Shape Analysis Framework Research Articles

ShapeAXI: Shape Analysis Explainability and Interpretability.

ShapeAXI represents a cutting-edge framework for shape analysis that leverages a multi-view approach, capturing 3D objects from diverse viewpoints and subsequently analyzing them via 2D Convolutional Neural Networks (CNNs). We implement an automatic N-fold cross-validation process and aggregate the results across all folds. This ensures insightful explainability heat-maps for each class across every shape, enhancing interpretability and contributing to a more nuanced understanding of the underlying phenomena. We demonstrate the versatility of ShapeAXI through two targeted classification experiments. The first experiment categorizes condyles into healthy and degenerative states. The second, more intricate experiment, engages with shapes extracted from CBCT scans of cleft patients, efficiently classifying them into four severity classes. This innovative application not only aligns with existing medical research but also opens new avenues for specialized cleft patient analysis, holding considerable promise for both scientific exploration and clinical practice. The rich insights derived from ShapeAXI's explainability images reinforce existing knowledge and provide a platform for fresh discovery in the fields of condyle assessment and cleft patient severity classification. As a versatile and interpretative tool, ShapeAXI sets a new benchmark in 3D object interpretation and classification, and its groundbreaking approach hopes to make significant contributions to research and practical applications across various domains. ShapeAXI is available in our GitHub repository https://github.com/DCBIA-OrthoLab/ShapeAXI.

Longitudinal Elastic Shape Analysis of Brain Subcortical Structures

AbstractBackgroundOver the past 30 years, MRI has become a ubiquitous tool for accurately visualizing the development of the brain’s subcortical structures. However, the quantification of subcortical structures is still in its infancy due to challenges in shape extraction, representation, and modeling.MethodA simple and efficient framework of longitudinal elastic shape analysis (LESA) is developed for subcortical structures. By employing the elastic shape analysis of static surfaces using the square root normal field (SRNF) representations, we computed shape summaries of surfaces and represented the complex subcortical structures using a small number of basis functions. Further, we statistically modeled the sparse longitudinal data to predict the spatial‐temporal shape changes of subcortical structures. We also conducted the temporal registration using the square‐root velocity function (SRVF) framework to quantify the accelerated aging of Alzheimer’s disease (AD).ResultWe applied LESA to analyze longitudinal neuroimaging data sets and showcased its wide applications in estimating continuous shape trajectories, building life‐span growth patterns, and comparing shape differences among different groups. Using the Alzheimer’s Disease Neuroimaging Initiative (ADNI) data, we found that from 60 to 90 years old, the AD group always had the largest left ventricle (LV) surface area and the smallest left hippocampus (LHC) surface area, followed by the MCI and NC groups. The enlargement of LV and the atrophy of LHC had different speeds among different groups. Most of the shape‐changing differences between the AD and NC groups were visible in the anterior and posterior ends of LV. The atrophy of LHC happened mainly at the posterior end with aging. The AD group had the sharpest posterior end, and the NC group deformed the least with aging. Compared with the NC group, the AD group showed an average accelerated enlargement of 2.91 years and accelerated shape change of 2.76 years in LV; an average accelerated atrophy of 2.56 years and accelerated shape change of 2.59 years in LHC. By constructing the life‐span growth trajectories, we observed increases in shape changes’ speed in both LV and LHC after 60 years old.ConclusionAD can significantly speed up the shape change of ventricle and hippocampus compared with normal aging.

Automated characterization and monitoring of material shape using Riemannian geometry

Shape affects both the physical and chemical properties of a material. Characterizing the roughness, convexity, and general geometry of a material can yield information on its catalytic efficiency, solubility, elasticity, porosity, and overall effectiveness in the application of interest. However, material shape can be defined in a multitude of conflicting ways where different aspects of a material’s geometry are emphasized over others, leading to bespoke measures of shape that are not easily generalizable. In this paper, we explore the use of Riemannian geometry in the analysis of shape, and show that a Riemannian geometric framework for shape analysis is generalizable, computationally scalable, and can be directly integrated into common data analysis methods. In this framework, material shapes are abstracted as points on a Riemannian manifold. This information can be used to construct statistical moments (e.g., means, variances) and perform tasks such as dimensionality reduction and statistical process control. We provide a practical introduction to the mathematics of shape analysis through Riemannian geometry and illustrate its application on a manufactured/mined granular material dataset provided by Covia Corp. We show that the Riemannian framework can be used to automatically extract and quantify the shape of granular materials in a statistically rigorous manner.

Elastic shape analysis of brain structures for predictive modeling of PTSD.

It is well-known that morphological features in the brain undergo changes due to traumatic events and associated disorders such as post-traumatic stress disorder (PTSD). However, existing approaches typically offer group-level comparisons, and there are limited predictive approaches for modeling behavioral outcomes based on brain shape features that can account for heterogeneity in PTSD, which is of paramount interest. We propose a comprehensive shape analysis framework representing brain sub-structures, such as the hippocampus, amygdala, and putamen, as parameterized surfaces and quantifying their shape differences using an elastic shape metric. Under this metric, we compute shape summaries (mean, covariance, PCA) of brain sub-structures and represent individual brain shapes by their principal scores under a shape-PCA basis. These representations are rich enough to allow visualizations of full 3D structures and help understand localized changes. In order to validate the elastic shape analysis, we use the principal components (PCs) to reconstruct the brain structures and perform further evaluation by performing a regression analysis to model PTSD and trauma severity using the brain shapes represented via PCs and in conjunction with auxiliary exposure variables. We apply our method to data from the Grady Trauma Project (GTP), where the goal is to predict clinical measures of PTSD. The framework seamlessly integrates accurate morphological features and other clinical covariates to yield superior predictive performance when modeling PTSD outcomes. Compared to vertex-wise analysis and other widely applied shape analysis methods, the elastic shape analysis approach results in considerably higher reconstruction accuracy for the brain shape and reveals significantly greater predictive power. It also helps identify local deformations in brain shapes associated with PTSD severity.

LESA: Longitudinal Elastic Shape Analysis of Brain Subcortical Structures

Over the past 30 years, magnetic resonance imaging has become a ubiquitous tool for accurately visualizing the change and development of the brain’s subcortical structures (e.g., hippocampus). Although subcortical structures act as information hubs of the nervous system, their quantification is still in its infancy due to many challenges in shape extraction, representation, and modeling. Here, we develop a simple and efficient framework of longitudinal elastic shape analysis (LESA) for subcortical structures. Integrating ideas from elastic shape analysis of static surfaces and statistical modeling of sparse longitudinal data, LESA provides a set of tools for systematically quantifying changes of longitudinal subcortical surface shapes from raw structure MRI data. The key novelties of LESA include: (i) it can efficiently represent complex subcortical structures using a small number of basis functions and (ii) it can accurately delineate the spatiotemporal shape changes of the human subcortical structures. We applied LESA to analyze three longitudinal neuroimaging datasets and showcase its wide applications in estimating continuous shape trajectories, building life-span growth patterns, and comparing shape differences among different groups. In particular, with the Alzheimer’s Disease Neuroimaging Initiative (ADNI) data, we found that Alzheimer’s Disease (AD) can significantly speed the shape change of the lateral ventricle and the hippocampus from 60 to 75 years olds compared with normal aging. Supplementary materials for this article are available online.

Feasibility of a longitudinal statistical atlas model to study aortic growth in congenital heart disease

Studying anatomical shape progression over time is of utmost importance to refine our understanding of clinically relevant processes. These include vascular remodeling, such as aortic dilation, which is particularly important in some congenital heart defects (CHD). A novel methodological framework for three-dimensional shape analysis has been applied for the first time in a CHD scenario, i.e., bicuspid aortic valve (BAV) disease, the most common CHD. Three-dimensional aortic shapes (n = 94) reconstructed from cardiovascular magnetic resonance imaging (MRI) data as surface meshes represented the input for a longitudinal atlas model, using multiple scans over time (n = 2–4 per patient). This model relies on diffeomorphism transformations in the absence of point-to-point correspondence, and on the right combination of initialization, estimation and registration parameters. We computed the shape trajectory of an average disease progression in our cohort, as well as time-dependent parameters, geometric variations and the average shape of the population. Results cover a spatiotemporal spectrum of visual and numerical information that can be further used to run clinical associations. This proof-of-concept study demonstrates the feasibility of applying advanced statistical shape models to track disease progression and stratify patients with CHD.

Abstract P377: Collagen Remodeling Of Spontaneously Hypertensive Rats Undergoing Quinapril Treatment Measured By Three Dimensional Shape Analysis

Introduction: Myocardial collagen structure is a complex three dimensional (3D) hierarchy, organizing cardiomyocytes and providing structural integrity. Changes in collagen organization have a crucial role in cardiac remodeling. However, quantitative methods for the analysis of collagen structures are still lacking. Herein, we apply our collagen shape analysis framework to compare within-sheetlet collagen (both endomysial collagen surrounding myocytes and perimysial cords) against sheetlet-surface collagen (perimysial meshwork collagen deposited between sheetlets) in spontaneously hypertensive rats undergoing quinapril treatment. Hypothesis: The different functional properties of endomysial and perimysial meshwork collagen are linked to their structure and will have markedly different features quantifiable by 3D shape analysis. Methods: Rat myocardium was stained for collagen (picrosirius red), and the left ventricular free wall imaged using extended volume confocal microscopy (0.4 μm 3 voxel size). The 3D image volumes were segmented into within-sheetlet and sheetlet-surface regions. Shape analysis was performed using eigenanalysis of the inertia matrix to derive 3D morphological (elongation and flatness) parameters, characterizing the collagen shape in the two regions. Results: Shape analysis revealed marked differences between within-sheetlet and sheetlet-surface collagen organization. Long-term quinapril treatment resulted in notable differences in within-sheetlet collagen shape in rats at 14 months of age (mo) compared with 24 mo. At 14 mo, the within-sheetlet collagen is a mix of sheet-, ribbon-, and rod-like structures (elongation: 0.43, flatness: 0.05), but by 24 mo this has remodeled to predominately thin rod-like structures (elongation: 0.03, flatness: 0.35). The sheetlet-surface collagen consisted of thick sheet-like structures in both 14 and 24 mo rats (elongation: 0.83 vs 0.79, flatness: 0.13 vs 0.09). Conclusion: Shape analysis in 3D showed marked differences between within-sheetlet and sheetlet-surface collagen structure, as well as differences between early treatment and late treatment of diseased hearts.

IsoExplorer: an isosurface-driven framework for 3D shape analysis of biomedical volume data.

The high-resolution scanning devices developed in recent decades provide biomedical volume datasets that support the study of molecular structure and drug design. Isosurface analysis is an important tool in these studies, and the key is to construct suitable description vectors to support subsequent tasks, such as classification and retrieval. Traditional methods based on handcrafted features are insufficient for dealing with complex structures, while deep learning-based approaches have high memory and computation costs when dealing directly with volume data. To address these problems, we propose IsoExplorer, an isosurface-driven framework for 3D shape analysis of biomedical volume data. We first extract isosurfaces from volume data and split them into individual 3D shapes according to their connectivity. Then, we utilize octree-based convolution to design a variational autoencoder model that learns the latent representations of the shape. Finally, these latent representations are used for low-dimensional isosurface representation and shape retrieval. We demonstrate the effectiveness and usefulness of IsoExplorer via isosurface similarity analysis, shape retrieval of real-world data, and comparison with existing methods.Graphic abstract

A network-based framework for shape analysis enables accurate characterization of leaf epidermal cells

Cell shape is crucial for the function and development of organisms. Yet, versatile frameworks for cell shape quantification, comparison, and classification remain underdeveloped. Here, we introduce a visibility graph representation of shapes that facilitates network-driven characterization and analyses across shapes encountered in different domains. Using the example of complex shape of leaf pavement cells, we show that our framework accurately quantifies cell protrusions and invaginations and provides additional functionality in comparison to the contending approaches. We further show that structural properties of the visibility graphs can be used to quantify pavement cell shape complexity and allow for classification of plants into their respective phylogenetic clades. Therefore, the visibility graphs provide a robust and unique framework to accurately quantify and classify the shape of different objects.

Tail-Heaviness, Asymmetry, and Profitability Forecasting by Quantile Regression

We show that quantile regression is better than ordinary-least-squares (OLS) regression in forecasting profitability for a range of profitability measures following the conventional setup of the accounting literature, including the mean absolute forecast error (MAFE) evaluation criterion. Moreover, we perform both a simulated-data and an archival-data analysis to examine how the forecasting performance of quantile regression against OLS changes with the shape of the profitability distribution. Considering the MAFE and mean squared forecast error (MSFE) criteria together, we see that the quantile regression is more accurate relative to OLS when the profitability to be forecast has a heavier-tailed distribution. In addition, the asymmetry of the profitability distribution has either a U-shape or an inverted-U-shape effect on the forecasting accuracy of quantile regression. An application of the distributional shape analysis framework to cash flow forecasting demonstrates the usefulness of the framework beyond profitability forecasting, providing additional empirical evidence on the positive effect of tail-heaviness and supporting the notion of an inverted-U-shape effect of asymmetry. This paper was accepted by Shiva Rajgopal, accounting.

Shape analysis via inconsistent surface registration.

In this work, we develop a framework for shape analysis using inconsistent surface mapping. Traditional landmark-based geometric morphometr- ics methods suffer from the limited degrees of freedom, while most of the more advanced non-rigid surface mapping methods rely on a strong assumption of the global consistency of two surfaces. From a practical point of view, given two anatomical surfaces with prominent feature landmarks, it is more desirable to have a method that automatically detects the most relevant parts of the two surfaces and finds the optimal landmark-matching alignment between these parts, without assuming any global 1-1 correspondence between the two surfaces. Our method is capable of solving this problem using inconsistent surface registration based on quasi-conformal theory. It further enables us to quantify the dissimilarity of two shapes using quasi-conformal distortion and differences in mean and Gaussian curvatures, thereby providing a natural way for shape classification. Experiments on Platyrrhine molars demonstrate the effectiveness of our method and shed light on the interplay between function and shape in nature.

Handling confounding variables in statistical shape analysis - application to cardiac remodelling.

Statistical shape analysis is a powerful tool to assess organ morphologies and find shape changes associated to a particular disease. However, imbalance in confounding factors, such as demographics might invalidate the analysis if not taken into consideration. Despite the methodological advances in the field, providing new methods that are able to capture complex and regional shape differences, the relationship between non-imaging information and shape variability has been overlooked. We present a linear statistical shape analysis framework that finds shape differences unassociated to a controlled set of confounding variables. It includes two confounding correction methods: confounding deflation and adjustment. We applied our framework to a cardiac magnetic resonance imaging dataset, consisting of the cardiac ventricles of 89 triathletes and 77 controls, to identify cardiac remodelling due to the practice of endurance exercise. To test robustness to confounders, subsets of this dataset were generated by randomly removing controls with low body mass index, thus introducing imbalance. The analysis of the whole dataset indicates an increase of ventricular volumes and myocardial mass in athletes, which is consistent with the clinical literature. However, when confounders are not taken into consideration no increase of myocardial mass is found. Using the downsampled datasets, we find that confounder adjustment methods are needed to find the real remodelling patterns in imbalanced datasets.

3D Shape Modeling and Analysis of Retinal Microvasculature in OCT-Angiography Images.

3D optical coherence tomography angiography (OCT-A) is a novel and non-invasive imaging modality for analyzing retinal diseases. The studies of microvasculature in 2D en face projection images have been widely implemented, but comprehensive 3D analysis of OCT-A images with rich depth-resolved microvascular information is rarely considered. In this paper, we propose a robust, effective, and automatic 3D shape modeling framework to provide a high-quality 3D vessel representation and to preserve valuable 3D geometric and topological information for vessel analysis. Effective vessel enhancement and extraction steps by means of curvelet denoising and optimally oriented flux (OOF) filtering are first designed to produce 3D microvascular networks. Afterwards, a novel 3D data representation of OCT-A microvasculature is reconstructed via advanced mesh reconstruction techniques. Based on the 3D surfaces, shape analysis is established to extract novel shape-based microvascular area distortion via the Laplace-Beltrami eigen-projection. The extracted feature is integrated into a graph-cut segmentation system to categorize large vessels and small capillaries for more precise shape analysis. The proposed framework is validated on a dedicated repeated scan dataset including 260 volume images and shows high repeatability. Statistical analysis using the surface area biomarker is performed on small capillaries to avoid the effect of tailing artifact from large vessels. It shows significant differences ( ) between DR stages on 100 subjects in a OCTA-DR dataset. The proposed shape modeling and analysis framework opens the possibility for further investigating OCT-A microvasculature in a new perspective.

Shape Analysis of Framed Space Curves

In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed planar shapes, endowed with a natural metric, is isometric to an infinite-dimensional Grassmann manifold via the so-called square root transform. This result facilitates efficient shape comparison by virtue of explicit descriptions of Grassmannian geodesics. In this paper, we extend this shape analysis framework to treat shapes of framed space curves. By considering framed curves, we are able to generalize the square root transform by using quaternionic arithmetic and properties of the Hopf fibration. Under our coordinate transformation, the space of closed framed curves corresponds to an infinite-dimensional complex Grassmannian. This allows us to describe geodesics in framed curve space explicitly. We are also able to produce explicit geodesics between closed, unframed space curves by studying the action of the loop group of the circle on the Grassmann manifold. We apply our results to compute means for collections of space curves and to perform statistical analysis of circular DNA molecule shapes.

Visualization and Outlier Detection for Multivariate Elastic Curve Data.

We propose a new method for the construction and visualization of geometrically-motivated boxplot displays for elastic curve data. We use a recent shape analysis framework, based on the square-root velocity function representation of curves, to extract different sources of variability from elastic curves, which include location, scale, shape, orientation and parametrization. We then focus on constructing separate displays for these various components using the Riemannian geometry of their representation spaces. This involves computation of a median, two quartiles, and two extremes based on geometric considerations. The outlyingness of an elastic curve is also defined separately based on each of the five components. We evaluate the proposed methods using multiple simulations, and then focus our attention on real data applications. In particular, we study variability in (a) 3D spirals, (b) handwritten signatures, (c) 3D fibers from diffusion tensor magnetic resonance imaging, and (d) trajectories of the Lorenz system.

A particle shape extraction and evaluation method using a deep convolutional neural network and digital image processing

Acquiring particle shapes from raw particle images with complex backgrounds is a prerequisite for particle shape evaluation, yet it is a challenging task. In this study, a systematic framework for particle extraction and shape analysis is developed using a deep convolutional neural network (lightweight U-net) and digital image processing. First, raw images of particles are cropped and labeled manually to train the neural network. Then, the well-trained network is employed to extract particle projections from images of arbitrary size with complex backgrounds. Next, the particle boundaries are separated and smoothed using the improved erosion and flood filling method and the B-spline curve technique. Finally, the shapes of the extracted particles are evaluated and compared with the shape data obtained from the manually extracted particles. The shape distributions from these two approaches are found to be well correlated, illustrating the reliability and capability of the proposed algorithms.

Elastic Statistical Shape Analysis of Biological Structures with Case Studies: A Tutorial.

We describe a recent framework for statistical shape analysis of curves and show its applicability to various biological datasets. The presented methods are based on a functional representation of shape called the square-root velocity function and a closely related elastic metric. The main benefit of this approach is its invariance to reparameterization (in addition to the standard shape-preserving transformations of translation, rotation and scale), and ability to compute optimal registrations (point correspondences) across objects. Building upon the defined distance between shapes, we additionally describe tools for computing sample statistics including the mean and covariance. Based on the covariance structure, one can also explore variability in shape samples via principal component analysis. Finally, the estimated mean and covariance can be used to define Wrapped Gaussian models on the shape space, which are easy to sample from. We present multiple case studies on various biological datasets including (1) leaf outlines, (2) internal carotid arteries, (3) Diffusion Tensor Magnetic Resonance Imaging fiber tracts, (4) Glioblastoma Multiforme tumors, and (5) vertebrae in mice. We additionally provide a MATLAB package that can be used to produce the results given in this manuscript.

Development of a pulse reconstruction routine for Transistor-Reset Preamplifiers for (n,xn) measurements using HPGe detectors at the n_TOF facility at CERN

The knowledge of neutron induced cross-sections is essential both for applications as well as for the refinement of nuclear model parameters. Amongst other reactions, (n,xn) are equally important, however the determination of such cross-sections could be limited by current experimental techniques. In this respect, feasibility studies have been performed at the n_TOF facility at CERN concerning the study of (n,xn) reactions by detecting the prompt γ radiation emitted from the residual nuclei an experimental technique which required the use of a Transistor Reset Preamplifier (TRP) set-up. The digital acquisition system at n_TOF is accompanied by a pulse shape analysis framework which lacked the capability of dealing with TRP, therefore a new reconstruction routine had to be developed. A brief description and the first preliminary results are presented.

Conformal initialization for shape analysis applications in SlicerSALT.

Shape analysis is an important method used in neuroimaging research community due to its potential to precisely locate morphological changes between healthy and pathological structures. A popular shape analysis framework in the neuroimaging community is based on the encoding surface locations as spherical harmonics for a representation called SPHARM-PDM.1 The SPHARM-PDM pipeline takes a set of brain segmentation of a single brain structure (for example, hippocampus) as input and converts them into a corresponding spherical harmonic description (SPHARM), which is then sampled into triangulated surface (SPHARM-PDM). At present, the SPHARM-PDM pipeline utilizes an area-preserving optimization of the spherical mapping based on an initial heat-equation based mapping of the surface mesh to the unit sphere. In the case of objects with complex shape, this initial spherical mapping suffers from a high degree of mapping distortion that cannot always be corrected by the following optimization procedure. Here we proposed the use of an alternative initialization based on a conformal flattening.2 This method adopts a bijective angle preserving conformal flattening scheme to replace the heat equation mapping scheme as initialization for use in the SPHARM-PDM pipeline. After quantitative measures of shape calculated from various complex structures, we concluded that in most cases, the new pipeline produced dramatically better results than the old pipeline. The main contribution of this paper is a command line tool based on the Slicer Execution Model, which merges the conformal flattening into the SPHARM-PDM pipeline for use in the SALT shape analysis toolbox.

Adaptive O-CNN

We present an Adaptive Octree-based Convolutional Neural Network (Adaptive O-CNN) for efficient 3D shape encoding and decoding. Different from volumetric-based or octree-based CNN methods that represent a 3D shape with voxels in the same resolution, our method represents a 3D shape adaptively with octants at different levels and models the 3D shape within each octant with a planar patch. Based on this adaptive patch-based representation, we propose an Adaptive O-CNN encoder and decoder for encoding and decoding 3D shapes. The Adaptive O-CNN encoder takes the planar patch normal and displacement as input and performs 3D convolutions only at the octants at each level, while the Adaptive O-CNN decoder infers the shape occupancy and subdivision status of octants at each level and estimates the best plane normal and displacement for each leaf octant. As a general framework for 3D shape analysis and generation, the Adaptive O-CNN not only reduces the memory and computational cost, but also offers better shape generation capability than the existing 3D-CNN approaches. We validate Adaptive O-CNN in terms of efficiency and effectiveness on different shape analysis and generation tasks, including shape classification, 3D autoencoding, shape prediction from a single image, and shape completion for noisy and incomplete point clouds.