Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well.<<ETX>> <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> The shape variation displayed by a class of objects can be represented as probability density function, allowing us to determine plausible and implausible examples of the class. Given a training set of example shapes we can align them into a common co-ordinate frame and use kernel-based density estimation techniques to represent this distribution. Such an estimate is complex and expensive, so we generate a simpler approximation using a mixture of gaussians. We show how to calculate the distribution, and how it can be used in image search to locate examples of the modelled object in new images. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> The authors consider elastic image registration based on a set of corresponding anatomical point landmarks and approximating thin-plate splines. This approach is an extension of the original interpolating thin-plate spline approach and allows to take into account landmark localization errors. The extension is important for clinical applications since landmark extraction is always prone to error. The authors' approach is based on a minimizing functional and can cope with isotropic as well as anisotropic landmark errors. In particular, in the latter case it is possible to include different types of landmarks, e.g., unique point landmarks as well as arbitrary edge points. Also, the scheme is general with respect to the image dimension and the order of smoothness of the underlying functional. Optimal affine transformations as well as interpolating thin-plate splines are special cases of this scheme. To localize landmarks the authors use a semi-automatic approach which is based on three-dimensional (3-D) differential operators. Experimental results are presented for two-dimensional as well as 3-D tomographic images of the human brain. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> Abstract This paper deals with the registration of geometric shapes. Our primary contribution is the use of a simple and robust shape representation (distance functions) for global-to-local alignment. We propose a rigid-invariant variational framework that can deal as well with local non-rigid transformations. To this end, the registration map consists of a linear motion model and a local deformations field, incrementally recovered. In order to demonstrate the performance of the selected representation a simple criterion is considered, the sum of square differences. Empirical validation and promising results were obtained on examples that exhibit large global motion as well as important local deformations and arbitrary topological changes. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> This paper presents a methodology and algorithm for generating diffeomorphisms of the sphere onto itself, given the displacements of a finite set of template landmarks. Deformation maps are constructed by integration of velocity fields that minimize a quadratic smoothness energy under the specified landmark constraints. We present additional formulations of this problem which incorporate a given error variance in the positions of the landmarks. Finally, some experimental results are presented. This work has application in brain mapping, where surface data is typically mapped to the sphere as a common coordinate system. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> In the paper, we study the problem of optimal matching of two generalized functions (distributions) via a diffeomorphic transformation of the ambient space. In the particular case of discrete distributions (weighted sums of Dirac measures), we provide a new algorithm to compare two arbitrary unlabelled sets of points, and show that it behaves properly in limit of continuous distributions on sub-manifolds. As a consequence, the algorithm may apply to various matching problems, such as curve or surface matching (via a sub-sampling), or mixings of landmark and curve data. As the solution forbids high energy solutions, it is also robust towards addition of noise and the technique can be used for nonlinear projection of datasets. We present 2D and 3D experiments. <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> Correlation is a very effective way to align intensity images. We extend the correlation technique to point set registration using a method we call kernel correlation. Kernel correlation is an affinity measure, and it is also a function of the point set entropy. We define the point set registration problem as finding the maximum kernel correlation configuration of the the two point sets to be registered. The new registration method has intuitive interpretations, simple to implement algorithm and easy to prove convergence property. Our method shows favorable performance when compared with the iterative closest point (ICP) and EM-ICP methods. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> Challenges to accurate registration come from three factors -presence of background clutter, occlusion of the pattern being registered and changes in feature values across images. To address these concerns, we propose a robust probabilistic estimation approach predicated on representations of the object model and the target image using a kernel density estimate. These representations are then matched in the space of density functions using a correlation measure, termed the Kernel Density Correlation (KDC) measure. A popular metric which has been widely used by previous image registration approaches is the Mutual Information (MI) metric. We compare the proposed KDC metric with the MI metric to highlight its better robustness to occlusions and random background clutter-this is a consequence of the fact that the KDC measure forms a re-descending M-estimator. Another advantage of the proposed metric is that the registration problem can be efficiently solved using a variational optimization algorithm. We show that this algorithm is an iteratively reweighted least squares (IRLS) algorithm and prove its convergence properties. The efficacy of the proposed algorithm is demonstrated by its application on standard stereo registration data-sets and real tracking sequences. <s> BIB008 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> In this paper, a new framework for brain warping via landmark matching is proposed using implicit representations or the level set method. We demonstrate this powerful technique by matching landmark curves identified on brain surfaces. Each landmark curve to be matched is represented by the intersection of the zero level sets of two level set functions. A diffeomorphic mapping from the template to the target image is then generated by solving a non-linear Euler transport equation with a semi-Lagrangian formulation for the corresponding level set functions. This representation technique has applications in mapping growth processes, recovering intra-operative brain deformation, and integrating and comparing brain imaging data across subjects and groups. <s> BIB009 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> This paper presents a novel approach to feature-based brain image warping, by using a hybrid implicit/explicit framework, which unifies many prior approaches in a common framework. In the first step, we develop links between image warping and the level-set method, and we formulate the fundamental mathematics required for this hybrid implicit/explicit approach. In the second step, we incorporate the large-deformation models into these formulations, leading to a complete and elegant treatment of anatomical structure matching. In this latest approach, exact matching of anatomy is achieved by comparing the target to the warped source structure under the forward mapping and the source to the warped target structure under the backward mapping. Because anatomy is represented nonparametrically, a path is constructed linking the source to the target structure without prior knowledge of their point correspondence. The final point correspondence is constructed based on the linking path with the minimal energy. Intensity-similarity measures can be naturally incorporated in the same framework as landmark constraints by combining them in the gradient descent body forces. We illustrate the approach with two applications: (1) tensor-based morphometry of the corpus callosum in autistic children; and (2) matching cortical surfaces to measure the profile of cortical anatomic variation. In summary, the new mathematical techniques introduced here contribute fundamentally to the mapping of brain structure and its variation and provide a framework that unites feature and intensity-based image registration techniques. <s> BIB010 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> Estimating a meaningful average or mean shape from a set of shapes represented by unlabeled point-sets is a challenging problem since, usually this involves solving for point correspondence under a non-rigid motion setting. In this paper, we propose a novel and robust algorithm that is capable of simultaneously computing the mean shape from multiple unlabeled point-sets (represented by finite mixtures) and registering them nonrigidly to this emerging mean shape. This algorithm avoids the correspondence problem by minimizing the Jensen-Shannon (JS) divergence between the point sets represented as finite mixtures. We derive the analytic gradient of the cost function namely, the JS-divergence, in order to efficiently achieve the optimal solution. The cost function is fully symmetric with no bias toward any of the given shapes to be registered and whose mean is being sought. Our algorithm can be especially useful for creating atlases of various shapes present in images as well as for simultaneously (rigidly or non-rigidly) registering 3D range data sets without having to establish any correspondence. We present experimental results on non-rigidly registering 2D as well as 3D real data (point sets). <s> BIB011 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> We present a novel, variational and statistical approach for shape registration. Shapes of interest are implicitly embedded in a higher-dimensional space of distance transforms. In this implicit embedding space, registration is formulated in a hierarchical manner: the mutual information criterion supports various transformation models and is optimized to perform global registration; then, a B-spline-based incremental free form deformations (IFFD) model is used to minimize a sum-of-squared-differences (SSD) measure and further recover a dense local nonrigid registration field. The key advantage of such framework is twofold: 1) it naturally deals with shapes of arbitrary dimension (2D, 3D, or higher) and arbitrary topology (multiple parts, closed/open) and 2) it preserves shape topology during local deformation and produces local registration fields that are smooth, continuous, and establish one-to-one correspondences. Its invariance to initial conditions is evaluated through empirical validation, and various hard 2D/3D geometric shape registration examples are used to show its robustness to noise, severe occlusion, and missing parts. We demonstrate the power of the proposed framework using two applications: one for statistical modeling of anatomical structures, another for 3D face scan registration and expression tracking. We also compare the performance of our algorithm with that of several other well-known shape registration algorithms <s> BIB012 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> There exists a large body of literature on shape matching and registration in medical image analysis. However, most of the previous work is focused on matching particular sets of features--point-sets, lines, curves and surfaces. In this work, we forsake specific geometric shape representations and instead seek probabilistic representations-- specifically Gaussian mixture models--of shapes. We evaluate a closed-form distance between two probabilistic shape representations for the general case where the mixture models differ in variance and the number of components. We then cast non-rigid registration as a deformable density matching problem. In our approach, we take one mixture density onto another by deforming the component centroids via a thin-plate spline (TPS) and also minimizing the distance with respect to the variance parameters. We validate our approach on synthetic and 3D arterial tree data and evaluate it on 3D hippocampal shapes. <s> BIB013 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> We introduce a new approximation approach for landmark-based elastic image registration using Gaussian elastic body splines (GEBS). We formulate an extended energy functional related to the Navier equation under Gaussian forces which allows to individually weight the landmarks according to their localization uncertainties. These uncertainties are characterized either by scalar weights or by weight matrices representing isotropic or anisotropic errors. Since the approach is based on a physical deformation model, cross-effects in elastic deformations can be taken into account. Moreover, with Gaussian forces we have a free parameter to control the locality of the transformation for improved registration of local geometric image differences. We demonstrate the applicability of our scheme based on analytic experiments, 3D CT images from the Truth Cube experiment, as well as 2D MR images of the brain. From the experiments it turned out that the new approximating GEBS approach achieves more accurate registration results in comparison to previously proposed interpolating GEBS as well as interpolating and approximating TPS. <s> BIB014 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> Point set registration is a key component in many computer vision tasks. The goal of point set registration is to assign correspondences between two sets of points and to recover the transformation that maps one point set to the other. Multiple factors, including an unknown nonrigid spatial transformation, large dimensionality of point set, noise, and outliers, make the point set registration a challenging problem. We introduce a probabilistic method, called the Coherent Point Drift (CPD) algorithm, for both rigid and nonrigid point set registration. We consider the alignment of two point sets as a probability density estimation problem. We fit the Gaussian mixture model (GMM) centroids (representing the first point set) to the data (the second point set) by maximizing the likelihood. We force the GMM centroids to move coherently as a group to preserve the topological structure of the point sets. In the rigid case, we impose the coherence constraint by reparameterization of GMM centroid locations with rigid parameters and derive a closed form solution of the maximization step of the EM algorithm in arbitrary dimensions. In the nonrigid case, we impose the coherence constraint by regularizing the displacement field and using the variational calculus to derive the optimal transformation. We also introduce a fast algorithm that reduces the method computation complexity to linear. We test the CPD algorithm for both rigid and nonrigid transformations in the presence of noise, outliers, and missing points, where CPD shows accurate results and outperforms current state-of-the-art methods. <s> BIB015 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> This paper proposes a novel and robust approach to the groupwise point-sets registration problem in the presence of large amounts of noise and outliers. Each of the point sets is represented by a mixture of Gaussians and the point-sets registration is treated as a problem of aligning the multiple mixtures. We develop a novel divergence measure which is defined between any arbitrary number of probability distributions based on L2 distance, and we call this new divergence measure "Generalized L2-divergence ". We derive a closed-form expression for the Generalized-L2 divergence between multiple Gaussian mixtures, which in turn leads to a computationally efficient registration algorithm. This new algorithm has an intuitive interpretation, is simple to implement and exhibits inherent statistical robustness. Experimental results indicate that our algorithm achieves very good performance in terms of both robustness and accuracy. <s> BIB016 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> Optical imaging offers the possibility to study and image biology at molecular scale in small animal with applications such as oncology or gene expression studies. New device enable cinematic acquisitions but in some cases signals could suffer from weak spatial information. The aim of this paper is to improve localization of the luminescent data for a freely moving animal using motion information obtained with an additional camera. To this end, we propose an approach that performs a non-rigid registration in the scene video and use the corresponding motion vectors to filter the optical one. Due to the lack of contrast and texture, our method introduces silhouette constraints and landmarks on the mouse skin within a variation framework. Motion is represented using thin-plate-splines and the objective function is optimized using a gradient conjugate descent approach. Experimental results demonstrate the potentials of the proposed framework. <s> BIB017 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> Groupwise image registration algorithms seek to establish dense correspondences between sets of images. Typically, they involve iteratively improving the registration between each image and an evolving mean. A variety of methods have been proposed, which differ in their choice of objective function, representation of deformation field, and optimization methods. Given the complexity of the task, the final accuracy is significantly affected by the choices made for each component. Here, we present a groupwise registration algorithm which can take advantage of the statistics of both the image intensities and the range of shapes across the group to achieve accurate matching. By testing on large sets of images (in both 2D and 3D), we explore the effects of using different image representations and different statistical shape constraints. We demonstrate that careful choice of such representations can lead to significant improvements in overall performance. <s> BIB018 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> In this paper, we present a unified framework for the rigid and nonrigid point set registration problem in the presence of significant amounts of noise and outliers. The key idea of this registration framework is to represent the input point sets using Gaussian mixture models. Then, the problem of point set registration is reformulated as the problem of aligning two Gaussian mixtures such that a statistical discrepancy measure between the two corresponding mixtures is minimized. We show that the popular iterative closest point (ICP) method and several existing point set registration methods in the field are closely related and can be reinterpreted meaningfully in our general framework. Our instantiation of this general framework is based on the the L2 distance between two Gaussian mixtures, which has the closed-form expression and in turn leads to a computationally efficient registration algorithm. The resulting registration algorithm exhibits inherent statistical robustness, has an intuitive interpretation, and is simple to implement. We also provide theoretical and experimental comparisons with other robust methods for point set registration. <s> BIB019 </s> Deformable Medical Image Registration: A Survey <s> 3) Methods That Infer Only the Spatial Transformation: <s> We introduce a labeled point set registration algorithm based on a family of novel information-theoretic measures derived as a generalization of the well-known Shannon entropy. This generalization, known as the Havrda-Charvat-Tsallis entropy, permits a fine-tuning between solution types of varying degrees of robustness of the divergence measure between multiple point sets. A variant of the traditional free-form deformation approach, known as directly manipulated free-form deformation, is used to model the transformation of the registration solution. We provide an overview of its open source implementation based on the Insight Toolkit of the National Institutes of Health. Characterization of the proposed framework includes comparison with other state of the art kernel-based methods and demonstration of its utility for lung registration via labeled point set representation of lung anatomy. <s> BIB020
The aim of these methods is to estimate the spatial transformation that, when applied, will align the two sets of landmarks and . These methods do not aim to explicitly establish correspondences between the two landmark sets. The output of the algorithm is the spatial transformation that relates the two point sets and not an explicit assignment of the every point to a point . Two different classes of methods can be distinguished according to whether the correspondences are known or not. The case of known correspondences is briefly presented for completeness reasons. We focus more on the case of unknown correspondences because it is more challenging and there is a number of recent algorithms that have been proposed to tackle the problem. a) Known Correspondences: Two categories of methods should be considered (see Fig. 2 ). The first one assumes that the correspondences are known in an exact or inexact way. This problem is known as exact or inexact landmark matching. In the exact case, a smooth transformation is sought so that the correspondences are respected exactly or a regularization energy is optimized under correspondence constraints. In the inexact case, a compromise between matching and smoothing the deformation is preferred. Procrustes analysis is a popular method for shape analysis and is useful when homologies between point-sets are given BIB018 ,  -  . In Procrustes analysis, a least-squares distance is minimized. Given the correspondences, a solution that consists of translating, rotating and scaling can be analytically calculated BIB002 . Given the correspondences, one may estimate nonrigid transformations by adopting an interpolation strategy (Section II-B). Radial basis functions are able to produce dense deformation fields for any spatial distribution of points. Moreover, approximating splines are able to account for the uncertainty in the estimated correspondences BIB003 , BIB014 . Guo et al. has presented a solution for both the exact and inexact landmark matching problems for the case of diffeomorphic deformations  . Glaunes et al. have extended this method to the case where the domain is a sphere BIB005 . b) Unknown Correspondences: The second subclass opts to estimate the transformation without concerning itself with the establishment of correspondences. These methods are more robust to missing correspondences and outliers. One may distinguish two different subclasses depending on the nature of the transformation that is estimated. The methods that belong in the first category estimate a global linear transformation, while the methods in the second category estimate a nonrigid transformation. For the description of some of the methods that belong to the first class, we refer the reader to  . Because the aim of this review is to describe the recent advances in deformable registration, we are going to focus here on the second class of methods. The estimation of nonrigid transformations was achieved through the use of alternative representations of the geometric information. One possibility is to represent the point sets as probability distributions. In this case, the nonrigid transformation is estimated by minimizing a distance measure between the two distributions. Glaunes et al. BIB006 extended the large diffeomorphic deformation framework in the case of distributions and unlabeled point sets. Point sets were modeled as a weighted sum of Dirac measures and a kernel-based error measure was used. Tsin and Kanade BIB007 proposed to register point sets based on a measure called kernel correlation. The proposed measure is proportional to the correlation of two kernel density estimates. Singh et al. presented a similar approach based on kernel density correlation BIB008 . Gaussian mixture models (GMMs) are a common way to model distributions. Jian et al. BIB019 modeled each point set using GMMs and used a distance to compare them. Myronenko and Song BIB015 recast registration as a probability density estimation problem. The points of the first set were considered as the centroids of the GMMs which were fitted to the data (or points of the second set) by likelihood maximization. Special care was taken so that the centroids move in a coherent way. Roy et al. BIB013 modeled each feature of each shape as GMM. A mixture model was used to represent the shape by assuming that features are independent and identically distributed. A closed-form distance between the two distributions was used. Wang et al. used a similar model to tackle the problem of the simultaneous registration of multiple point sets BIB011 . Jensen-Shannon divergence was used as the similarity metric. The drawback of this approach was that the problem could not be solved in closed-form. Instead, a computationally and memory demanding estimation based on the law of large numbers was required. In a subsequent work, Wang et al. BIB016 alleviated this shortcoming by using the generalized -divergence that allows for a closed-form solution. Tustison et al. also used a GMM with the difference that the Gaussians were not isotropic BIB020 . The Havrda-Charvat-Tsallis (HCT) divergence was used to compare the two distributions. Another way to perform nonrigid registration of shapes and points without caring to establish correspondences is by adopting a representation of the geometric information based on the use of signed distance functions. In this case, the geometric primitives (e.g., landmarks or shapes) are assigned to zero distance, while the rest of the image elements are assigned a signed value based on their euclidean distance from the geometric primitives. Based on this representation, the optimal transformation can be estimated by performing standard intensity-based registration. Paragios et al. embedded shapes to the higher dimensional space defined by the signed distance transform and register them by evaluating the sum of squared differences criterion over a narrow band around the shapes BIB004 . Huang et al. used the same shape representation and investigated the use of Mutual Information to globally align them BIB012 . The sum of squared differences was used to nonrigidly register the shapes. Savinaud et al. represented both silhouettes and landmarks using the Euclidean distance transform BIB017 . Leow et al. used implicit representation to tackle brain warping BIB009 , BIB010 . Leow et al. formulated the energy minimization problem as a curve evolution problem motivated by the geodesic active contours BIB001 .