Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> The decomposition of deformations by principal warps is demonstrated. The method is extended to deal with curving edges between landmarks. This formulation is related to other applications of splines current in computer vision. How they might aid in the extraction of features for analysis, comparison, and diagnosis of biological and medical images in indicated. > <s> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> The thin-plate spline, a general-purpose interpolant for labelled point data, ties the geometry of image deformation to the classic biometric algebra of quadratic forms. The technique thus helps not only in the production of biomedical atlases—averaged or normative images of particular structures or configurations of parts—but also in the understanding of specimen-by-specimen variability around those atlases. This report summarizes the statistics of thirteen landmark points in midsagittal MRI images of nine normal adult human brains, produces and evaluates averaged images with the aid of those statistics, and pursues some implications for stereotaxy, for region-tracing, and for the understanding of averaged image structures. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Abstract The problem of deforming a given spatial shape is treated. There are many examples of applications in visual computing: fitting surfaces to sampled data points in space, correction of distortions in tomographic imaging, modeling of free-form geometric shapes, and animating metamorphoses of geometric objects. Our solution warps the space surrounding the given shape with the effect of deforming the embedded shape, too, with a function derived with scattered data interpolation methods from the displacements of a finite set of control points that can be placed arbitrarily and adaptively. We present algorithms implementing this idea on parametric surfaces and rasterized volume data. <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Abstract The human face is an elastic object. A natural paradigm for representing facial expressions is to form a complete 3D model of facial muscles and tissues. However, determining the actual parameter values for synthesizing and animating facial expressions is tedious; evaluating these parameters for facial expression analysis out of gray-level images is ahead of the state of the art in computer vision. Using only 2D face images and a small number of anchor points, we show that the method of radial basis functions provides a powerful mechanism for processing facial expressions. Although constructed specifically for facial expressions, our method is applicable to other elastic objects as well. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> We construct a new class of positive definite and compactly supported radial functions which consist of a univariate polynomial within their support. For given smoothness and space dimension it is proved that they are of minimal degree and unique up to a constant factor. Finally, we establish connections between already known functions of this kind. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedC k smoothness, there is a function inC k (ℝ n ), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degreeB-splines. <s> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Medical image registration algorithms invariably assume that the objects in the images can be treated as a single rigid body. In practice, some parts of a patient, usually bony structures, may move as rigid bodies, while others may deform. To address this, we have developed a new technique that allows identified objects in the image to move as rigid bodies, while the remainder smoothly deforms. The transformation is based on a radial basis function solution with a basis function modified by use of distance transforms based on rigid structures within the image. These distance transforms are also used to form an underlying transformation which ensures an interpolating solution. The resulting deformation technique is valid in any dimension, subject to the choice of the basis function. We demonstrate this algorithm in two dimensions on illustrative images as well as on sagittal magnetic resonance images collected from a volunteer. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> We consider error estimates for interpolation by a special class of compactly supported radial basis functions. These functions consist of a univariate polynomial within their support and are of minimal degree depending on space dimension and smoothness. Their associated “native” Hilbert spaces are shown to be norm-equivalent to Sobolev spaces. Thus we can derive approximation orders for functions from Sobolev spaces which are comparable to those of thin-plate-spline interpolation. Finally, we investigate the numerical stability of the interpolation process. <s> BIB008 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Many image matching schemes are based on mapping coordinate locations, such as the locations of landmarks, in one image to corresponding locations in a second image. A new approach to this mapping (coordinate transformation), called the elastic body spline (EBS), is described. The spline is based on a physical model of a homogeneous, isotropic three-dimensional (3-D) elastic body. The model can approximate the way that some physical objects deform. The EBS as well as the affine transformation, the thin plate spline [1], [2] and the volume spline [3] are used to match 3-D magnetic resonance images (MRI's) of the breast that are used in the diagnosis and evaluation of breast cancer. These coordinate transformations are evaluated with different types of deformations and different numbers of corresponding (paired) coordinate locations. In all but one of the cases considered, using the EBS yields more similar images than the other methods. <s> BIB009 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Single photon emission computed tomography (SPECT) imaging with /sup 201/Tl or /sup 99m/Tc agent is used to assess the location or the extent of myocardial infarction or ischemia. A method is proposed to decrease the effect of operator variability in the visual or quantitative interpretation of scintigraphic myocardial perfusion studies. To effect this, the patient's myocardial images (target cases) are registered automatically over a template image, utilizing a nonrigid transformation. The intermediate steps are: 1) Extraction of feature points in both stress and rest three-dimensional (3-D) images. The images are resampled in a polar geometry to detect edge points, which in turn are filtered by the use of a priori constraints. The remaining feature points are assumed to be points on the edges of the left ventricular myocardium. 2) Registration of stress and rest images with a global affine transformation. The matching method is an adaptation of the iterative closest point algorithm. 3) Registration and morphological matching of both stress and rest images on a template using a nonrigid local spline transformation following a global affine transformation. 4) Resampling of both stress and rest images in the geometry of the template. Optimization of the method was performed on a database of 40 pairs of stress and rest images selected to obtain a wide variation of images and abnormalities. Further testing was performed on 250 cases selected from the same database on the basis of the availability of angiographic results and patient stratification. <s> BIB010 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Abstract Common elastic registration schemes based on landmarks and radial basis functions (RBFs) such as thin-plate splines or multiquadrics are global. Here, we introduce radial basis functions with compact support for elastic registration of medical images which have an improved locality, i.e. which allow to constrain elastic deformations to image parts where required. We give the theoretical background of these basis functions and compare them with other basis functions w.r.t. locality, solvability, and efficiency. A detailed comparison with the Gaussian as well as conditions for preserving topology is given. An important property of the used RBFs (Wendland's ψ-functions) is that they are positive definite. Therefore, in comparison to the use of the truncated Gaussian, the solvability of the resulting system of equations is always guaranteed. We demonstrate the applicability of our approach for synthetic as well as for 2D and 3D tomographic images. <s> BIB011 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Radial basis functions are well-known and successful tools for the interpolation of data in many dimensions. Several radial basis functions of compact support that give rise to nonsingular interpolation problems have been proposed, and in this paper we study a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature. <s> BIB012 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Landmark-based thin-plate spline image registration is one of the most commonly used methods for non-rigid medical image registration and anatomical shape analysis. It is well known that this method does not produce a unique correspondence between two images away from the landmark locations because interchanging the role of source and target landmarks does not produce forward and reverse transformations that are inverses of each other. In this paper, we present two new image registration algorithms that minimize the thin-plate spline bending energy and the inverse consistency error--the error between the forward and the inverse of the reverse transformation. The landmarkbased consistent thin-plate spline algorithm registers images given a set of corresponding landmarks while the intensity-based consistent thinplate spline algorithm uses both corresponding landmarks and image intensities. Results are presented that demonstrate that using landmark and intensity information to jointly estimate the forward and reverse transformations provides better correspondence than using landmarks or intensity alone. <s> BIB013 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <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> BIB014 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> We propose a simple and efficient method to interpolate landmark matching by a non-ambiguous mapping (a diffeomorphism). This method is based on spline interpolation, and on recent techniques developed for the estimation of flows of diffeomorphisms. Experimental results show interpolations of remarkable quality. Moreover, the method provides a Riemannian distance on sets of landmarks (with fixed cardinality), which can be defined intrinsically, without refering to diffeomorphisms. The numerical implementation is simple and efficient, based on an energy minimization by gradient descent. This opens important perspectives for shape analysis, applications in medical imaging, or computer graphics <s> BIB015 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> A new approach is presented for elastic registration of medical images, and is applied to magnetic resonance images of the brain. Experimental results demonstrate remarkably high accuracy in superposition of images from different subjects, thus enabling very precise localization of morphological characteristics in population studies. There are two major novelties in the proposed algorithm. First, it uses an attribute vector, i.e. a set of geometric moment invariants that is defined on each voxel in an image, to reflect the underlying anatomy at different scales. The attribute vector, if rich enough, can distinguish between different parts of an image, which helps establish anatomical correspondences in the deformation procedure. This is a fundamental deviation of our method, referred to as HAMMER, from other volumetric deformation methods, which are typically based on maximizing image similarity. Second, in order to avoid being trapped by local minima, i.e. suboptimal poor matches, HAMMER uses a successive approximation of the energy function being optimized by lower dimensional energy functions, which are constructed to have significantly fewer local minima. This is achieved by hierarchically selecting features that have distinct attribute vectors, thus drastically reducing ambiguity in finding correspondence. A number of experiments in this paper have demonstrated excellent performance. <s> BIB016 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> This work presents a framework for non-rigid registration which extends and generalizes a previously developed technique by Rueckert et al. [1]. We combine multi-resolution optimization with free-form deformations (FFDs) based on multi-level B-splines to simulate a non-uniform control point distribution. We have applied this to a number of different medical registration tasks to demonstrate its wide applicability, including interventional MRI brain tissue deformation compensation, breathing motion compensation in liver MRI, intra-modality inter-modality registration of pre-operative brain MRI to CT electrode implant data, and inter-subject registration of brain MRI. Our results demonstrate that the new algorithm can successfully register images with an improved performance, while achieving a significant reduction in run-time. <s> BIB017 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> The thin plate spline (TPS) is an effective tool for modeling coordinate transformations that has been applied successfully in several computer vision applications. Unfortunately the solution requires the inversion of a p × p matrix, where p is the number of points in the data set, thus making it impractical for large scale applications. As it turns out, a surprisingly good approximate solution is often possible using only a small subset of corresponding points. We begin by discussing the obvious approach of using the subsampled set to estimate a transformation that is then applied to all the points, and we show the drawbacks of this method. We then proceed to borrow a technique from the machine learning community for function approximation using radial basis functions (RBFs) and adapt it to the task at hand. Using this method, we demonstrate a significant improvement over the naive method. One drawback of this method, however, is that is does not allow for principal warp analysis, a technique for studying shape deformations introduced by Bookstein based on the eigenvectors of the p × p bending energy matrix. To address this, we describe a third approximation method based on a classic matrix completion technique that allows for principal warp analysis as a by-product. By means of experiments on real and synthetic data, we demonstrate the pros and cons of these different approximations so as to allow the reader to make an informed decision suited to his or her application. <s> BIB018 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> This paper introduces a novel groupwise data-driven algorithm for non-rigid registration. The motivation behind the algorithm is to enable the analysis of groups of registered images; to this end, the algorithm automatically constructs a low-dimensional, common representation of the warp fields. We demonstrate the algorithm on an example set of 2D medical images, and show that we can obtain good registration across the set, with automatic detection and correction of misaligned examples, whilst still maintaining a low-dimensional representation. <s> BIB019 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Nonrigid registration of medical images is important for a number of applications such as the creation of population averages, atlas-based segmentation, or geometric correction of functional magnetic resonance imaging (IMRI) images to name a few. In recent years, a number of methods have been proposed to solve this problem, one class of which involves maximizing a mutual information (Ml)-based objective function over a regular grid of splines. This approach has produced good results but its computational complexity is proportional to the compliance of the transformation required to register the smallest structures in the image. Here, we propose a method that permits the spatial adaptation of the transformation's compliance. This spatial adaptation allows us to reduce the number of degrees of freedom in the overall transformation, thus speeding up the process and improving its convergence properties. To develop this method, we introduce several novelties: 1) we rely on radially symmetric basis functions rather than B-splines traditionally used to model the deformation field; 2) we propose a metric to identify regions that are poorly registered and over which the transformation needs to be improved; 3) we partition the global registration problem into several smaller ones; and 4) we introduce a new constraint scheme that allows us to produce transformations that are topologically correct. We compare the approach we propose to more traditional ones and show that our new algorithm compares favorably to those in current use. <s> BIB020 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Groupwise nonrigid registrations of medical images define dense correspondences across a set of images, defined by a continuous deformation field that relates each target image in the group to some reference image. These registrations can be automatic, or based on the interpolation of a set of user-defined landmarks, but in both cases, quantifying the normal and abnormal structural variation across the group of imaged structures implies analysis of the set of deformation fields. We contend that the choice of representation of the deformation fields is an integral part of this analysis. This paper presents methods for constructing a general class of multi-dimensional diffeomorphic representations of deformations. We demonstrate, for the particular case of the polyharmonic clamped-plate splines, that these representations are suitable for the description of deformations of medical images in both two and three dimensions, using a set of two-dimensional annotated MRI brain slices and a set of three-dimensional segmented hippocampi with optimized correspondences. The class of diffeomorphic representations also defines a non-Euclidean metric on the space of patterns, and, for the case of compactly supported deformations, on the corresponding diffeomorphism group. In an experimental study, we show that this non-Euclidean metric is superior to the usual ad hoc Euclidean metrics in that it enables more accurate classification of legal and illegal variations. <s> BIB021 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> This paper describes a groupwise, non-rigid registration algorithm to simultaneously register all subjects in a population to a common reference (or natural) coordinate system, which is defined to be the average of the population. This natural coordinate system is calculated implicitly by constraining the sum of all deformations from itself to each subject to be zero. To do this, the gradient projection method for constrained optimization is applied to maximize the similarity between the images, subject to the constraint being satisfied. The algorithm has been tested on a group of 19 brain MR images acquired from a population of subjects with schizophrenia. <s> BIB022 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> In this paper we develop a spatio-temporal registration algorithm for cardiac MR image sequences. The algorithm has the ability to correct any spatial misalignment between the images caused by global differences in the acquisition of the image sequences and by local shape differences. In addition it has the ability to correct temporal misalignment caused by differences in the length of the cardiac cycles and by differences in the dynamic properties of the hearts. The algorithm uses a 4D deformable transformation model which is separated into spatial and temporal components. The registration method was qualitatively evaluated by visual inspection and by measuring the overlap and surface distance of anatomical regions. The results demonstrate that a significant improvement in the alignment of the image sequences is achieved by the use of the deformable transformation model. <s> BIB023 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> We introduce a new point-based elastic registration scheme for medical images which is based on elastic body splines (EBS). Since elastic body splines result from a physical model in form of analytical solutions of the Navier equation these splines describe elastic deformations of physical objects. This property is advantageous in medical registration applications, in which the geometric differences between the images are often caused by physical deformations of human tissue due to surgical interventions or pathological processes. In this contribution we introduce a new class of elastic body splines which is based on Gaussian forces (GEBS). ::: ::: By varying the standard deviation of the Gaussian forces our new approach is well suited to cope with local as well as global differences in the images. This is in contrast to the previous EBS approach where polynomial and rational forces have been used. We demonstrate the performance of our new approach by presenting two different kinds of experiments. First, we demonstrate that this approach well approximates deformations given by an analytic solution of the Navier equation. Second, we apply our approach to pre- and postsurgical tomographic images of the human brain. It turns out that the new EBS approach well models the physical deformation behavior of tissues and in the case of local deformations performs significantly better than the previous EBS. <s> BIB024 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Transformation functions play a major role in nonrigid image registration. In this paper, the characteristics of thin-plate spline (TPS), multiquadric (MQ), piecewise linear (PL), and weighted mean (WM) transformations are explored and their performances in nonrigid image registration are compared. TPS and MQ are found to be most suitable when the set of control-point correspondences is not large (fewer than a thousand) and variation in spacing between the control points is not large. When spacing between the control points varies greatly, PL is found to produce a more accurate registration than TPS and MQ. When a very large set of control points is given and the control points contain positional inaccuracies, WM is preferred over TPS, MQ, and PL because it uses an averaging process that smoothes the noise and does not require the solution of a very large system of equations. Use of transformation functions in the detection of incorrect correspondences is also discussed. <s> BIB025 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Geodesic spline interpolation is a simple and efficient approach for landmark matching by nonambiguous mappings (diffeomorphisms), combining classic spline interpolation and flows of diffeomorphisms. Here, we extend the method to incorporate the estimation of a affine transformation, yielding a consistent and numerically stable algorithm. A theoretical justification is provided by studying the existence of the global minimum of the energy. <s> BIB026 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> We examine non-rigid image registration by knotpoint mat-break ching. We consider registering two images, each with a set of knotpoints marked, where one of the images is to be registered to the other by a nonlinear warp so that the knotpoints on the template image are exactly aligned with the corresponding knotpoints on the reference image. We explore two approaches for computing the registration by the Geodesic Interpolating Spline. First, we describe a method which exploits the structure of the problem in order to permit efficient optimization and second, we outline an approach using the framework of classical mechanics. <s> BIB027 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> In this paper we propose a diffeomorphic non-rigid registration algorithm based on free-form deformations (FFDs) which are modelled by B-splines. In contrast to existing non-rigid registration methods based on FFDs the proposed diffeomorphic non-rigid registration algorithm based on free-form deformations (FFDs) which are modelled by B-splines. To construct a diffeomorphic transformation we compose a sequence of free-form deformations while ensuring that individual FFDs are one-to-one transformations. We have evaluated the algorithm on 20 normal brain MR images which have been manually segmented into 67 anatomical structures. Using the agreement between manual segmentation and segmentation propagation as a measure of registration quality we have compared the algorithm to an existing FFD registration algorithm and a modified FFD registration algorithm which penalises non-diffeomorphic transformations. The results show that the proposed algorithm generates diffeomorphic transformations while providing similar levels of performance as the existing FFD registration algorithm in terms of registration accuracy. <s> BIB028 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Common tools based on landmarks in medical image elastic registration are Thin Plate Spline (TPS) and Compact Support Radial Basis Function (CSRBF). TPS forces the corresponding landmarks to exactly match each other and minimizes the bending energy of the whole image. However, in real application, such scheme would deform the image globally when deformation is only local. CSRBF needs manually determine the support size, although its deformation is limited local. Therefore, to limit the effect of the deformation, new Compact Support Thin Plate Spline algorithm (CSTPS) is approached, analyzed and applied. Such new approach gains optimal mutual information, which shows its registration result satisfactory. The experiments also show it can apply in both local and global elastic registration. <s> BIB029 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> The focus of medical image registration has shifted to nonrigid registration. It is also a core component in computational neuroanatomy. In this paper, we propose an efficient registration framework which has feature of multiresolution and multigrid. In contrast to the existing registration algorithms. Free-Form Deformations based NURBS (Nonuniform Rational B Spline) are used to acquire nonrigid transformation. This can provide a competitive alternative to Free-Form Deformations based B spline on flexibility and accuracy. To our knowledge, it is the first report of the use of this mathematical tool for medical image registration. Subdivision of NURBS is extended to 3D and is used in hierarchical optimization to speed up the registration and avoid local minima. The performance of this method is numerically evaluated on simulated images and real images. Compared with the registration method using uniform Free-Form Deformations based B spline, our method can successfully register images with improved performance. <s> BIB030 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <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> BIB031 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> This paper presents a new nonrigid monomodality image registration algorithm based on B-splines. The deformation is described by a cubic B-spline field and found by minimizing the energy between a reference image and a deformed version of a floating image. To penalize noninvertible transformation, we propose two different constraints on the Jacobian of the transformation and its derivatives. The problem is modeled by an inequality constrained optimization problem which is efficiently solved by a combination of the multipliers method and the L-BFGS algorithm to handle the large number of variables and constraints of the registration of 3-D images. Numerical experiments are presented on magnetic resonance images using synthetic deformations and atlas based segmentation. <s> BIB032 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Image registration aims at estimating a consistent mapping between two images. Common techniques consist in choosing arbitrarily one image as a reference image and the other one as a floating image, thus leading to the estimation of inconsistent mappings. We present a symmetric formulation of the registration problem that maps the two images in a common coordinate system halfway between them. This framework has been considered to devise an efficient strategy for mapping a large set of images in a common coordinate system. Some results are presented in the context of 3-D nonrigid brain MR image registration for the construction of average brain templates. <s> BIB033 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> The work in this paper presents a non-rigid registration approach using proposed radial basis function (RBF). The RBF is based on locally constrained cosines. The proposed function is designed to overcome the weaknesses, observed in previous RBFs. The criteria to evaluate the accuracy of transformation functions are also proposed in this paper. Results of the proposed RBF are analyzed and compared with the existing RBFs, based on the proposed evaluation criteria and some other similarity measures. We demonstrate the applicability of proposed approach for registration of medical images and image warping. <s> BIB034 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> A consistent image registration formulation with B-spline deformation model is proposed. This formulation avoids the computation of the inverse of a deformation in the iterative optimization and allows the analytical computation of both gradient and Hessian of the cost function. Because of the formulation, the algorithm is computationally efficient and could potentially produce more accurate registration results. Experiments show that the proposed algorithm produces promising results while keeping the estimated deformations consistent. <s> BIB035 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Many Radial Basis Function (RBF)-based transformations are used to model the deformations in image registration, and they have different topology preservation properties. This paper compares analytically and experimentally the topology preservation performance of compact-support thin-plate Spline (CSTPS), locally constrained cosine (Cos), Wendland, Gaussian, Buhmann and Wu functions in landmark-based image registration. In addition, the topology preservation characteristics of thin-plate Spline (TPS) and elastic body Spline (EBS)-based transformations are compared for global-support deformations. The comparative results show that, for local deformation CSTPS and Buhmann preserve topology better than others. The Cos and Gaussian functions could easily produce topology violations for relatively dense-landmark matching. For global-support transformations, CSTPS, Wendland @j"3","1, Buhmann, Wu and TPS outperformed others because they preserve topology better. The Cos, Gaussian and EBS functions perform poorly because folds and tears of the deformation surface occur easily. With very large support, CSTPS produces similar results as TPS, and Wendland @j"3","1 has similar performance with Wu functions. Also, Cos and Gaussian performed similarly in this case. In the experiments, these theoretical results are evaluated extensively using transformations on random point sets, artificial images, and medical images. <s> BIB036 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> In a recent paper, Siddiqui et al. introduced a kernel function to be used as a radial basis function (RBF) in image registration tasks. This function is mainly designed so that the resulting deformation is fairly distributed inside its support. The important property of positive definiteness is checked in the paper erroneously, so that the conclusions inferred are wrong. In this communication, we discuss this point and some other methodological errors in the formulation. In addition, we provide some insights into the importance of positive definiteness, concluding that this property may not be critical, or may even be worthless, in certain interpolation problems. <s> BIB037 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> A registration method for motion estimation in dynamic medical imaging data is proposed. Registration is performed directly on the dynamic image, thus avoiding a bias towards a specifically chosen reference time point. Both spatial and temporal smoothness of the transformations are taken into account. Optionally, cyclic motion can be imposed, which can be useful for visualization (viewing the segmentation sequentially) or model building purposes. The method is based on a 3D (2D + time) or 4D (3D + time) free-form B-spline deformation model, a similarity metric that minimizes the intensity variances over time and constrained optimization using a stochastic gradient descent method with adaptive step size estimation. The method was quantitatively compared with existing registration techniques on synthetic data and 3D + t computed tomography data of the lungs. This showed subvoxel accuracy while delivering smooth transformations, and high consistency of the registration results. Furthermore, the accuracy of semi-automatic derivation of left ventricular volume curves from 3D + t computed tomography angiography data of the heart was evaluated. On average, the deviation from the curves derived from the manual annotations was approximately 3%. The potential of the method for other imaging modalities was shown on 2D + t ultrasound and 2D + t magnetic resonance images. The software is publicly available as an extension to the registration package elastix. <s> BIB038 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Purpose: Four-dimensional computed tomography (4D CT) can provide patient-specific motion information for radiotherapy planning and delivery. Motion estimation in 4D CT is challenging due to the reduced image quality and the presence of artifacts. We aim to improve the robustness of deformable registration applied to respiratory-correlated imaging of the lungs, by using a global problem formulation and pursuing a restrictive parametrization for the spatiotemporal deformation model. <s> BIB039 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Thin-plate splines are probably the most often used technique for landmark-based registration with radial basis functions. However, a disadvantage is that the locality of the transformation cannot be controlled. We introduce an extension of thin-plate splines which enables to control the spatial influence of acting forces and thus the locality of the transformation. Our approach is based on a new type of splines which we derived as closed-form solution of the biharmonic partial differential equation under certain forces. We have applied our approach to synthetic and real image data and have compared the results with standard thin-plate splines registration. <s> BIB040 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Non-rigid image registration using free-form deformations (FFD) is a widely used technique in medical image registration. The balance between robustness and accuracy is controlled by the control point grid spacing and the amount of regularization. In this paper, we revisit the classic FFD registration approach and propose a sparse representation for FFDs using the principles of compressed sensing. The sparse free-form deformation model (SFFD) can capture fine local details such as motion discontinuities without sacrificing robustness. We demonstrate the capabilities of the proposed framework to accurately estimate smooth as well as discontinuous deformations in 2D and 3D image sequences. Compared to the classic FFD approach, a significant increase in registration accuracy can be observed in natural images (61%) as well as in cardiac MR images (53%) with discontinuous motions. <s> BIB041 </s> Deformable Medical Image Registration: A Survey <s> 1) Radial Basis Functions: <s> Image registration is in principle a symmetric problem. Nonetheless, most intensity-based non-rigid algorithms are asymmetric. In this paper, we propose a novel symmetric deformable registration algorithm formulated in a Markov Random Fields framework where both images are let to deform towards a common domain that lies halfway between two image domains. A grid-based deformation model is employed and the latent variables correspond to the displacements of the grid-nodes towards both image domains. First-order interactions between the unknown variables model standard smoothness priors. Efficient linear programming is consider to recover the optimal solution. The discrete nature of our algorithm allows the handling of both mono- and multi-modal registration problems. Promising experimental results demonstrate the potentials of our approach. <s> BIB042
One of the most important families of interpolation strategies is that of radial basis functions (RBFs), where the value at an interpolation point is given as function of its distance from the known sample , or (7) Zagorchev and Goshtasby presented an evaluation study comparing RBFs used as transformation functions in nonrigid image registration in BIB025 . More recently, Yang et al. BIB036 presented an analysis with respect to the ability of RBFs to preserve topology. An important property of RBFs is that they are able to interpolate a deformation field from irregularly placed known values. A common property of most RBFs, that are described in this section, is their global support. Knowing the displacement at one point influences the values of points in the whole image domain. As a consequence, interpolation in sparsely populated areas is feasible. On the other hand, this behavior is undesirable when seeking local transformations. In order to counter it, sufficient landmarks are required in the regions of interest. Bookstein proposed the use of thin-plate splines (TPS) for image registration in BIB001 , BIB002 . TPS minimize the bending energy assuming infinite boundary conditions. The solution is given in a closed-form and its uniqueness is guaranteed in most cases. Nonetheless, TPS, as proposed by Bookstein, are known to exhibit certain shortcomings. The transformation from one image domain to another is not inverse consistent BIB013 . Moreover, their support is global, which hinders the recovery of local image warpings BIB029 - BIB040 . Furthermore, TPS do not take into consideration possible errors in the estimation of the displacements in the landmark positions BIB014 . Lastly, as the number of points increases, the interpolation becomes computationally demanding BIB018 . A number of researchers have worked to lessen the importance of these shortcomings BIB013 - BIB018 . In BIB013 , Johnson and Christensen tackled the inverse inconsistency problem. They considered the minimization of the bending energy under cyclic boundary conditions in an effort to account for the great consistency error that they observed in the boundary of the images. Additionally, a term that penalizes the consistency error was introduced in the objective function to render the registration inverse consistent. Li et al. coped with the problem of the global nature of TPS in BIB029 . TPS were constructed in such a way that their support is restricted locally. In a subsequent work, Yang et al.  defined the support of each point in an adaptive way by taking into consideration the distribution of the points in the image domain. These approaches BIB029 ,  were based on heuristics and a truncation of the original basis, to limit the influence of the control points. Rohr and Wörz BIB040 introduced a variant of TPS which assumes that the forces that act at the landmarks, also influence the region around them. These forces are described by a Gaussian function of the radial distance from the landmark instead of a Dirac delta function as in the classical TPS. The parametrization of the forces by the standard deviation of the Gaussian function allows for the control of the locality of the transformation. Rohr et al. BIB014 proposed to take into consideration the landmark localization error when estimating the dense deformation field through the use of approximating thin-plate splines. The authors proposed to weaken the interpolation constraint and estimate the transformation by minimizing a functional that weights the approximation error according to the (isotropic or anisotropic) landmark position estimation error. The approximation problem admits an analytical solution that consists of the same basis functions as the interpolation problem. Three ways to address the computational problems related with the presence of a great number of landmarks were studied by Donato and Belongie BIB018 . The straightforward approach of sub-sampling the points was compared to two more elaborated ones that use either a subset of the basis functions or a matrix approximation technique based on the Nyström method. The more sophisticated methods were shown to outperform the naive approach in terms of mean squared error. The matrix approximation method was also shown to be useful when principal warp analysis was taken into account. Marsland and Twining BIB021 , BIB019 employed Clamped-Plate splines for groupwise registration and groupwise analysis of deformable registrations. Clamped-plate splines minimize the same energy as TPS though under specific boundary conditions. Camion and Younes introduced goedesic interpolating splines (GIS) following the LDDMM framework BIB015 . The dense deformation field that results from the interpolation with these splines is diffeomorphic. Younes extended this method to combine GIS with affine transformations in BIB026 while two ways to calculate them were presented by Mills et al. BIB027 . Ruprecht et al. have proposed another family of RBFs, that of multi-quadratics, that has global support BIB003 . Little et al. extended this approach to cope with the presence of a rigid object BIB007 . Arad et al. BIB004 suggested the use of Gaussian functions to parametrize the deformation. The choice of an appropriate Gaussian kernel allows for the control of their spatial influence. By choosing a small size for the Gaussian kernel, their influence can be greatly restricted and thus local displacements may be recovered. A recent example of the use of this deformation model in brain registration can be found in BIB016 . Zagorchev and Goshtasby BIB025 investigated the use of the normalized weighted average of sparse displacements to create dense deformation fields. Despite the global support of the control points, the locality of the transformation can be adapted by choosing an appropriate weighting function. In medical image analysis, the presence of different anatomical structures characterized by different properties and the subsequent need to recover local deformations render the previous models not well suited. To cope successfully with such cases, interpolation methods where control points have spatially limited influence are appropriate. Fornefett et al. BIB011 investigated the use of Wendland functions BIB005 , BIB008 that exhibit the desired locality property, for deformable registration. Other local support radial basis functions include the smooth Wu functions BIB006 and the functions proposed by Buhmann BIB012 . Rohde et al. BIB020 applied the Wu functions in image registration and derived bounds for the basis function's coefficients so that the Jacobian of the computed transformation remains positive. More recently, Siddiqui et al. BIB034 defined a new model based on the cosine function. Contrary to what is claimed in the paper, the new model is not positive definite BIB037 . A realvalued, continuously differentiable function is called positive definite on a neighborhood of the origin, if it is zero for the origin and greater than zero for the rest of the points in the neighborhood. The positive definiteness of the functions is important because it guarantees that the system of linear equations, that arises when estimating the coefficients of the interpolation problem, is solvable for all possible sets of pairs of corresponding landmarks in the two image domains, which are not colinear in and coplanar in BIB011 . Lowitzsch  introduced a class of RBFs that are vector-valued analogues of the Wendland functions BIB005 , BIB008 . This class of RBFs provide interpolated displacement fields that are divergence free. Yang et al. BIB036 compared the previous locally constrained radial basis functions by using transformations on random point sets, artificial images and medical images. 2) Elastic Body Splines: Splines, though mainly inspired by interpolation and approximation theory, may also be inspired by physical models. Such is the case of Elastic Body Splines (EBS), which were introduced by Davis et al. BIB009 . These splines are solutions of the Navier-Cauchy equilibrium equation for a homogeneous isotropic elastic body subjected to forces. When the force field that drives the registration based on the landmark correspondences is given as a radial symmetric function of the distance from the landmark, one can solve the equation analytically. Kohlrausch et al. BIB024 extended the previous work by considering forces that are given as a Gaussian function of the distance from the landmark (Gaussian EBS). The size of the kernel of the Gaussian can be used to parametrize the compactness of the model's support. As a result, the transformation model can cope better with local deformations. An analytic solution for the equilibrium equation also exists for this type of force field. Wörz and Rohr extended Gaussian EBS in BIB031 . Instead of opting for an exact interpolation, an approximation strategy was employed to account for errors in the landmark displacements. The PDE was extended to incorporate Gaussian forces that were weighted by the localization uncertainty. The uncertainties, depending on their isotropic or anisotropic nature, were represented as either scalar weights or matrices. An analytic solution was obtained for the extended equation. 3) Free Form Deformations: Free-form deformations (FFDs) is one of the most common types of transformation models in medical image registration. A rectangular grid is superimposed on the image (size ) that gets deformed under the influence of the control points. The dense deformation is given as a summation of tensor products of univariate splines. FFDs were first popularized in the computer graphics community  ,  but gained wide acceptance in the medical image analysis community when coupled with cubic-splines BIB010 - BIB032 . The displacement field is given as (8) where and . represents the th basis function of the -spline and denotes displacement. This transformation model is simple and can efficiently provide smooth deformations. Moreover, it requires few degrees of freedom to describe local deformations. While in general the transformations that result from cubic -spline FFDs are smooth, the preservation of topology is not guaranteed. Rueckert et al. BIB028 imposed the hard constraints proven in  to produce diffeomorphic deformation fields. The required condition is that the maximum displacement should not be greater than four tenths of the grid spacing. Preservation of topology may also be ensured through the use of soft constraints (see Section II-D1). Many extensions of FFDs have been proposed in the literature. While FFDs are usually uniform, nonuniform approaches have been proposed. Schnabel et al. BIB017 proposed to use multilevel B-splines. In this case, the transformation was given as a summation of the individual transformations of each level. The authors proposed to assign to every control point a status, either active or passive, in order to simulate a nonuniform control point distribution. Active control points were allowed to move, while passive control points remained fixed. Wang and Jiang BIB030 employed nonuniform rational -splines (NURBS) to perform medical image registration in an adaptive focus manner. Shi et al. BIB041 used the multi-level B-splines model of BIB017 while imposing that only a sparse subset of the control points is active. Noblet et al. BIB033 presented a symmetric extension of FFDs. The authors assumed that both images deform toward a common domain under the influence of two isomorphic grids. The common domain was assumed to be in an equal distance from the source and the target. Given the parametric nature of the transformation, this results in constraining the displacements of the corresponding nodes in the two grids to sum to zero. Moreover, in order to calculate the mapping from one image domain to the other, the respective estimated mappings toward the common domain should be invertible. Feng et al. BIB035 proposed an inverse consistent method based on FFDs. The proposed method did not require the inversion of the deformation field. It examined how well the composition of the two transformations mapped back to the image domain. Sotiras and Paragios BIB042 used a similar model to BIB033 . The two models differed in the way the invertibility of the mappings was guaranteed, and the fact that in BIB042 , the registration problem was formulated as a discrete labeling one. FFDs have been extended to tackle multiple-image registration where hard constraints are employed to define a reference domain BIB022 - BIB038 . Moreover, the transformation model has been extended to the spatio-temporal domain where -splines are also used for the temporal axis BIB023 - BIB039 .