature energy functionals without sacrificing accuracy for efficiency. Unlike the existing approaches based on operator splitting and the Augmented Lagrangian method (ALM), no artificial parameters are introduced in our formulation, which guarantees the robustness of the proposed algorithm. Meanwhile, we adopt the domain decomposition method to promote parallel computing and use the fine-to-coarse structure to accelerate convergence. Numerical experiments are presented on image denoising, CT, and MRI reconstruction problems to demonstrate the superiority of our method in preserving geometric structures and fine details. The proposed method is also shown effective in dealing with large-scale image processing problems by recovering an image of size $1024\times 1024$ within 40s, while the ALM-based method requires around 200s.

3D medical images consist of voxels with points, edges, faces, and volumes. A fascinating question is how to compute the shortest basis of the first homology group of a 3D image. The fastest time complexity known for this question is O($n^{\omega}+n^{2}$g), where n is the size of voxels and $\omega \lt$ 2.3728639 is a quantity so that two n×n matrices can be multiplied in O($n^{\omega}$) time. But it is still slow in practical applications. We first construct a hexahedral mesh of an arbitrary domain of a 3D image and second propose an approximate algorithm with time complexity O($n^{\omega}$) to calculate the minimal homology basis for the 3D images. Experiments show that our approximate algorithm is very close to the exact algorithm. We demonstrate the effectiveness of our algorithm in segmenting the semicircular canals, the organ with complex topology.

The Voronoi diagram-based dual-front scheme is known as a powerful and efficient technique for addressing the image segmentation and domain partitioning problems. In the basic formulation of existing dual-front approaches, the evolving contour can be considered as the interfaces of adjacent Voronoi regions. Among these dual-front models, a crucial ingredient is regarded as the geodesic metrics by which the geodesic distances and the corresponding Voronoi diagram can be estimated. In this paper, we introduce a new dual-front model based on asymmetric quadratic metrics. These metrics considered are built by the integration of the image features and a vector field derived from the evolving contour. The use of the asymmetry enhancement can reduce the risk for the segmentation contours being stuck at false positions, especially when the initial curves are far away from the target boundaries or the images have complicated intensity distributions. Moreover, the proposed dual-front model can be applied for image segmentation in conjunction with various region-based homogeneity terms. The numerical experiments on both synthetic and real images show that the proposed dual-front model indeed achieves encouraging results.

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.

Selective segmentation methods involve incorporating user input to partition an image into a foreground and background. These methods are often sensitive to some aspect of the user input in a counter intuitive manner, making their use in practice difficult. The most robust methods often involve laborious refinement on the part of the user, and sometimes editing/supervision. The proposed method reduces the burden of the user by simplifying the requirements in the input. Specifically, the fitting term does not depend on a distance function, and so no selection parameter is introduced. Instead, we consider how the user input relates to some general intensity fitting term to ensure the approach is less sensitive to the decisions or intuition of the user. We give comparisons to existing approaches to show the advantages of the new selective segmentation model.

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however, first, no analysis has been done yet, and second, all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while nonzero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem---one based on the variational Split-Bregman idea and three based on direct solution of the discretize-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.

Automated detection of blood vessel structures is becoming of crucial interest for better management of vascular disease. In this paper, we propose a new infinite active contour model that uses hybrid region information of the image to approach this problem. More specifically, an infinite perimeter regularizer, provided by using L(2) Lebesgue measure of the γ -neighborhood of boundaries, allows for better detection of small oscillatory (branching) structures than the traditional models based on the length of a feature's boundaries (i.e., H(1) Hausdorff measure). Moreover, for better general segmentation performance, the proposed model takes the advantage of using different types of region information, such as the combination of intensity information and local phase based enhancement map. The local phase based enhancement map is used for its superiority in preserving vessel edges while the given image intensity information will guarantee a correct feature's segmentation. We evaluate the performance of the proposed model by applying it to three public retinal image datasets (two datasets of color fundus photography and one fluorescein angiography dataset). The proposed model outperforms its competitors when compared with other widely used unsupervised and supervised methods. For example, the sensitivity (0.742), specificity (0.982) and accuracy (0.954) achieved on the DRIVE dataset are very close to those of the second observer's annotations.

Image denoising has been a research topic deeply investigated within the last two decades. Excellent results have been obtained by using such models as the total variation (TV) minimization by Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259–268], which involves solving a second order PDE. In more recent years some effort has been made [Y.-L. You and M. Kaveh, IEEE Trans. Image Process., 9 (2000), pp. 1723–1730; M. Lysaker, S. Osher, and X.-C. Tai, IEEE Trans. Image Process., 13 (2004), pp. 1345–1357; M. Lysaker, A. Lundervold, and X.-C. Tai, IEEE Trans. Image Process., 12 (2003), pp. 1579–1590; Y. Chen, S. Levine, and M. Rao, SIAM J. Appl. Math., 66 (2006), pp. 1383–1406] in improving these results by using higher order models, particularly to avoid the staircase effect inherent to the solution of the TV model. However, the construction of stable numerical schemes for the resulting PDEs arising from the minimization of such high order models has proved to be very difficult due to high nonlinearity and stiffness. In this paper, we study a curvature-based energy minimizing model [W. Zhu and T. F. Chan, Image Denoising Using Mean Curvature, preprint, http://www.math.nyu.edu/ wzhu/], for which one has to solve a fourth order PDE. For this model we develop two new algorithms: a stabilized fixed point method and, based upon this, an efficient nonlinear multigrid (MG) algorithm. We will show numerical experiments to demonstrate the very good performance of our MG algorithm.

Variational techniques for gray-scale image denoising have been deeply investigated for many years; however, little research has been done for the vector-valued denoising case and the very few existent works are all based on total-variation regularization. It is known that total-variation models for denoising gray-scaled images suffer from staircasing effect and there is no reason to suggest this effect is not transported into the vector-valued models. High-order models, on the contrary, do not present staircasing. In this paper, we introduce three high-order and curvature-based denoising models for vector-valued images. Their properties are analyzed and a fast multigrid algorithm for the numerical solution is provided. AMS subject classifications: 68U10, 65F10, 65K10.

In this paper, we present two related multigrid algorithms for multiphase image segmentation. Algorithm I solves the model by Vese-Chan. We first generalize our recently developed multigrid method to this multiphase segmentation model (MG1); we also give a local Fourier analysis for the local smoother which leads to a new and more effective smoother. Although MG1 is found many magnitudes faster than the fast method of additive operator splitting (AOS), both algorithms are not robust with regard to the initial guess. To overcome this dependence on the initial guess, we consider a hierarchical segmentation model which achieves multiphase segmentation by repeated use of the Chan-Vese two-phase model; our Algorithm II solves this model by a multigrid algorithm (MG2). Numerical experiments show that both algorithms are efficient and in particular MG2 is more robust than MG1 with respect to initial guesses. AMS subject classifications: 68U10, 65F10, 65K10.

Abstract In this paper we consider the parallelization of the generation and iterative solution of coupled linear systems modelling the interaction of an acoustic field in a fluid medium with an elastic structure immersed in the fluid. The particular case studied is that of a hollow steel sphere in water. The aim of the work is to speed up the generation and solution of the systems. We describe the methods used, which involve special sparse storage arrangements and a novel application of a sparse approximate inverse preconditioning technique, and present results showing that the methods are very effective in terms of speeding up the generation and iterative solution of the systems. Copyright © 2007 John Wiley & Sons, Ltd.