Shape-based nonrigid correspondence with application to heart motion analysis

Left atrial evaluation by cardiovascular magnetic resonance: sensitive and unique biomarkers

Left atrial (LA) imaging is still not routinely used for diagnosis and risk stratification, although recent studies have emphasized its importance as an imaging biomarker. Cardiovascular magnetic resonance is able to evaluate LA structure and function, metrics that serve as early indicators of disease, and provide prognostic information, e.g. regarding diastolic dysfunction, and atrial fibrillation (AF). MR angiography defines atrial anatomy, useful for planning ablation procedures, and also for characterizing atrial shapes and sizes that might predict cardiovascular events, e.g. stroke. Long-axis cine images can be evaluated to define minimum, maximum, and pre-atrial contraction LA volumes, and ejection fractions (EFs). More modern feature tracking of these cine images provides longitudinal LA strain through the cardiac cycle, and strain rates. Strain may be a more sensitive marker than EF and can predict post-operative AF, AF recurrence after ablation, outcomes in hypertrophic cardiomyopathy, stratification of diastolic dysfunction, and strain correlates with atrial fibrosis. Using high-resolution late gadolinium enhancement (LGE), the extent of fibrosis in the LA can be estimated and post-ablation scar can be evaluated. The LA LGE method is widely available, its reproducibility is good, and validations with voltage-mapping exist, although further scan-rescan studies are needed, and consensus regarding atrial segmentation is lacking. Using LGE, scar patterns after ablation in AF subjects can be reproducibly defined. Evaluation of 'pre-existent' atrial fibrosis may have roles in predicting AF recurrence after ablation, predicting new-onset AF and diastolic dysfunction in patients without AF. LA imaging biomarkers are ready to enter into diagnostic clinical practice.

Deformation Based Curved Shape Representation

In this paper, we introduce a deformation based representation space for curved shapes in . Given an ordered set of points sampled from a curved shape, the proposed method represents the set as an element of a finite dimensional matrix Lie group. Variation due to scale and location are filtered in a preprocessing stage, while shapes that vary only in rotation are identified by an equivalence relationship. The use of a finite dimensional matrix Lie group leads to a similarity metric with an explicit geodesic solution. Subsequently, we discuss some of the properties of the metric and its relationship with a deformation by least action. Furthermore, invariance to reparametrization or estimation of point correspondence between shapes is formulated as an estimation of sampling function. Thereafter, two possible approaches are presented to solve the point correspondence estimation problem. Finally, we propose an adaptation of k-means clustering for shape analysis in the proposed representation space. Experimental results show that the proposed representation is robust to uninformative cues, e.g., local shape perturbation and displacement. In comparison to state of the art methods, it achieves a high precision on the Swedish and the Flavia leaf datasets and a comparable result on MPEG-7, Kimia99 and Kimia216 datasets.

Incorporating population-level variability in orthopedic biomechanical analysis: a review

Effectively addressing population-level variability within orthopedic analyses requires robust data sets that span the target population and can be greatly facilitated by statistical methods for incorporating such data into functional biomechanical models. Data sets continue to be disseminated that include not just anatomical information but also key mechanical data including tissue or joint stiffness, gait patterns, and other inputs relevant to analysis of joint function across a range of anatomies and physiologies. Statistical modeling can be used to establish correlations between a variety of structural and functional biometrics rooted in these data and to quantify how these correlations change from health to disease and, finally, to joint reconstruction or other clinical intervention. Principal component analysis provides a basis for effectively and efficiently integrating variability in anatomy, tissue properties, joint kinetics, and kinematics into mechanistic models of joint function. With such models, bioengineers are able to study the effects of variability on biomechanical performance, not just on a patient-specific basis but in a way that may be predictive of a larger patient population. The goal of this paper is to demonstrate the broad use of statistical modeling within orthopedics and to discuss ways to continue to leverage these techniques to improve biomechanical understanding of orthopedic systems across populations.

Shape variation in outline shapes

A general morphometric method for describing shape variation in a sample consisting of landmarks and multiple outline shapes is developed in this article. A distance metric is developed for such data and is used to embed the data in a low-dimensional Euclidean space. The Euclidean space is used to generate summary statistics such as mean and principal shape variation which are implicitly represented in the original space using elements of the sample. A new distance metric for outline shapes is proposed based on Procrustes distance that does not require the extraction of discrete points along the curve. The outline distance metric can be naturally combined with distances between landmarks. A method for aligning outlines and multiple outlines is developed that minimizes the distance metric. The method is compared with semilandmarks on synthetic data and 2 real data sets. Outline methods produce useful and valid results when suitably constrained by landmarks and are useful visualization aids, but questions remain about their suitability for answering biological questions until appropriate distance metrics can be biologically validated.

A Mixture of Transformed Hidden Markov Models for elastic motion estimation

Elastic motion is a nonrigid motion constrained only by some degree of smoothness and continuity. Consequently, elastic motion estimation by explicit feature matching actually contains two correlated subproblems: shape registration and motion tracking, which account for spatial smoothness and temporal continuity, respectively. If we ignore their interrelationship, solving each of them alone will be rather challenging, especially when the cluttered features are involved. To integrate them into a probabilistic model, one straightforward approach is to draw the dependence between their hidden states. With regard to their separated states, there are, however, two different explanations of motion which are still made under the individual constraint of smoothness or continuity. Each one can be error-prone, and their coupling causes error propagation. Therefore, it is highly desirable to design a probabilistic model in which a unified state is shared by the two subproblems. This paper is intended to propose such a model, i.e., a Mixture of Transformed Hidden Markov Models (MTHMM), where a unique explanation of motion is made simultaneously under the spatiotemporal constraints. As a result, the MTHMM could find a coherent global interpretation of elastic motion from local cluttered edge features, and experiments show its robustness under ambiguities, data missing, and outliers.

Quantification of carotid vessel wall and plaque thickness change using 3D ultrasound images

Quantitative measurements of carotid plaque burden progression or regression are important in monitoring patients and in evaluation of new treatment options. 3D ultrasound (US) has been used to monitor the progression or regression of carotid artery plaques. This paper reports on the development and application of a method used to analyze changes in carotid plaque morphology from 3D US. The technique used is evaluated using manual segmentations of the arterial wall and lumen from 3D US images acquired in two imaging sessions. To reduce the effect of segmentation variability, segmentation was performed five times each for the wall and lumen. The mean wall and lumen surfaces, computed from this set of five segmentations, were matched on a point-by-point basis, and the distance between each pair of corresponding points served as an estimate of the combined thickness of the plaque, intima, and media (vessel-wall-plus-plaque thickness or VWT). The VWT maps associated with the first and the second US images were compared and the differences of VWT were obtained at each vertex. The 3D VWT and VWT-Change maps may provide important information for evaluating the location of plaque progression in relation to the localized disturbances of flow pattern, such as oscillatory shear, and regression in response to medical treatments.

Simultaneous nonrigid registration of multiple point sets and atlas construction

Groupwise registration of a set of shapes represented by unlabeled point sets is a challenging problem since, usually, this involves solving for point correspondence in a nonrigid motion setting. In this paper, we propose a novel and robust algorithm that is capable of simultaneously computing the mean shape, represented by a probability density function, from multiple unlabeled point sets(represented by finite-mixture models), and registering them nonrigidly to this emerging mean shape. This algorithm avoids the correspondence problem by minimizing the Jensen-Shannon (JS) divergence between the point sets represented as finite mixtures of Gaussian densities. We motivate the use of the JS divergence by pointing out its close relationship to hypothesis testing. Essentially,minimizing the JS divergence is asymptotically equivalent to maximizing the likelihood ratio formed from a probability density of the pooled point sets and the product of the probability densities of the individual point sets. We derive the analytic gradient of the cost function, namely, the JS-divergence, in order to efficiently achieve the optimal solution. The cost function is fully symmetric, with no bias toward any of the given shapes to be registered and whose mean is being sought. A by-product of the registration process is a probabilistic atlas, which is defined as the convex combination of the probability densities of the input point sets being aligned. Our algorithm can be especially useful for creating atlases of various shapes present in images and for simultaneously (rigidly or nonrigidly)registering 3D range data sets (in vision and graphics applications), without having to establish any correspondence. We present experimental results on nonrigidly registering 2D and 3D real and synthetic data (point sets).

A methodology for validating a 3D imaging modality for brain AVM delineation: application to 3DRA

A general methodology is described to validate a 3D imaging modality with respect to 2D digital subtracted angiography (DSA) for brain AVMs (BAVM) delineation. It relies on the assessment of the statistical compatibility of the radiosurgical target delineated in 3D with its delineations in 2D. This methodology is demonstrated through a preliminary evaluation of 3D rotational angiography (3DRA). Generally speaking, BAVM delineation cannot be performed on 3DRA alone. However, in our study, 3DRA showed similar performances to DSA for rather easy cases, and even better for three patients. Conversely, three problematic cases are identified and discussed.

Characterizing spatio-temporal patterns for disease discrimination in cardiac echo videos

Disease-specific understanding of echocardiographic sequences requires accurate characterization of spatio-temporal motion patterns. In this paper we present a method of automatic extraction and matching of spatio-temporal patterns from cardiac echo videos. Specifically, we extract cardiac regions (chambers and walls) using a variation of multiscale normalized cuts that combines motion estimates from deformable models with image intensity. We then derive spatio-temporal trajectories of region measurements such as wall motion, volume and thickness. The region trajectories are then matched to infer the similarities in disease labels of patients. Validation results on patient data sets collected from many hospitals are presented.

Deformable density matching for 3D non-rigid registration of shapes

There exists a large body of literature on shape matching and registration in medical image analysis. However, most of the previous work is focused on matching particular sets of features--point-sets, lines, curves and surfaces. In this work, we forsake specific geometric shape representations and instead seek probabilistic representations--specifically Gaussian mixture models--of shapes. We evaluate a closed-form distance between two probabilistic shape representations for the general case where the mixture models differ in variance and the number of components. We then cast non-rigid registration as a deformable density matching problem. In our approach, we take one mixture density onto another by deforming the component centroids via a thin-plate spline (TPS) and also minimizing the distance with respect to the variance parameters. We validate our approach on synthetic and 3D arterial tree data and evaluate it on 3D hippocampal shapes.

Statistical analyses of brain surfaces using Gaussian random fields on 2-D manifolds

Interest in the morphometric analysis of the brain and its subregions has recently intensified because growth or degeneration of the brain in health or illness affects not only the volume but also the shape of cortical and subcortical brain regions, and new image processing techniques permit detection of small and highly localized perturbations in shape or localized volume, with remarkable precision. An appropriate statistical representation of the shape of a brain region is essential, however, for detecting, localizing, and interpreting variability in its surface contour and for identifying differences in volume of the underlying tissue that produce that variability across individuals and groups of individuals. Our statistical representation of the shape of a brain region is defined by a reference region for that region and by a Gaussian random field (GRF) that is defined across the entire surface of the region. We first select a reference region from a set of segmented brain images of healthy individuals. The GRF is then estimated as the signed Euclidean distances between points on the surface of the reference region and the corresponding points on the corresponding region in images of brains that have been coregistered to the reference. Correspondences between points on these surfaces are defined through deformations of each region of a brain into the coordinate space of the reference region using the principles of fluid dynamics. The warped, coregistered region of each subject is then unwarped into its native space, simultaneously bringing into that space the map of corresponding points that was established when the surfaces of the subject and reference regions were tightly coregistered. The proposed statistical description of the shape of surface contours makes no assumptions, other than smoothness, about the shape of the region or its GRF. The description also allows for the detection and localization of statistically significant differences in the shapes of the surfaces across groups of subjects at both a fine and coarse scale. We demonstrate the effectiveness of these statistical methods by applying them to study differences in shape of the amygdala and hippocampus in a large sample of normal subjects and in subjects with attention deficit/hyperactivity disorder (ADHD).

Recognition of shapes by editing their shock graphs

This paper presents a novel framework for the recognition of objects based on their silhouettes. The main idea is to measure the distance between two shapes as the minimum extent of deformation necessary for one shape to match the other. Since the space of deformations is very high-dimensional, three steps are taken to make the search practical: 1) define an equivalence class for shapes based on shock-graph topology, 2) define an equivalence class for deformation paths based on shock-graph transitions, and 3) avoid complexity-increasing deformation paths by moving toward shock-graph degeneracy. Despite these steps, which tremendously reduce the search requirement, there still remain numerous deformation paths to consider. To that end, we employ an edit-distance algorithm for shock graphs that finds the optimal deformation path in polynomial time. The proposed approach gives intuitive correspondences for a variety of shapes and is robust in the presence of a wide range of visual transformations. The recognition rates on two distinct databases of 99 and 216 shapes each indicate highly successful within category matches (100 percent in top three matches), which render the framework potentially usable in a range of shape-based recognition applications.

Unsupervised learning of an atlas from unlabeled point-sets

One of the key challenges in deformable shape modeling is the problem of estimating a meaningful average or mean shape from a set of unlabeled shapes. We present a new joint clustering and matching algorithm that is capable of computing such a mean shape from multiple shape samples which are represented by unlabeled point-sets. An iterative bootstrap process is used wherein multiple shape sample point-sets are nonrigidly deformed to the emerging mean shape, with subsequent estimation of the mean shape based on these nonrigid alignments. The process is entirely symmetric with no bias toward any of the original shape sample point-sets. We believe that this method can be especially useful for creating atlases of various shapes present in medical images. We have applied the method to create mean shapes from nine hand-segmented 2D corpus callosum data sets and 10 hippocampal 3D point-sets.

Evaluation of 3D correspondence methods for model building

The correspondence problem is of high relevance in the construction and use of statistical models. Statistical models are used for a variety of medical application, e.g. segmentation, registration and shape analysis. In this paper, we present comparative studies in three anatomical structures of four different correspondence establishing methods. The goal in all of the presented studies is a model-based application. We have analyzed both the direct correspondence via manually selected landmarks as well as the properties of the model implied by the correspondences, in regard to compactness, generalization and specificity. The studied methods include a manually initialized subdivision surface (MSS) method and three automatic methods that optimize the object parameterization: SPHARM, MDL and the covariance determinant (DetCov) method. In all studies, DetCov and MDL showed very similar results. The model properties of DetCov and MDL were better than SPHARM and MSS. The results suggest that for modeling purposes the best of the studied correspondence method are MDL and DetCov.

Automatic segmentation of echocardiographic sequences by active appearance motion models

A novel extension of active appearance models (AAMs) for automated border detection in echocardiographic image sequences is reported. The active appearance motion model (AAMM) technique allows fully automated robust and time-continuous delineation of left ventricular (LV) endocardial contours over the full heart cycle with good results. Nonlinear intensity normalization was developed and employed to accommodate ultrasound-specific intensity distributions. The method was trained and tested on 16-frame phase-normalized transthoracic four-chamber sequences of 129 unselected infarct patients, split randomly into a training set (n = 65) and a test set (n = 64). Borders were compared to expert drawn endocardial contours. On the test set, fully automated AAMM performed well in 97% of the cases (average distance between manual and automatic landmark points was 3.3 mm, comparable to human interobserver variabilities). The ultrasound-specific intensity normalization proved to be of great value for good results in echocardiograms. The AAMM was significantly more accurate than an equivalent set of two-dimensional AAMs.

Estimation of 3-D left ventricular deformation from medical images using biomechanical models

The quantitative estimation of regional cardiac deformation from three-dimensional (3-D) image sequences has important clinical implications for the assessment of viability in the heart wall. We present here a generic methodology for estimating soft tissue deformation which integrates image-derived information with biomechanical models, and apply it to the problem of cardiac deformation estimation. The method is image modality independent. The images are segmented interactively and then initial correspondence is established using a shape-tracking approach. A dense motion field is then estimated using a transversely isotropic, linear-elastic model, which accounts for the muscle fiber directions in the left ventricle. The dense motion field is in turn used to calculate the deformation of the heart wall in terms of strain in cardiac specific directions. The strains obtained using this approach in open-chest dogs before and after coronary occlusion, exhibit a high correlation with strains produced in the same animals using implanted markers. Further, they show good agreement with previously published results in the literature. This proposed method provides quantitative regional 3-D estimates of heart deformation.

A minimum description length approach to statistical shape modeling

We describe a method for automatically building statistical shape models from a training set of example boundaries/surfaces. These models show considerable promise as a basis for segmenting and interpreting images. One of the drawbacks of the approach is, however, the need to establish a set of dense correspondences between all members of a set of training shapes. Often this is achieved by locating a set of "landmarks" manually on each training image, which is time consuming and subjective in two dimensions and almost impossible in three dimensions. We describe how shape models can be built automatically by posing the correspondence problem as one of finding the parameterization for each shape in the training set. We select the set of parameterizations that build the "best" model. We define "best" as that which minimizes the description length of the training set, arguing that this leads to models with good compactness, specificity and generalization ability. We show how a set of shape parameterizations can be represented and manipulated in order to build a minimum description length model. Results are given for several different training sets of two-dimensional boundaries, showing that the proposed method constructs better models than other approaches including manual landmarking-the current gold standard. We also show that the method can be extended straightforwardly to three dimensions.

Parameter space warping: shape-based correspondence between morphologically different objects

This paper presents a novel and comprehensive method for the automated determination of correspondences between two morphologically different two-dimensional (2-D) or three-dimensional (3-D) objects. Correspondences are determined by warping parametric representations of the objects to be matched. The warp is guided by the minimization of a similarity criterion function that measures features related to structural correspondence, including Euclidian point-to-point distance and differences in normals and curvature. The method uses a continuous harmonic parameterization for both the object and the warp, which provides: 1) a high degree of computational efficiency; 2) robust extraction of differential features, not subject to discretization errors or noise amplification in differentiation; 3) direct formulation of constraints to avoid overlaps in the resulting correspondence set; and 4) a scale-space paradigm of object shape and warp. The new method does not search for individual landmarks, but operates with a complete, integrated representation of the object geometry. The method was tested on 2-D and 3-D objects with substantial shape differences. Results demonstrated substantial improvements of 2%-33% in correspondence accuracy and 15%-59% in correspondence quality compared with direct registration methods.

An adaptive-focus statistical shape model for segmentation and shape modeling of 3-D brain structures

This paper presents a deformable model for automatically segmenting brain structures from volumetric magnetic resonance (MR) images and obtaining point correspondences, using geometric and statistical information in a hierarchical scheme. Geometric information is embedded into the model via a set of affine-invariant attribute vectors, each of which characterizes the geometric structure around a point of the model from a local to a global scale. The attribute vectors, in conjunction with the deformation mechanism of the model, warranty that the model not only deforms to nearby edges, as is customary in most deformable surface models, but also that it determines point correspondences based on geometric similarity at different scales. The proposed model is adaptive in that it initially focuses on the most reliable structures of interest, and gradually shifts focus to other structures as those become closer to their respective targets and, therefore, more reliable. The proposed techniques have been used to segment boundaries of the ventricles, the caudate nucleus, and the lenticular nucleus from volumetric MR images.