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

Relationship Between Isokinetic Lower-Limb Joint Strength, Isometric Time Force Characteristics, and Leg-Spring Stiffness in Recreational Runners

Neuromuscular characteristics, such as lower-limb joint strength and the ability to rapidly generate force, may play an important role in leg-spring stiffness regulation. This study aimed to investigate the relationship between isokinetic knee and ankle joint peak torque (PT), the force-time characteristics of isometric mid-thigh pull (IMTP), and leg stiffness (K_leg)/vertical stiffness (K_vert) in recreationally trained runners. Thirty-one male runners were recruited and underwent three separate tests. In the first session, the body composition, K_leg, and K_vert at running speeds of 12 and 14 km⋅h^–1 were measured. In the second session, isokinetic knee and ankle joint PT at 60°⋅s^–1 were tested. The force-time characteristics of the IMTP were evaluated in the final session. Pearson’s product-moment correlations, with the Benjamini–Hochberg correction procedure, showed that the knee flexor concentric and eccentric and extensor concentric PT (r = 0.473–0.654, p < 0.05) were moderate to largely correlated with K_leg and K_vert at 12 and 14 km⋅h^–1. The knee extensor eccentric PT (r = 0.440, p = 0.050) was moderately correlated with the 14 km⋅h^–1K_vert. The ankle plantar flexor concentric and dorsiflexor eccentric PT (r = 0.506–0.571, p < 0.05) were largely correlated with K_leg at 12 km⋅h^–1. The ankle plantar flexor concentric and eccentric and dorsiflexor eccentric PT (r = 0.436–0.561, p < 0.05) were moderate to largely correlated with K_vert at 12 and 14 km⋅h^–1. For IMTP testing, high correlation was only found between the IMPT peak force (PF) and K_vert at 14 km⋅h^–1 (r = 0.510, p = 0.014). Thus, superior leg-spring stiffness in recreational runners may be related to increased knee and ankle joint strength, eccentric muscular capacity, and maximal force production.

A Hierarchical Curve-Based Approach to the Analysis of Manifold Data

One of the data structures generated by medical imaging technology is high resolution point clouds representing anatomical surfaces. Stereophotogrammetry and laser scanning are two widely available sources of this kind of data. A standardised surface representation is required to provide a meaningful correspondence across different images as a basis for statistical analysis. Point locations with anatomical definitions, referred to as landmarks, have been the traditional approach. Landmarks can also be taken as the starting point for more general surface representations, often using templates which are warped on to an observed surface by matching landmark positions and subsequent local adjustment of the surface. The aim of the present paper is to provide a new approach which places anatomical curves at the heart of the surface representation and its analysis. Curves provide intermediate structures which capture the principal features of the manifold (surface) of interest through its ridges and valleys. As landmarks are often available these are used as anchoring points, but surface curvature information is the principal guide in estimating the curve locations. The surface patches between these curves are relatively flat and can be represented in a standardised manner by appropriate surface transects to give a complete surface model. This new approach does not require the use of a template, reference sample or any external information to guide the method and, when compared with a surface based approach, the estimation of curves is shown to have improved performance. In addition, examples involving applications to mussel shells and human faces show that the analysis of curve information can deliver more targeted and effective insight than the use of full surface information.

Skeletal Shape Correspondence Through Entropy

We present a novel approach for improving the shape statistics of medical image objects by generating correspondence of skeletal points. Each object's interior is modeled by an s-rep, i.e., by a sampled, folded, two-sided skeletal sheet with spoke vectors proceeding from the skeletal sheet to the boundary. The skeleton is divided into three parts: the up side, the down side, and the fold curve. The spokes on each part are treated separately and, using spoke interpolation, are shifted along that skeleton in each training sample so as to tighten the probability distribution on those spokes' geometric properties while sampling the object interior regularly. As with the surface/boundary-based correspondence method of Cates et al., entropy is used to measure both the probability distribution tightness and the sampling regularity, here of the spokes' geometric properties. Evaluation on synthetic and real world lateral ventricle and hippocampus data sets demonstrate improvement in the performance of statistics using the resulting probability distributions. This improvement is greater than that achieved by an entropy-based correspondence method on the boundary points.

Entropy-based particle correspondence for shape populations

PURPOSE

Statistical shape analysis of anatomical structures plays an important role in many medical image analysis applications such as understanding the structural changes in anatomy in various stages of growth or disease. Establishing accurate correspondence across object populations is essential for such statistical shape analysis studies.

METHODS

In this paper, we present an entropy-based correspondence framework for computing point-based correspondence among populations of surfaces in a groupwise manner. This robust framework is parameterization-free and computationally efficient. We review the core principles of this method as well as various extensions to deal effectively with surfaces of complex geometry and application-driven correspondence metrics.

RESULTS

We apply our method to synthetic and biological datasets to illustrate the concepts proposed and compare the performance of our framework to existing techniques.

CONCLUSIONS

Through the numerous extensions and variations presented here, we create a very flexible framework that can effectively handle objects of various topologies, multi-object complexes, open surfaces, and objects of complex geometry such as high-curvature regions or extremely thin features.

Fitting Skeletal Object Models Using Spherical Harmonics Based Template Warping

We present a scheme that propagates a reference skeletal model (s-rep) into a particular case of an object, thereby propagating the initial shape-related layout of the skeleton-to-boundary vectors, called spokes. The scheme represents the surfaces of the template as well as the target objects by spherical harmonics and computes a warp between these via a thin plate spline. To form the propagated s-rep, it applies the warp to the spokes of the template s-rep and then statistically refines. This automatic approach promises to make s-rep fitting robust for complicated objects, which allows s-rep based statistics to be available to all. The improvement in fitting and statistics is significant compared with the previous methods and in statistics compared with a state-of-the-art boundary based method.

Registration of challenging pre-clinical brain images

The size and complexity of brain imaging studies in pre-clinical populations are increasing, and automated image analysis pipelines are urgently required. Pre-clinical populations can be subjected to controlled interventions (e.g., targeted lesions), which significantly change the appearance of the brain obtained by imaging. Existing systems for registration (the systematic alignment of scans into a consistent anatomical coordinate system), which assume image similarity to a reference scan, may fail when applied to these images. However, affine registration is a particularly vital pre-processing step for subsequent image analysis which is assumed to be an effective procedure in recent literature describing sophisticated techniques such as manifold learning. Therefore, in this paper, we present an affine registration solution that uses a graphical model of a population to decompose difficult pairwise registrations into a composition of steps using other members of the population. We developed this methodology in the context of a pre-clinical model of stroke in which large, variable hyper-intense lesions significantly impact registration performance. We tested this technique systematically in a simulated human population of brain tumour images before applying it to pre-clinical models of Parkinson's disease and stroke.

A Parameterization Based Correspondence Method for PDM Building

Place-march of corresponding landmarks is one of the major factors influencing 3D Points Distribution Model (PDM) quality. In this study, we propose a semi-automatic correspondence method based on surface parameterization theory. All the training sets are mapped into a spherical domain previously. The rotation transformation of training samples is regarded as spherical rotation of their maps. We solve it by comparing the density distribution of surface map of training sample with respect to the reference model. Simultaneously, the corresponding landmarks across the whole training set are marketed depending on the spherical coordinates on parameter domain. In this paper, we also compared the corresponding results with two constraint conditions of spherical conformal mapping: 3 datum points constrain and zero-mass constrain. Experimental results are given for left lung training sets of 3D shapes. The mean result with the 3 datum points constraint and the zero mass-center constraint was 21.65 mm and 20.19 mm respectively.

3D quantification of mandibular asymmetry using the SPHARM-PDM tool box

PURPOSE

Pretreatment diagnosis of mandibular asymmetry in orthognathic surgery patients can be improved by quantitative shape modeling and analysis. The UNC SPHARM-PDM (University of North Carolina Spherical Harmonics-Point Distribution Model) toolbox was applied to a cohort of patients and the results were evaluated.

METHODS

Three-dimensional (3D) virtual surface models are constructed from CBCT scans of each patient in the cohort by segmentation. Mirroring on a sagittal arbitrary plane is used to flip the left and right sides of each image. An automatic voxel-based registration on the cranial base is used to align the volume and its mirror for comparison. SPHARM-PDM is used to compute correspondent models for each hemimandible and the mirror of the contralateral side. Procrustes analysis was used to evaluate discrepancies between each pair of models to assess asymmetry. Mandibular asymmetry was also located and quantified by computing corresponding surface distances between each hemimandible (left and right sides) and the mirror of the contralateral side.

RESULTS

There were no statistically significant differences in surrogates for mandibular asymmetry assessment based on right or the left side mirroring. Those surrogates are the rotational and translational differences between each hemimandible and the mirror of the contralateral side in 3 planes of space (the absolute values of Procrustes registration output in 6 degrees of freedom). Absolute and signed distance maps between each hemimandible and the mirror of the contralateral side located and quantified areas of asymmetry diagnosis for each patient. Even though mandibular condyle asymmetry was observed in 8% of the cases and mandibular asymmetry along areas of the ramus and mandibular corpus was noted in 17.8% of the cases, the remaining 74.2% showed generalized morphological and positional asymmetry at the condyle, the ramus and mandibular corpus.

CONCLUSION

Three-dimensional diagnosis of mandibular asymmetry revealed the complex involvement of morphological components of the mandible and the heterogeneous nature of this clinical condition. SPHARM-PDM has a promising role in the individual diagnosis and quantification of mandibular asymmetry.

General multivariate linear modeling of surface shapes using SurfStat

Although there are many imaging studies on traditional ROI-based amygdala volumetry, there are very few studies on modeling amygdala shape variations. This paper presents a unified computational and statistical framework for modeling amygdala shape variations in a clinical population. The weighted spherical harmonic representation is used to parameterize, smooth out, and normalize amygdala surfaces. The representation is subsequently used as an input for multivariate linear models accounting for nuisance covariates such as age and brain size difference using the SurfStat package that completely avoids the complexity of specifying design matrices. The methodology has been applied for quantifying abnormal local amygdala shape variations in 22 high functioning autistic subjects.

Building 3-D statistical shape models by direct optimization

Statistical shape models are powerful tools for image interpretation and shape analysis. A simple, yet effective, way of building such models is to capture the statistics of sampled point coordinates over a training set of example shapes. However, a major drawback of this approach is the need to establish a correspondence across the training set. In 2-D, a correspondence is often defined using a set of manually placed 'landmarks' and linear interpolation to sample the shape in between. Such annotation is, however, time-consuming and subjective, particularly when extended to 3-D. In this paper, we show that it is possible to establish a dense correspondence across the whole training set automatically by treating correspondence as an optimization problem. The objective function we use for the optimization is based on the minimum description length principle, which we argue is a criterion that leads to models with good compactness, specificity, and generalization ability. We manipulate correspondence by reparameterizing each training shape. We describe an explicit representation of reparameterization for surfaces in 3-D that makes it impossible to generate an illegal (i.e., not one-to-one) correspondence. We also describe several large-scale optimization strategies for model building, and perform a detailed analysis of each approach. Finally, we derive quantitative measures of model quality, allowing meaningful comparison between models built using different methods. Results are given for several different training sets of 3-D shapes, which show that the minimum description length models perform significantly better than other approaches.

Shape registration by optimally coding shapes

This paper formulates shape registration as an optimal coding problem. It employs a set of landmarks to establish the correspondence between shapes, and assumes that the best correspondence can be achieved when the polygons formed by the landmarks optimally code all the shape contours, i.e., obtain their minimum description length (MDL). This is different from previous MDL-based shape registration methods, which code the landmark locations. In this paper, each contour is discretized to be a set of points to make the coding feasible, and a number of strategies are adopted to tackle the difficult optimization problem involved. The resulting algorithm, called CAP, is able to yield statistical shape model with better quality in terms of model generalization error, which is demonstrated on both synthetic and biomedical shapes.

Shape registration by simultaneously optimizing representation and transformation

This paper proposes a novel approach that achieves shape registration by optimizing shape representation and transformation simultaneously, which are modeled by a constrained Gaussian Mixture Model (GMM) and a regularized thin plate spline respectively. The problem is formulated within a Bayesian framework and solved by an expectation-maximum (EM) algorithm. Compared with the popular methods based on landmarks-sliding, its advantages include: (1) It can naturally deal with shapes of complex topologies and 3D dimension; (2) It is more robust against data noise; (3) The registration performance is better in terms of the generalization error of the resultant statistical shape model. These are demonstrated on both synthetic and biomedical shapes.

Multiple sclerosis medical image analysis and information management

Magnetic resonance imaging (MRI) has become a central tool for patient management, as well as research, in multiple sclerosis (MS). Measurements of disease burden and activity derived from MRI through quantitative image analysis techniques are increasingly being used. There are many complexities and challenges in building computerized processing pipelines to ensure efficiency, reproducibility, and quality control for MRI scans from MS patients. Such paradigms require advanced image processing and analysis technologies, as well as integrated database management systems to ensure the most utility for clinical and research purposes. This article reviews pipelines available for quantitative clinical MRI research in MS, including image segmentation, registration, time-series analysis, performance validation, visualization techniques, and advanced medical imaging software packages. To address the complex demands of the sequential processes, the authors developed a workflow management system that uses a centralized database and distributed computing system for image processing and analysis. The implementation of their system includes a web-form-based Oracle database application for information management and event dispatching, and multiple modules for image processing and analysis. The seamless integration of processing pipelines with the database makes it more efficient for users to navigate complex, multistep analysis protocols, reduces the user's learning curve, reduces the time needed for combining and activating different computing modules, and allows for close monitoring for quality-control purposes. The authors' system can be extended to general applications in clinical trials and to routine processing for image-based clinical research.

Compensating for intraoperative soft-tissue deformations using incomplete surface data and finite elements

Image-guided liver surgery requires the ability to identify and compensate for soft tissue deformation in the organ. The predeformed state is represented as a complete three-dimensional surface of the organ, while the intraoperative data is a range scan point cloud acquired from the exposed liver surface. The first step is to rigidly align the coordinate systems of the intraoperative and preoperative data. Most traditional rigid registration methods minimize an error metric over the entire data set. In this paper, a new deformation-identifying rigid registration (DIRR) is reported that identifies and aligns minimally deformed regions of the data using a modified closest point distance cost function. Once a rigid alignment has been established, deformation is accounted for using a linearly elastic finite element model (FEM) and implemented using an incremental framework to resolve geometric nonlinearities. Boundary conditions for the incremental formulation are generated from intraoperatively acquired range scan surfaces of the exposed liver surface. A series of phantom experiments is presented to assess the fidelity of the DIRR and the combined DIRR/FEM approaches separately. The DIRR approach identified deforming regions in 90% of cases under conditions of realistic surgical exposure. With respect to the DIRR/FEM algorithm, subsurface target errors were correctly located to within 4 mm in phantom experiments.

Mindboggle: automated brain labeling with multiple atlases

Background

To make inferences about brain structures or activity across multiple individuals, one first needs to determine the structural correspondences across their image data. We have recently developed Mindboggle as a fully automated, feature-matching approach to assign anatomical labels to cortical structures and activity in human brain MRI data. Label assignment is based on structural correspondences between labeled atlases and unlabeled image data, where an atlas consists of a set of labels manually assigned to a single brain image. In the present work, we study the influence of using variable numbers of individual atlases to nonlinearly label human brain image data.

Methods

Each brain image voxel of each of 20 human subjects is assigned a label by each of the remaining 19 atlases using Mindboggle. The most common label is selected and is given a confidence rating based on the number of atlases that assigned that label. The automatically assigned labels for each subject brain are compared with the manual labels for that subject (its atlas). Unlike recent approaches that transform subject data to a labeled, probabilistic atlas space (constructed from a database of atlases), Mindboggle labels a subject by each atlas in a database independently.

Results

When Mindboggle labels a human subject's brain image with at least four atlases, the resulting label agreement with coregistered manual labels is significantly higher than when only a single atlas is used. Different numbers of atlases provide significantly higher label agreements for individual brain regions.

Conclusion

Increasing the number of reference brains used to automatically label a human subject brain improves labeling accuracy with respect to manually assigned labels. Mindboggle software can provide confidence measures for labels based on probabilistic assignment of labels and could be applied to large databases of brain images.

Statistical validation of brain tumor shape approximation via spherical harmonics for image-guided neurosurgery

RATIONALE AND OBJECTIVES

Surgical planning now routinely uses both two-dimensional (2D) and three-dimensional (3D) models that integrate data from multiple imaging modalities, each highlighting one or more aspects of morphology or function. We performed a preliminary evaluation of the use of spherical harmonics (SH) in approximating the 3D shape and estimating the volume of brain tumors of varying characteristics.

MATERIALS AND METHODS

Magnetic resonance (MR) images from five patients with brain tumors were selected randomly from our MR-guided neurosurgical practice. Standardized mean square reconstruction errors (SMSRE) by tumor volume were measured. Validation metrics for comparing performances of the SH method against segmented contours (SC) were the dice similarity coefficient (DSC) and standardized Euclidean distance (SED) measure.

RESULTS

Tumor volume range was 22,413-85,189 mm3, and range of number of vertices in triangulated models was 3674-6544. At SH approximations with degree of at least 30, SMSRE were within 1.66 x 10(-5) mm(-1). Summary measures yielded a DSC range of 0.89-0.99 (pooled median, 0.97 and significantly >0.7; P < .001) and an SED range of 0.0002-0.0028 (pooled median, 0.0005).

CONCLUSION

3D shapes of tumors may be approximated by using SH for neurosurgical applications.

Mindboggle: a scatterbrained approach to automate brain labeling

Mindboggle (http://www.binarybottle.com/mindboggle.html) is a fully automated, feature matching approach to label cortical structures and activity anatomically in human brain MRI data. This approach does not assume that the existence of component structures and their relative spatial relationship is preserved from brain to brain, but instead disassembles a labeled atlas and reassembles its pieces to match corresponding pieces in an unlabeled subject brain before labeling. Mindboggle: (1) converts linearly coregistered subject and atlas MRI data into sulcus pieces, (2) matches each atlas piece with a combination of subject pieces by minimizing a cost function, (3) transforms atlas label boundaries to the matching subject pieces, (4) warps atlas labels to their transformed boundaries, and (5) propagates labels to fill remaining gaps in a mask derived from the subject brain. We compared Mindboggle with four registration methods: linear registration, and nonlinear registration using SPM2, AIR, and ANIMAL. Automated labeling by all of the nonlinear methods was found to be at least comparable with linear registration. Mindboggle outperformed every other method, as measured by the agreement between overlapping atlas labels and manually assigned subject labels, with respect to the union or the intersection of voxels. After applying the same procedure that Mindboggle uses to fill a subject's segmented gray matter mask with labels (step 5), the results of the other methods improved. However, after performing a one-way ANOVA (and Tukey's honestly significant difference criterion) in a multiple comparison between the results obtained by the different methods, Mindboggle was still found to be the only nonlinear method whose labeling performance was significantly better than that of linear registration or SPM2. Further advantages to Mindboggle include a high degree of robustness against image artifacts, poor image quality, and incomplete brain data. We tested the latter hypothesis by conducting all of the tests again, this time registering the atlas to an artificially lesioned version of itself, and found that Mindboggle was the only method whose performance did not degrade significantly as the lesion size increased.

The shape operator for differential analysis of images

This work provides a new technique for surface oriented volumetric image analysis. The method makes no assumptions about topology, instead constructing a local neighborhood from image information, such as a segmentation or edge map, to define a surface patch. Neighborhood constructions using extrinsic and intrinsic distances are given. This representation allows one to estimate differential properties directly from the image's Gauss map. We develop a novel technique for this purpose which estimates the shape operator and yields both principal directions and curvatures. Only first derivatives need be estimated, making the method numerically stable. We show the use of these measures for multi-scale classification of image structure by the mean and Gaussian curvatures. Finally, we propose to register image volumes by surface curvature. This is particularly useful when geometry is the only variable. To illustrate this, we register binary segmented data by surface curvature, both rigidly and non-rigidly. A novel variant of Demons registration, extensible for use with differentiable similarity metrics, is also applied for deformable curvature-driven registration of medical images.

Statistical shape analysis of neuroanatomical structures based on medial models

Knowledge about the biological variability of anatomical objects is essential for statistical shape analysis and discrimination between healthy and pathological structures. This paper describes a novel approach that incorporates the variability of an object population into the generation of a characteristic 3D shape model. The proposed shape representation is a coarse-scale sampled medial description derived from a fine-scale spherical harmonics (SPHARM) boundary description. This medial description is composed of a net of medial samples (m-rep) with fixed graph properties. The medial model is computed automatically from a predefined shape space using pruned 3D Voronoi skeletons. A new method determines the stable medial branching topology from the shape space. An intrinsic coordinate system and an implicit correspondence between shapes is defined on the medial manifold. Several studies of biological structures clearly demonstrate that the novel representation has the promise to describe shape changes in a natural and intuitive way. A new medial shape similarity study of group differences between monozygotic and dizygotic twins in lateral ventricle shape demonstrates the meaningful and powerful representation of local and global form.

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.