Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <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> BIB001 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> A food plate package having a bottom plate and a top cover for the temporary storage of prepared food products, the top cover being convex-oval, with support ribs integrally formed in the cover. The support ribs provide a raised flat surface to stably support another food plate package when stacking one on top of the other. Channels are provided between the support ribs for an air space between adjacent packages when stacked and the subsequent dissipation of overly high temperatures in the central portion. <s> BIB002 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> ▪ Abstract This paper reviews literature, current concepts and approaches in computational anatomy (CA). The model of CA is a Grenander deformable template, an orbit generated from a template under groups of diffeomorphisms. The metric space of all anatomical images is constructed from the geodesic connecting one anatomical structure to another in the orbit. The variational problems specifying these metrics are reviewed along with their associated Euler-Lagrange equations. The Euler equations of motion derived by Arnold for the geodesics in the group of divergence-free volume-preserving diffeomorphisms of incompressible fluids are generalized for the larger group of diffeomorphisms used in CA with nonconstant Jacobians. Metrics that accommodate photometric variation are described extending the anatomical model to incorporate the construction of neoplasm. Metrics on landmarked shapes are reviewed as well as Joshi's diffeomorphism metrics, Bookstein's thin-plate spline approximate-metrics, and Kendall's aff... <s> BIB003 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> In a previous paper, it was proposed to see the deformations of a common pattern as the action of an infinite dimensional group. We show in this paper that this approac h can be applied numerically for pattern matching in image analysis of digital images. Using Lie group ideas, we construct a distance between deformations defined through a metric given the cost of infinitesimal deformations. Then we propose a numerical scheme to solve a variational problem involving this distance and leading to a sub-optimal gradient pattern matching. Its links with fluid models are established. <s> BIB004 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper examine the Euler-Lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in Dupuis et al. (1998) and Trouve (1995) in which two images I 0, I 1 are given and connected via the diffeomorphic change of coordinates I 0???1=I 1 where ?=?1 is the end point at t= 1 of curve ? t , t?[0, 1] satisfying .? t =v t (? t ), t? [0,1] with ?0=id. The variational problem takes the form ::: ::: $$\mathop {\arg {\text{m}}in}\limits_{\upsilon :\dot \phi _t = \upsilon _t \left( {\dot \phi } \right)} \left( {\int_0^1 {\left\| {\upsilon _t } \right\|} ^2 {\text{d}}t + \left\| {I_0 \circ \phi _1^{ - 1} - I_1 } \right\|_{L^2 }^2 } \right),$$ ::: ::: where ?v t? V is an appropriate Sobolev norm on the velocity field v t(·), and the second term enforces matching of the images with ?·?L 2 representing the squared-error norm. ::: ::: In this paper we derive the Euler-Lagrange equations characterizing the minimizing vector fields v t, t?[0, 1] assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms. We describe the implementation of the Euler equations using semi-lagrangian method of computing particle flows and show the solutions for various examples. As well, we compute the metric distance on several anatomical configurations as measured by ?0 1?v t? V dt on the geodesic shortest paths. <s> BIB005 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <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> BIB006 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper constructs metrics on the space of images {\cal I} defined as orbits under group actions {\cal G}. The groups studied include the finite dimensional matrix groups and their products, as well as the infinite dimensional diffeomorphisms examined in Trouve (1999, Quaterly of Applied Math.) and Dupuis et al. (1998). Quaterly of Applied Math.). Left-invariant metrics are defined on the product {\cal G} \times {\cal I} thus allowing the generation of transformations of the background geometry as well as the image values. Examples of the application of such metrics are presented for rigid object matching with and without signature variation, curves and volume matching, and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another. <s> BIB007 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <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> BIB008 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <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> 5) Flows of Diffeomorphisms: <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> 5) Flows of Diffeomorphisms: <s> This paper proposes a method to match diffusion tensor magnetic resonance images (DT-MRIs) through the large deformation diffeomorphic metric mapping of vector fields, focusing on the fiber orientations, considered as unit vector fields on the image volume. We study a suitable action of diffeomorphisms on such vector fields, and provide an extension of the Large Deformation Diffeomorphic Metric Mapping framework to this type of dataset, resulting in optimizing for geodesics on the space of diffeomorphisms connecting two images. Existence of the minimizers under smoothness assumptions on the compared vector fields is proved, and coarse to fine hierarchical strategies are detailed, to reduce both ambiguities and computation load. This is illustrated by numerical experiments on DT-MRI heart images. <s> BIB011 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> We present a new method for computing an optimal deformation between two arbitrary surfaces embedded in Euclidean 3-dimensional space. Our main contribution is in building a norm on the space of surfaces via representation by currents of geometric measure theory. Currents are an appropriate choice for representations because they inherit natural transformation properties from differential forms. We impose a Hilbert space structure on currents, whose norm gives a convenient and practical way to define a matching functional. Using this Hilbert space norm, we also derive and implement a surface matching algorithm under the large deformation framework, guaranteeing that the optimal solution is a one-to-one regular map of the entire ambient space. We detail an implementation of this algorithm for triangular meshes and present results on 3D face and medical image data. <s> BIB012 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper proposes a method to match diffusion tensor magnetic resonance images (DT-MRI) through the large deformation diffeomorphic metric mapping of tensor fields on the image volume, resulting in optimizing for geodesics on the space of diffeomorphisms connecting two diffusion tensor images. A coarse to fine multi-resolution and multikernel- width scheme is detailed, to reduce both ambiguities and computation load. This is illustrated by numerical experiments on DT-MRI brain and images. <s> BIB013 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> We develop a novel Lagrangian reference frame diffeomorphic image and landmark registration method. The algorithm uses the fixed Langrangian reference frame to define the map between coordinate systems, but also generates and stores the inverse map from the Eulerian to the Lagrangian frame. Computing both maps allows facile computation of both Eulerian and Langrangian quantities. We apply this algorithm to estimating a putative evolutionary change of coordinates between a population of chimpanzee and human cortices. Inter-species functional homologues fix the map explicitly, where they are known, while image similarities guide the alignment elsewhere. This map allows detailed study of the volumetric change between chimp and human cortex. Instead of basing the inter-species study on a single species atlas, we diffeomorphically connect the mean shape and intensity templates for each group. The human statistics then map diffeomorphically into the space of the chimpanzee cortex providing a comparison between species. The population statistics show a significant doubling of the relative prefrontal lobe size in humans, as compared to chimpanzees. <s> BIB014 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> In this article, we focus on the computation of statistics of invertible geometrical deformations (i.e., diffeomorphisms), based on the generalization to this type of data of the notion of principal logarithm. Remarkably, this logarithm is a simple 3D vector field, and is well-defined for diffeomorphisms close enough to the identity. This allows to perform vectorial statistics on diffeomorphisms, while preserving the invertibility constraint, contrary to Euclidean statistics on displacement fields. We also present here two efficient algorithms to compute logarithms of diffeomorphisms and exponentials of vector fields, whose accuracy is studied on synthetic data. Finally, we apply these tools to compute the mean of a set of diffeomorphisms, in the context of a registration experiment between an atlas an a database of 9 T1 MR images of the human brain. <s> BIB015 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> Nonrigid medical image registration between images that are linked by an invertible transformation is an inherently symmetric problem. The transformation that registers the image pair should ideally be the inverse of the transformation that registers the pair with the order of images interchanged. This property is referred to as symmetry in registration or inverse consistent registration. However, in practical estimation, the available registration algorithms have tended to produce inverse inconsistent transformations when the template and target images are interchanged. In this paper, we propose two novel cost functions in the large deformation diffeomorphic framework that are inverse consistent. These cost functions have symmetric data-attachment terms; in the first, the matching error is measured at all points along the flow between template and target, and in the second, matching is enforced only at the midpoint of the flow between the template and target. We have implemented these cost functions and present experimental results to validate their inverse consistent property and registration accuracy. <s> BIB016 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> Spatial normalization is a crucial step in assessing patterns of neuroanatomical structure and function associated with health and disease. Errors that occur during spatial normalization can influence hypothesis testing due to the dimensionalities of mapping algorithms and anatomical manifolds (landmarks, curves, surfaces, volumes) used to drive the mapping algorithms. The primary aim of this paper is to improve statistical inference using multiple anatomical manifolds and large deformation diffeomorphic metric mapping (LDDMM) algorithms. We propose that combining information generated by the various manifolds and algorithms improves the reliability of hypothesis testing. We used this unified approach to assess variation in the thickness of the cingulate gyrus in subjects with schizophrenia and healthy comparison subjects. Three different LDDMM algorithms for mapping landmarks, curves and triangulated meshes were used to transform thickness maps of the cingulate surfaces into an atlas coordinate system. We then tested for group differences by combining the information from the three types of anatomical manifolds and LDDMM mapping algorithms. The unified approach provided reliable statistical results and eliminated ambiguous results due to surface mismatches. Subjects with schizophrenia had non-uniform cortical thinning over the left and right cingulate gyri, especially in the anterior portion, as compared to healthy comparison subjects. <s> BIB017 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper presents a series of applications of the Jacobi evolution equations along geodesics in groups of diffeomorphisms. We describe, in particular, how they can be used to perform implementable gradient descent algorithms for image matching, in several situations, and illustrate this with 2D and 3D experiments. We also discuss parallel translation in the group, and its projection on shape manifolds, and focus in particular on an implementation of these equations using iterated Jacobi fields. <s> BIB018 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> Diffeomorphic image registration, where images are aligned using diffeomorphic warps, is a popular subject for research in medical image analysis. We introduce a novel algorithm for computing diffeomorphic warps that solves the Euler equations on the diffeomorphism group explicitly, based on a discretisation of the Hamiltonian, rather than using an optimiser. The result is an algorithm that is many times faster than those considered previously. <s> BIB019 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper describes DARTEL, which is an algorithm for diffeomorphic image registration. It is implemented for both 2D and 3D image registration and has been formulated to include an option for estimating inverse consistent deformations. Nonlinear registration is considered as a local optimisation problem, which is solved using a Levenberg-Marquardt strategy. The necessary matrix solutions are obtained in reasonable time using a multigrid method. A constant Eulerian velocity framework is used, which allows a rapid scaling and squaring method to be used in the computations. DARTEL has been applied to intersubject registration of 471 whole brain images, and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects. <s> BIB020 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper focuses on the issue of translating the relative variation of one shape with respect to another in a template centered representation. The context is the theory of Diffeomorphic Pattern Matching which provides a representation of the space of shapes of objects, including images and point sets, as an infinite dimensional Riemannian manifold which is acted upon by groups of diffeomorphisms. We discuss two main options for achieving our goal; the first one is the parallel translation, based on the Riemannian metric; the second one, based on the group action, is the coadjoint transport. These methods are illustrated with 3D experiments. <s> BIB021 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> One of the most challenging problems in modern neuroimaging is detailed characterization of neurodegeneration. Quantifying spatial and longitudinal atrophy patterns is an important component of this process. These spatiotemporal signals will aid in discriminating between related diseases, such as frontotemporal dementia (FTD) and Alzheimer's disease (AD), which manifest themselves in the same at-risk population. Here, we develop a novel symmetric image normalization method (SyN) for maximizing the cross-correlation within the space of diffeomorphic maps and provide the Euler-Lagrange equations necessary for this optimization. We then turn to a careful evaluation of our method. Our evaluation uses gold standard, human cortical segmentation to contrast SyN's performance with a related elastic method and with the standard ITK implementation of Thirion's Demons algorithm. The new method compares favorably with both approaches, in particular when the distance between the template brain and the target brain is large. We then report the correlation of volumes gained by algorithmic cortical labelings of FTD and control subjects with those gained by the manual rater. This comparison shows that, of the three methods tested, SyN's volume measurements are the most strongly correlated with volume measurements gained by expert labeling. This study indicates that SyN, with cross-correlation, is a reliable method for normalizing and making anatomical measurements in volumetric MRI of patients and at-risk elderly individuals. <s> BIB022 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> One of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Over the last decade, our group has progressively developed several approaches for this problem, all related to the Riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act. Several important shape evolution equations that are now used routinely in applications have emerged over time. Our goal in this paper is to provide an overview of these equations, placing them in their theoretical context, and giving examples of applications in which they can be used. We introduce the required theoretical background before discussing several classes of equations of increasingly complexity. These equations include energy minimizing evolutions deriving from Riemannian gradient descent, geodesics, parallel transport and Jacobi fields. <s> BIB023 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> Computing, visualizing and interpreting statistics on shapes like curves or surfaces is a real challenge with many applications ranging from medical image analysis to computer graphics. Modeling such geometrical primitives with currents avoids to base the comparison between primitives either on a selection of geometrical measures (like length, area or curvature) or on the assumption of point-correspondence. This framework has been used relevantly to register brain surfaces or to measure geometrical invariants. However, while the state-of-the-art methods efficiently perform pairwise registrations, new numerical schemes are required to process groupwise statistics due to an increasing complexity when the size of the database is growing. In this paper, we propose a Matching Pursuit Algorithm for currents, which allows us to approximate, at any desired accuracy, the mean and modes of a population of geometrical primitives modeled as currents. This leads to a sparse representation of the currents, which offers a way to visualize, and hence to interpret, such statistics. More importantly, this tool allows us to build atlases from a population of currents, based on a rigorous statistical model. In this model, data are seen as deformations of an unknown template perturbed by random currents. A Maximum A Posteriori approach is used to estimate consistently the template, the deformations of this template to each data and the residual perturbations. Statistics on both the deformations and the residual currents provide a complete description of the geometrical variability of the structures. Eventually, this framework is generic and can be applied to a large range of anatomical data. We show the relevance of our approach by describing the variability of population of sulcal lines, surfaces of internal structures of the brain and white matter fiber bundles. Complementary experiments on simulated data show the potential of the method to give anatomical characterization of pathologies in the context of supervised learning. <s> BIB024 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper proposes a new registration method for the in vivo quantification of cardiac deformation from a sequence of possibly noisy images. Our algorithm has been applied to 3D ultrasound (3D-US) images, which currently give a reasonable spatial and time resolution, but suffer from significant acquisition noise. Therefore, this modality requires the design of a robust strategy to quantify motion and deformation with the clinical aim of better quantifying cardiac function e.g. in heart failure. In the proposed method, referred to as Large Diffeomorphic Free Form Deformation (LDFFD), the displacement field at each time step is computed from a smooth non-stationary velocity field, thus imposing a coupling between the transformations at successive time steps. Our contribution is to extend this framework to the estimation of motion and deformation in an image sequence. Similarity is captured for the entire image sequence using an extension of the pairwise mutual information metric. The LDFFD algorithm is applied here to recover longitudinal strain curves from healthy and Left-Bundle Branch Block (LBBB) subjects. Strain curves for the healthy subjects are in accordance with the literature. For the LBBB patient, strain quantified before and after Cardiac Resynchronization Therapy show a clear improvement of cardiac function in this subject, in accordance with clinical observations. <s> BIB025 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> Computational Anatomy aims for the study of variability in anatomical structures from images. Variability is encoded by the spatial transformations existing between anatomical images and a template selected as reference. In the absence of a more justified model for inter-subject variability, transformations are considered to belong to a convenient family of diffeomorphisms which provides a suitable mathematical setting for the analysis of anatomical variability. One of the proposed paradigms for diffeomorphic registration is the Large Deformation Diffeomorphic Metric Mapping (LDDMM). In this framework, transformations are characterized as end points of paths parameterized by time-varying flows of vector fields defined on the tangent space of a Riemannian manifold of diffeomorphisms and computed from the solution of the non-stationary transport equation associated to these flows. With this characterization, optimization in LDDMM is performed on the space of non-stationary vector field flows resulting into a time and memory consuming algorithm. Recently, an alternative characterization of paths of diffeomorphisms based on constant-time flows of vector fields has been proposed in the literature. With this parameterization, diffeomorphisms constitute solutions of stationary ODEs. In this article, the stationary parameterization is included for diffeomorphic registration in the LDDMM framework. We formulate the variational problem related to this registration scenario and derive the associated Euler-Lagrange equations. Moreover, the performance of the non-stationary vs the stationary parameterizations in real and simulated 3D-MRI brain datasets is evaluated. Compared to the non-stationary parameterization, our proposal provides similar results in terms of image matching and local differences between the diffeomorphic transformations while drastically reducing memory and time requirements. <s> BIB026 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <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> BIB027 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> In the framework of large deformation diffeomorphic metric mapping (LDDMM), we present a practical methodology to integrate prior knowledge about the registered shapes in the regularizing metric. Our goal is to perform rich anatomical shape comparisons from volumetric images with the mathematical properties offered by the LDDMM framework. We first present the notion of characteristic scale at which image features are deformed. We then propose a methodology to compare anatomical shape variations in a multi-scale fashion, i.e., at several characteristic scales simultaneously. In this context, we propose a strategy to quantitatively measure the feature differences observed at each characteristic scale separately. After describing our methodology, we illustrate the performance of the method on phantom data. We then compare the ability of our method to segregate a group of subjects having Alzheimer's disease and a group of controls with a classical coarse to fine approach, on standard 3D MR longitudinal brain images. We finally apply the approach to quantify the anatomical development of the human brain from 3D MR longitudinal images of pre-term babies. Results show that our method registers accurately volumetric images containing feature differences at several scales simultaneously with smooth deformations. <s> BIB028 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> The Large Deformation Diffeomorphic Metric Mapping framework constitutes a widely used and mathematically well-founded setup for registration in medical imaging. At its heart lies the notion of the regularization kernel, and the choice of kernel greatly affects the results of registrations. This paper presents an extension of the LDDMM framework allowing multiple kernels at multiple scales to be incorporated in each registration while preserving many of the mathematical properties of standard LDDMM. On a dataset of landmarks from lung CT images, we show by example the influence of the kernel size in standard LDDMM, and we demonstrate how our framework, LDDKBM, automatically incorporates the advantages of each scale to reach the same accuracy as the standard method optimally tuned with respect to scale. The framework, which is not limited to landmark data, thus removes the need for classical scale selection. Moreover, by decoupling the momentum across scales, it promises to provide better interpolation properties, to allow sparse descriptions of the total deformation, to remove the tradeoff between match quality and regularity, and to allow for momentum based statistics using scale information. <s> BIB029 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper presents a nonlinear image registration algorithm based on the setting of Large Deformation Diffeomorphic Metric Mapping (LDDMM), but with a more efficient optimisation scheme — both in terms of memory required and the number of iterations required to reach convergence. Rather than perform a variational optimisation on a series of velocity fields, the algorithm is formulated to use a geodesic shooting procedure, so that only an initial velocity is estimated. A Gauss–Newton optimisation strategy is used to achieve faster convergence. The algorithm was evaluated using freely available manually labelled datasets, and found to compare favourably with other inter-subject registration algorithms evaluated using the same data. <s> BIB030 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> The alignment and normalization of individual brain structures is a prerequisite for group-level analyses of structural and functional neuroimaging data. The techniques currently available are either based on volume and/or surface attributes, with limited insight regarding the consistent alignment of anatomical landmarks across individuals. This article details a global, geometric approach that performs the alignment of the exhaustive sulcal imprints (cortical folding patterns) across individuals. This DIffeomorphic Sulcal-based COrtical (DISCO) technique proceeds to the automatic extraction, identification and simplification of sulcal features from T1-weighted Magnetic Resonance Image (MRI) series. These features are then used as control measures for fully-3-D diffeomorphic deformations. Quantitative and qualitative evaluations show that DISCO correctly aligns the sulcal folds and gray and white matter volumes across individuals. The comparison with a recent, iconic diffeomorphic approach (DARTEL) highlights how the absence of explicit cortical landmarks may lead to the misalignment of cortical sulci. We also feature DISCO in the automatic design of an empirical sulcal template from group data. We also demonstrate how DISCO can efficiently be combined with an image-based deformation (DARTEL) to further improve the consistency and accuracy of alignment performances. Finally, we illustrate how the optimized alignment of cortical folds across subjects improves sensitivity in the detection of functional activations in a group-level analysis of neuroimaging data. <s> BIB031 </s> Deformable Medical Image Registration: A Survey <s> 5) Flows of Diffeomorphisms: <s> This paper describes the variational formulation of template matching problems of computational anatomy (CA); introduces the EPDiff evolution equation in the context of an analogy between CA and fluid dynamics; discusses the singular solutions for the EPDiff equation and explains why these singular solutions exist (singular momentum map). Then it draws the consequences of EPDiff for outline matching problem in CA and gives numerical examples. <s> BIB032
Flows of diffeomorphisms have also been proposed to model the deformation. In this case, the deformation is modeled by considering its velocity over time according to the Lagrange transport equation BIB001 , BIB002 , BIB004 . The regularization term constrains the velocity field to be smooth (6) is a norm on the space of smooth velocity vector fields defined as , where is a differential operator and is the norm of square integrable functions. Choosing a kernel associated with allows for the modeling of different types of spatial regularization BIB002 . While most often a single Gaussian kernel is used BIB028 , it is possible to use multiple kernels and smooth the deformations adaptively at different scales BIB028 , BIB029 . Lastly, the fact that the velocity field varies over time allows for the estimation of large deformations BIB005 . This framework, known as large deformation diffeomorphic metric mapping (LDDMM), allows for the definition of a distance between images or sets of points  , BIB006 . The distance between these elements is defined as a geodesic, according to a metric, that connects them and can be used for studies of anatomical variability BIB003 . A number of theoretical aspects of this framework and especially the ones related with computational analysis were further developed in BIB007 - BIB021 . The interested reader is referred to BIB023 for an overview of its evolution and the corresponding equations. The LDDMM framework has been extended to solve a number of problems. Among its extensions, one may cite volume registration for scalar BIB005 , BIB016 - BIB030 vector- BIB011 and tensor-valued data BIB013 , point-matching  , point-matching on spheres BIB008 , matching sets of unlabeled points BIB009 - BIB010 , shape-matching BIB028 , BIB012 , curve-mapping BIB017 - BIB024 , and hybrid registration BIB014 , BIB031 . Even though the LDDMM framework provides diffeomorphic transformations, it is not symmetric. To encode the symmetric property a number of approaches have been proposed BIB016 , BIB022 , BIB018 . Beg and Khan BIB016 focused on providing symmetric data terms. Younes BIB018 also discussed ways to render the alignment process symmetric while Avants et al. BIB022 presented a symmetric LDDMM registration process driven by cross-correlation. The mathematical rigor of the LDDMM framework comes at an important cost. The fact that the velocity field has to be integrated over time results in high computational and memory demands. Moreover, the gradient descent scheme that is usually employed to solve the optimization problem of the geodesic path estimation converges slowly BIB030 . More efficient optimization techniques for the LDDMM have been investigated in BIB030 , BIB019 , BIB032 . Cotter and Holm presented an approach that involves a particle mesh method in BIB032 . Marsland and McLachlan BIB019 formulated the problem in a PDE framework and used a particle method to solve for the diffeomorphism. More recently, Ashburner and Friston BIB030 gave a Gauss-Newton implementation of the algorithm in BIB032 . These approaches were based on the fact that the initial velocity field is sufficient to calculate the intermediate and final deformations. In other words, the diffeomorphism is parametrized by the initial velocity field. These calculations are possible by reformulating the initial boundary problem to an initial value one. The initial conditions comprise the initial velocity and the starting position. The optimization opts to estimate the initial velocity field that best aligns the images. This approach is known as geodesic shooting. An alternative way to efficiently calculate diffeomorphisms involves the simplification of the problem by decreasing its degrees of freedom. Stationary velocity fields BIB015 have been used towards this direction. Despite being limited with respect to the diffeomorphisms that they can capture, stationary velocity fields are a common choice among many researchers BIB020 - BIB025 . Hernandez et al. followed this approach and used stationary Ordinary Differential Equations (ODEs) in the LDDMM framework BIB026 . Ashburner BIB020 assumed the velocity field to be constant over time in order to propose a fast diffeomorphic image registration that was based on either membrane, bending or linear elastic energy. The solution was estimated through integration over time by composing successive solutions. Given an even number of steps, this was performed efficiently by a scaling and squaring approach BIB027 ,  . Furthermore, the exponential of the flow field was used to guarantee that the inferred mapping is diffeomorphic. The energy was optimized using the Levenberg-Marquardt algorithm coupled with a full multi-grid approach to efficiently compute its update step.