• Corpus ID: 239016104

On the Statistical Analysis of Complex Tree-shaped 3D Objects

@article{Wang2021OnTS,
  title={On the Statistical Analysis of Complex Tree-shaped 3D Objects},
  author={Guan Wang and Hamid Laga and Anuj Srivastava},
  journal={ArXiv},
  year={2021},
  volume={abs/2110.08693}
}
How can one analyze detailed 3D biological objects, such as neurons and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and computing geodesic deformations between the shapes of such tree-like 3D objects. A hierarchical organization of subtrees characterizes these objects – each subtree has the main branch with some side branches attached – and one needs to match these structures… 

References

SHOWING 1-10 OF 39 REFERENCES
The Shape Space of 3D Botanical Tree Models
TLDR
An algorithm for generating novel 3D tree model variations from existing ones via geometric and structural blending and the application of the framework in reflection symmetry analysis and symmetrization of botanical trees is demonstrated.
Statistical Modeling of the 3D Geometry and Topology of Botanical Trees
TLDR
A framework for statistical modeling of the 3D geometry and topology of botanical trees is proposed and it is shown how to use this framework for computing statistical summaries, e.g. the mean and modes of variations, of a population ofBotanical trees.
Geometries on Spaces of Treelike Shapes
TLDR
It is shown that the new metric QED has nice geometric properties which facilitate statistical analysis, such as existence and local uniqueness of geodesics and averages, while TED, on the other hand, has algorithmic advantages, while it does not share the geometric strongpoints of QED.
Tree-Space Statistics and Approximations for Large-Scale Analysis of Anatomical Trees
TLDR
This paper takes advantage of a very large dataset (N=8016) to obtain computable approximations, under the assumption that the data trees parametrize the relevant parts of tree-space well and illustrates how the structure and geometry of airway trees vary across a population.
Elastic Shape Analysis of Three-Dimensional Objects
TLDR
A comprehensive framework for analyzing shapes of spherical objects, i.e., objects that are embeddings of a unit sphere in ℝ is developed, including tools for quantifying shape differences, optimally deforming shapes into each other, summarizing shape samples, extracting principal modes of shape variability, and modeling shape variability associated with populations.
Toward a Theory of Statistical Tree-Shape Analysis
TLDR
A shape space framework for tree-shapes and study metrics on the shape space, TED and QED is constructed and it is shown that the new metric QED has nice geometric properties that are needed for statistical analysis.
Shape Analysis of Elastic Curves in Euclidean Spaces
TLDR
This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric and demonstrates a wrapped probability distribution for capturing shapes of planar closed curves.
Landmark-free statistical analysis of the shape of plant leaves.
TLDR
A statistical model based on the Squared Root Velocity Function (SRVF) representation and the Riemannian elastic metric of Srivastava et al. (2011) to model the observed continuous variability in the shape of plant leaves is proposed.
Landmark‐Guided Elastic Shape Analysis of Spherically‐Parameterized Surfaces
TLDR
The core contribution is the re‐formulation of Kurtek et al.'s approach as a constrained optimization over all possible re‐parameterizations of the surfaces, using a sparse set of corresponding landmarks, to establish full surface registration and geodesic deformation between two surfaces.
Elastic Shape Matching of Parameterized Surfaces Using Square Root Normal Fields
TLDR
This paper defines a new methodology for shape analysis of parameterized surfaces, and introduces a novel representation of surfaces termed square root normal fields or SRNFs, which results in more natural shape matchings and has some theoretical advantages over previous methods.
...
1
2
3
4
...