Diffeomorphic Mapping Research Articles

Stabilizing 3D in vivo intravital microscopy images with an iteratively refined soft-tissue model for immunology experiments

We describe a set of new algorithms and a software tool, StabiTissue, for stabilizing in vivo intravital microscopy images that suffer from soft-tissue background movement. Because these images lack predetermined anchors and are dominated by noise, we use a pixel weighted image alignment together with a correction for nonlinear tissue deformations. We call this correction a poor man׳s diffeomorphic map since it ascertains the nonlinear regions of the image without resorting to a full integral equation method. To determine the quality of the image stabilization, we developed an ensemble sampling method that quantifies the coincidence between image pairs from randomly distributed image regions. We obtain global stabilization alignment through an iterative constrained simulated annealing optimization procedure. To show the accuracy of our algorithm with existing software, we measured the misalignment error rate in datasets taken from two different organs and compared the results to a similar and popular open-source solution. Present open-source stabilization software tools perform poorly because they do not treat the specific needs of the IV-2pM datasets with soft-tissue deformation, speckle noise, full 5D inter- and intra-stack motion error correction, and undefined anchors. In contrast, the results of our tests demonstrate that our method is more immune to noise and provides better performance for datasets’ possessing nonlinear tissue deformations. As a practical application of our software, we show how our stabilization improves cell tracking, where the presence of background movement would degrade track information. We also provide a qualitative comparison of our software with other open-source libraries/applications. Our software is freely available at the open source repository http://sourceforge.net/projects/stabitissue/.

Diffeomorphic Metric Landmark Mapping Using Stationary Velocity Field Parameterization

Large deformation diffeomorphic metric mapping (LDDMM) has been shown as an effective computational paradigm to measure anatomical variability. However, its time-varying vector field parameterization of diffeomorphism flow leads to computationally expensive implementation, as well as some theoretical issues in metric based shape analysis, e.g. high order metric approximation via Baker---Campbell---Hausdorff (BCH) formula. To address these problems, we study the role of stationary vector field parameterization in context of LDDMM. Under this setting registration is formulated as finding the Lie group exponential path with minimal energy in Riemannian manifold of diffeomorphisms bringing two shapes together. Accurate derivation of Euler---Lagrange equation shows that optimal vector field for landmark matching is associated with singular momenta at landmark trajectories in whole time domain, and a new momentum optimization scheme is proposed to solve the variational problem. Length of group exponential path is also proposed as an alternative shape metric to geodesic distance, and pair-wise metrics among a population are computed through an approximation method via BCH formula which only needs registrations to a template. The proposed methods have been tested on both synthesized data and real database. Compared to non-stationary parameterization, this method can achieve comparable registration accuracy in significantly reduced time. Second order metric approximation by this method also improves significantly over first order, which can not be achieved by non-stationary parameterization. Correlation between the two shape metrics is also investigated, and their statistical power in clinical study compared.

Multiresolution Diffeomorphic Mapping for Cortical Surfaces.

Due to the convoluted folding pattern of the cerebral cortex, accurate alignment of cortical surfaces remains challenging. In this paper, we present a multiresolution diffeomorphic surface mapping algorithm under the framework of large deformation diffeomorphic metric mapping (LDDMM). Our algorithm takes advantage of multiresolution analysis (MRA) for surfaces and constructs cortical surfaces at multiresolution. This family of multiresolution surfaces are used as natural sparse priors of the cortical anatomy and provide the anchor points where the parametrization of deformation vector fields is supported. This naturally constructs tangent bundles of diffeomorphisms at different resolution levels and hence generates multiresolution diffeomorphic transformation. We show that our construction of multiresolution LDDMM surface mapping can potentially reduce computational cost and improves the mapping accuracy of cortical surfaces.

Simultaneous Longitudinal Registration with Group-Wise Similarity Prior.

Here we present an algorithm for the simultaneous registration of N longitudinal image pairs such that information acquired by each pair is used to constrain the registration of each other pair. More specifically, in the geodesic shooting setting for Large Deformation Diffeomorphic Metric Mappings (LDDMM) an average of the initial momenta characterizing the N transformations is maintained throughout and updates to individual momenta are constrained to be similar to this average. In this way, the N registrations are coupled and explore the space of diffeomorphisms as a group, the variance of which is constrained to be small. Our approach is motivated by the observation that transformations learned from images in the same diagnostic category share characteristics. The group-wise consistency prior serves to strengthen the contribution of the common signal among the N image pairs to the transformation for a specific pair, relative to features particular to that pair. We tested the algorithm on 57 longitudinal image pairs of Alzheimer's Disease patients from the Alzheimer's Disease Neuroimaging Initiative and evaluated the ability of the algorithm to produce momenta that better represent the long-term biological processes occurring in the underlying anatomy. We found that for many image pairs, momenta learned with the group-wise prior better predict a third time point image unobserved in the registration.

Studying ventricular abnormalities in mild cognitive impairment with hyperbolic Ricci flow and tensor-based morphometry

Mild Cognitive Impairment (MCI) is a transitional stage between normal aging and dementia and people with MCI are at high risk of progression to dementia. MCI is attracting increasing attention, as it offers an opportunity to target the disease process during an early symptomatic stage. Structural magnetic resonance imaging (MRI) measures have been the mainstay of Alzheimer's disease (AD) imaging research, however, ventricular morphometry analysis remains challenging because of its complicated topological structure. Here we describe a novel ventricular morphometry system based on the hyperbolic Ricci flow method and tensor-based morphometry (TBM) statistics. Unlike prior ventricular surface parameterization methods, hyperbolic conformal parameterization is angle-preserving and does not have any singularities. Our system generates a one-to-one diffeomorphic mapping between ventricular surfaces with consistent boundary matching conditions. The TBM statistics encode a great deal of surface deformation information that could be inaccessible or overlooked by other methods. We applied our system to the baseline MRI scans of a set of MCI subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI: 71 MCI converters vs. 62 MCI stable). Although the combined ventricular area and volume features did not differ between the two groups, our fine-grained surface analysis revealed significant differences in the ventricular regions close to the temporal lobe and posterior cingulate, structures that are affected early in AD. Significant correlations were also detected between ventricular morphometry, neuropsychological measures, and a previously described imaging index based on fluorodeoxyglucose positron emission tomography (FDG-PET) scans. This novel ventricular morphometry method may offer a new and more sensitive approach to study preclinical and early symptomatic stage AD.

Individualized diffeomorphic mapping of brains with large cortical infarcts

Whole brain mapping of stroke patients with large cortical infarcts is not trivial due to the complexity of infarcts’ anatomical location and appearance in magnetic resonance image. In this study, we proposed an individualized diffeomorphic mapping framework for solving this problem. This framework is based on our recent work of large deformation diffeomorphic metric mapping (LDDMM) in Du et al. (2011) and incorporates anatomical features, such as sulcal/gyral curves, cortical surfaces, brain intensity image, and masks of infarcted regions, in order to align a normal brain to the brain of stroke patients. We applied this framework to synthetic data and data of stroke patients and validated the mapping accuracy in terms of the alignment of gyral/sulcal curves, sulcal regions, and brain segmentation. Our results revealed that this framework provided comparable mapping results for stroke patients and healthy controls, suggesting the importance of incorporating individualized anatomical features in whole brain mapping of brains with large cortical infarcts.

Gauss–Newton inspired preconditioned optimization in large deformation diffeomorphic metric mapping

In this work, we propose a novel preconditioned optimization method in the paradigm of Large Deformation Diffeomorphic Metric Mapping (LDDMM). The preconditioned update scheme is formulated for the non-stationary and the stationary parameterizations of diffeomorphisms, yielding three different LDDMM methods. The preconditioning matrices are inspired in the Hessian approximation used in Gauss–Newton method. The derivatives are computed using Frechet differentials. Thus, optimization is performed in a Sobolev space, in contrast to optimization in L2 commonly used in non-rigid registration literature. The proposed LDDMM methods have been evaluated and compared with their respective implementations of gradient descent optimization. Evaluation has been performed using real and simulated images from the Non-rigid Image Registration Evaluation Project (NIREP). The experiments conducted in this work reported that our preconditioned LDDMM methods achieved a performance similar or superior to well-established-in-literature gradient descent non-stationary LDDMM in the great majority of cases. Moreover, preconditioned optimization showed a substantial reduction in the execution time with an affordable increase of the memory usage per iteration. Additional experiments reported that optimization using Frechet differentials should be preferable to optimization using L2 differentials.

Comparing fractional anisotropy in patients with childhood-onset schizophrenia, their healthy siblings, and normal volunteers through DTI.

Diffusion tensor imaging is a neuroimaging method that quantifies white matter (WM) integrity and brain connectivity based on the diffusion of water in the brain. White matter has been hypothesized to be of great importance in the development of schizophrenia as part of the dysconnectivity model. Childhood-onset schizophrenia (COS), is a rare, severe form of the illness that resembles poor outcome adult-onset schizophrenia. We hypothesized that COS would be associated with WM abnormalities relative to a sample of controls. To evaluate WM integrity in this population 39 patients diagnosed with COS, 39 of their healthy (nonpsychotic) siblings, and 50 unrelated healthy volunteers were scanned using a diffusion tensor imaging (DTI) sequence during a 1.5 T MRI acquisition. Each DTI scan was processed via atlas-based analysis using a WM parcellation map, and diffeomorphic mapping that shapes a template atlas to each individual subject space. Fractional anisotropy (FA), a measure of WM integrity was averaged over each of the 46 regions of the atlas. Eleven WM regions were examined based on previous reports of WM growth abnormalities in COS. Of those regions, patients with COS, and their healthy siblings had significantly lower mean FA in the left and right cuneus as compared to the healthy volunteers (P < .005). Together, these findings represent the largest DTI study in COS to date, and provide evidence that WM integrity is significantly impaired in COS. Shared deficits in their healthy siblings might result from increased genetic risk.

Generalization of Zernike polynomials for regular portions of circles and ellipses.

Zernike polynomials are commonly used to represent the wavefront phase on circular optical apertures, since they form a complete and orthonormal basis on the unit circle. Here, we present a generalization of this Zernike basis for a variety of important optical apertures. On the contrary to ad hoc solutions, most of them based on the Gram-Schmidt orthonormalization method, here we apply the diffeomorphism (mapping that has a differentiable inverse mapping) that transforms the unit circle into an angular sector of an elliptical annulus. In this way, other apertures, such as ellipses, rings, angular sectors, etc. are also included as particular cases. This generalization, based on in-plane warping of the basis functions, provides a unique solution and what is more important, it guarantees a reasonable level of invariance of the mathematical properties and the physical meaning of the initial basis functions. Both, the general form and the explicit expressions for most common, elliptical and annular apertures are provided.

Sliding mode based feedback linearizing controller for grid connected multiple fuel cells scenario

The energy demand is growing at a fast rate around the globe. The conventional power generation technologies may not meet the environmental constraints. Hence, a Solid Oxide Fuel Cell based power generation unit is considered as distributed generating source here, in this paper. An integrated operation of two units is done with a master–slave control strategy. Voltage source converters are used to interface nonconventional energy sources with grid, in general. However, these systems are highly nonlinear and coupled systems. Hence conventional PI controller will not give satisfactory performance for changing operating conditions and uncertain load changes. Hence, feedback linearization scheme has been formulated, which uses nonlinear transformation by means of diffeomorphism mapping to convert nonlinear models into linear models. Sliding mode control technique has been incorporated in the feedback linearized control to address the issues of uncertainty and to add robustness to the control algorithm. The developed control algorithm is tested for different dispatches, load uncertainties, decoupling of control variables and finally the variable grid frequency within a small band.

Multimodal MRI-based Imputation of the Aβ+ in Early Mild Cognitive Impairment.

ObjectiveThe primary goal of this study was to identify brain atrophy from structural MRI (magnetic resonance imaging) and cerebral blood flow (CBF) patterns from arterial spin labeling perfusion MRI that are best predictors of the Aβ-burden, measured as composite 18F-AV45-PET (positron emission tomography) uptake, in individuals with early mild cognitive impairment (MCI). Furthermore, another objective was to assess the relative importance of imaging modalities in classification of Aβ+/Aβ− early MCI.MethodsSixty-seven Alzheimer's Disease Neuroimaging Initiative (ADNI)-GO/2 participants with early MCI were included. Voxel-wise anatomical shape variation measures were computed by estimating the initial diffeomorphic mapping momenta from an unbiased control template. CBF measures normalized to average motor cortex CBF were mapped onto the template space. Using partial least squares regression, we identified the structural and CBF signatures of Aβ after accounting for normal cofounding effects of age, gender, and education.Results18F-AV45-positive early MCIs could be identified with 83% classification accuracy, 87% positive predictive value, and 84% negative predictive value by multidisciplinary classifiers combining demographics data, ApoE ε4-genotype, and a multimodal MRI-based Aβ score.InterpretationMultimodal MRI can be used to predict the amyloid status of early-MCI individuals. MRI is a very attractive candidate for the identification of inexpensive and noninvasive surrogate biomarkers of Aβ deposition. Our approach is expected to have value for the identification of individuals likely to be Aβ+ in circumstances where cost or logistical problems prevent Aβ detection using cerebrospinal fluid analysis or Aβ-PET. This can also be used in clinical settings and clinical trials, aiding subject recruitment and evaluation of treatment efficacy. Imputation of the Aβ-positivity status could also complement Aβ-PET by identifying individuals who would benefit the most from this assessment.

Detection of Time-Varying Structures by Large Deformation Diffeomorphic Metric Mapping to Aid Reading of High-Resolution CT Images of the Lung

ObjectivesTo evaluate the accuracy of advanced non-linear registration of serial lung Computed Tomography (CT) images using Large Deformation Diffeomorphic Metric Mapping (LDDMM).MethodsFifteen cases of lung cancer with serial lung CT images (interval: 62.2±26.9 days) were used. After affine transformation, three dimensional, non-linear volume registration was conducted using LDDMM with or without cascading elasticity control. Registration accuracy was evaluated by measuring the displacement of landmarks placed on vessel bifurcations for each lung segment. Subtraction images and Jacobian color maps, calculated from the transformation matrix derived from image warping, were generated, which were used to evaluate time-course changes of the tumors.ResultsThe average displacement of landmarks was 0.02±0.16 mm and 0.12±0.60 mm for proximal and distal landmarks after LDDMM transformation with cascading elasticity control, which was significantly smaller than 3.11±2.47 mm and 3.99±3.05 mm, respectively, after affine transformation. Emerged or vanished nodules were visualized on subtraction images, and enlarging or shrinking nodules were displayed on Jacobian maps enabled by highly accurate registration of the nodules using LDDMM. However, some residual misalignments were observed, even with non-linear transformation when substantial changes existed between the image pairs.ConclusionsLDDMM provides accurate registration of serial lung CT images, and temporal subtraction images with Jacobian maps help radiologists to find changes in pulmonary nodules.

Spatially-varying metric learning for diffeomorphic image registration: a variational framework.

This paper introduces a variational strategy to learn spatially-varying metrics on large groups of images, in the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. Spatially-varying metrics we learn not only favor local deformations but also correlated deformations in different image regions and in different directions. In addition, metric parameters can be efficiently estimated using a gradient descent method. We first describe the general strategy and then show how to use it on 3D medical images with reasonable computational ressources. Our method is assessed on the 3D brain images of the LPBA40 dataset. Results are compared with ANTS-SyN and LDDMM with spatially-homogeneous metrics.

Deformation field correction for spatial normalization of PET images using a population-derived partial least squares model.

Spatial normalization of positron emission tomography (PET) images is essential for population studies, yet work on anatomically accurate PET-to-PET registration is limited. We present a method for the spatial normalization of PET images that improves their anatomical alignment based on a deformation correction model learned from structural image registration. To generate the model, we first create a population-based PET template with a corresponding structural image template. We register each PET image onto the PET template using deformable registration that consists of an affine step followed by a diffeomorphic mapping. Constraining the affine step to be the same as that obtained from the PET registration, we find the diffeomorphic mapping that will align the structural image with the structural template. We train partial least squares (PLS) regression models within small neighborhoods to relate the PET intensities and deformation fields obtained from the diffeomorphic mapping to the structural image deformation fields. The trained model can then be used to obtain more accurate registration of PET images to the PET template without the use of a structural image. A cross validation based evaluation on 79 subjects shows that our method yields more accurate alignment of the PET images compared to deformable PET-to-PET registration as revealed by 1) a visual examination of the deformed images, 2) a smaller error in the deformation fields, and 3) a greater overlap of the deformed anatomical labels with ground truth segmentations.

Human brain atlas for automated region of interest selection in quantitative susceptibility mapping: Application to determine iron content in deep gray matter structures

The purpose of this paper is to extend the single-subject Eve atlas from Johns Hopkins University, which currently contains diffusion tensor and T1-weighted anatomical maps, by including contrast based on quantitative susceptibility mapping. The new atlas combines a “deep gray matter parcellation map” (DGMPM) derived from a single-subject quantitative susceptibility map with the previously established “white matter parcellation map” (WMPM) from the same subject's T1-weighted and diffusion tensor imaging data into an MNI coordinate map named the “Everything Parcellation Map in Eve Space,” also known as the “EvePM.” It allows automated segmentation of gray matter and white matter structures. Quantitative susceptibility maps from five healthy male volunteers (30 to 33years of age) were coregistered to the Eve Atlas with AIR and Large Deformation Diffeomorphic Metric Mapping (LDDMM), and the transformation matrices were applied to the EvePM to produce automated parcellation in subject space. Parcellation accuracy was measured with a kappa analysis for the left and right structures of six deep gray matter regions. For multi-orientation QSM images, the Kappa statistic was 0.85 between automated and manual segmentation, with the inter-rater reproducibility Kappa being 0.89 for the human raters, suggesting “almost perfect” agreement between all segmentation methods. Segmentation seemed slightly more difficult for human raters on single-orientation QSM images, with the Kappa statistic being 0.88 between automated and manual segmentation, and 0.85 and 0.86 between human raters. Overall, this atlas provides a time-efficient tool for automated coregistration and segmentation of quantitative susceptibility data to analyze many regions of interest. These data were used to establish a baseline for normal magnetic susceptibility measurements for over 60 brain structures of 30- to 33-year-old males. Correlating the average susceptibility with age-based iron concentrations in gray matter structures measured by Hallgren and Sourander (1958) allowed interpolation of the average iron concentration of several deep gray matter regions delineated in the EvePM.

Sobolev Metrics on the Manifold of All Riemannian Metrics

On the manifold $\mathcal{M}(M)$ of all Riemannian metrics on a compact manifold $M$, one can consider the natural $L^2$-metric as described first by D.G. Ebin, The manifold of Riemannian metrics. In this paper we consider variants of this metric, which in general are of higher order. We derive the geodesic equations; we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.

Computational analysis of LDDMM for brain mapping.

One goal of computational anatomy (CA) is to develop tools to accurately segment brain structures in healthy and diseased subjects. In this paper, we examine the performance and complexity of such segmentation in the framework of the large deformation diffeomorphic metric mapping (LDDMM) registration method with reference to atlases and parameters. First we report the application of a multi-atlas segmentation approach to define basal ganglia structures in healthy and diseased kids' brains. The segmentation accuracy of the multi-atlas approach is compared with the single atlas LDDMM implementation and two state-of-the-art segmentation algorithms—Freesurfer and FSL—by computing the overlap errors between automatic and manual segmentations of the six basal ganglia nuclei in healthy subjects as well as subjects with diseases including ADHD and Autism. The high accuracy of multi-atlas segmentation is obtained at the cost of increasing the computational complexity because of the calculations necessary between the atlases and a subject. Second, we examine the effect of parameters on total LDDMM computation time and segmentation accuracy for basal ganglia structures. Single atlas LDDMM method is used to automatically segment the structures in a population of 16 subjects using different sets of parameters. The results show that a cascade approach and using fewer time steps can reduce computational complexity as much as five times while maintaining reliable segmentations.

Higher-Order Momentum Distributions and Locally Affine LDDMM Registration

To achieve sparse parametrizations that allow intuitive analysis, we aim to represent deformation with a basis containing interpretable elements, and we wish to use elements that have the description capacity to represent the deformation compactly. To accomplish this, we introduce in this paper higher-order momentum distributions in the large deformation diffeomorphic metric mapping (LDDMM) registration framework. While the zeroth-order moments previously used in LDDMM only describe local displacement, the first-order momenta that are proposed here represent a basis that allows local description of affine transformations and subsequent compact description of nontranslational movement in a globally nonrigid deformation. The resulting representation contains directly interpretable information from both mathematical and modeling perspectives. We develop the mathematical construction of the registration framework with higher-order momenta, we show the implications for sparse image registration and deformation description, and we provide examples of how the parametrization enables registration with a very low number of parameters. The capacity and interpretability of the parametrization using higher-order momenta lead to natural modeling of articulated movement, and the method promises to be useful for quantifying ventricle expansion and progressing atrophy during Alzheimer's disease.

Explicit B-spline regularization in diffeomorphic image registration

Diffeomorphic mappings are central to image registration due largely to their topological properties and success in providing biologically plausible solutions to deformation and morphological estimation problems. Popular diffeomorphic image registration algorithms include those characterized by time-varying and constant velocity fields, and symmetrical considerations. Prior information in the form of regularization is used to enforce transform plausibility taking the form of physics-based constraints or through some approximation thereof, e.g., Gaussian smoothing of the vector fields [a la Thirion's Demons (Thirion, 1998)]. In the context of the original Demons' framework, the so-called directly manipulated free-form deformation (DMFFD) (Tustison et al., 2009) can be viewed as a smoothing alternative in which explicit regularization is achieved through fast B-spline approximation. This characterization can be used to provide B-spline “flavored” diffeomorphic image registration solutions with several advantages. Implementation is open source and available through the Insight Toolkit and our Advanced Normalization Tools (ANTs) repository. A thorough comparative evaluation with the well-known SyN algorithm (Avants et al., 2008), implemented within the same framework, and its B-spline analog is performed using open labeled brain data and open source evaluation tools.

Robust Diffeomorphic Mapping via Geodesically Controlled Active Shapes

This paper presents recent advances in the use of diffeomorphic active shapes which incorporate the conservation laws of large deformation diffeomorphic metric mapping. The equations of evolution satisfying the conservation law are geodesics under the diffeomorphism metric and therefore termed geodesically controlled diffeomorphic active shapes (GDAS). Our principal application in this paper is on robust diffeomorphic mapping methods based on parameterized surface representations of subcortical template structures. Our parametrization of the GDAS evolution is via the initial momentum representation in the tangent space of the template surface. The dimension of this representation is constrained using principal component analysis generated from training samples. In this work, we seek to use template surfaces to generate segmentations of the hippocampus with three data attachment terms: surface matching, landmark matching, and inside-outside modeling from grayscale T1 MR imaging data. This is formulated as an energy minimization problem, where energy describes shape variability and data attachment accuracy, and we derive a variational solution. A gradient descent strategy is employed in the numerical optimization. For the landmark matching case, we demonstrate the robustness of this algorithm as applied to the workflow of a large neuroanatomical study by comparing to an existing diffeomorphic landmark matching algorithm.