Skeletons, Object Shape, Statistics.

Objects and object complexes in 3D, as well as those in 2D, have many possible representations. Among them skeletal representations have special advantages and some limitations. For the special form of skeletal representation called "s-reps," these advantages include strong suitability for representing slabular object populations and statistical applications on these populations. Accomplishing these statistical applications is best if one recognizes that s-reps live on a curved shape space. Here we will lay out the definition of s-reps, their advantages and limitations, their mathematical properties, methods for fitting s-reps to single- and multi-object boundaries, methods for measuring the statistics of these object and multi-object representations, and examples of such applications involving statistics. While the basic theory, ideas, and programs for the methods are described in this paper and while many applications with evaluations have been produced, there remain many interesting open opportunities for research on comparisons to other shape representations, new areas of application and further methodological developments, many of which are explicitly discussed here.

RNNSLAM: Reconstructing the 3D colon to visualize missing regions during a colonoscopy.

Colonoscopy is the gold standard for pre-cancerous polyps screening and treatment. The polyp detection rate is highly tied to the percentage of surveyed colonic surface. However, current colonoscopy technique cannot guarantee that all the colonic surface is well examined because of incomplete camera orientations and of occlusions. The missing regions can hardly be noticed in a continuous first-person perspective. Therefore, a useful contribution would be an automatic system that can compute missing regions from an endoscopic video in real-time and alert the endoscopists when a large missing region is detected. We present a novel method that reconstructs dense chunks of a 3D colon in real time, leaving the unsurveyed part unreconstructed. The method combines a standard SLAM system with a depth and pose prediction network to achieve much more robust tracking and less drift. It addresses the difficulties for colonoscopic images of existing simultaneous localization and mapping (SLAM) systems and end-to-end deep learning methods.

Fitting unbranching skeletal structures to objects.

Representing an object by a skeletal structure can be powerful for statistical shape analysis if there is good correspondence of the representations within a population. Many anatomic objects have a genus-zero boundary and can be represented by a smooth unbranching skeletal structure that can be discretely approximated. We describe how to compute such a discrete skeletal structure ("d-s-rep") for an individual 3D shape with the desired correspondence across cases. The method involves fitting a d-s-rep to an input representation of an object's boundary. A good fit is taken to be one whose skeletally implied boundary well approximates the target surface in terms of low order geometric boundary properties: (1) positions, (2) tangent fields, (3) various curvatures. Our method involves a two-stage framework that first, roughly yet consistently fits a skeletal structure to each object and second, refines the skeletal structure such that the shape of the implied boundary well approximates that of the object. The first stage uses a stratified diffeomorphism to produce topologically non-self-overlapping, smooth and unbranching skeletal structures for each object of a population. The second stage uses loss terms that measure geometric disagreement between the skeletally implied boundary and the target boundary and avoid self-overlaps in the boundary. By minimizing the total loss, we end up with a good d-s-rep for each individual shape. We demonstrate such d-s-reps for various human brain structures. The framework is accessible and extensible by clinical users, researchers and developers as an extension of SlicerSALT, which is based on 3D Slicer.

A Novel Method for High-Dimensional Anatomical Mapping of Extra-Axial Cerebrospinal Fluid: Application to the Infant Brain.

Cerebrospinal fluid (CSF) plays an essential role in early postnatal brain development. Extra-axial CSF (EA-CSF) volume, which is characterized by CSF in the subarachnoid space surrounding the brain, is a promising marker in the early detection of young children at risk for neurodevelopmental disorders. Previous studies have focused on global EA-CSF volume across the entire dorsal extent of the brain, and not regionally-specific EA-CSF measurements, because no tools were previously available for extracting local EA-CSF measures suitable for localized cortical surface analysis. In this paper, we propose a novel framework for the localized, cortical surface-based analysis of EA-CSF. The proposed processing framework combines probabilistic brain tissue segmentation, cortical surface reconstruction, and streamline-based local EA-CSF quantification. The quantitative analysis of local EA-CSF was applied to a dataset of typically developing infants with longitudinal MRI scans from 6 to 24 months of age. There was a high degree of consistency in the spatial patterns of local EA-CSF across age using the proposed methods. Statistical analysis of local EA-CSF revealed several novel findings: several regions of the cerebral cortex showed reductions in EA-CSF from 6 to 24 months of age, and specific regions showed higher local EA-CSF in males compared to females. These age-, sex-, and anatomically-specific patterns of local EA-CSF would not have been observed if only a global EA-CSF measure were utilized. The proposed methods are integrated into a freely available, open-source, cross-platform, user-friendly software tool, allowing neuroimaging labs to quantify local extra-axial CSF in their neuroimaging studies to investigate its role in typical and atypical brain development.

NON-EUCLIDEAN, CONVOLUTIONAL LEARNING ON CORTICAL BRAIN SURFACES.

In recent years there have been many studies indicating that multiple cortical features, extracted at each surface vertex, are promising in the detection of various neurodevelopmental and neurodegenerative diseases. However, with limited datasets, it is challenging to train stable classifiers with such high-dimensional surface data. This necessitates a feature reduction that is commonly accomplished via regional volumetric morphometry from standard brain atlases. However, current regional summaries are not specific to the given age or pathology that is studied, which runs the risk of losing relevant information that can be critical in the classification process. To solve this issue, this paper proposes a novel data-driven approach by extending convolutional neural networks (CNN) for use on non-Euclidean manifolds such as cortical surfaces. The proposed network learns the most powerful features and brain regions from the extracted large dimensional feature space; thus creating a new feature space in which the dimensionality is reduced and feature distributions are better separated. We demonstrate the usability of the proposed surface-CNN framework in an example study classifying Alzheimers disease patients versus normal controls. The high performance in the cross-validation diagnostic results shows the potential of our proposed prediction system.

SlicerSALT: Shape AnaLysis Toolbox.

SlicerSALT is an open-source platform for disseminating state-of-the-art methods for performing statistical shape analysis. These methods are developed as 3D Slicer extensions to take advantage of its powerful underlying libraries. SlicerSALT itself is a heavily customized 3D Slicer package that is designed to be easy to use for shape analysis researchers. The packaged methods include powerful techniques for creating and visualizing shape representations as well as performing various types of analysis.

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.

A Segmentation Editing Framework Based on Shape Change Statistics.

Segmentation is a key task in medical image analysis because its accuracy significantly affects successive steps. Automatic segmentation methods often produce inadequate segmentations, which require the user to manually edit the produced segmentation slice by slice. Because editing is time-consuming, an editing tool that enables the user to produce accurate segmentations by only drawing a sparse set of contours would be needed. This paper describes such a framework as applied to a single object. Constrained by the additional information enabled by the manually segmented contours, the proposed framework utilizes object shape statistics to transform the failed automatic segmentation to a more accurate version. Instead of modeling the object shape, the proposed framework utilizes shape change statistics that were generated to capture the object deformation from the failed automatic segmentation to its corresponding correct segmentation. An optimization procedure was used to minimize an energy function that consists of two terms, an external contour match term and an internal shape change regularity term. The high accuracy of the proposed segmentation editing approach was confirmed by testing it on a simulated data set based on 10 in-vivo infant magnetic resonance brain data sets using four similarity metrics. Segmentation results indicated that our method can provide efficient and adequately accurate segmentations (Dice segmentation accuracy increase of 10%), with very sparse contours (only 10%), which is promising in greatly decreasing the work expected from the user.

Non-Euclidean classification of medically imaged objects via s-reps.

Classifying medically imaged objects, e.g., into diseased and normal classes, has been one of the important goals in medical imaging. We propose a novel classification scheme that uses a skeletal representation to provide rich non-Euclidean geometric object properties. Our statistical method combines distance weighted discrimination (DWD) with a carefully chosen Euclideanization which takes full advantage of the geometry of the manifold on which these non-Euclidean geometric object properties (GOPs) live. Our method is evaluated via the task of classifying 3D hippocampi between schizophrenics and healthy controls. We address three central questions. 1) Does adding shape features increase discriminative power over the more standard classification based only on global volume? 2) If so, does our skeletal representation provide greater discriminative power than a conventional boundary point distribution model (PDM)? 3) Especially, is Euclideanization of non-Euclidean shape properties important in achieving high discriminative power? Measuring the capability of a method in terms of area under the receiver operator characteristic (ROC) curve, we show that our proposed method achieves strongly better classification than both the classification method based on global volume alone and the s-rep-based classification method without proper Euclideanization of non-Euclidean GOPs. We show classification using Euclideanized s-reps is also superior to classification using PDMs, whether the PDMs are first Euclideanized or not. We also show improved performance with Euclideanized boundary PDMs over non-linear boundary PDMs. This demonstrates the benefit that proper Euclideanization of non-Euclidean GOPs brings not only to s-rep-based classification but also to PDM-based classification.

Entropy-based Correspondence Improvement of Interpolated Skeletal Models.

Statistical analysis of shape representations relies on having good correspondence across a population. Improving correspondence yields improved statistics. Point distribution models (PDMs) are often used to represent object boundaries. Skeletal representations (s-reps) model object widths and boundary directions as well as boundary positions, so they should yield better correspondence. We present two methods: one for continuously interpolating a discretely-sampled skeletal model and one for improving correspondence by using this interpolation to shift skeletal samples to new positions. The interpolation operates by an extension of the mathematics of medial structures. As with Cates' boundary-based method, we evaluate correspondence in terms of regularity and shape-feature population entropies. Evaluation on both synthetic and real data shows that our method both improves correspondence of s-rep models fit to segmented lateral ventricles and that the combined boundary-and-skeletal PDMs implied by these optimized s-reps have better correspondence than optimized boundary PDMs.

Skeletal shape correspondence via entropy minimization.

PURPOSE: Improving the shape statistics of medical image objects by generating correspondence of interior skeletal points. DATA: Synthetic objects and real world lateral ventricles segmented from MR images. METHODS: Each object's interior is modeled by a skeletal representation called the s-rep, which is a quadrilaterally sampled, folded 2-sided skeletal sheet with spoke vectors proceeding from the sheet to the boundary. The skeleton is divided into three parts: up-side, down-side and fold-curve. The spokes on each part are treated separately and, using spoke interpolation, are shifted along their skeletal parts 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-based correspondence method of Cates et al., entropy is used to measure both the probability distribution tightness and sampling regularity. The spokes' geometric properties are skeletal position, spoke length and spoke direction. The properties used to measure the regularity are the volumetric subregions bounded by the spokes, their quadrilateral sub-area and edge lengths on the skeletal surface and on the boundary. RESULTS: Evaluation on synthetic and real world lateral ventricles demonstrated improvement in the performance of statistics using the resulting probability distributions, as compared to methods based on boundary models. The evaluation measures used were generalization, specificity, and compactness. CONCLUSIONS: S-rep models with the proposed improved correspondence provide significantly enhanced statistics as compared to standard boundary models.

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.

Geometric-feature-based spectral graph matching in pharyngeal surface registration.

Fusion between an endoscopic movie and a CT can aid specifying the tumor target volume for radiotherapy. That requires a deformable pharyngeal surface registration between a 3D endoscope reconstruction and a CT segmentation. In this paper, we propose to use local geometric features for deriving a set of initial correspondences between two surfaces, with which an association graph can be constructed for registration by spectral graph matching. We also define a new similarity measurement to provide a meaningful way for computing inter-surface affinities in the association graph. Our registration method can deal with large non-rigid anatomical deformation, as well as missing data and topology change. We tested the robustness of our method with synthetic deformations and showed registration results on real data.

Local metric learning in 2D/3D deformable registration with application in the abdomen.

In image-guided radiotherapy (IGRT) of disease sites subject to respiratory motion, soft tissue deformations can affect localization accuracy. We describe the application of a method of 2D/3D deformable registration to soft tissue localization in abdomen. The method, called registration efficiency and accuracy through learning a metric on shape (REALMS), is designed to support real-time IGRT. In a previously developed version of REALMS, the method interpolated 3D deformation parameters for any credible deformation in a deformation space using a single globally-trained Riemannian metric for each parameter. We propose a refinement of the method in which the metric is trained over a particular region of the deformation space, such that interpolation accuracy within that region is improved. We report on the application of the proposed algorithm to IGRT in abdominal disease sites, which is more challenging than in lung because of low intensity contrast and nonrespiratory deformation. We introduce a rigid translation vector to compensate for nonrespiratory deformation, and design a special region-of-interest around fiducial markers implanted near the tumor to produce a more reliable registration. Both synthetic data and actual data tests on abdominal datasets show that the localized approach achieves more accurate 2D/3D deformable registration than the global approach.

2D/3D Image Registration using Regression Learning.

In computer vision and image analysis, image registration between 2D projections and a 3D image that achieves high accuracy and near real-time computation is challenging. In this paper, we propose a novel method that can rapidly detect an object's 3D rigid motion or deformation from a 2D projection image or a small set thereof. The method is called CLARET (Correction via Limited-Angle Residues in External Beam Therapy) and consists of two stages: registration preceded by shape space and regression learning. In the registration stage, linear operators are used to iteratively estimate the motion/deformation parameters based on the current intensity residue between the target projec-tion(s) and the digitally reconstructed radiograph(s) (DRRs) of the estimated 3D image. The method determines the linear operators via a two-step learning process. First, it builds a low-order parametric model of the image region's motion/deformation shape space from its prior 3D images. Second, using learning-time samples produced from the 3D images, it formulates the relationships between the model parameters and the co-varying 2D projection intensity residues by multi-scale linear regressions. The calculated multi-scale regression matrices yield the coarse-to-fine linear operators used in estimating the model parameters from the 2D projection intensity residues in the registration. The method's application to Image-guided Radiation Therapy (IGRT) requires only a few seconds and yields good results in localizing a tumor under rigid motion in the head and neck and under respiratory deformation in the lung, using one treatment-time imaging 2D projection or a small set thereof.

Multi-object analysis of volume, pose, and shape using statistical discrimination.

One goal of statistical shape analysis is the discrimination between two populations of objects. Whereas traditional shape analysis was mostly concerned with single objects, analysis of multi-object complexes presents new challenges related to alignment and pose. In this paper, we present a methodology for discriminant analysis of multiple objects represented by sampled medial manifolds. Non-euclidean metrics that describe geodesic distances between sets of sampled representations are used for alignment and discrimination. Our choice of discriminant method is the distance-weighted discriminant because of its generalization ability in high-dimensional, low sample size settings. Using an unbiased, soft discrimination score, we associate a statistical hypothesis test with the discrimination results. We explore the effectiveness of different choices of features as input to the discriminant analysis, using measures like volume, pose, shape, and the combination of pose and shape. Our method is applied to a longitudinal pediatric autism study with 10 subcortical brain structures in a population of 70 subjects. It is shown that the choices of type of global alignment and of intrinsic versus extrinsic shape features, the latter being sensitive to relative pose, are crucial factors for group discrimination and also for explaining the nature of shape change in terms of the application domain.

Shape-correlated Deformation Statistics for Respiratory Motion Prediction in 4D Lung.

4D image-guided radiation therapy (IGRT) for free-breathing lungs is challenging due to the complicated respiratory dynamics. Effective modeling of respiratory motion is crucial to account for the motion affects on the dose to tumors. We propose a shape-correlated statistical model on dense image deformations for patient-specic respiratory motion estimation in 4D lung IGRT. Using the shape deformations of the high-contrast lungs as the surrogate, the statistical model trained from the planning CTs can be used to predict the image deformation during delivery verication time, with the assumption that the respiratory motion at both times are similar for the same patient. Dense image deformation fields obtained by diffeomorphic image registrations characterize the respiratory motion within one breathing cycle. A point-based particle optimization algorithm is used to obtain the shape models of lungs with group-wise surface correspondences. Canonical correlation analysis (CCA) is adopted in training to maximize the linear correlation between the shape variations of the lungs and the corresponding dense image deformations. Both intra- and inter-session CT studies are carried out on a small group of lung cancer patients and evaluated in terms of the tumor location accuracies. The results suggest potential applications using the proposed method.

A SHAPE-NAVIGATED IMAGE DEFORMATION MODEL FOR 4D LUNG RESPIRATORY MOTION ESTIMATION.

Intensity modulated radiation therapy (IMRT) for cancers in the lung remains challenging due to the complicated respiratory dynamics. We propose a shape-navigated dense image deformation model to estimate the patient-specific breathing motion using 4D respiratory correlated CT (RCCT) images. The idea is to use the shape change of the lungs, the major motion feature in the thorax image, as a surrogate to predict the corresponding dense image deformation from training.To build the statistical model, dense diffeomorphic deformations between images of all other time points to the image at end expiration are calculated, and the shapes of the lungs are automatically extracted. By correlating the shape variation with the temporally corresponding image deformation variation, a linear mapping function that maps a shape change to its corresponding image deformation is calculated from the training sample. Finally, given an extracted shape from the image at an arbitrary time point, its dense image deformation can be predicted from the pre-computed statistics.The method is carried out on two patients and evaluated in terms of the tumor and lung estimation accuracies. The result shows robustness of the model and suggests its potential for 4D lung radiation treatment planning.