Link: http://dx.doi.org/10.13140/RG.2.2.14238.45126 

Abstract: The central curve of a tubular object, known as the spine, provides a compact and informative representation of the object’s geometry, making it highly valuable for shape analysis, deformation modeling, segmentation, and quantitative measurements. However, extracting the spine remains challenging due to the geometric complexity of real-world tubular structures. A possible solution is to leverage machine learning techniques to learn from well-fitted models and predict the spines of unseen samples. Yet, such approaches typically require large training datasets, which are often unavailable in practice. This work introduces a pipeline for generating synthetic training data using the Elliptical Tube Representation framework (ETRep), enabling the training of deep neural networks for spine prediction. The effectiveness of this approach is demonstrated through its application to the extraction of the spine of hippocampal structures.

Link: doi.org/10.1080/10618600.2025.2535600 

ETRep R package: doi.org/10.32614/CRAN.package.ETRep 

GitHub: github.com/MohsenTaheriShalmani/Elliptical_Tubes

Abstract: A key challenge for object representations is defining shape spaces that contain only geometrically valid objects, excluding those that are self-intersecting or otherwise invalid. Such shape spaces inherently ensure that Fréchet means of object populations do not locally self-intersect. We show how to produce a shape space guaranteeing no local self-intersections for specific but important cases where objects are represented by swept elliptical disks. This representation can model a variety of anatomic objects, such as the colon and hippocampus. Our approach for computing geodesic paths in this shape space enables detailed comparisons of structural variations between groups, such as patients and controls. The guarantee is met by constraining the shape space using the Relative Curvature Condition (RCC) of swept regions. This study introduces the Elliptical Tube Representation (ETRep) framework to provide a systematic approach to ensure valid mean shapes, effectively addressing the challenges of complex non-convex spaces while adhering to the RCC. The ETRep shape space incorporates an intrinsic distance metric defined based on the skeletal coordinate system of the shape space. The proposed methodology is applied to statistical shape analysis, facilitating the development of both global and partial hypothesis testing methods, which were employed to investigate hippocampal structures in early Parkinson’s disease.

Link: hdl.handle.net/11250/3133161 

Abstract: Statistical shape analysis has emerged as a crucial tool for medical researchers and clinicians to study medical objects such as brain subcortical structures. The insights gained from such analyses hold immense potential for diagnoses and enhancing our understanding of various diseases, particularly neurological disorders. This thesis explores three important areas of statistical shape analysis, which are detailed in three separate papers: “Statistical Analysis of Locally Parameterized Shapes,” “Fitting Discrete Swept Skeletal Structures to Slabular Objects,” and “The Mean Shape under the Relative Curvature Condition.” The innovative approaches discussed in these papers offer a fresh perspective for representing complex shapes, enabling more nuanced analysis and interpretation. Central to this work is the discussion surrounding the introduction of robust skeletal representations for establishing correspondences for a class of swept regions called slabular objects and providing proper mathematical methodologies supporting the statistical objectives, such as hypothesis testing and classification. The proposed skeletal models are alignment-independent and invariant to the act of Euclidean similarity transformations of translation, rotations, and scaling. Damon’s criterion of the relative curvature condition (RCC) is an essential factor for valid swept skeletal structures. This work extensively discusses fitting skeletal models, defining shape space, and calculating the mean shape for such models following the RCC. The efficacy of the proposed methodology is underscored through rigorous examinations, both visually and statistically. These methodologies are specifically applied to medical contexts, focusing on analyzing subcortical structures. Synthetic and actual datasets serve for validation, facilitating a comprehensive comparison with existing skeletal representations. This work highlights the resilience and adaptability of innovative approaches, paving the way for further medical research and diagnostic endeavors.

Link: doi.org/10.1080/10618600.2022.2116445 

Abstract: In statistical shape analysis, the establishment of correspondence and defining shape representation are crucial steps for hypothesis testing to detect and explain local dissimilarities between two groups of objects. Most commonly used shape representations are based on object properties that are either extrinsic or noninvariant to rigid transformation. Shape analysis based on noninvariant properties is biased because the act of alignment is necessary, and shape analysis based on extrinsic properties could be misleading. Besides, a mathematical explanation of the type of dissimilarity, for example, bending, twisting, stretching, etc., is desirable. This work proposes a novel hierarchical shape representation based on invariant and intrinsic properties to detect and explain locational dissimilarities by using local coordinate systems. The proposed shape representation is also superior for shape deformation and simulation. The power of the method is demonstrated on the hypothesis testing of simulated data as well as the left hippocampi of patients with Parkinson’s disease and controls. Supplementary materials for this article are available online.

Link: hdl.handle.net/11250/2680212 

Abstract: The purpose of this work is to study structural differences of the left hippocampus between patients with Parkinson’s disease (PD) and a healthy control group (CG) based on shape models like skeletal representation (s-rep) and spherical harmonics point distribution model (SPHARM-PDM). We apply a permutation test on the s-reps of CG and PD to detect significant differences between the means of their geometric object properties (GOPs). We also introduce a parametric test for s-rep, constructed on the multivariate Hotelling’s T2 test. We discuss different methods of alignment, their impact on the result, and propose the elimination algorithm to have an adequate alignment. To make the test independent of the alignment, we propose a method based on distance matrices. We explain possible approaches to define the mean and variation of directional data, including principal nested spheres (PNS), and principal geodesic analysis (PGA). Besides, we propose a non-linear PGA (NLPGA) on the rotating tangent space of the unit sphere. Finally, we discuss the results of the hypothesis tests and show there are statistically significant differences between PD and CG.