Volume Tensor Estimation Using a Virtual Line Grid: Study of a Developing Pheasant Brain

Authors

  • Jiri Janacek Institute of Physiology, ASCR, v.v.i.
  • Daniel Jirak Institute for Clinical and Experimental Medicine and Institute of Biophysics and Informatics, First Faculty of Medicine, Charles University

DOI:

https://doi.org/10.5566/ias.1701

Keywords:

bird brain, development, Fakir probe, magnetic resonance imaging, volume Minkowski tensor

Abstract

The volume tensor provides a robust estimate of the shape and orientation of an object in space. In this paper, we introduce Fakir method for estimating the tensor of an object in 3D data set based on the intersections of objects boundary with virtual lines. We calculate the precision of shape estimates by predicting the variance of estimators of integrals based on systematic sampling. To demonstrate the ability of the Fakir method, we measure changes in shape and orientation of compartments in the pheasant brain during development.

References

Barbier JE (1860). Note sur problème de l'aiguille et le jeu du joint couvert, Journal de Mathématiques Pures et Appliquès 5:273--87.

Beyrand V, Voeten DFAE, Bureš S, Fernandez V, Janáček J, Jirák D, Rauhut O, Tafforeau P (2019). Multiphase progenetic development shaped the brain of flying archosaurs. Sci Rep 9.

Cruz-Orive LM (1997). Stereology of single objects,

Journal of Microscopy 186:93--107.

Dryden IL, Koloydenko A, Zhou D (2009). Non-Euclidean statistics for covariance matrices, with application to diffusion tensor imaging. Annal Appl Statist 3:1102–1123.

Hlawka E (1950). Über integrale auf konvexen körpern I.

Monatsh für Math 54 1-36.

Hug D, Schneider R, Schuster R (2008) Local stereology of tensors of convex bodies. Methodol Comput Appl Prob 16:263–282.

Janáček J (1999). Errors of spatial grids estimators of volume and surface area, Acta Stereol 18:389-396.

Janáček J (2006). Variance of periodic measure of bounded set with random position. Comment Math Univ Carolinae 47:473--82.

Janáček J (2008). An asymptotics of variance of the lattice points count. Czechosl Math J 58:751-758.

Janáček J, Jirák D (2019). Variance of the isotropic uniform systematic sampling. submitted to IAS.

Janáček J, Kubínová L (2010) Variances of length and surface area estimates by spatial grids: preliminary study. Image Anal Stereol, 29:45-52.

Jensen EBV, Ziegel JF (2014). Integral geometry of tensor valuations. Adv Appl Math 41:482–509.

Jirák D, Janáček J, Kear BP (2015). A combined MR and CT study for precise quantitative analysis of the avian brain. Sci Rep 5.

Karten HJ, Brzozowska-Prechtl A, Lovell PV, Tang DD, Mello CV, Wang H, Mitra PP (2013). Digital atlas of the Zebra Finch (Taeniopygia guttata) brain: a high-resolution photo atlas. J Comp Neurology 521:3702-3715.

O’Keefe M (1992). Cubic cylinder packings. Crystalography A48:879-884.

Kendall DG (1948). On the number of lattice points inside the random oval. Q J Math Oxford Ser 19 (3):1--26.

Kubínová L, Janáček J (1998). Estimating surface area by the isotropic fakir method from thick slices cut in an arbitrary direction. J Microsc 191:201-211.

Kubínová L, Janáček J, Guilak F, Opatrný Z (1999). Comparison of several digital and stereological methods for estimating surface area and volume of cells studied by confocal microscopy. Cytometry 36:85-95.

Matheron G (1965). Les variables regionalisees et leur estimation. Masson et cie.

Rafati AH, Ziegel JF, Nyengaard JR, Jensen EBV (2016) Stereological estimation of particle shape and orientation from volume tensors. J Microsc 263:229-237.

Santaló LA (1976). Integral Geometry and Geometric Probability. Reading: Addison-Wesley.

Schröder-Turk GE, Mickel W, Kapfer SC, Klatt MA, Schaller FM, Hoffmann MJF, Kleppmann N, Armstrong P, Inayat A, Hug D, Reichelsdorfer M, Peukert W, Schwieger W, Mecke K (2011). Minkowski tensor shape analysis of cellular, granular and porous structures. Adv Materials 23: 2535-2553.

Ziegel JF, Nyengaard JR, Jensen EBV (2015) Estimating particle shape and orientation using volume tensors. Scand J Statist 42: 813-831.

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Published

2019-12-13

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Section

Original Research Paper

How to Cite

Janacek, J., & Jirak, D. (2019). Volume Tensor Estimation Using a Virtual Line Grid: Study of a Developing Pheasant Brain. Image Analysis and Stereology, 38(3), 255-260. https://doi.org/10.5566/ias.1701

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