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44
N.: A LogEuclidean framework for statistics on diffeomorphisms
 In: Proc. MICCAI’06. (2006) 924–931
"... Abstract. 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 welldefined for ..."
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Cited by 101 (44 self)
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Abstract. 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 welldefined 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. 1
Generalized TensorBased Morphometry of HIV/AIDS Using Multivariate Statistics on Deformation Tensors
"... Abstract—This paper investigates the performance of a new multivariate method for tensorbased morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical temp ..."
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Cited by 43 (10 self)
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Abstract—This paper investigates the performance of a new multivariate method for tensorbased morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor) of these deformations, as is common, we retain the full deformation tensors and apply a manifold version of Hotelling’s 2 test to them, in a LogEuclidean domain. In 2D and 3D magnetic resonance imaging (MRI) data from 26 HIV/AIDS patients and 14 matched healthy subjects, we compared multivariate tensor analysis versus univariate tests of simpler tensorderived indices: the Jacobian determinant, the trace, geodesic anisotropy, and eigenvalues of the deformation tensor, and the angle of rotation of its eigenvectors. We detected consistent, but more extensive patterns of structural abnormalities, with multivariate tests on the full tensor manifold. Their improved power was established by analyzing cumulativevalue plots using false discovery rate (FDR) methods, appropriately controlling for false positives. This increased detection sensitivity may empower drug trials and largescale studies of disease that use tensorbased morphometry. Index Terms—Brain, image analysis, Lie groups, magnetic resonance imaging (MRI), statistics. I.
Deformable medical image registration: A survey
 IEEE TRANSACTIONS ON MEDICAL IMAGING
, 2013
"... Deformable image registration is a fundamental task in medical image processing. Among its most important applications, one may cite: i) multimodality fusion, where information acquired by different imaging devices or protocols is fused to facilitate diagnosis and treatment planning; ii) longitudin ..."
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Cited by 34 (1 self)
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Deformable image registration is a fundamental task in medical image processing. Among its most important applications, one may cite: i) multimodality fusion, where information acquired by different imaging devices or protocols is fused to facilitate diagnosis and treatment planning; ii) longitudinal studies, where temporal structural or anatomical changes are investigated; and iii) population modeling and statistical atlases used to study normal anatomical variability. In this paper, we attempt to give an overview of deformable registration methods, putting emphasis on the most recent advances in the domain. Additional emphasis has been given to techniques applied to medical images. In order to study image registration methods in depth, their main components are identified and studied independently. The most recent techniques are presented in a systematic fashion. The contribution of this paper is to provide an extensive account of registration techniques in a systematic manner.
DTREFinD: Diffusion Tensor Registration with Exact FiniteStrain Differential
 IEEE Transactions on Medical Imaging, In
, 2009
"... Abstract—In this paper, we propose the DTREFinD algorithm for the diffeomorphic nonlinear registration of diffusion tensor images. Unlike scalar images, deforming tensor images requires choosing both a reorientation strategy and an interpolation scheme. Current diffusion tensor registration algorit ..."
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Cited by 25 (9 self)
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Abstract—In this paper, we propose the DTREFinD algorithm for the diffeomorphic nonlinear registration of diffusion tensor images. Unlike scalar images, deforming tensor images requires choosing both a reorientation strategy and an interpolation scheme. Current diffusion tensor registration algorithms that use full tensor information face difficulties in computing the differential of the tensor reorientation strategy and consequently, these methods often approximate the gradient of the objective function. In the case of the finitestrain (FS) reorientation strategy, we borrow results from the pose estimation literature in computer vision to derive an analytical gradient of the registration objective function. By utilizing the closedform gradient and the velocity field representation of one parameter subgroups of diffeomorphisms, the resulting registration algorithm is diffeomorphic and fast. We contrast the algorithm with a traditional FS alternative
Spherical Demons: Fast Diffeomorphic LandmarkFree Surface Registration
 IEEE TRANSACTIONS ON MEDICAL IMAGING. 29(3):650–668, 2010
, 2010
"... We present the Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizors for the modified Demons objective function can be efficiently approximated on the sphere using iterative smoothing. B ..."
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Cited by 24 (5 self)
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We present the Spherical Demons algorithm for registering two spherical images. By exploiting spherical vector spline interpolation theory, we show that a large class of regularizors for the modified Demons objective function can be efficiently approximated on the sphere using iterative smoothing. Based on one parameter subgroups of diffeomorphisms, the resulting registration is diffeomorphic and fast. The Spherical Demons algorithm can also be modified to register a given spherical image to a probabilistic atlas. We demonstrate two variants of the algorithm corresponding to warping the atlas or warping the subject. Registration of a cortical surface mesh to an atlas mesh, both with more than 160k nodes requires less than 5 minutes when warping the atlas and less than 3 minutes when warping the subject on a Xeon 3.2GHz single processor machine. This is comparable to the fastest nondiffeomorphic landmarkfree surface registration algorithms. Furthermore, the accuracy of our method compares favorably to the popular FreeSurfer registration algorithm. We validate the technique in two different applications that use registration to transfer segmentation labels onto a new image: (1) parcellation of invivo cortical surfaces and (2) Brodmann area localization in exvivo cortical surfaces.
Statistical properties of Jacobian maps and the realization of unbiased largedeformation nonlinear image registration
 IEEE Trans. Med. Imaging
"... Abstract—Maps of local tissue compression or expansion are often computed by comparing magnetic resonance imaging (MRI) scans using nonlinear image registration. The resulting changes are commonly analyzed using tensorbased morphometry to make inferences about anatomical differences, often based on ..."
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Cited by 19 (5 self)
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Abstract—Maps of local tissue compression or expansion are often computed by comparing magnetic resonance imaging (MRI) scans using nonlinear image registration. The resulting changes are commonly analyzed using tensorbased morphometry to make inferences about anatomical differences, often based on the Jacobian map, which estimates local tissue gain or loss. Here, we provide rigorous mathematical analyses of the Jacobian maps, and use themto motivate a new numerical method to construct unbiased nonlinear image registration. First, we argue that logarithmic transformation is crucial for analyzing Jacobian values representing morphometric differences. We then examine the statistical distributions of logJacobian maps by defining the Kullback–Leibler (KL) distance on material density functions arising in continuummechanical models. With this framework, unbiased image registration can be constructed by quantifying
A multiscale kernel bundle for LDDMM: Towards sparse deformation description across space and scales
 IN: IPMI. LNCS
, 2011
"... The Large Deformation Diffeomorphic Metric Mapping framework constitutes a widely used and mathematically wellfounded 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 pap ..."
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Cited by 15 (7 self)
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The Large Deformation Diffeomorphic Metric Mapping framework constitutes a widely used and mathematically wellfounded 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.
Leftinvariant Riemannian elasticity: a distance on shape diffeomorphisms
, 2006
"... Abstract. In intersubject registration, one often lacks a good model of the transformation variability to choose the optimal regularization. Some works attempt to model the variability in a statistical way, but the reintroduction in a registration algorithm is not easy. In [1], we interpreted the ..."
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Cited by 15 (5 self)
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Abstract. In intersubject registration, one often lacks a good model of the transformation variability to choose the optimal regularization. Some works attempt to model the variability in a statistical way, but the reintroduction in a registration algorithm is not easy. In [1], we interpreted the elastic energy as the distance of the GreenSt Venant strain tensor to the identity. By changing the Euclidean metric for a more suitable Riemannian one, we defined a consistent statistical framework to quantify the amount of deformation. In particular, the mean and the covariance matrix of the strain tensor could be efficiently computed from a population of nonlinear transformations and introduced as parameters in a Mahalanobis distance to measure the statistical deviation from the observed variability. This statistical Riemannian elasticity was able to handle anisotropic deformations but its isotropic stationary version was locally inverseconsistent. In this paper, we investigate how to modify the Riemannian elasticity to make it globally inverse consistent. This allows to define a leftinvariant ”distance ” between shape diffeomorphisms that we call the leftinvariant Riemannian elasticity. Such a closed form energy on diffeomorphisms can optimize it directly without relying on a time and memory consuming numerical optimization of the geodesic path. 1
Triangular Springs for Modeling NonLinear Membranes
"... This paper provides a formal connexion between springs and continuum mechanics in the context of onedimensional and twodimensional elasticity. In a first stage, the equivalence between tensile springs and the finite element discretization of stretching energy on planar curves is established. Furth ..."
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Cited by 14 (2 self)
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This paper provides a formal connexion between springs and continuum mechanics in the context of onedimensional and twodimensional elasticity. In a first stage, the equivalence between tensile springs and the finite element discretization of stretching energy on planar curves is established. Furthermore, when considering a quadratic strain function of stretch, we introduce a new type of springs called tensile biquadratic springs. In a second stage, we extend this equivalence to nonlinear membranes (St VenantKirchhoff materials) on triangular meshes leading to triangular biquadratic and quadratic springs. Those tensile and angular springs produce isotropic deformations parameterized by Young modulus and Poisson ratios on unstructured meshes in an efficient and simple way. For a specific choice of the Poisson ratio, 0.3, we show that regular springmass models may be used realistically to simulate a membrane behavior. Finally, the different spring formulations are tested in pure traction and cloth simulation experiments.
A nonlinear elastic shape averaging approach
 SIAM Journal on Imaging Sciences
, 2008
"... Abstract. A physically motivated approach is presented to compute a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elast ..."
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Cited by 11 (6 self)
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Abstract. A physically motivated approach is presented to compute a shape average of a given number of shapes. An elastic deformation is assigned to each shape. The shape average is then described as the common image under all elastic deformations of the given shapes, which minimizes the total elastic energy stored in these deformations. The underlying nonlinear elastic energy measures the local change of length, area, and volume. It is invariant under rigid body motions, and isometries are local minimizers. The model is relaxed involving a further energy which measures how well the elastic deformation image of a particular shape matches the average shape, and a suitable shape prior can be considered for the shape average. Shapes are represented via their edge sets, which also allows for an application to averaging image morphologies described via ensembles of edge sets. To make the approach computationally tractable, sharp edges are approximated via phase fields, and a corresponding variational phase field model is derived. Finite elements are applied for the spatial discretization, and a multiscale alternating minimization approach allows the efficient computation of shape averages in 2D and 3D. Various applications, e. g. averaging the shape of feet or human organs, underline the qualitative properties of the presented approach.