Large Deformation Diffeomorphic Metric Curve Mapping

Deformable image registration with strategic integration pyramid framework for brain MRI

Medical image registration plays a crucial role in medical imaging, with a wide range of clinical applications. In this context, brain MRI registration is commonly used in clinical practice for accurate diagnosis and treatment planning. In recent years, deep learning-based deformable registration methods have achieved remarkable results. However, existing methods have not been flexible and efficient in handling the feature relationships of anatomical structures at different levels when dealing with large deformations. To address this limitation, we propose a novel strategic integration registration network based on the pyramid structure. Our strategy mainly includes two aspects of integration: fusion of features at different scales, and integration of different neural network structures. Specifically, we design a CNN encoder and a Transformer decoder to efficiently extract and enhance both global and local features. Moreover, to overcome the error accumulation issue inherent in pyramid structures, we introduce progressive optimization iterations at the lowest scale for deformation field generation. This approach more efficiently handles the spatial relationships of images while improving accuracy. We conduct extensive evaluations across multiple brain MRI datasets, and experimental results show that our method outperforms other deep learning-based methods in terms of registration accuracy and robustness.

Hybrid transformer and convolution iteratively optimized pyramid network for brain large deformation image registration

In recent years, the pyramid-based encoder-decoder network architecture has become a popular solution to the problem of large deformation image registration due to its excellent multi-scale deformation field prediction ability. However, there are two main limitations in existing research: one is that it over-focuses on the fusion of multi-layer deformation sub-fields on the decoding path, while ignoring the impact of feature encoders on network performance; the other is the lack of specialized design for the characteristics of feature maps at different scales. To this end, we propose an innovative hybrid Transformer and convolution iteratively optimized pyramid network for large deformation brain image registration. Specifically, four encoder variants are designed to study the impact of different structures on the performance of the pyramid registration network. Secondly, the Swin-Transformer module is combined with the convolution iterative strategy, and each layer of the decoder is carefully designed according to the semantic information characteristics of different decoding layers. Extensive experimental results on three public brain magnetic resonance imaging datasets show that our method has the highest registration accuracy compared with 9 cutting-edge registration methods, which fully verifies the effectiveness and application potential of our model design.

DeST-OT: Alignment of spatiotemporal transcriptomics data

Spatially resolved transcriptomics (SRT) measures mRNA transcripts at thousands of locations within a tissue slice, revealing spatial variations in gene expression and cell types. SRT has been applied to tissue slices from multiple time points during the development of an organism. We introduce developmental spatiotemporal optimal transport (DeST-OT), a method to align spatiotemporal transcriptomics data using optimal transport (OT). DeST-OT uses semi-relaxed OT to model cellular growth, death, and differentiation processes. We also derive a growth distortion metric and a migration metric to quantify the plausibility of spatiotemporal alignments. DeST-OT outperforms existing methods on the alignment of spatiotemporal transcriptomics data from developing mouse kidney and axolotl brain. DeST-OT estimated growth rates also provide insights into the gene expression programs governing the growth and differentiation of cells over space and time.

GroupMorph: Medical Image Registration via Grouping Network With Contextual Fusion

Pyramid-based deformation decomposition is a promising registration framework, which gradually decomposes the deformation field into multi-resolution subfields for precise registration. However, most pyramid-based methods directly produce one subfield per resolution level, which does not fully depict the spatial deformation. In this paper, we propose a novel registration model, called GroupMorph. Different from typical pyramid-based methods, we adopt the grouping-combination strategy to predict deformation field at each resolution. Specifically, we perform group-wise correlation calculation to measure the similarities of grouped features. After that, n groups of deformation subfields with different receptive fields are predicted in parallel. By composing these subfields, a deformation field with multi-receptive field ranges is formed, which can effectively identify both large and small deformations. Meanwhile, a contextual fusion module is designed to fuse the contextual features and provide the inter-group information for the field estimator of the next level. By leveraging the inter-group correspondence, the synergy among deformation subfields is enhanced. Extensive experiments on four public datasets demonstrate the effectiveness of GroupMorph. Code is available at https://github.com/TVayne/GroupMorph.

Unsupervised lung CT image registration via stochastic decomposition of deformation fields

We address the problem of lung CT image registration, which underpins various diagnoses and treatments for lung diseases. The main crux of the problem is the large deformation that the lungs undergo during respiration. This physiological process imposes several challenges from a learning point of view. In this paper, we propose a novel training scheme, called stochastic decomposition, which enables deep networks to effectively learn such a difficult deformation field during lung CT image registration. The key idea is to stochastically decompose the deformation field, and supervise the registration by synthetic data that have the corresponding appearance discrepancy. The stochastic decomposition allows for revealing all possible decompositions of the deformation field. At the learning level, these decompositions can be seen as a prior to reduce the ill-posedness of the registration yielding to boost the performance. We demonstrate the effectiveness of our framework on Lung CT data. We show, through extensive numerical and visual results, that our technique outperforms existing methods.

Integrative spatial analysis reveals a multi-layered organization of glioblastoma

Glioma contains malignant cells in diverse states. Here, we combine spatial transcriptomics, spatial proteomics, and computational approaches to define glioma cellular states and uncover their organization. We find three prominent modes of organization. First, gliomas are composed of small local environments, each typically enriched with one major cellular state. Second, specific pairs of states preferentially reside in proximity across multiple scales. This pairing of states is consistent across tumors. Third, these pairwise interactions collectively define a global architecture composed of five layers. Hypoxia appears to drive the layers, as it is associated with a long-range organization that includes all cancer cell states. Accordingly, tumor regions distant from any hypoxic/necrotic foci and tumors that lack hypoxia such as low-grade IDH-mutant glioma are less organized. In summary, we provide a conceptual framework for the organization of cellular states in glioma, highlighting hypoxia as a long-range tissue organizer.

Space-feature measures on meshes for mapping spatial transcriptomics

Advances in the development of largely automated microscopy methods such as MERFISH for imaging cellular structures in mouse brains are providing spatial detection of micron resolution gene expression. While there has been tremendous progress made in the field of Computational Anatomy (CA) to perform diffeomorphic mapping technologies at the tissue scales for advanced neuroinformatic studies in common coordinates, integration of molecular- and cellular-scale populations through statistical averaging via common coordinates remains yet unattained. This paper describes the first set of algorithms for calculating geodesics in the space of diffeomorphisms, what we term space-feature-measure LDDMM, extending the family of large deformation diffeomorphic metric mapping (LDDMM) algorithms to accommodate a space-feature action on marked particles which extends consistently to the tissue scales. It leads to the derivation of a cross-modality alignment algorithm of transcriptomic data to common coordinate systems attached to standard atlases. We represent the brain data as geometric measures, termed as space-feature measures supported by a large number of unstructured points, each point representing a small volume in space and carrying a list of densities of features elements of a high-dimensional feature space. The shape of space-feature measure brain spaces is measured by transforming them by diffeomorphisms. The metric between these measures is obtained after embedding these objects in a linear space equipped with the norm, yielding a so-called "chordal metric".

Multiscale unsupervised network for deformable image registration

BACKGROUND

Deformable image registration (DIR) plays an important part in many clinical tasks, and deep learning has made significant progress in DIR over the past few years.

OBJECTIVE

To propose a fast multiscale unsupervised deformable image registration (referred to as FMIRNet) method for monomodal image registration.

METHODS

We designed a multiscale fusion module to estimate the large displacement field by combining and refining the deformation fields of three scales. The spatial attention mechanism was employed in our fusion module to weight the displacement field pixel by pixel. Except mean square error (MSE), we additionally added structural similarity (ssim) measure during the training phase to enhance the structural consistency between the deformed images and the fixed images.

RESULTS

Our registration method was evaluated on EchoNet, CHAOS and SLIVER, and had indeed performance improvement in terms of SSIM, NCC and NMI scores. Furthermore, we integrated the FMIRNet into the segmentation network (FCN, UNet) to boost the segmentation task on a dataset with few manual annotations in our joint leaning frameworks. The experimental results indicated that the joint segmentation methods had performance improvement in terms of Dice, HD and ASSD scores.

CONCLUSIONS

Our proposed FMIRNet is effective for large deformation estimation, and its registration capability is generalizable and robust in joint registration and segmentation frameworks to generate reliable labels for training segmentation tasks.

STalign: Alignment of spatial transcriptomics data using diffeomorphic metric mapping

Spatial transcriptomics (ST) technologies enable high throughput gene expression characterization within thin tissue sections. However, comparing spatial observations across sections, samples, and technologies remains challenging. To address this challenge, we develop STalign to align ST datasets in a manner that accounts for partially matched tissue sections and other local non-linear distortions using diffeomorphic metric mapping. We apply STalign to align ST datasets within and across technologies as well as to align ST datasets to a 3D common coordinate framework. We show that STalign achieves high gene expression and cell-type correspondence across matched spatial locations that is significantly improved over landmark-based affine alignments. Applying STalign to align ST datasets of the mouse brain to the 3D common coordinate framework from the Allen Brain Atlas, we highlight how STalign can be used to lift over brain region annotations and enable the interrogation of compositional heterogeneity across anatomical structures. STalign is available as an open-source Python toolkit at https://github.com/JEFworks-Lab/STalign and as Supplementary Software with additional documentation and tutorials available at https://jef.works/STalign .

Alignment of spatial transcriptomics data using diffeomorphic metric mapping

Spatial transcriptomics (ST) technologies enable high throughput gene expression characterization within thin tissue sections. However, comparing spatial observations across sections, samples, and technologies remains challenging. To address this challenge, we developed STalign to align ST datasets in a manner that accounts for partially matched tissue sections and other local non-linear distortions using diffeomorphic metric mapping. We apply STalign to align ST datasets within and across technologies as well as to align ST datasets to a 3D common coordinate framework. We show that STalign achieves high gene expression and cell-type correspondence across matched spatial locations that is significantly improved over landmark-based affine alignments. Applying STalign to align ST datasets of the mouse brain to the 3D common coordinate framework from the Allen Brain Atlas, we highlight how STalign can be used to lift over brain region annotations and enable the interrogation of compositional heterogeneity across anatomical structures. STalign is available as an open-source Python toolkit at https://github.com/JEFworks-Lab/STalign and as supplementary software with additional documentation and tutorials available at https://jef.works/STalign.

Intra-Patient Lung CT Registration through Large Deformation Decomposition and Attention-Guided Refinement

Deformable lung CT image registration is an essential task for computer-assisted interventions and other clinical applications, especially when organ motion is involved. While deep-learning-based image registration methods have recently achieved promising results by inferring deformation fields in an end-to-end manner, large and irregular deformations caused by organ motion still pose a significant challenge. In this paper, we present a method for registering lung CT images that is tailored to the specific patient being imaged. To address the challenge of large deformations between the source and target images, we break the deformation down into multiple continuous intermediate fields. These fields are then combined to create a spatio-temporal motion field. We further refine this field using a self-attention layer that aggregates information along motion trajectories. By leveraging temporal information from a respiratory cycle, our proposed methods can generate intermediate images that facilitate image-guided tumor tracking. We evaluated our approach extensively on a public dataset, and our numerical and visual results demonstrate the effectiveness of the proposed method.

An optical flow based left-invariant metric for natural gradient descent in affine image registration

Accurate spatial alignment is essential for any population neuroimaging study, and affine (12 parameter linear/translation) or rigid (6 parameter rotation/translation) alignments play a major role. Here we consider intensity based alignment of neuroimages using gradient based optimization, which is a problem that continues to be important in many other areas of medical imaging and computer vision in general. A key challenge is robustness. Optimization often fails when transformations have components with different characteristic scales, such as linear versus translation parameters. Hand tuning or other scaling approaches have been used, but efficient automatic methods are essential for generalizing to new imaging modalities, to specimens of different sizes, and to big datasets where manual approaches are not feasible. To address this we develop a left invariant metric on these two matrix groups, based on the norm squared of optical flow induced on a template image. This metric is used in a natural gradient descent algorithm, where gradients (covectors) are converted to perturbations (vectors) by applying the inverse of the metric to define a search direction in which to update parameters. Using a publicly available magnetic resonance neuroimage database, we show that this approach outperforms several other gradient descent optimization strategies. Due to left invariance, our metric needs to only be computed once during optimization, and can therefore be implemented with negligible computation time.

Functional magnetic resonance imaging progressive deformable registration based on a cascaded convolutional neural network

Background

Intersubject registration of functional magnetic resonance imaging (fMRI) is necessary for group analysis. Accurate image registration can significantly improve the results of statistical analysis. Traditional methods are achieved by using high-resolution structural images or manually extracting functional information. However, structural alignment does not necessarily lead to functional alignment, and manually extracting functional features is complicated and time-consuming. Recent studies have shown that deep learning-based methods can be used for deformable image registration.

Methods

We proposed a deep learning framework with a three-cascaded multi-resolution network (MR-Net) to achieve deformable image registration. MR-Net separately extracts the features of moving and fixed images via a two-stream path, predicts a sub-deformation field, and is cascaded three times. The moving and fixed images' deformation field is composed of all sub-deformation fields predicted by the MR-Net. We imposed large smoothness constraints on all sub-deformation fields to ensure their smoothness. Our proposed architecture can complete the progressive registration process to ensure the topology of the deformation field.

Results

We implemented our method on the 1000 Functional Connectomes Project (FCP) and Eyes Open Eyes Closed fMRI datasets. Our method increased the peak t values in six brain functional networks to 19.8, 17.8, 15.0, 16.4, 17.0, and 13.2. Compared with traditional methods [i.e., FMRIB Software Library (FSL) and Statistical Parametric Mapping (SPM)] and deep learning networks [i.e., VoxelMorph (VM) and Volume Tweening Network (VTN)], our method improved 47.58%, 11.88%, 18.60%, and 15.16%, respectively.

Conclusions

Our three-cascaded MR-Net can achieve statistically significant improvement in functional consistency across subjects.

Down-sampling template curve to accelerate LDDMM-curve with application to shape analysis of the corpus callosum

Large deformation diffeomorphic metric mapping for curve (LDDMM-curve) has been widely used in deformation based statistical shape analysis of the mid-sagittal corpus callosum. A main limitation of LDDMM-curve is that it is time-consuming and computationally complex. In this study, down-sampling strategies for accelerating LDDMM-curve are investigated and tested on two large datasets, one on Alzheimer's disease (155 Alzheimer's disease, 325 mild cognitive impairment and 185 healthy controls) and the other on first-episode schizophrenia (92 first-episode schizophrenia and 106 healthy controls). For both datasets a variety of down-sampling factors are tested in terms of registration accuracy, registration speed, and most importantly disease-related patterns. Experimental results revealed that down-sampling template curve by a factor of 2 can significantly reduce the running time of LDDMM-curve without sacrificing the registration accuracy. Also, the disease-induced patterns, or more specifically the group comparison results, were almost identical before and after down-sampling. It is also shown that there was no need to down-sample the target population curves but only the single template curve of the study of interest. Comprehensive analyses were conducted.

A Deformation-Based Shape Study of the Corpus Callosum in First Episode Schizophrenia

Background: Previous first-episode schizophrenia (FES) studies have reported abnormalities in the volume and mid-sagittal size of the corpus callosum (CC), but findings have been inconsistent. Besides, the CC shape has rarely been analyzed in FES. Therefore, in this study, we investigated FES-related CC shape abnormalities using 198 participants [92 FES patients and 106 healthy controls (HCs)]. Methods: We conducted statistical shape analysis of the mid-sagittal CC curve in a large deformation diffeomorphic metric mapping framework. The CC was divided into the genu, body, and splenium (gCC, bCC, and sCC) to target the key CC sub-regions affected by the FES pathology. Gender effects have been investigated. Results: There were significant area differences between FES and HC in the entire CC and gCC but not in bCC nor sCC. In terms of the localized shape morphometrics, significant region-specific shape inward-deformations were detected in the superior portion of gCC and the anterosuperior portion of bCC in FES. These global area and local shape morphometric abnormalities were restricted to female FES but not male FES. Conclusions: gCC was significantly affected in the neuropathology of FES and this finding was specific to female FES. This study suggests that gCC may be a key sub-region that is vulnerable to the neuropathology of FES, specifically in female patients. The morphometrics of gCC may serve as novel and efficient biomarkers for screening female FES patients.

A Large Deformation Diffeomorphic Framework for Fast Brain Image Registration via Parallel Computing and Optimization

In this paper, we proposed an efficient approach for large deformation diffeomorphic metric mapping (LDDMM) for brain images by utilizing GPU-based parallel computing and a mixture automatic step size estimation method for gradient descent (MAS-GD). We systematically evaluated the proposed approach in terms of two matching cost functions, including the Sum of Squared Differences (SSD) and the Cross-Correlation (CC). The registration accuracy and computational efficiency on two datasets inducing respective 120 and 1,560 registration maps were evaluated and compared between CPU-based LDDMM-SSD and GPU-based LDDMM-SSD both utilizing backtracking line search for gradient descent (BLS-GD), GPU-based LDDMM (BLS-GD) and GPU-based LDDMM (MAS-GD) with each of the two matching cost functions being used. In addition, we compared our GPU-based LDDMM-CC (MAS-GD) with another widely-used state-of-the-art image registration algorithm, the symmetric diffeomorphic image registration with CC (SyN-CC). The GPU-based LDDMM-SSD was about 94 times faster than the CPU-based version (8.78 mins versus 828.35 mins) without sacrificing the Dice accuracy (0.8608 versus 0.8609). The computational time of LDDMM with MAS-GD for SSD and CC were shorter than that of LDDMM with BLS-GD (5.29 mins versus 8.78 mins for SSD and 6.69 mins versus 65.87 mins for CC), and the corresponding Dice scores were higher, especially for CC (0.8672 versus 0.8633). Compared with SyN-CC, the proposed algorithm, GPU-based LDDMM-CC (MAS-GD) had a higher registration accuracy (0.8672 versus 0.8612 and 0.7585 versus 0.7537 for the two datasets) and less computational time (6.80 mins versus 25.97 mins and 6.58 mins versus 26.23 mins for the two datasets).

Multimodal cross-registration and quantification of metric distortions in marmoset whole brain histology using diffeomorphic mappings

Whole brain neuroanatomy using tera-voxel light-microscopic data sets is of much current interest. A fundamental problem in this field is the mapping of individual brain data sets to a reference space. Previous work has not rigorously quantified in-vivo to ex-vivo distortions in brain geometry from tissue processing. Further, existing approaches focus on registering unimodal volumetric data; however, given the increasing interest in the marmoset model for neuroscience research and the importance of addressing individual brain architecture variations, new algorithms are necessary to cross-register multimodal data sets including MRIs and multiple histological series. Here we present a computational approach for same-subject multimodal MRI-guided reconstruction of a series of consecutive histological sections, jointly with diffeomorphic mapping to a reference atlas. We quantify the scale change during different stages of brain histological processing using the Jacobian determinant of the diffeomorphic transformations involved. By mapping the final image stacks to the ex-vivo post-fixation MRI, we show that (a) tape-transfer assisted histological sections can be reassembled accurately into 3D volumes with a local scale change of 2.0 ± 0.4% per axis dimension; in contrast, (b) tissue perfusion/fixation as assessed by mapping the in-vivo MRIs to the ex-vivo post fixation MRIs shows a larger median absolute scale change of 6.9 ± 2.1% per axis dimension. This is the first systematic quantification of local metric distortions associated with whole-brain histological processing, and we expect that the results will generalize to other species. These local scale changes will be important for computing local properties to create reference brain maps.

Analysis of shape data: From landmarks to elastic curves

Proliferation of high-resolution imaging data in recent years has led to sub-stantial improvements in the two popular approaches for analyzing shapes of data objects based on landmarks and/or continuous curves. We provide an expository account of elastic shape analysis of parametric planar curves representing shapes of two-dimensional (2D) objects by discussing its differences, and its commonalities, to the landmark-based approach. Particular attention is accorded to the role of reparameterization of a curve, which in addition to rotation, scaling and translation, represents an important shape-preserving transformation of a curve. The transition to the curve-based approach moves the mathematical setting of shape analysis from finite-dimensional non-Euclidean spaces to infinite-dimensional ones. We discuss some of the challenges associated with the infinite-dimensionality of the shape space, and illustrate the use of geometry-based methods in the computation of intrinsic statistical summaries and in the definition of statistical models on a 2D imaging dataset consisting of mouse vertebrae. We conclude with an overview of the current state-of-the-art in the field.

Simplifying Transforms for General Elastic Metrics on the Space of Plane Curves

In the shape analysis approach to computer vision problems, one treats shapes as points in an infinite-dimensional Riemannian manifold, thereby facilitating algorithms for statistical calculations such as geodesic distance between shapes and averaging of a collection of shapes. The performance of these algorithms depends heavily on the choice of the Riemannian metric. In the setting of plane curve shapes, attention has largely been focused on a two-parameter family of first order Sobolev metrics, referred to as elastic metrics. They are particularly useful due to the existence of simplifying coordinate transformations for particular parameter values, such as the well-known square-root velocity transform. In this paper, we extend the transformations appearing in the existing literature to a family of isometries, which take any elastic metric to the flat L ² metric. We also extend the transforms to treat piecewise linear curves and demonstrate the existence of optimal matchings over the diffeomorphism group in this setting. We conclude the paper with multiple examples of shape geodesics for open and closed curves. We also show the benefits of our approach in a simple classification experiment.

CT temporal subtraction improves early detection of bone metastases compared to SPECT

OBJECTIVES

To compare observer performance of detecting bone metastases between bone scintigraphy, including planar scan and single-photon emission computed tomography, and computed tomography (CT) temporal subtraction (TS).

METHODS

Data on 60 patients with cancer who had undergone CT (previous and current) and bone scintigraphy were collected. Previous CT images were registered to the current ones by large deformation diffeomorphic metric mapping; the registered previous images were subtracted from the current ones to produce TS. Definitive diagnosis of bone metastases was determined by consensus between two radiologists. Twelve readers independently interpreted the following pairs of examinations: NM-pair, previous and current CTs and bone scintigraphy, and TS-pair, previous and current CTs and TS. The readers assigned likelihood levels to suspected bone metastases for diagnosis. Sensitivity, number of false positives per patient (FPP), and reading time for each pair of examinations were analysed for evaluating observer performance by performing the Wilcoxon signed-rank test. Figure-of-merit (FOM) was calculated using jackknife alternative free-response receiver operating characteristic analysis.

RESULTS

The sensitivity of TS was significantly higher than that of bone scintigraphy (54.3% vs. 41.3%, p = 0.006). FPP with TS was significantly higher than that with bone scintigraphy (0.189 vs. 0.0722, p = 0.003). FOM of TS tended to be better than that of bone scintigraphy (0.742 vs. 0.691, p = 0.070).

CONCLUSION

Sensitivity of TS in detecting bone metastasis was significantly higher than that of bone scintigraphy, but still limited to 54%. TS might be superior to bone scintigraphy for early detection of bone metastasis.

KEY POINTS

• Computed tomography temporal subtraction was helpful in early detection of bone metastases. • Sensitivity for bone metastasis was higher for computed tomography temporal subtraction than for bone scintigraphy. • Figure-of-merit of computed tomography temporal subtraction was better than that of bone scintigraphy.

Spatiotemporal analysis for detection of pre-symptomatic shape changes in neurodegenerative diseases: Initial application to the GENFI cohort

Brain atrophy as measured from structural MR images, is one of the primary imaging biomarkers used to track neurodegenerative disease progression. In diseases such as frontotemporal dementia or Alzheimer's disease, atrophy can be observed in key brain structures years before any clinical symptoms are present. Atrophy is most commonly captured as volume change of key structures and the shape changes of these structures are typically not analysed despite being potentially more sensitive than summary volume statistics over the entire structure. In this paper we propose a spatiotemporal analysis pipeline based on Large Diffeomorphic Deformation Metric Mapping (LDDMM) to detect shape changes from volumetric MRI scans. We applied our framework to a cohort of individuals with genetic variants of frontotemporal dementia and healthy controls from the Genetic FTD Initiative (GENFI) study. Our method, take full advantage of the LDDMM framework, and relies on the creation of a population specific average spatiotemporal trajectory of a relevant brain structure of interest, the thalamus in our case. The residuals from each patient data to the average spatiotemporal trajectory are then clustered and studied to assess when presymptomatic mutation carriers differ from healthy control subjects. We found statistical differences in shape in the anterior region of the thalamus at least five years before the mutation carrier subjects develop any clinical symptoms. This region of the thalamus has been shown to be predominantly connected to the frontal lobe, consistent with the pattern of cortical atrophy seen in the disease.

ESTIMATING DIFFEOMORPHIC MAPPINGS BETWEEN TEMPLATES AND NOISY DATA: VARIANCE BOUNDS ON THE ESTIMATED CANONICAL VOLUME FORM

Anatomy is undergoing a renaissance driven by the availability of large digital data sets generated by light microscopy. A central computational task is to map individual data volumes to standardized templates. This is accomplished by regularized estimation of a diffeomorphic transformation between the coordinate systems of the individual data and the template, building the transformation incrementally by integrating a smooth flow field. The canonical volume form of this transformation is used to quantify local growth, atrophy, or cell density. While multiple implementations exist for this estimation, less attention has been paid to the variance of the estimated diffeomorphism for noisy data. Notably, there is an infinite dimensional unobservable space defined by those diffeomorphisms which leave the template invariant. These form the stabilizer subgroup of the diffeomorphic group acting on the template. The corresponding flat directions in the energy landscape are expected to lead to increased estimation variance. Here we show that a least-action principle used to generate geodesics in the space of diffeomor-phisms connecting the subject brain to the template removes the stabilizer. This provides reduced-variance estimates of the volume form. Using simulations we demonstrate that the asymmetric large deformation diffeomorphic mapping methods (LDDMM), which explicitly incorporate the asymmetry between idealized template images and noisy empirical images, provide lower variance estimators than their symmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for the variances in the limit of small deformations. Analytical results are shown for the Jacobian in terms of perturbations of the vector fields and divergence of the vector field.

Multiscale Frame-Based Kernels for Large Deformation Diffeomorphic Metric Mapping

We present a set of multiscale frame-based kernels that can be used to construct diffeomorphic transformation in the large deformation diffeomorphic metric mapping (LDDMM) framework. We construct multiscale kernels via compact wavelet frames that are equipped with the hierarchical multiresolution analysis. We show that these kernels under certain conditions can form reproducing kernel Hilbert spaces of smooth velocity fields and hence can be used to generate multiscale diffeomorphic transformation for LDDMM. As a proof of concept, we incorporate these kernels in the LDDMM framework. We show the improvement of whole brain mapping accuracy using the LDDMM with frame-based kernels in comparison to that obtained using the LDDMM with Gaussian kernels. Moreover, we evaluate whole brain mapping accuracy of the LDDMM with frame-based kernels against that obtained from the 14 brain mapping methods given by Klein et al.. Our results suggest that the LDDMM with frame-based kernels has the potential to outperform the 14 brain mapping methods for whole brain mapping.

Analysis of mitochondrial shape dynamics using large deformation diffeomorphic metric curve matching

Mitochondrial shape changes are essential to mitochondrial functions. Quantification of mitochondrial shape changes is essential to understanding related physiology and disease mechanisms. In this study, we proposed a new automated pipeline for quantifying the shape changing patterns of mitochondria in the framework of large deformation diffeomorphic metric mapping for curve. We validated the accuracy of our pipeline on 32 mitochondria data, each having 6 sequential time-lapse frames. The contour of each mitochondrion is modeled by a curve consisting of a set of landmark points ranging from 39 to 358, with the moving distance between every two consecutive frames quantified for each localized point. The sensitivity of the proposed pipeline, with respect to different curve discretization, was investigated, with high robustness established. In addition, we quantified the uncertainty level of the proposed pipeline using 10 fixed mitochondria data with 6 time frames as well, with the mean between-frame moving distance found to be smaller than 28 nm for a majority of the 10 fixed mitochondria data. This indicates that the proposed pipeline has a very low level of uncertainty. The encouraging results from this work suggest that the proposed pipeline is potentially a powerful tool for quantifying shape dynamics, both globally and locally, of a variety of cellular components.

Computational anatomy and diffeomorphometry: A dynamical systems model of neuroanatomy in the soft condensed matter continuum

The nonlinear systems models of computational anatomy that have emerged over the past several decades are a synthesis of three significant areas of computational science and biological modeling. First is the algebraic model of biological shape as a Riemannian orbit, a set of objects under diffeomorphic action. Second is the embedding of anatomical shapes into the soft condensed matter physics continuum via the extension of the Euler equations to geodesic, smooth flows with inverses, encoding divergence for the compressibility of atrophy and expansion of growth. Third, is making human shape and form a metrizable space via geodesic connections of coordinate systems. These three themes place our formalism into the modern data science world of personalized medicine supporting inference of high-dimensional anatomical phenotypes for studying neurodegeneration and neurodevelopment. The dynamical systems model of growth and atrophy that emerges is one which is organized in terms of forces, accelerations, velocities, and displacements, with the associated Hamiltonian momentum and the diffeomorphic flow acting as the state, and the smooth vector field the control. The forces that enter the model derive from external measurements through which the dynamical system must flow, and the internal potential energies of structures making up the soft condensed matter. We examine numerous examples on growth and atrophy. This article is categorized under: Analytical and Computational Methods > Computational Methods Laboratory Methods and Technologies > Imaging Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

A Riemannian Framework for Linear and Quadratic Discriminant Analysis on the Tangent Space of Shapes

We present a Riemannian framework for linear and quadratic discriminant classification on the tangent plane of the shape space of curves. The shape space is infinite dimensional and is constructed out of square root velocity functions of curves. We introduce the notion of mean and covariance of shape-valued random variables and samples from a tangent space to the pre-shapes (invariant to translation and scaling) and then extend it to the full shape space (rotational invariance). The shape observations from the population are approximated by coefficients of a Fourier basis of the tangent space. The algorithms for linear and quadratic discriminant analysis are then defined using reduced dimensional features obtained by projecting the original shape observations on to the truncated Fourier basis. We show classification results on synthetic data and shapes of cortical sulci, corpus callosum curves, as well as facial midline curve profiles from patients with fetal alcohol syndrome (FAS).

Parametric Surface Diffeomorphometry for Low Dimensional Embeddings of Dense Segmentations and Imagery

In the field of Computational Anatomy, biological form (including our focus, neuroanatomy) is studied quantitatively through the action of the diffeomorphism group on example anatomies - a technique called diffeomorphometry. Here we design an algorithm within this framework to pass from dense objects common in neuromaging studies (binary segmentations, structural images) to a sparse representation defined on the surface boundaries of anatomical structures, and embedded into the low dimensional coordinates of a parametric model. Our main new contribution is to introduce an expanded group action to simultaneously deform surfaces through direct mapping of points, as well as images through functional composition with the inverse. This allows us to index the diffeomorphisms with respect to two-dimensional surface geometries like subcortical gray matter structures, but explicitly map onto cost functions determined by noisy 3-dimensional measurements. We consider models generated from empirical covariance of training data, as well as bandlimited (Laplace-Beltrami eigenfunction) models when no such data is available. We show applications to noisy or anomalous segmentations, and other typical problems in neuroimaging studies. We reproduce statistical results detecting changes in Alzheimer's disease, despite dimensionality reduction. Lastly we apply our algorithm to the common problem of segmenting subcortical structures from T1 MR images.

A computational framework to characterize and compare the geometry of coronary networks

This work presents a computational framework to perform a systematic and comprehensive assessment of the morphometry of coronary arteries from in vivo medical images. The methodology embraces image segmentation, arterial vessel representation, characterization and comparison, data storage, and finally analysis. Validation is performed using a sample of 48 patients. Data mining of morphometric information of several coronary arteries is presented. Results agree to medical reports in terms of basic geometric and anatomical variables. Concerning geometric descriptors, inter-artery and intra-artery correlations are studied. Data reported here can be useful for the construction and setup of blood flow models of the coronary circulation. Finally, as an application example, similarity criterion to assess vasculature likelihood based on geometric features is presented and used to test geometric similarity among sibling patients. Results indicate that likelihood, measured through geometric descriptors, is stronger between siblings compared with non-relative patients. Copyright © 2016 John Wiley & Sons, Ltd.

Diffeomorphic Surface Registration with Atrophy Constraints

Diffeomorphic registration using optimal control on the diffeomorphism group and on shape spaces has become widely used since the development of the large deformation diffeomorphic metric mapping (LDDMM) algorithm. More recently, a series of algorithms involving sub-Riemannian constraints have been introduced in which the velocity fields that control the shapes in the LDDMM framework are constrained in accordance with a specific deformation model. Here, we extend this setting by considering, for the first time, inequality constraints in order to estimate surface deformations that only allow for atrophy, introducing for this purpose an algorithm that uses the augmented Lagrangian method. We prove the existence of solutions of the associated optimal control problem and the consistency of our approximation scheme. These developments are illustrated by numerical experiments on simulated and real data.

Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D'Arcy Thompson

The Computational Anatomy project is the morphome-scale study of shape and form, which we model as an orbit under diffeomorphic group action. Metric comparison calculates the geodesic length of the diffeomorphic flow connecting one form to another. Geodesic connection provides a positioning system for coordinatizing the forms and positioning their associated functional information. This article reviews progress since the Euler-Lagrange characterization of the geodesics a decade ago. Geodesic positioning is posed as a series of problems in Hamiltonian control, which emphasize the key reduction from the Eulerian momentum with dimension of the flow of the group, to the parametric coordinates appropriate to the dimension of the submanifolds being positioned. The Hamiltonian viewpoint provides important extensions of the core setting to new, object-informed positioning systems. Several submanifold mapping problems are discussed as they apply to metamorphosis, multiple shape spaces, and longitudinal time series studies of growth and atrophy via shape splines.

Abnormal surface morphology of the central sulcus in children with attention-deficit/hyperactivity disorder

The central sulcus (CS) divides the primary motor and somatosensory areas, and its three-dimensional (3D) anatomy reveals the structural changes of the sensorimotor regions. Attention-deficit/hyperactivity disorder (ADHD) is a neurodevelopmental disorder that is associated with sensorimotor and executive function deficits. However, it is largely unknown whether the morphology of the CS alters due to inappropriate development in the ADHD brain. Here, we employed the sulcus-based morphometry approach to investigate the 3D morphology of the CS in 42 children whose ages spanned from 8.8 to 13.5 years (21 with ADHD and 21 controls). After automatic labeling of each CS, we computed seven regional shape metrics for each CS, including the global average length, average depth, maximum depth, average span, surface area, average cortical thickness, and local sulcal profile. We found that the average depth and maximum depth of the left CS as well as the average cortical thickness of bilateral CS in the ADHD group were significantly larger than those in the healthy children. Moreover, significant between-group differences in the sulcal profile had been found in middle sections of the CSs bilaterally, and these changes were positively correlated with the hyperactivity-impulsivity scores in the children with ADHD. Altogether, our results provide evidence for the abnormity of the CS anatomical morphology in children with ADHD due to the structural changes in the motor cortex, which significantly contribute to the clinical symptomatology of the disorder.

Estimating dense cardiac 3D motion using sparse 2D tagged MRI cross-sections

In this work, we describe a new method, an extension of the Large Deformation Diffeomorphic Metric Mapping to estimate three-dimensional deformation of tagged Magnetic Resonance Imaging Data. Our approach relies on performing non-rigid registration of tag planes that were constructed from set of initial reference short axis tag grids to a set of deformed tag curves. We validated our algorithm using in-vivo tagged images of normal mice. The mapping allows us to compute root mean square distance error between simulated tag curves in a set of long axis image planes and the acquired tag curves in the same plane. Average RMS error was 0.31 ± 0.36(SD) mm, which is approximately 2.5 voxels, indicating good matching accuracy.

Highly Symmetric and Congruently Tiled Meshes for Shells and Domes

We describe the generation of all possible shell and dome shapes that can be uniquely meshed (tiled) using a single type of mesh face (tile), and following a single meshing (tiling) rule that governs the mesh (tile) arrangement with maximal vertex, edge and face symmetries. Such tiling arrangements or congruently tiled meshed shapes, are frequently found in chemical forms (fullerenes or Bucky balls, crystals, quasi-crystals, virus nano shells or capsids), and synthetic shapes (cages, sports domes, modern architectural facades). Congruently tiled meshes are both aesthetic and complete, as they support maximal mesh symmetries with minimal complexity and possess simple generation rules. Here, we generate congruent tilings and meshed shape layouts that satisfy these optimality conditions. Further, the congruent meshes are uniquely mappable to an almost regular 3D polyhedron (or its dual polyhedron) and which exhibits face-transitive (and edge-transitive) congruency with at most two types of vertices (each type transitive to the other). The family of all such congruently meshed polyhedra create a new class of meshed shapes, beyond the well-studied regular, semi-regular and quasi-regular classes, and their duals (platonic, Catalan and Johnson). While our new mesh class is infinite, we prove that there exists a unique mesh parametrization, where each member of the class can be represented by two integer lattice variables, and moreover efficiently constructable.

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.

Large deformation diffeomorphic registration of diffusion-weighted imaging data

Registration plays an important role in group analysis of diffusion-weighted imaging (DWI) data. It can be used to build a reference anatomy for investigating structural variation or tracking changes in white matter. Unlike traditional scalar image registration where spatial alignment is the only focus, registration of DWI data requires both spatial alignment of structures and reorientation of local signal profiles. As such, DWI registration is much more complex and challenging than scalar image registration. Although a variety of algorithms has been proposed to tackle the problem, most of them are restricted by the diffusion model used for registration, making it difficult to fit to the registered data a different model. In this paper we describe a method that allows any diffusion model to be fitted after registration for subsequent multifaceted analysis. This is achieved by directly aligning DWI data using a large deformation diffeomorphic registration framework. Our algorithm seeks the optimal coordinate mapping by simultaneously considering structural alignment, local signal profile reorientation, and deformation regularization. Our algorithm also incorporates a multi-kernel strategy to concurrently register anatomical structures at different scales. We demonstrate the efficacy of our approach using in vivo data and report detailed qualitative and quantitative results in comparison with several different registration strategies.

Detection of time-varying structures by large deformation diffeomorphic metric mapping to aid reading of high-resolution CT images of the lung

Objectives

To evaluate the accuracy of advanced non-linear registration of serial lung Computed Tomography (CT) images using Large Deformation Diffeomorphic Metric Mapping (LDDMM).

Methods

Fifteen 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.

Results

The 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.

Conclusions

LDDMM provides accurate registration of serial lung CT images, and temporal subtraction images with Jacobian maps help radiologists to find changes in pulmonary nodules.

Deformable medical image registration: a survey

Deformable image registration is a fundamental task in medical image processing. Among its most important applications, one may cite: 1) multi-modality fusion, where information acquired by different imaging devices or protocols is fused to facilitate diagnosis and treatment planning; 2) longitudinal studies, where temporal structural or anatomical changes are investigated; and 3) 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.

Model-driven harmonic parameterization of the cortical surface: HIP-HOP

In the context of inter subject brain surface matching, we present a parameterization of the cortical surface constrained by a model of cortical organization. The parameterization is defined via an harmonic mapping of each hemisphere surface to a rectangular planar domain that integrates a representation of the model. As opposed to previous landmark-based registration methods we do not match folds between individuals but instead optimize the fit between cortical sulci and specific iso-coordinate axis in the model. This strategy overcomes some limitation to sulcus-based registration techniques such as topological variability in sulcal landmarks across subjects. Experiments on 62 subjects with manually traced sulci are presented and compared with the result of the Freesurfer software. The evaluation involves a measure of dispersion of sulci with both angular and area distortions. We show that the model-based strategy can lead to a natural, efficient and very fast (less than 5 min per hemisphere) method for defining inter subjects correspondences. We discuss how this approach also reduces the problems inherent to anatomically defined landmarks and open the way to the investigation of cortical organization through the notion of orientation and alignment of structures across the cortex.

An invariant shape representation using the anisotropic Helmholtz equation

Analyzing geometry of sulcal curves on the human cortical surface requires a shape representation invariant to Euclidean motion. We present a novel shape representation that characterizes the shape of a curve in terms of a coordinate system based on the eigensystem of the anisotropic Helmholtz equation. This representation has many desirable properties: stability, uniqueness and invariance to scaling and isometric transformation. Under this representation, we can find a point-wise shape distance between curves as well as a bijective smooth point-to-point correspondence. When the curves are sampled irregularly, we also present a fast and accurate computational method for solving the eigensystem using a finite element formulation. This shape representation is used to find symmetries between corresponding sulcal shapes between cortical hemispheres. For this purpose, we automatically generate 26 sulcal curves for 24 subject brains and then compute their invariant shape representation. Left-right sulcal shape symmetry as measured by the shape representation's metric demonstrates the utility of the presented invariant representation for shape analysis of the cortical folding pattern.

Joint T1 and brain fiber log-demons registration using currents to model geometry

We present an extension of the diffeomorphic Geometric Demons algorithm which combines the iconic registration with geometric constraints. Our algorithm works in the log-domain space, so that one can efficiently compute the deformation field of the geometry. We represent the shape of objects of interest in the space of currents which is sensitive to both location and geometric structure of objects. Currents provides a distance between geometric structures that can be defined without specifying explicit point-to-point correspondences. We demonstrate this framework by registering simultaneously T1 images and 65 fiber bundles consistently extracted in 12 subjects and compare it against non-linear T1, tensor, and multi-modal T1 + Fractional Anisotropy (FA) registration algorithms. Results show the superiority of the Log-domain Geometric Demons over their purely iconic counterparts.

Large deformation diffeomorphic registration of diffusion-weighted images with explicit orientation optimization

We seek to compute a diffeomorphic map between a pair of diffusion-weighted images under large deformation. Unlike existing techniques, our method allows any diffusion model to be fitted after registration for subsequent multifaceted analysis. This is achieved by directly aligning the diffusion-weighted images using a large deformation diffeomorphic registration framework formulated from an optimal control perspective. Our algorithm seeks the optimal coordinate mapping by simultaneously considering structural alignment, local fiber reorientation, and deformation regularization. Our algorithm also incorporates a multi-kernel strategy to concurrently register anatomical structures of different scales. We demonstrate the efficacy of our approach using in vivo data and report on detailed qualitative and quantitative results in comparison with several different registration strategies.