Inference on 3D Procrustes Means: Tree Bole Growth, Rank Deficient Diffusion Tensors and Perturbation Models

Huckemann, Stephan

doi:10.1111/j.1467-9469.2010.00724.x

Cited by 36 publications

(38 citation statements)

References 31 publications

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“…This approach is often called a perturbation model (Goodall, 1991). It is wellknown that a perturbation model introduces a bias in the estimation of the geodesic mean unless the distribution is isotropic (Kent and Mardia, 1997;Le, 1998;Huckemann, 2011a). In this paper in Section 4, we use the von Mises-Fisher distribution and in Section 5 the perturbation model.…”

Section: Rigid Rotation Modelmentioning

confidence: 99%

Analysis of Rotational Deformations From Directional Data

Schulz¹,

Jung²,

Huckemann³

et al. 2015

Journal of Computational and Graphical Statistics

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This paper discusses a novel framework to analyze rotational deformations of real 3D objects. The rotational deformations such as twisting or bending have been observed as the major variation in some medical applications, where the features of the deformed 3D objects are directional data. We propose modeling and estimation of the global deformations in terms of generalized rotations of directions. The proposed method can be cast as a generalized small circle fitting on the unit sphere. We also discuss the estimation of descriptors for more complex deformations composed of two simple deformations. The proposed method can be used for a number of different 3D object models. Two analyses of 3D object data are presented in detail: one using skeletal representations in medical image analysis as well as one from biomechanical gait analysis of the knee joint. Supplementary Materials are available online.

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Section: Rigid Rotation Modelmentioning

confidence: 99%

Analysis of Rotational Deformations From Directional Data

Schulz¹,

Jung²,

Huckemann³

et al. 2015

Journal of Computational and Graphical Statistics

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“…Intuition from the Euclidean setting suggests that if the points are randomly sampled from a well-behaved probability distribution on a space M of dimension d + 1, then the random variable that is the barycenter ought not be confined to a particular subspace of dimension d or less, if the distribution is generic. While this intuition has been made rigorous when M is a manifold [5,12,14,15], it can fail when M has certain types of singularities, as we demonstrate here for an open book O: a space obtained by gluing disjoint copies of a half-space along their boundary hyperplanes; see Section 1 for precise definitions.…”

mentioning

confidence: 99%

Sticky central limit theorems on open books

Hotz¹,

Huckemann²,

Le³

et al. 2013

Ann. Appl. Probab.

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Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fr\'{e}chet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension $1$ and hence measure $0$) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine. We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky).Comment: Published in at http://dx.doi.org/10.1214/12-AAP899 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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“…Table 1 are based on the distributions of suitable estimates As our growth model can be viewed as a perturbation (or generative) model, it is not clear that estimation via a Procrustes-type algorithm will recover the modelled 'true' anisotropic growth, at least asymptotically. In fact, it is well known from shape analysis that, under general error models, Procrustes means are usually inconsistent with centres of perturbation models; see Lele (1993), Le (1998), Kent and Mardia (1997), Huckemann (2011) and Devilliers et al (2017). Indeed, our simulations based on real distortions in Section 3.1 indicate towards minor inconsistency, namely that true anisotropic growth is slightly overestimated and this effect wanes with increased anisotropy.…”

Section: Introductionmentioning

confidence: 68%

Detecting Anisotropy in Fingerprint Growth

Markert

Krehl

Gottschlich

et al. 2019

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary From infancy to adulthood, human growth is anisotropic, much more along the proximal–distal axis (height) than along the medial–lateral axis (width), particularly at extremities. Detecting and modelling the rate of anisotropy in fingerprint growth facilitate the use of children's fingerprints for long‐term biometric identification. Using standard fingerprint scanners, anisotropic growth is highly overshadowed by the varying distortions created by each imprint, and it seems that this difficulty has hampered to date the development of suitable methods, detecting anisotropy, let alone designing models. We provide a tool chain to detect statistically anisotropy in planar shape and its preferred axis. For this we develop a new anisotropic growth model with a Procrustes‐type algorithm and a new parametric and non‐parametric neighbourhood hypothesis test, tunable to measurement accuracy. In application to fingerprint growth, we require only a standard fingerprint scanner and a minutiae matcher. Taking into account realistic distortions caused by pressing fingers on scanners, our simulations based on real data indicate that, for example, already in rather small samples (56 matches) we can significantly detect proximal–distal growth if it exceeds medial–lateral growth by only around 5%. Our method is well applicable to future data sets of child fingerprint time series. We provide an implementation of our algorithms and tests with matched minutiae pattern data.

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Inference on 3D Procrustes Means: Tree Bole Growth, Rank Deficient Diffusion Tensors and Perturbation Models

Cited by 36 publications

References 31 publications

Analysis of Rotational Deformations From Directional Data

Analysis of Rotational Deformations From Directional Data

Sticky central limit theorems on open books

Detecting Anisotropy in Fingerprint Growth

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