scholarly journals Smoothing splines on Riemannian manifolds, with applications to 3D shape space

2020 ◽  
Author(s):  
Kwang‐Rae Kim ◽  
Ian L. Dryden ◽  
Huiling Le ◽  
Katie E. Severn
Keyword(s):  
Riemannian Manifolds ◽  
Smoothing Splines ◽  
Shape Space ◽  
3D Shape

C1 interpolating Bézier path on Riemannian manifolds, with applications to 3D shape space

2019 ◽  
Vol 348 ◽  
pp. 371-384
Author(s):  
Chafik Samir ◽  
Ines Adouani
Keyword(s):  
Riemannian Manifolds ◽  
Shape Space ◽  
3D Shape

scholarly journals A Kernel Regression Procedure in the 3D Shape Space with an Application to Online Sales of Children’s Wear

2019 ◽  
Vol 34 (2) ◽  
pp. 236-252
Author(s):  
Gregorio Quintana-Ortí ◽  
Amelia Simó
Keyword(s):  
Kernel Regression ◽  
Shape Space ◽  
3D Shape ◽  
Regression Procedure ◽  
Online Sales

scholarly journals 4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data

2021 ◽  
Author(s):  
HAMID LAGA ◽  
Marcel Padilla ◽  
Ian H. Jermyn ◽  
Sebastian Kurtek ◽  
Mohammed Bennamoun ◽  
...  
Keyword(s):  
Statistical Analysis ◽  
Building Blocks ◽  
Shape Space ◽  
Spatiotemporal Variability ◽  
3D Shape ◽  
Execution Rates ◽  
And Performance ◽  
Body Shapes ◽  
Spatiotemporal Registration ◽  
Shape Data

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of subjects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered onto each others. Also, different deforming subjects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus, they need to be temporally aligned onto each others. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemmanian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories, or 1D curves, embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the L2 metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a Euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates, (2) the computation of geodesics between 4D surfaces, (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces, and (4) the synthesis of random 4D surfaces. We demonstrate the utility and performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

scholarly journals 4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data

2021 ◽  
Author(s):  
HAMID LAGA ◽  
Marcel Padilla ◽  
Ian H. Jermyn ◽  
Sebastian Kurtek ◽  
Mohammed Bennamoun ◽  
...  
Keyword(s):  
Statistical Analysis ◽  
Building Blocks ◽  
Shape Space ◽  
Spatiotemporal Variability ◽  
3D Shape ◽  
Execution Rates ◽  
And Performance ◽  
Body Shapes ◽  
Spatiotemporal Registration ◽  
Shape Data

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of subjects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered onto each others. Also, different deforming subjects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus, they need to be temporally aligned onto each others. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemmanian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories, or 1D curves, embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the L2 metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a Euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates, (2) the computation of geodesics between 4D surfaces, (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces, and (4) the synthesis of random 4D surfaces. We demonstrate the utility and performance of the proposed framework using 4D facial surfaces and 4D human body shapes.

scholarly journals Postnatal ontogeny of the femur in fossorial and semi‐aquatic water voles in the 3D shape space

2021 ◽  
Author(s):  
Ana Filipa Durão ◽  
Francesc Muñoz‐Muñoz ◽  
Jacint Ventura
Keyword(s):  
Shape Space ◽  
Postnatal Ontogeny ◽  
3D Shape

scholarly journals Spline Positional Encoding for Learning 3D Implicit Signed Distance Fields

2021 ◽  
Author(s):  
Peng-Shuai Wang ◽  
Yang Liu ◽  
Yu-Qi Yang ◽  
Xin Tong
Keyword(s):  
Dimensional Space ◽  
Point Clouds ◽  
Shape Space ◽  
Shape Reconstruction ◽  
Multilayer Perceptrons ◽  
3D Shape ◽  
Signed Distance ◽  
Distance Fields ◽  
3D Point Clouds ◽  
3D Shape Reconstruction

Multilayer perceptrons (MLPs) have been successfully used to represent 3D shapes implicitly and compactly, by mapping 3D coordinates to the corresponding signed distance values or occupancy values. In this paper, we propose a novel positional encoding scheme, called Spline Positional Encoding, to map the input coordinates to a high dimensional space before passing them to MLPs, which help recover 3D signed distance fields with fine-scale geometric details from unorganized 3D point clouds. We verified the superiority of our approach over other positional encoding schemes on tasks of 3D shape reconstruction and 3D shape space learning from input point clouds. The efficacy of our approach extended to image reconstruction is also demonstrated and evaluated.

Higher-order smoothing splines versus least squares problems on Riemannian manifolds

2010 ◽  
Vol 16 (1) ◽  
pp. 121-148 ◽  
Author(s):  
L. Machado ◽  
F. Silva Leite ◽  
K. Krakowski
Keyword(s):  
Least Squares ◽  
Riemannian Manifolds ◽  
Smoothing Splines ◽  
Higher Order ◽  
Least Squares Problems

scholarly journals Optimization Methods on Riemannian Manifolds and Their Application to Shape Space

2012 ◽  
Vol 22 (2) ◽  
pp. 596-627 ◽  
Author(s):  
Wolfgang Ring ◽  
Benedikt Wirth
Keyword(s):  
Riemannian Manifolds ◽  
Optimization Methods ◽  
Shape Space

Shape-space smoothing splines for planar landmark data

Biometrika ◽  
2007 ◽  
Vol 94 (3) ◽  
pp. 513-528 ◽  
Author(s):  
A. Kume ◽  
I. L. Dryden ◽  
H. Le
Keyword(s):  
Smoothing Splines ◽  
Shape Space ◽  
Landmark Data

Practical electron tomography

1995 ◽  
Vol 53 ◽  
pp. 742-743
Author(s):  
C.L. Woodcock
Keyword(s):  
Electron Tomography ◽  
Tomographic Reconstruction ◽  
Three Dimensional ◽  
3D Shape ◽  
Computational Resources ◽  
Shape Of Particles

Despite the potential of the technique, electron tomography has yet to be widely used by biologists. This is in part related to the rather daunting list of equipment and expertise that are required. Thanks to continuing advances in theory and instrumentation, tomography is now more feasible for the non-specialist. One barrier that has essentially disappeared is the expense of computational resources. In view of this progress, it is time to give more attention to practical issues that need to be considered when embarking on a tomographic project. The following recommendations and comments are derived from experience gained during two long-term collaborative projects.Tomographic reconstruction results in a three dimensional description of an individual EM specimen, most commonly a section, and is therefore applicable to problems in which ultrastructural details within the thickness of the specimen are obscured in single micrographs. Information that can be recovered using tomography includes the 3D shape of particles, and the arrangement and dispostion of overlapping fibrous and membranous structures.

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