A Small Circle of Best Fit for Spherical Data and Areas of Vulcanism

Mardia, Kanti V.; Gadsden, R. J.

doi:10.2307/2346963

Cited by 45 publications

(29 citation statements)

References 4 publications

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“…As will be pointed out later and may be guessed from Fig. 3c, the estimation of the axis of rotation based on directional vectors has a close relationship with the estimation of small circles on the unit sphere, which was studied in various contexts (Mardia and Gadsden, 1977;Rivest, 1999;Jung et al, 2011).…”

Section: S-repmentioning

confidence: 86%

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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: S-repmentioning

confidence: 86%

“…The problem (7) is precisely the fitting of concentric (small) circles. Therefore, numerical algorithms for (7) are generalized algorithms of the well-studied fitting of small circles (Mardia and Gadsden, 1977;Rivest, 1999;Jung et al, 2011Jung et al, , 2012 and are discussed in the Appendix.…”

Section: Estimationmentioning

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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“…That of θ 13 is proportional to exp(a 13 cos θ 13 + b 13 sin θ 13 ) cos θ 13 and we use a random walk Metropolis update for this variable, with a perturbation uniformly distributed on [−0.1, 0.1]. The latter distribution has been studied in Mardia & Gadsden (1977) but with no discussion on how to simulate from it.…”

Section: Priors and Markov Chain Monte Carlo Updating For A Rotation mentioning

confidence: 99%

Bayesian alignment using hierarchical models, with applications in protein bioinformatics

Green¹,

Mardia²

2006

Biometrika

153

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SUMMARYAn important problem in shape analysis is to match configurations of points in space filtering out some geometrical transformation. In this paper we introduce hierarchical models for such tasks, in which the points in the configurations are either unlabelled, or have at most a partial labelling constraining the matching, and in which some points may only appear in one of the configurations. We derive procedures for simultaneous inference about the matching and the transformation, using a Bayesian approach. Our model is based on a Poisson process for hidden true point locations; this leads to considerable mathematical simplification and efficiency of implementation. We find a novel use for classical distributions from directional statistics in a conditionally conjugate specification for the case where the geometrical transformation includes an unknown rotation. Throughout, we focus on the case of affine or rigid motion transformations. Under a broad parametric family of loss functions, an optimal Bayesian point estimate of the matching matrix can be constructed, that depends only on a single parameter of the family.Our methods are illustrated by two applications from bioinformatics. The first problem is of matching protein gels in two dimensions, and the second consists of aligning active sites of proteins in three dimensions. In the latter case, we also use information related to the grouping of the amino acids. We discuss some open problems and suggest directions for future work.

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“…They arise on many different situations on geological data such as planar attitudes measured on certain superimposed folding systems (Ramsay, 1962), conical folds at tappering ends 5 of cylindrical folding systems (Cruden and Charlesworth, 1972), paleomagnetic data directions restoration (Pueyo et al, 2003;Mochales et al, 2016), linear attitudes measured on folding systems that condition them to a constant angular displacement from the fold axis or segments of hotspot traces (Mardia and Gadsden, 1977) and transform faults. As noticed by Mulchrone et al (2013), 10 the most popular free available directional data analysis applications either do not offer the functionality of small circle fitting or provide methods which may struggle to estimate the best fit cone when dealing with complex distributions.…”

Section: Introductionmentioning

confidence: 99%

“…So far the axes for these small circles have been adjusted either by visual inspection using π-diagrams or by iterative numerical methods (Ramsay, 1962;Mardia 15 and Gadsden, 1977;Bingham and Mardia, 1978;Mulchrone et al, 2013). Until present, there is no reliable graphical method for estimating best fit small circles.…”

Section: Introductionmentioning

confidence: 99%

Methods and applications for geological directional data analysis

Correia¹

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Autorizo a reprodução e divulgação total ou parcial deste trabalho, por qualquer meio convencional ou eletrônico, para fins de estudo e pesquisa, desde que citada a fonte. As equipes da Seção de Pós-Graduação, Graduação, Biblioteca, Gráfica e demais funcionários do Instituto de Geociências;A CAPES pelo apoio financeiro através de bolsa de mestrado;Esta dissertação foi escrita em L A T E X com a classe IAGTESE."We have no idea, now, of who or what the inhabitants of our future might be. In that sense, we have no future. Not in the sense that our grandparents had a future, or thought they did. Fully imagined cultural futures were the luxury of another day, one in which 'now' was of some greater duration. For us, of course, things can change so abruptly, so violently, so profoundly, that futures like our grandparents' have insufficient 'now' to stand on. We have no future because our present is too volatile. ... We have only risk management. The spinning of the given moment's scenarios. Pattern recognition" William Gibson, Pattern Recognition ResumoOpenStereo foi desenvolvido originalmente para preencher uma lacuna entre aplicativos de análise para geologia estrutural, como um software livre, gratuito e multi-plataforma.Ao longo dos anos ele adquiriu um grande número de usuários, com citações regulares.Este trabalho objetivou a reestruturação do OpenStereo como um todo, mudando-o para uma nova estrutura de interface gráfica e construíndo-o do zero visando desempenho, estabilidade e facilidade de manutenção e extensão. Diversas novas funcionalidades foram incluídas tais como projetos, conversão de notação de atitudes, ajuste de pequenos círculos, extração de atitudes de modelos tridimensionais e conversão de shapefiles de linhas para dados circulares. A pesquisa gerou dois subprodutos principais: um novo método gráfico para ajuste de pequenos círculos e a biblioteca de análise de dados estruturais Auttitude.Palavras chave: Geologia Estrutural, Dados Direcionais, Software AbstractOpenStereo was originally developed to fill a gap among software packages for structural geology analysis, as a free open source cross-platform software. Over the years it has acquired a great number of users, with regular citations. This work aimed to restructure OpenStereo as a whole, changing to a new graphical interface framework and building it from the ground up for speed, stability, ease of maintenance and extension. Many new functionalities were also included, such as project management, structural attitudes notation handling, small circle fitting, extractions of attitudes from three-dimensional models and conversion of lines shapefiles to circular data. The research involved had two main byproducts, a new graphical method for small circle data fitting and a directional data analysis library, Auttitude.

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A Small Circle of Best Fit for Spherical Data and Areas of Vulcanism

Cited by 45 publications

References 4 publications

Analysis of Rotational Deformations From Directional Data

Analysis of Rotational Deformations From Directional Data

Bayesian alignment using hierarchical models, with applications in protein bioinformatics

Methods and applications for geological directional data analysis

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