arXiv subjects

Jörn Schulz

Publications and source records attributed to Jörn Schulz.

5 recordsLinked to original sources

arXiv2026

Model Fitting and Analysis of the Discrete Swept Skeletal Representation for Ellipsoidal Objects

Statistical shape analysis of quasi-ellipsoidal objects, such as groups of hippocampi, is crucial for advancing medical research by aiding in the diagnosis and understanding of various diseases. This work presents a novel object representation method, termed the Locally Parameterized Discrete Swept Skeletal Representation (LPDSSRep). We discuss the model fitting and analysis techniques for this representation, which rely on boundary division and surface flattening. The quality of the model fitting is assessed based on the symmetry and tidiness of the skeletal structure and the volume of the implied boundary. 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.

stat.ME↗

arXiv2024

The Mean Shape under the Relative Curvature Condition

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 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 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.

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arXiv2021

Non-Euclidean Analysis of Joint Variations in Multi-Object Shapes

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 multi-block 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).

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arXiv2021

Statistical analysis of locally parameterized shapes

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.

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arXiv2017

Small sphere distributions for directional data with application to medical imaging

We propose new small-sphere distributional families for modeling multivariate directional data on $(\mathbb{S}^{p-1})^K$ for $p \ge 3$ and $K \ge 1$. In a special case of univariate directions in $\Re^3$, the new densities model random directions on $\mathbb{S}^2$ with a tendency to vary along a small circle on the sphere, and with a unique mode on the small circle. The proposed multivariate densities enable us to model association among multivariate directions, and are useful in medical imaging, where multivariate directions are used to represent shape and shape changes of 3-dimensional objects. When the underlying objects are rotationally deformed under noise, for instance, twisted and/or bend, corresponding directions tend to follow the proposed small-sphere distributions. The proposed models have several advantages over other methods analyzing small-circle-concentrated data, including inference procedures on the association and small-circle fitting. We demonstrate the use of the proposed multivariate small-sphere distributions in analyses of skeletally-represented object shapes and human knee gait data.

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