Papers by Mohsen Taheri Shalmani
A key challenge for object representations is defining shape spaces that contain only geometrical... more 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. Supplementary materials for this article are available online.
The Mean Shape under the Relative Curvature Condition
arXiv (Cornell University), Apr 1, 2024
Guaranteeing that Fréchet means of object populations do not locally self-intersect or are thereb... more Guaranteeing that Fréchet means of object populations do not locally self-intersect or are thereby affected is a serious challenge for object representations because the objects' shape space typically includes elements corresponding to geometrically invalid objects. We show how to produce a shape space guaranteeing no local selfintersections 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 of 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.
Journal of Computational and Graphical Statistics, 2024
Guaranteeing that Fréchet means of object populations do not locally self-intersect or are thereb... more Guaranteeing that Fréchet means of object populations do not locally self-intersect or are thereby affected is a serious challenge for object representations because the objects' shape space typically includes elements corresponding to geometrically invalid objects. We show how to produce a shape space guaranteeing no local selfintersections 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 of 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.
Research Square (Research Square), May 18, 2023
Statistical analysis of the skeletal structure of slabular objects like groups of hippocampi is v... more Statistical analysis of the skeletal structure of slabular objects like groups of hippocampi is valuable for medical researchers as it can be useful for diagnoses and understanding diseases. This work proposes a novel object representation based on model fitting and analysis of the locally parameterized discrete swept skeletal representation of such entities where the model fitting procedure is based on boundary division and surface flattening. The goodness of the model fitting is demonstrated according to the skeletal symmetry, the volume of the implied boundary, and skeletal perturbation. The power of the method is demonstrated by visual inspection and statistical analysis of a synthetic and an actual data set in comparison with an available skeletal representation.
Carolina Digital Repository (University of North Carolina at Chapel Hill), 2022
Objects and object complexes in 3D, as well as those in 2D, have many possible representations. A... more 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.
arXiv (Cornell University), Aug 18, 2021
The alignment of shapes has been a crucial step in statistical shape analysis, for example, in ca... more The alignment of shapes has been a crucial step in statistical shape analysis, for example, in calculating mean shape, detecting locational differences between two shape populations, and classification. Procrustes alignment is the most commonly used method and state of the art. In this work, we uncover that alignment might seriously affect the statistical analysis. For example, alignment can induce false shape differences and lead to misleading results and interpretations. We propose a novel hierarchical shape parameterization based on local coordinate systems. The local parameterized shapes are translation and rotation invariant. Thus, the inherent alignment problems from the commonly used global coordinate system for shape representation can be avoided using this parameterization. The new parameterization is also superior for shape deformation and simulation. The method's power is demonstrated on the hypothesis testing of simulated data as well as the left hippocampi of patients with Parkinson's disease and controls.
Frontiers in Computer Science
Objects and object complexes in 3D, as well as those in 2D, have many possible representations. A... more 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 interest...
arXiv (Cornell University), Sep 6, 2021
This paper considers joint analysis of multiple functionally related structures in classification... more This paper considers joint analysis of multiple functionally related structures in classification tasks. In particular, our method developed is driven by how functionally correlated brain structures vary together between autism and control groups. To do so, we devised a method based on a novel combination of (1) non-Euclidean statistics that can faithfully represent non-Euclidean data in Euclidean spaces and (2) a non-parametric integrative analysis method that can decompose multi-block Euclidean data into joint, individual, and residual structures. We find that the resulting joint structure is effective, robust, and interpretable in recognizing the underlying patterns of the joint variation of multiblock non-Euclidean data. We verified the method in classifying the structural shape data collected from cases that developed and did not develop into Autistic Spectrum Disorder (ASD).
The purpose of this work is to study structural differences of the left hippocampus between patie... more The purpose of this work is to study structural differences of the left hippocampus between patients with Parkinson’s disease (PD) and healthy control group (CG) based on shape models like skeletal representation (s-rep) and spherical harmonics point distribution model (SPHARMPDM). 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 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 from the alignment, we propose a method according to distance matrices. We explain possible approaches to define 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 rotating tangen...
Thesis Chapters by Mohsen Taheri Shalmani
University of Stavanger, 2024
Statistical shape analysis has emerged as a crucial tool for medical researchers and clinicians t... more 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.
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Papers by Mohsen Taheri Shalmani
Thesis Chapters by Mohsen Taheri Shalmani