Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> 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. <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Matching of locally variant data to an explicit 3-dimensional pictorial model is developed for X-ray computed tomography scans of the human brain, where the model is a voxel representation of an anatomical human brain atlas. The matching process is 3-dimensional without any preference given to the slicing plane. After global alignment the brain atlas is deformed like a piece of rubber, without tearing or folding. Deformation proceeds step-by-step in a coarse-to-fine strategy, increasing the local similarity and global coherence. The assumption underlying this approach is that all normal brains, at least at a certain level of representation, have the same topological structure, but may differ in shape details. Results show that we can account for these differences. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In the current work we integrate well established techniques from finite deformation continuum mechanics with concepts from pattern recognition and image processing to develop a new finite element (FE) tool that combines image-based data with mechanics. Results track the deformation of material continua in the presence of unknown forces and/or material properties by using image-based data to provide the additional required information. The deformation field is determined from a variational problem that combines both the mechanics and models of the imaging sensors. A nonlinear FE approach is used to approximate the solution of the coupled problem. Results can be applied to (1) track the motion of deforming material and/or, (2) morphological warping of template images or patterns. 2D example results are provided for problems of the second type. One of the present examples was motivated primarily by a problem in medical imaging--mapping intersubject geometrical differences in human anatomical structures--with specific results given for the mapping 2D slices of the human distal femur based on X-ray computed tomographic images. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> A general automatic approach is presented for accommodating local shape variation when mapping a two-dimensional (2-D) or three-dimensional (3-D) template image into alignment with a topologically similar target image. Local shape variability is accommodated by applying a vector-field transformation to the underlying material coordinate system of the template while constraining the transformation to be smooth (globally positive definite Jacobian). Smoothness is guaranteed without specifically penalizing large-magnitude deformations of small subvolumes by constraining the transformation on the basis of a Stokesian limit of the fluid-dynamical Navier-Stokes equations. This differs fundamentally from quadratic penalty methods, such as those based on linearized elasticity or thin-plate splines, in that stress restraining the motion relaxes over time allowing large-magnitude deformations. Kinematic nonlinearities are inherently necessary to maintain continuity of structures during large-magnitude deformations, and are included in all results. After initial global registration, final mappings are obtained by numerically solving a set of nonlinear partial differential equations associated with the constrained optimization problem. Automatic regridding is performed by propagating templates as the nonlinear transformations evaluated on a finite lattice become singular. Application of the method to intersubject registration of neuroanatomical structures illustrates the ability to account for local anatomical variability. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper offers a new fast algorithm for non-rigid Viscous Fluid Registration of medical images that is at least an order of magnitude faster than the previous method by Christensen et al. [4]. The core algorithm in the fluid registration method is based on a linear elastic deformation of the velocity field of the fluid. Using the linearity of this deformation we derive a convolution filter which we use in a scalespace framework. We also demonstrate that the ’demon’-based registration method of Thirion [13] can be seen as an approximation to the fluid registration method and point to possible problems. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> The development of algorithms for the spatial transformation and registration of tomographic brain images is a key issue in several clinical and basic science medical applications, including computer-aided neurosurgery, functional image analysis, and morphometrics. This paper describes a technique for the spatial transformation of brain images, which is based on elastically deformable models. A deformable surface algorithm is used to find a parametric representation of the outer cortical surface and then to define a map between corresponding cortical regions in two brain images. Based on the resulting map, a three-dimensional elastic warping transformation is then determined, which brings two images into register. This transformation models images as inhomogeneous elastic objects which are deformed into registration with each other by external force fields. The elastic properties of the images can vary from one region to the other, allowing more variable brain regions, such as the ventricles, to deform more freely than less variable ones. Finally, the framework of prestrained elasticity is used to model structural irregularities, and in particular the ventricular expansion occurring with aging or diseases, and the growth of tumors. Performance measurements are obtained using magnetic resonance images. <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper presents diffeomorphic transformations of three-dimensional (3-D) anatomical image data of the macaque occipital lobe and whole brain cryosection imagery and of deep brain structures in human brains as imaged via magnetic resonance imagery. These transformations are generated in a hierarchical manner, accommodating both global and local anatomical detail. The initial low-dimensional registration is accomplished by constraining the transformation to be in a low-dimensional basis. The basis is defined by the Green's function of the elasticity operator placed at predefined locations in the anatomy and the eigenfunctions of the elasticity operator. The high-dimensional large deformations are vector fields generated via the mismatch between the template and target-image volumes constrained to be the solution of a Navier-Stokes fluid model. As part of this procedure, the Jacobian of the transformation is tracked, insuring the generation of diffeomorphisms. It is shown that transformations constrained by quadratic regularization methods such as the Laplacian, biharmonic, and linear elasticity models, do not ensure that the transformation maintains topology and, therefore, must only be used for coarse global registration. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> The purpose of this paper is to present a survey of recent (published in 1993 or later) publications concerning medical image registration techniques. These publications will be classified according to a model based on nine salient criteria, the main dichotomy of which is extrinsic versus intrinsic methods. The statistics of the classification show definite trends in the evolving registration techniques, which will be discussed. At this moment, the bulk of interesting intrinsic methods is based on either segmented points or surfaces, or on techniques endeavouring to use the full information content of the images involved. <s> BIB008 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In this paper, we present the concept of diffusing models to perform image-to-image matching. Having two images to match, the main idea is to consider the objects boundaries in one image as semi-permeable membranes and to let the other image, considered as a deformable grid model, diffuse through these interfaces, by the action of effectors situated within the membranes. We illustrate this concept by an analogy with Maxwell's demons. We show that this concept relates to more traditional ones, based on attraction, with an intermediate step being optical flow techniques. We use the concept of diffusing models to derive three different non-rigid matching algorithms, one using all the intensity levels in the static image, one using only contour points, and a last one operating on already segmented images. Finally, we present results with synthesized deformations and real medical images, with applications to heart motion tracking and three-dimensional inter-patients matching. <s> BIB009 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> A computational framework has evolved in which individual anatomies are modeled as warped versions of a canonical representation of the anatomy. To realize this framework, the method of elastic matching was invented for determining the spatial mapping between a three-dimensional image pair in which one image volume is modeled as an elastic continuum that is deformed to match the appearance of the second volume. This chapter describes the basic concepts underlying the original method and their probabilistic generalization within a Bayesian framework. For brain warping, a number of effective similarity metrics exist, whereas there is comparatively little known about the kinds of prior models that are specifically suited to the problem (and its applications) or how their influence on the solution, relative to the likelihood, should be determined. The resolution of these issues directly impacts the viability of any matching procedure, Bayesian or otherwise, but is best addressed within a probabilistic setting. To reduce the likelihood of false matches that arise when the local variations in anatomy are very large in magnitude, a standard approach is to solve the problem at different spatial scales. <s> BIB010 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> The "Demons Algorithm" in increasingly used for non-rigid registrationof 3D medical images. However, if it is fast an d usually accu- rate, the algorithm is based onin tuitive ideas about image registration and it is difficult to predict when it will fail and why. We show in this paper that this algorithm canbe considered as anapproximationof a sec- ond order gradient descent on the sum of square of intensity differences criterion. We also reformulate Gaussian and physical model regulariza- tions as minimization problems. Experimental results on synthetic and 3D Ultrasound images show that this formalization helps identifying the weak points of the algorithm and offers new research openings. <s> BIB011 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper describes two new atlas-based methods of 2D single modality non-rigid registration using the combined power of physical and statistical shape models. The transformations are constrained to be consistent with the physical properties of deformable elastic solids in the first method and those of viscous fluids in the second, to maintain smoothness and continuity. A Bayesian formulation, based on each physical model, an intensity similarity measure, and statistical shape information embedded in corresponding boundary points, is employed to derive more accurate and robust approaches to non-rigid registration. A dense set of forces arises from the intensity similarity measure to accommodate complex anatomical details. A sparse set of forces constrains consistency with statistical shape models derived from a training set. A number of experiments were performed on both synthetic and real medical images of the brain and heart to evaluate the approaches. It is shown that statistical boundary shape information significantly augments and improves physical model-based non-rigid registration and the two methods we present each have advantages under different conditions. <s> BIB012 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper presents a new method for image registration based on jointly estimating the forward and reverse transformations between two images while constraining these transforms to be inverses of one another. This approach produces a consistent set of transformations that have less pairwise registration error, i.e., better correspondence, than traditional methods that estimate the forward and reverse transformations independently. The transformations are estimated iteratively and are restricted to preserve topology by constraining them to obey the laws of continuum mechanics. The transformations are parameterized by a Fourier series to diagonalize the covariance structure imposed by the continuum mechanics constraints and to provide a computationally efficient numerical implementation. Results using a linear elastic material constraint are presented using both magnetic resonance and X-ray computed tomography image data. The results show that the joint estimation of a consistent set of forward and reverse transformations constrained by linear-elasticity give better registration results than using either constraint alone or none at all. <s> BIB013 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Radiological images are increasingly being used in healthcare and medical research. There is, consequently, widespread interest in accurately relating information in the different images for diagnosis, treatment and basic science. This article reviews registration techniques used to solve this problem, and describes the wide variety of applications to which these techniques are applied. Applications of image registration include combining images of the same subject from different modalities, aligning temporal sequences of images to compensate for motion of the subject between scans, image guidance during interventions and aligning images from multiple subjects in cohort studies. Current registration algorithms can, in many cases, automatically register images that are related by a rigid body transformation (i.e. where tissue deformation can be ignored). There has also been substantial progress in non-rigid registration algorithms that can compensate for tissue deformation, or align images from different subjects. Nevertheless many registration problems remain unsolved, and this is likely to continue to be an active field of research in the future. <s> BIB014 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper presents an original method for three-dimensional elastic registration of multimodal images. We propose to make use of a scheme that iterates between correcting for intensity differences between images and performing standard monomodal registration. The core of our contribution resides in providing a method that finds the transformation that maps the intensities of one image to those of another. It makes the assumption that there are at most two functional dependencies between the intensities of structures present in the images to register, and relies on robust estimation techniques to evaluate these functions. We provide results showing successful registration between several imaging modalities involving segmentations, T1 magnetic resonance (MR), T2 MR, proton density (PD) MR and computed tomography (CT). We also argue that our intensity modeling may be more appropriate than mutual information (MI) in the context of evaluating high-dimensional deformations, as it puts more constraints on the parameters to be estimated and, thus, permits a better search of the parameter space. <s> BIB015 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper presents a new image registration algorithm that accommodates locally large nonlinear deformations. The algorithm concurrently estimates the forward and reverse transformations between a pair of images while minimizing the inverse consistency error between the transformations. It assumes that the two images to be registered contain topologically similar objects and were collected using the same imaging modality. The large deformation transformation from one image to the other is accommodated by concatenating a sequence of small deformation transformations. Each incremental transformation is regularized using a linear elastic continuum mechanical model. Results of ten 2D and twelve 3D MR image registration experiments are presented that tested the algorithm’s performance on real brain shapes. For these experiments, the inverse consistency error was reduced on average by 50 times in 2D and 30 times in 3D compared to the viscous fluid registration algorithm. <s> BIB016 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> A variational method for nonrigid registration of multimodal image data is presented. A suitable deformation will be determined via the minimization of a morphological, i.e., contrast invariant, matching functional along with an appropriate regularization energy. The aim is to correlate the morphologies of a template and a reference image under the deformation. Mathematically, the morphology of images can be described by the entity of level sets of the image and hence by its Gauss map. A class of morphological matching functionals is presented which measure the defect of the template Gauss map in the deformed state with respect to the deformed Gauss map of the reference image. The problem is regularized by considering a nonlinear elastic regularization energy. Existence of homeomorphic, minimizing deformation is proved under assumptions on the class of admissible deformations. With respect to actual medical applications, suitable generalizations of the matching energies and the boundary conditions are pre... <s> BIB017 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Abstract We propose a multimodal free-form registration algorithm based on maximization of mutual information. The warped image is modeled as a viscous fluid that deforms under the influence of forces derived from the gradient of the mutual information registration criterion. Parzen windowing is used to estimate the joint intensity probability of the images to be matched. The method is evaluated for non-rigid inter-subject registration of MR brain images. The accuracy of the method is verified using simulated multi-modal MR images with known ground truth deformation. The results show that the root mean square difference between the recovered and the ground truth deformation is smaller than 1 voxel. We illustrate the application of the method for atlas-based brain tissue segmentation in MR images in case of gross morphological differences between atlas and patient images. <s> BIB018 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper aims to present a review of recent as well as classic image registration methods. Image registration is the process of overlaying images (two or more) of the same scene taken at different times, from different viewpoints, and/or by different sensors. The registration geometrically align two images (the reference and sensed images). The reviewed approaches are classified according to their nature (areabased and feature-based) and according to four basic steps of image registration procedure: feature detection, feature matching, mapping function design, and image transformation and resampling. Main contributions, advantages, and drawbacks of the methods are mentioned in the paper. Problematic issues of image registration and outlook for the future research are discussed too. The major goal of the paper is to provide a comprehensive reference source for the researchers involved in image registration, regardless of particular application areas. q 2003 Elsevier B.V. All rights reserved. <s> BIB019 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> As image registration becomes more and more central to many biomedical imaging applications, the efficiency of the algorithms becomes a key issue. Image registration is classically performed by optimizing a similarity criterion over a given spatial transformation space. Even if this problem is considered as almost solved for linear registration, we show in this paper that some tools that have recently been developed in the field of vision-based robot control can outperform classical solutions. The adequacy of these tools for linear image registration leads us to revisit non-linear registration and allows us to provide interesting theoretical roots to the different variants of Thirion's demons algorithm. This analysis predicts a theoretical advantage to the symmetric forces variant of the demons algorithm. We show that, on controlled experiments, this advantage is confirmed, and yields a faster convergence. <s> BIB020 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In this paper, we first propose a new subdivision of the image information axis used for the classification of nonrigid registration algorithms. Namely, we introduce the notion of iconic feature based (IFB) algorithms, which lie between geometrical and standard intensity based algorithms for they use both an intensity similarity measure and a geometrical distance. Then we present a new registration energy for IFB registration that generalizes some of the existing techniques. We compare our algorithm with other registration approaches, and show the advantages of this energy. Besides, we also present a fast technique for the computation of local statistics between images, which turns out to be useful on pairs of images having a complex, nonstationary relationship between their intensities, as well as an hybrid regularization scheme mixing elastic and fluid components. The potential of the algorithm is finally demonstrated on a clinical application, namely deep brain stimulation of a Parkinsonian patient. Registration of pre- and immediate postoperative MR images allow to quantify the range of the deformation due to pneumocephalus over the entire brain, thus yielding to measurement of the deformation around the preoperatively computed stereotactic targets. <s> BIB021 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In this paper, several vision-based robot control methods are classified following an analogy with well known minimization methods. Comparing the rate of convergence between minimization algorithms helps us to understand the difference of performance of the control schemes. In particular, it is shown that standard vision-based control methods have in general low rates of convergence. Thus, the performance of vision-based control could be improved using schemes which perform like the Newton minimization algorithm that has a high convergence rate. Unfortunately, the Newton minimization method needs the computation of second derivatives that can be ill-conditioned causing convergence problems. In order to solve these problems, this paper proposes two new control schemes based on efficient second-order minimization techniques. <s> BIB022 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Abstract Multi-subject non-rigid registration algorithms using dense deformation fields often encounter cases where the transformation to be estimated has a large spatial variability. In these cases, linear stationary regularization methods are not sufficient. In this paper, we present an algorithm that uses a priori information about the nature of imaged objects in order to adapt the regularization of the deformations. We also present a robustness improvement that gives higher weight to those points in images that contain more information. Finally, a fast parallel implementation using networked personal computers is presented. In order to improve the usability of the parallel software by a clinical user, we have implemented it as a grid service that can be controlled by a graphics workstation embedded in the clinical environment. Results on inter-subject pairs of images show that our method can take into account the large variability of most brain structures. The registration time for images of size 256 × 256 × 124 is 5 min on 15 standard PCs. A comparison of our non-stationary visco-elastic smoothing versus solely elastic or fluid regularizations shows that our algorithm converges faster towards a more optimal solution in terms of accuracy and transformation regularity. <s> BIB023 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> This paper presents a new approach to inverse consistent image registration. A uni-directional algorithm is developed using symmetric cost functionals and regularizers. Instead of enforcing inverse consistency using an additional penalty that penalizes inconsistency error, the new algorithm directly models the backward mapping by inverting the forward mapping. The resulting minimization problem can then be solved uni-directionally involving only the forward mapping, without optimizing in the backward direction. Lastly, we evaluated the algorithm by applying it to the serial MRI scans of a clinical case of semantic dementia. The statistical distributions of the local volume change (Jacobian) maps were examined by considering the Kullback-Liebler distances on the material density functions. Contrary to common belief, the values of any non-trivial Jacobian map do not follow a log-normal distribution with zero mean. Statistically significant differences were detected between consistent versus inconsistent matching when permutation tests were performed on the resulting deformation maps <s> BIB024 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In inter-subject registration, one often lacks a good model of the transformation variability to choose the optimal regularization. Some works attempt to model the variability in a statistical way, but the re-introduction in a registration algorithm is not easy. In this paper, we interpret the elastic energy as the distance of the Green-St Venant strain tensor to the identity, which reflects the deviation of the local deformation from a rigid transformation. By changing the Euclidean metric for a more suitable Riemannian one, we define a consistent statistical framework to quantify the amount of deformation. In particular, the mean and the covariance matrix of the strain tensor can be consistently and efficiently computed from a population of non-linear transformations. These statistics are then used as parameters in a Mahalanobis distance to measure the statistical deviation from the observed variability, giving a new regularization criterion that we called the statistical Riemannian elasticity. This new criterion is able to handle anisotropic deformations and is inverse-consistent. Preliminary results show that it can be quite easily implemented in a non-rigid registration algorithms. <s> BIB025 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Registration using models of compressible viscous fluids has not found the general application of some other techniques (e.g., free-form-deformation (FFD)) despite its ability to model large diffeomorphic deformations. We report on a multi-resolution fluid registration algorithm which improves on previous work by (a) directly solving the Navier–Stokes equation at the resolution of the images, (b) accommodating image sampling anisotropy using semi-coarsening and implicit smoothing in a full multi-grid (FMG) solver and (c) exploiting the inherent multi-resolution nature of FMG to implement a multi-scale approach. Evaluation is on five magnetic resonance (MR) breast images subject to six biomechanical deformation fields over 11 multi-resolution schemes. Quantitative assessment is by tissue overlaps and target registration errors and by registering using the known correspondences rather than image features to validate the fluid model. Context is given by comparison with a validated FFD algorithm and by application to images of volunteers subjected to large applied deformation. The results show that fluid registration of 3D breast MR images to sub-voxel accuracy is possible in minutes on a 1.6 GHz Linux-based Athlon processor with coarse solutions obtainable in a few tens of seconds. Accuracy and computation time are comparable to FFD techniques validated for this application. <s> BIB026 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Maps of local tissue compression or expansion are often computed by comparing magnetic resonance imaging (MRI) scans using nonlinear image registration. The resulting changes are commonly analyzed using tensor-based morphometry to make inferences about anatomical differences, often based on the Jacobian map, which estimates local tissue gain or loss. Here, we provide rigorous mathematical analyses of the Jacobian maps, and use themto motivate a new numerical method to construct unbiased nonlinear image registration. First, we argue that logarithmic transformation is crucial for analyzing Jacobian values representing morphometric differences. We then examine the statistical distributions of log-Jacobian maps by defining the Kullback-Leibler (KL) distance on material density functions arising in continuum-mechanical models. With this framework, unbiased image registration can be constructed by quantifying the symmetric KL-distance between the identity map and the resulting deformation. Implementation details, addressing the proposed unbiased registration as well as the minimization of symmetric image matching functionals, are then discussed and shown to be applicable to other registration methods, such as inverse consistent registration. In the results section, we test the proposed framework, as well as present an illustrative application mapping detailed 3-D brain changes in sequential magnetic resonance imaging scans of a patient diagnosed with semantic dementia. Using permutation tests, we show that the symmetrization of image registration statistically reduces skewness in the log-Jacobian map. <s> BIB027 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Nonparametric image registration algorithms use deformation fields to define nonrigid transformations relating two images. Typically, these algorithms operate by successively solving linear systems of partial differential equations. These PDE systems arise by linearizing the Euler-Lagrange equations associated with the minimization of a functional defined to contain an image similarity term and a regularizer. Iterative linear system solvers can be used to solve the linear PDE systems, but they can be extremely slow. Some faster techniques based on Fourier methods, multigrid methods, and additive operator splitting, exist for solving the linear PDE systems for specific combinations of regularizers and boundary conditions. In this paper, we show that Fourier methods can be employed to quickly solve the linear PDE systems for every combination of standard regularizers (diffusion, curvature, elastic, and fluid) and boundary conditions (Dirichlet, Neumann, and periodic). <s> BIB028 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> We propose a non-parametric diffeomorphic image registration algorithm based on Thirion's demons algorithm. The demons algorithm can be seen as an optimization procedure on the entire space of displacement fields. The main idea of our algorithm is to adapt this procedure to a space of diffeomorphic transformations. In contrast to many diffeomorphic registration algorithms, our solution is computationally efficient since in practice it only replaces an addition of free form deformations by a few compositions. Our experiments show that in addition to being diffeomorphic, our algorithm provides results that are similar to the ones from the demons algorithm but with transformations that are much smoother and closer to the true ones in terms of Jacobians. <s> BIB029 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> We apply an information-theoretic cost metric, the symmetrized Kullback-Leibler (sKL) divergence, or J-divergence, to fluid registration of diffusion tensor images. The difference between diffusion tensors is quantified based on the sKL-divergence of their associated probability density functions (PDFs). Three-dimensional DTI data from 34 subjects were fluidly registered to an optimized target image. To allow large image deformations but preserve image topology, we regularized the flow with a large-deformation diffeomorphic mapping based on the kinematics of a Navier-Stokes fluid. A driving force was developed to minimize the J-divergence between the deforming source and target diffusion functions, while reorienting the flowing tensors to preserve fiber topography. In initial experiments, we showed that the sKL-divergence based on full diffusion PDFs is adaptable to higher-order diffusion models, such as high angular resolution diffusion imaging (HARDI). The sKL-divergence was sensitive to subtle differences between two diffusivity profiles, showing promise for nonlinear registration applications and multisubject statistical analysis of HARDI data. <s> BIB030 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Modern morphometric studies use non-linear image registration to compare anatomies and perform group analysis. Recently, log-Euclidean approaches have contributed to promote the use of such computational anatomy tools by permitting simple computations of statistics on a rather large class of invertible spatial transformations. In this work, we propose a non-linear registration algorithm perfectly fit for log-Euclidean statistics on diffeomorphisms. Our algorithm works completely in the log-domain, i.e. it uses a stationary velocity field. This implies that we guarantee the invertibility of the deformation and have access to the true inverse transformation. This also means that our output can be directly used for log-Euclidean statistics without relying on the heavy computation of the log of the spatial transformation. As it is often desirable, our algorithm is symmetric with respect to the order of the input images. Furthermore, we use an alternate optimization approach related to Thirion's demons algorithm to provide a fast non-linear registration algorithm. First results show that our algorithm outperforms both the demons algorithm and the recently proposed diffeomorphic demons algorithm in terms of accuracy of the transformation while remaining computationally efficient. <s> BIB031 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> The main limitation of the original demons image registration algorithm is the need to ensure that the grey level of corresponding voxels remains the same in both the images to register. This difficulty may be overcome by the use of more robust similarity measures, whenever they may be expressed as the sum over the entire image of a local energy. We demonstrate the convenience of a local optimisation strategy of this energy instead of the optimisation of the global metric, and suggest a methodology for the use of more general similarity measures. <s> BIB032 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In this paper, we propose a generic framework for inter-subject non-linear registration of 4D time-series images. In this framework, spatio-temporal registration is defined by mapping trajectories of physical points as opposed to spatial registration that solely aims at mapping homologous points. First, we determine the trajectories we want to register in each sequence using a motion tracking algorithm based on the Diffeomorphic Demons algorithm. Then, we perform simultaneously pairwise registrations of corresponding time-points with the constraint to map the same physical points over time. We show this trajectory registration can be formulated as a multichannel registration of 3D images. We solve it using the Diffeomorphic Demons algorithm extended to vector-valued 3D images. This framework is applied to the inter-subject non-linear registration of 4D cardiac CT sequences. <s> BIB033 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Whenever images taken at different times, from different viewpoints, and/or by different sensors need to be compared, merged, or integrated, image registration is required. Registration, also known as alignment, fusion, or warping, is the process of transforming data into a common reference frame. This book provides an overview of state-of-the-art registration techniques from theory to practice, plus numerous exercises designed to enhance readers understanding of the principles and mechanisms of the described techniques. It also provides, via a supplementary Web page, free access to FAIR.m, a package that is based on the MATLAB software environment, which enables readers to experiment with the proposed algorithms and explore the presented examples in more depth. Written from an interdisciplinary point of view, this book will appeal to mathematicians who want to learn more about image registration, medical imaging professionals who want to know more about and explore available imaging techniques, and computer scientists and engineers who want to understand the numerical schemes behind the techniques. The book is also appropriate for use as a course text at the advanced graduate level. Contents: FAIR Listings; FAIR Examples; List of Figures; List of Tables; Preface; Chapter 1: Introduction; Chapter 2: FAIR Concepts; Chapter 3: Image Interpolation; Chapter 4: Transforming Images by Parameterized Transformations; Chapter 5: Landmark-Based Registration; Chapter 6: Parametric Image Registration; Chapter 7: Distance Measures; Chapter 8: Regularization; Chapter 9: Nonparametric Image Registration; Chapter 10: Outlook; Bibliography; Symbols, Acronyms, Index. <s> BIB034 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> The scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in the MATLAB function expm. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Pade approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give a new rounding error analysis that shows the computed Pade approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Pade approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than the expm function in MATLAB 7.0 when the matrix norm exceeds 1, which can amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Pade approximation to the function $x \coth(x)$. This method is found to be essentially a variation of the standard one with weaker supporting error analysis. <s> BIB035 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> As the number of clinical applications requiring nonrigid image registration continues to grow, it is important to design registration algorithms that not only build on the best available theory, but also are computationally efficient. Thirion's Demons algorithm [1] estimates nonrigid deformations by successively estimating force vectors that drive the deformation toward alignment, and then smoothing the force vectors by convolution with a Gaussian kernel. It essentially approximates a deformation under diffusion regularization [2], and it is a popular choice of algorithm for nonrigid registration because of its linear computational complexity and ease of implementation. In this article, we show how the Demons algorithm can be generalized to handle other common regularizers, yielding O(n) algorithms that employ Gaussian convolution for elastic, fluid, and curvature registration. We compare the speed of the proposed algorithms with algorithms based on Fourier methods [3] for registering serial chest CT studies. <s> BIB036 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Thirion's Demons [1] is a popular algorithm for nonrigid image registration because of its linear computational complexity and ease of implementation. It approximately solves the diffusion registration problem [2] by successively estimating force vectors that drive the deformation toward alignment and smoothing the force vectors by Gaussian convolution. In this article, we show how the Demons algorithm can be generalized to allow image-driven locally adaptive regularization [3,4] in a manner that preserves both the linear complexity and ease of implementation of the original Demons algorithm. We show that the proposed algorithm exhibits lower target registration error and requires less computational effort than the original Demons algorithm on the registration of serial chest CT scans of patients with lung nodules. <s> BIB037 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In this paper, we propose the DT-REFinD algorithm for the diffeomorphic nonlinear registration of diffusion tensor images. Unlike scalar images, deforming tensor images requires choosing both a reorientation strategy and an interpolation scheme. Current diffusion tensor registration algorithms that use full tensor information face difficulties in computing the differential of the tensor reorientation strategy and consequently, these methods often approximate the gradient of the objective function. In the case of the finite-strain (FS) reorientation strategy, we borrow results from the pose estimation literature in computer vision to derive an analytical gradient of the registration objective function. By utilizing the closed-form gradient and the velocity field representation of one parameter subgroups of diffeomorphisms, the resulting registration algorithm is diffeomorphic and fast. We contrast the algorithm with a traditional FS alternative that ignores the reorientation in the gradient computation. We show that the exact gradient leads to significantly better registration at the cost of computation time. Independently of the choice of Euclidean or Log-Euclidean interpolation and sum of squared differences dissimilarity measure, the exact gradient achieves better alignment over an entire spectrum of deformation penalties. Alignment quality is assessed with a battery of metrics including tensor overlap, fractional anisotropy, inverse consistency and closeness to synthetic warps. The improvements persist even when a different reorientation scheme, preservation of principal directions, is used to apply the final deformations. <s> BIB038 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Tracking soft tissues in medical images using non-linear image registration algorithms requires methods that are fast and provide spatial transformations consistent with the biological characteristics of the tissues. LogDemons algorithm is a fast non-linear registration method that computes diffeomorphic transformations parameterised by stationary velocity fields. Although computationally efficient, its use for tissue tracking has been limited because of its ad-hoc Gaussian regularisation, which hampers the implementation of more biologically motivated regularisations. In this work, we improve the logDemons by integrating elasticity and incompressibility for soft-tissue tracking. To that end, a mathematical justification of demons Gaussian regularisation is proposed. Building on this result, we replace the Gaussian smoothing by an efficient elastic-like regulariser based on isotropic differential quadratic forms of vector fields. The registration energy functional is finally minimised under the divergence-free constraint to get incompressible deformations. As the elastic regulariser and the constraint are linear, the method remains computationally tractable and easy to implement. Tests on synthetic incompressible deformations showed that our approach outperforms the original logDemons in terms of elastic incompressible deformation recovery without reducing the image matching accuracy. As an application, we applied the proposed algorithm to estimate 3D myocardium strain on clinical cine MRI of two adult patients. Results showed that incompressibility constraint improves the cardiac motion recovery when compared to the ground truth provided by 3D tagged MRI. <s> BIB039 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> The demons algorithm is a fast non-parametric non-rigid registration method. In recent years great efforts ::: have been made to improve the approach; the state of the art version yields symmetric inverse-consistent largedeformation ::: diffeomorphisms. However, only limited work has explored inter-modal similarity metrics, with ::: no practical evaluation on multi-modality data. We present a diffeomorphic demons implementation using the ::: analytical gradient of Normalised Mutual Information (NMI) in a conjugate gradient optimiser. We report the ::: first qualitative and quantitative assessment of the demons for inter-modal registration. Experiments to spatially ::: normalise real MR images, and to recover simulated deformation fields, demonstrate (i) similar accuracy from ::: NMI-demons and classical demons when the latter may be used, and (ii) similar accuracy for NMI-demons on ::: T1w-T1w and T1w-T2w registration, demonstrating its potential in multi-modal scenarios. <s> BIB040 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> We present the Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizors for the modified Demons objective function can be efficiently approximated on the sphere using iterative smoothing. Based on one parameter subgroups of diffeomorphisms, the resulting registration is diffeomorphic and fast. The Spherical Demons algorithm can also be modified to register a given spherical image to a probabilistic atlas. We demonstrate two variants of the algorithm corresponding to warping the atlas or warping the subject. Registration of a cortical surface mesh to an atlas mesh, both with more than 160 k nodes requires less than 5 min when warping the atlas and less than 3 min when warping the subject on a Xeon 3.2 GHz single processor machine. This is comparable to the fastest nondiffeomorphic landmark-free surface registration algorithms. Furthermore, the accuracy of our method compares favorably to the popular FreeSurfer registration algorithm. We validate the technique in two different applications that use registration to transfer segmentation labels onto a new image 1) parcellation of in vivo cortical surfaces and 2) Brodmann area localization in ex vivo cortical surfaces. <s> BIB041 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In this paper, we propose a new large-deformation nonlinear image registration model in three dimensions, based on nonlinear elastic regularization and unbiased registration. Both the nonlinear elastic and the unbiased functionals are simplified introducing, in the modeling, a second unknown that mimics the Jacobian matrix of the displacement vector field, reducing 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 on the corresponding Euler-Lagrange equations. As a result, the new unbiased nonlinear elasticity model is computationally more efficient and easier to implement than the unbiased fluid registration. The model was tested using three-dimensional serial MRI images and shown to have some advantages for computational neuroimaging. <s> BIB042 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> In this paper, we present a new non-parametric combined segmentation and registration method. The shapes to be registered are implicitly modeled with level set functions and the problem is cast as an optimization one, combining a matching criterion based on the active contours without edges for segmentation (Chan and Vese, 2001) [8] and a nonlinear-elasticity-based smoother on the displacement vector field. This modeling is twofold: first, registration is jointly performed with segmentation since guided by the segmentation process; it means that the algorithm produces both a smooth mapping between the two shapes and the segmentation of the object contained in the reference image. Secondly, the use of a nonlinear-elasticity-type regularizer allows large deformations to occur, which makes the model comparable in this point with the viscous fluid registration method. In the theoretical minimization problem we introduce, the shapes to be matched are viewed as Ciarlet-Geymonat materials. We prove the existence of minimizers of the introduced functional and derive an approximated problem based on the Saint Venant-Kirchhoff stored energy for the numerical implementation and solved by an augmented Lagrangian technique. Several applications are proposed to demonstrate the potential of this method to both segmentation of one single image and to registration between two images. <s> BIB043 </s> Deformable Medical Image Registration: A Survey <s> 1) Elastic Body Models: a) Linear Models: <s> Image registration is one of the most challenging problems in image processing, where ill-posedness arises due to noisy data as well as nonuniqueness, and hence the choice of regularization is crucial. This paper presents hyperelasticity as a regularizer and introduces a new and stable numerical implementation. On one hand, hyperelastic registration is an appropriate model for large and highly nonlinear deformations, for which a linear elastic model needs to fail. On the other hand, the hyperelastic regularizer yields very regular and diffeomorphic transformations. While hyperelasticity might be considered as just an additional outstanding regularization option for some applications, it becomes inevitable for applications involving higher order distance measures like mass-preserving registration. The paper gives a short introduction to image registration and hyperelasticity. The hyperelastic image registration problem is phrased in a variational setting, and an existence proof is provided. The focus of th... <s> BIB044
In this case, the image under deformation is modeled as an elastic body. The Navier-Cauchy Partial Differential Equation (PDE) describes the deformation, or BIB008 where is the force field that drives the registration based on an image matching criterion, refers to the rigidity that quantifies the stiffness of the material and is Lamés first coefficient. Broit BIB001 first proposed to model an image grid as an elastic membrane that is deformed under the influence of two forces that compete until equilibrium is reached. An external force tries to deform the image such that matching is achieved while an internal one enforces the elastic properties of the material. Bajcsy and Kovacic BIB002 extended this approach in a hierarchical fashion where the solution of the coarsest scale is up-sampled and used to initialize the finer one. Linear registration was used at the lowest resolution. Gee and Bajscy BIB010 formulated the elastostatic problem in a variational setting. The problem was solved under the Bayesian paradigm allowing for the computation of the uncertainty of the solution as well as for confidence intervals. The finite element method (FEM) was used to infer the displacements for the element nodes, while an interpolation strategy was employed to estimate displacements elsewhere. The order of the interpolating or shape functions, determines the smoothness of the obtained result. Linear elastic models have also been used when registering brain images based on sparse correspondences. Davatzikos BIB006 first used geometric characteristics to establish a mapping between the cortical surfaces. Then, a global transformation was estimated by modeling the images as inhomogeneous elastic objects. Spatially-varying elasticity parameters were used to compensate for the fact that certain structures tend to deform more than others. In addition, a nonzero initial strain was considered so that some structures expand or contract naturally. In general, an important drawback of registration is that when source and target volumes are interchanged, the obtained transformation is not the inverse of the previous solution. In order to tackle this shortcoming, Christensen and Johnson BIB013 proposed to simultaneously estimate both forward and backward transformations, while penalizing inconsistent transformations by adding a constraint to the objective function. Linear elasticity was used as regularization constraint and Fourier series were used to parametrize the transformation. Leow et al. BIB024 took a different approach to tackle the inconsistency problem. Instead of adding a constraint that penalizes the inconsistency error, they proposed a unidirectional approach that couples the forward and backward transformation and provides inverse consistent transformations by construction. The coupling was performed by modeling the backward transformation as the inverse of the forward. This fact was also exploited during the optimization of the symmetric energy by only following the gradient direction of the forward mapping. He and Christensen BIB016 proposed to tackle large deformations in an inverse consistent framework by considering a sequence of small deformation transformations, each modeled by a linear elastic model. The problem was symmetrized by considering a periodic sequence of images where the first (or last) and middle image are the source and target respectively. The symmetric objective function thus comprised terms that quantify the difference between any two successive pairs of images. The inferred incremental transformation maps were concatenated to map one input image to another. b) Nonlinear Models: An important limitation of linear elastic models lies in their inability to cope with large deformations. In order to account for large deformations, nonlinear elastic models have been proposed. These models also guarantee the preservation of topology. Rabbitt et al. BIB003 modeled the deformable image based on hyperelastic material properties. The solution of the nonlinear equations was achieved by local linearization and the use of the Finite Element method. Pennec et al. BIB025 dropped the linearity assumption by modeling the deformation process through the St Venant-Kirchoff elasticity energy that extends the linear elastic model to the nonlinear regime. Moreover, the use of log-Euclidean metrics instead of Euclidean ones resulted in a Riemannian elasticity energy which is inverse consistent. Yanovsky et al. BIB042 proposed a symmetric registration framework based on the St Venant-Kirchoff elasticity. An auxiliary variable was added to decouple the regularization and the matching term. Symmetry was imposed by assuming that the Jacobian determinants of the deformation follow a zero mean, after log-transformation, log-normal distribution BIB027 . Droske and Rumpf BIB017 used an hyperelastic, polyconvex regularization term that takes into account the length, area and volume deformations. Le Guyader and Vese BIB043 presented an approach that combines segmentation and registration that is based on nonlinear elasticity. The authors used a polyconvex regularization energy based on the modeling of the images under deformation as Ciarlet-Geymonat materials  . Burger et al. BIB044 also used a polyconvex regularization term. The authors focused on the numerical implementation of the registration framework. They employed a discretize-then-optimize approach BIB034 that involved the partitioning voxels to 24 tetrahedra. 2) Viscous Fluid Flow Models: In this case, the image under deformation is modeled as a viscous fluid. The transformation is governed by the Navier-Stokes equation that is simplified by assuming a very low Reynold's number flow These models do not assume small deformations, and thus are able to recover large deformations BIB004 . The first term of the Navier-Stokes equation BIB014 , constrains neighboring points to deform similarly by spatially smoothing the velocity field. The velocity field is related to the displacement field as . The velocity field is integrated in order to estimate the displacement field. The second term allows structures to change in mass while and are the viscosity coefficients. Christensen et al. BIB004 modeled the image under deformation as a viscous fluid allowing for large magnitude nonlinear deformations. The PDE was solved for small time intervals and the complete solution was given by an integration over time. For each time interval a successive over-relaxation (SOR) scheme was used. To guarantee the preservation of topology, the Jacobian was monitored and each time its value fell under 0.5, the deformed image was regridded and a new one was generated to estimate a transformation. The final solution was the concatenation of all successive transformations occurring for each regridding step. In a subsequent work, Christensen et al. BIB007 presented a hierarchical way to recover the transformations for brain anatomy. Initially, global affine transformation was performed followed by a landmark transformation model. The result was refined by fluid transformation preceded by an elastic registration step. An important drawback of the earliest implementations of the viscous fluid models, that employed SOR to solve the equations, was computational inefficiency. To circumvent this shortcoming, Christensen et al. employed a massive parallel computer implementation in BIB004 . Bro-Nielsen and Gramkow BIB005 proposed a technique based on a convolution filter in scalespace. The filter was designed as the impulse response of the linear operator defined in its eigen-function basis. Crun et al. BIB026 proposed a multi-grid approach towards handling anisotropic data along with a multiresolution scheme opting for first recovering coarse velocity estimations and refining them in a subsequent step. Cahill et al. BIB028 showed how to use Fourier methods to efficiently solve the linear PDE system that arises from (3) for any boundary condition. Furthermore, Cahill et al. extended their analysis to show how these methods can be applied in the case of other regularizers (diffusion, curvature and elastic) under Dirichlet, Neumann, or periodic boundary conditions. Wang and Staib BIB012 used fluid deformation models in an atlas-enhanced registration setting while D'Agostino et al. tackled multi-modal registration with the use of such models in BIB018 . More recently, Chiang et al. BIB030 proposed an inverse consistent variant of fluid registration to register Diffusion Tensor images. Symmetrized Kullback-Leibler (KL) divergence was used as the matching criterion. Inverse consistency was achieved by evaluating the matching and regularization criteria towards both directions. 3) Diffusion Models: In this case, the deformation is modeled by the diffusion equation BIB019 Let us note that most of the algorithms, based on this transformation model and described in this section, do not explicitly state the (4) in their objective function. Nonetheless, they exploit the fact that the Gaussian kernel is the Green's function of the diffusion equation (4) (under appropriate initial and boundary conditions) to provide an efficient regularization step. Regularization is efficiently performed through convolutions with a Gaussian kernel. Thirion, inspired by Maxwell's Demons, proposed to perform image matching as a diffusion process BIB009 . The proposed algorithm iterated between two steps: 1) estimation of the demon forces for every demon (more precisely, the result of the application of a force during one iteration step, that is a displacement), and 2) update of the transformation based on the calculated forces. Depending on the way the demon positions are selected, the way the space of deformations is defined, the interpolation method that is used, and the way the demon forces are calculated, different variants can be obtained. The most suitable version for medical image analysis involved 1) selecting all image elements as demons, 2) calculating demon forces by considering the optical flow constraint, 3) assuming a nonparametric deformation model that was regularized by applying a Gaussian filter after each iteration, and 4) a trilinear interpolation scheme. The Gaussian filter can be applied either to the displacement field estimated at an iteration or the updated total displacement field. The bijectivity of the transformation was ensured by calculating for every point the difference between its initial position and the one that is reached after composing the forward with the backward deformation field, and redistributing the difference to each field. The bijectivity of the transformation can also be enforced by limiting the maximum length of the update displacement to half the voxel size and using composition to update the transformation. Variants for the contour-based registration and the registration between segmented images were also described in BIB009 . Most of the algorithms described in this section were inspired by the work of Thirion BIB009 and thus could alternatively be classified as "Demons approaches." These methods share the iterative approach that was presented in BIB009 that is, iterating between estimating the displacements and regularizing to obtain the transformation. This iterative approach results in increased computational efficiency. As it will be discussed later in this section, this feature led researchers to explore such strategies for different PDEs. The use of Demons, as initially introduced, was an efficient algorithm able to provide dense correspondences but lacked a sound theoretical justification. Due to the success of the algorithm, a number of papers tried to give theoretical insight into its workings. Fischer and Modersitzki  provided a fast algorithm for image registration. The result was given as the solution of linear system that results from the linearization of the diffusion PDE. An efficient scheme for its solution was proposed while a connection to the Thirion's Demons algorithm BIB009 was drawn. Pennec et al. BIB011 studied image registration as an energy minimization problem and drew the connection of the Demons algorithm with gradient descent schemes. Thirion's image force based on optical flow was shown to be equivalent with a second order gradient descent on the Sum of Square Differences (SSD) matching criterion. As for the regularization, it was shown that the convolution of the global transformation with a Gaussian kernel corresponds to a single step of a first order gradient descent of a functional that penalizes the remainder of the transformation after convolving it with a high-pass filter. Vercauteren et al. BIB020 adopted the alternate optimization framework that Cachier et al. BIB021 proposed, to relate symmetric Demons forces with the efficient second-order minimization (ESM) BIB022 . In this framework, an auxiliary variable was used to decouple the matching and regularization terms. Matching was performed by minimizing the data term through ESM optimization while regularization was achieved by Gaussian smoothing. In BIB029 , Vercauteren et al. proposed a variant of Thirion's algorithm endowed with the diffeomorphic property. In contrast to classical Demons approaches, in every iteration of the algorithm an update field is estimated. In order to estimate the current transformation, a compositional update rule is used between the previous estimate and the exponential map of the update field. The exponential map is efficiently calculated by using the scaling and squaring method BIB035 ,  and the composition of displacement fields. The exponentiation of the displacement field ensures the diffeomorphism of the mapping. To further facilitate the use of the Demons algorithm in anatomical computational studies, Vercauteren et al. BIB031 extended Demons to be symmetric. Initially, it was shown how the complete spatial transformation can be represented in the log-domain. Subsequently, a symmetric extension was provided by averaging the forward and backward forces that were computed separately. The efficiency of this two-step iterative strategy spurred research interest in seeking a mathematical justification of the smoothing step to allow for deformations bearing different physical properties BIB005 , BIB036 - BIB039 . Stefanescu et al. presented a way to perform adaptive smoothing by taking into account knowledge regarding the elasticity of tissues in BIB023 . A nonstationary diffusion filter was used to smooth less inside areas where greater deformations were expected and smooth more inside objects where coherence should be preserved. The authors also proposed to take into account the local image gradient content during smoothing. In areas with large image gradients where the local confidence for the established correspondences is higher, smoothing is scaled down. On the contrary, smoothing is scaled up in homogeneous areas. Cahill et al. BIB036 showed that curvature and fluid registration can be formulated as two coupled diffusion equations. Their stationary solution may be approached via successive Gaussian convolutions, thus yielding a Demons algorithm for these cases. In a subsequent work, Cahill et al. BIB037 showed how to extend the curvature regularization to consider local image gradient content. The authors proposed a coupled PDE system whose stationary solution can be attained by consecutive convolutions with the Green's function of the diffusion equation. In another example, Mansi et al. BIB039 introduced a physical constraint in the registration process to estimate the myocardium strain from Cine-MRI. The logDemons algorithm BIB031 was endowed with the incompressibility constraint by making the velocity field divergence-free. This was achieved by solving the Poisson equation under 0-Dirichlet boundary conditions within a subdomain of the image showing the myocardium. The earliest registration methods of this family of models used an SSD criterion to drive the matching. As a consequence, they were appropriate for mono-modal image registration. Subsequent approaches coped with the multi-modal registration problem. Guimond et al. BIB015 proposed a method that alternates between Demons based registration and intensity correction. Other efforts include the encoding of similarity metrics such as normalized mutual information by Tristán-Vega et al. BIB032 and Modat et al. BIB040 . The application of the Demons algorithm is not limited to scalar images and has been extended to multi-channel images BIB033 , diffusion tensor ones BIB038 , as well as different geometries BIB041 . Peyrat et al. used multi-channel Demons to register time-series of cardiac images by enforcing trajectory constraints in BIB033 . Each time instance was considered as a different channel while the estimated transformation between successive channels was considered as constraint. Yeo et al. BIB038 derived Demons forces from the squared difference between each element of the Log-Euclidean transformed tensors while taking into account the reorientation introduced by the transformation. Finally, the Demons framework was employed to register cortical surfaces parametrized as spheres by Yeo et al. BIB038 . To generalize Demons on the sphere, a method was introduced to measure the distance between two transformations and to regularize the transformation.