![Example for non-convexity of the quadratic elastic regularization functional S quad in (2.7). Values of the regularization functional are plotted for the transformations y c (x) = (cx 1 , cx 2 , x 3 ) and c ∈ [−1.5, 1.5]; see (2.8). The plot has a double-well shape and is thus not convex.](https://www.researchgate.net/profile/Lars-Ruthotto/publication/267143075/figure/fig3/AS:669433915797508@1536616891238/Example-for-non-convexity-of-the-quadratic-elastic-regularization-functional-S-quad-in_Q320.jpg)
![Registration results for an academic 2D example by Christensen [16]. The template image is registered to the reference using SSD distance and hyperelastic regularization functional. Final transformation (top row) and transformed template images (bottom row) are shown for different triangulations (left to right). Although both images are designed to be symmetrical, preferred directions are observable for M 2,bu and M 2,bu .](https://www.researchgate.net/profile/Lars-Ruthotto/publication/267143075/figure/fig5/AS:669433915797509@1536616891264/Registration-results-for-an-academic-2D-example-by-Christensen-16-The-template-image_Q320.jpg)
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Hyperelastic image registration. Theory, numerical methods, and applications
Abstract and Figures
Finding geometrical correspondences between two images, called image registration, is one of the numerous challenging problems in image processing. Commonly, image registration is phrased as a variational problem that is known to be ill-posed. Thus, regularization is used to ensure the existence of solutions, to introduce prior knowledge about the expected solution, and to increase the robustness against noise. In this thesis, we examine a regularization functional based on hyperelasticity. This work gives a comprehensive overview of theory, numerical methods, and applications of hyperelastic regularization functionals in image registration. We transfer existence results for polyconvex functionals from variational calculus to image registration. Thereby, we show that solutions to hyperelastic registration problems are guaranteed to be one-to-one and that the regularizer is well-suited for volume- and local rigidity constraints. We describe a numerical method that combines a Galerkin finite element approach with a multi-level image registration framework. The variational problem is solved in the space of continuous and piecewise linear transformations. Thus, our regularization functional is computed exactly and numerical solutions are one-to-one as guaranteed by the theory. We further give estimates for discretization errors and describe our implementation based on the registration toolbox Flexible Algorithms for Image Registration (FAIR). We outline the great potential of hyperelastic registration methods based on three applications from Positron Emission Tomography (PET), Echo-Planar Imaging (EPI) and Dynamic Contrast Enhanced Magnetic Resonance Imaging (MRI). This thesis establishes techniques from variational calculus and numerical analysis into image registration tools that are useful for applications in medical imaging. Our findings motivate further investigation and wider application of hyperelastic regularization techniques.
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... Here, compared with the conventional variational model for image registration, the proposed model contains not only an infinite dimensional variable but also a finite dimensional variable. To employ the direct method in the calculus of variations, we make a modification for [46] and introduce a product space to complete the proof. 3. Based on the generalized Gauss-Newton framework, we develop an iterative scheme to numerically solve the proposed model. ...
... Proof Firstly, by Lemma 1 in Sect. 3.3 of [46], there exist constants β 1 > 0 and γ 1 ∈ R such that for all y ∈ A it holds ...
... (22) Proof Firstly, by Lemma 2 in Sect. 3.3 of [46], we have ...
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.
... Here, compared with the conventional variational model for image registration, the proposed model contains not only an infinite dimensional variable but also a finite dimensional variable. To employ the direct method in the calculus of variations, we make a modification for [54] and introduce a product space to complete the proof. 3. Based on the generalized Gauss-Newton framework, we develop an iterative scheme to numerically solve the proposed model. ...
... Proof Firstly, by Lemma 1 in Section 3.3 of [54], there exist constants β 1 > 0 and γ 1 ∈ R such that for all y ∈ A it holds ...
... Proof Firstly, by Lemma 2 in Section 3.3 of [54], we have ...
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 [10], 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.
... This is the reason for introducing S as a Tikhonov regularizing term that 'convexifies' the functional in (1). Well-posedness of the regularized problem (1) depends naturally on the choice of D and S. In most of the standard settings, one can show existence of optimal solution, see [42]. On the other hand, showing uniqueness is rather difficult or even impossible. ...
... A natural solution would be to apply the projection operator. However, this approach may lead to instabilities as observed in [42]. Therefore, we follow the method proposed in [24]. ...
... Instead of discretizing det(·) in Ω c we calculate the approximated change of volume for each cell. This method is accurate up to second-order and allows to easily detect instabilities [42]. More precisely, let x ∈ Ω c and V (x) denote the cell centered at point x = (x 1 , x 2 ). ...
We establish a new framework for image registration, which is based on linear elasticity and optimal mass transportation theory. We combine these two arguments in order to obtain a PDE constrained optimization problem that is analytically investigated and further discretized with the finite difference method and solved by an inexact SQP algorithm. This requires to solve in each step a large sparse linear system, which has a saddle point form. Motivated by stability arguments we use a fully staggered grid for the discretization of the displacement vector field. Artificial and real world examples are presented to underline the numerical robustness of the method.
We propose a new nonlinear image registration model which is based on nonlinear elastic regularization and unbiased registration. The nonlinear elastic and the unbiased regularization terms are simpli-fied using the change of variables by introducing an unknown that ap-proximates the Jacobian matrix of the displacement field. This reduces the minimization to involve linear differential equations. In contrast to recently proposed unbiased fluid registration method, the new model is written in a unified variational form and is minimized using gradient descent. As a result, the new unbiased nonlinear elasticity model is com-putationally more efficient and easier to implement than the unbiased fluid registration. The unbiased large-deformation nonlinear elasticity method was tested using volumetric serial magnetic resonance images and shown to have some advantages for medical imaging applications.
The finite element method is the most powerful general-purpose technique for computing accurate solutions to partial differential equations. Understanding and Implementing the Finite Element Method is essential reading for those interested in understanding both the theory and the implementation of the finite element method for equilibrium problems. This book contains a thorough derivation of the finite element equations as well as sections on programming the necessary calculations, solving the finite element equations, and using a posteriori error estimates to produce validated solutions. Accessible introductions to advanced topics, such as multigrid solvers, the hierarchical basis conjugate gradient method, and adaptive mesh generation, are provided. Each chapter ends with exercises to help readers master these topics. Understanding and Implementing the Finite Element Method includes a carefully documented collection of MATLAB programs implementing the ideas presented in the book. Readers will benefit from a careful explanation of data structures and specific coding strategies and will learn how to write a finite element code from scratch. Students can use the MATLAB codes to experiment with the method and extend them in various ways to learn more about programming finite elements.
Motivated by the need to locate and identify objects in three dimensional CT images, an optimal registration method for matching two and three dimensional deformed images has been developed. This method was used to find optimal mappings between CT images and an atlas image of the same anatomy. Using these mappings, object boundaries from the atlas was superimposed on the CT images.^ A cost function of the form DEFORMATION-SIMILARITY is associated with each mapping between the two images. The mapping obtained by our registration process is optimal with respect to this cost function. The registration process simulates a model in which one of the images made from an elastic material is deformed until it matches the other image. The cross correlation function which measures the similarity between the two images serves as a potential function from which the forces required to deform the image are derived. The deformation part of the cost function is measured by the strain energy of the deformed image. Therefore, the cost function of a mapping is given in this model by the total energy of the elastic image.^ The optimal mapping is obtained by finding the equilibrium state of the elastic image, which by definition corresponds to a local minimum of the total energy. The equilibrium state is obtained by solving a set of partial differential equations taken from the linear theory of elasticity. These equations are solved iteratively using the finite differences approximation on a grid which describes the mapping.^ The image function in a spherical region around each grid point is described by its projections on a set of orthogonal functions. The cross correlation function between the image functions in two regions is computed from these projections which, serve as the components of a feature vector associated with the grid points. In each iteration step of the process, the values of the projections are modified according to the currently approximated deformation.^ The method was tested by registering several two and three dimensional image pairs. It can also be used to obtain the optimal mapping between two regions from a set of corresponding points (with and without error estimates) in these regions.
Image registration is one of the most challenging tasks within digital imaging, in particular in medical imaging. Typically, the underlying problems are high dimensional and demand for fast and efficient numerical schemes. Here, we propose a novel scheme for automatic image registration by introducing a specific regularizing term. The new scheme is called diffusion registration since its implementation is based on the solution of a diffusion type partial differential equation. The main ingredient for a fast implementation of the diffusion registration is the so-called additive (Operator Splitting (AOS) Scheme. The AOS-scheme is known to be as accurate as a conventional semi-implicit scheme and has a linear complexity with respect to the size of the images. We present a proof of these properties based purely on matrix analysis. The performance of the new scheme is demonstrated for a typical medical registration problem. It is worth noticing that the diffusion registration is extremely well-suited for a parallel implementation. Finally, we also draw a connection to Thirion’s demon based approach.
1. The direct method in the calculus of variations.- 2. Minimum problems for integral functionals.- 3. Relaxation.- 4. ?-convergence and K-convergence.- 5. Comparison with pointwise convergence.- 6. Some properties of ?-limits.- 7. Convergence of minima and of minimizers.- 8. Sequential characterization of ?-limits.- 9. ?-convergence in metric spaces.- 10. The topology of ?-convergence.- 11. ?-convergence in topological vector spaces.- 12. Quadratic forms and linear operators.- 13. Convergence of resolvents and G-convergence.- 14. Increasing set functions.- 15. Lower semicontinuous increasing functionals.- 16. -convergence of increasing set functional.- 17. The topology of -convergence.- 18. The fundamental estimate.- 19. Local functionals and the fundamental estimate.- 20. Integral representation of ?-limits.- 21. Boundary conditions.- 22. G-convergence of elliptic operators.- 23. Translation invariant functional.- 24. Homogenization.- 25. Some examples in homogenization.- Guide to the literature.
In this paper we present a new and general framework for image registration when having additional constraints on the transformation. We demonstrate that registration without constraints leads to arbitrary results depending on the regularization and in particular produces non-physical deformations. Having additional constraints based on the images introduces prior knowledge and contributes to reliability and uniqueness of the registration. In particular we consider recently proposed locally rigid transfor-mations as an example. In contrast to existing approaches, we propose a constraint optimization framework and do not rely on penalty methods that do not guarantee feasible solutions.