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Measuring Shape Relations Using r-Parallel Sets

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Abstract

Geometrical measurements of biological objects form the basis of many quantitative analyses. Hausdorff measures such as the volume and the area of objects are simple and popular descriptors of individual objects; however, for most biological processes, the interaction between objects cannot be ignored, and the shape and function of neighboring objects are mutually influential. In this paper, we present a theory on the geometrical interaction between objects inspired by K-functions for spatial point-processes. Our theory describes the relation between two objects: a reference and an observed object. We generate the r-parallel sets of the reference object, calculate the intersection between the r-parallel sets and the observed object, and define measures on these intersections. The measures are simple, like the volume or surface area, but describe further details about the shape of individual objects and their pairwise geometrical relation. Finally, we propose a summary-statistics. To evaluate these measures, we present a new segmentation of cell membrane, mitochondria, synapses, vesicles, and endoplasmic reticulum in a publicly available FIB-SEM 3D brain tissue data set and use our proposed method to analyze key biological structures herein.

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References

  1. Baddeley, A., Rubak, E., Turner, R.: Spatial Point Patterns: Methodology and Applications with R. CRC Press, Boca Raton (2015)

    Book  Google Scholar 

  2. Dixon, P.M.: Ripley’s \(k\) Function. Statistics Reference Online, Wiley, New York (2014)

    Book  Google Scholar 

  3. Fesce, R., Grohovaz, F., Valtorta, F., Meldolesi, J.: Neurotransmitter release: fusion or kiss-and-run? Trends Cell Biol. 4(1), 1–4 (1994)

    Article  Google Scholar 

  4. Gavet, Y., Fernandes, M., Debayle, J., Pinoli, J.C.: Dissimilarity criteria and their comparison for quantitative evaluation of image segmentation: application to human retina vessels. Mach. Vis. Appl. 25(8), 1953–1966 (2014)

    Article  Google Scholar 

  5. Gavrilov, N., Golyagina, I., Brazhe, A., Scimemi, A., Turlapov, V., Semyanov, A.: Astrocytic coverage of dendritic spines, dendritic shafts, and axonal boutons in hippocampal neuropil. Front. Cell. Neurosci. 12, 248 (2018)

    Article  Google Scholar 

  6. Hahn, U.: A studentized permutation test for the comparison of spatial point patterns. J. Am. Stat. Assoc. 107(498), 754–764 (2012). https://doi.org/10.1080/01621459.2012.688463

    Article  MathSciNet  MATH  Google Scholar 

  7. Hu, Y., Zhou, Q., Gao, X., Jacobson, A., Zorin, D., Panozzo, D.: Tetrahedral meshing in the wild. ACM Trans. Gr. 37(4), 60:1-60:14 (2018). https://doi.org/10.1145/3197517.3201353

    Article  Google Scholar 

  8. Khanmohammadi, M., Waagepetersen, R.P., Sporring, J.: Analysis of shape and spatial interaction of synaptic vesicles using data from focused ion beam scanning electron microscopy (FIB-SEM). Front. Neuroanatomy (2015). https://doi.org/10.3389/fnana.2015.00116

  9. Koenig, J., Ikeda, K.: Synaptic vesicles have two distinct recycling pathways. J. Cell Biol. 135(3), 797–808 (1996)

    Article  Google Scholar 

  10. Lucchi, A., Li, Y., Becker, C., Fua, P.: Electron microscopy dataset. https://cvlab.epfl.ch/data/data-em/. Accessed 14 Mar 2020

  11. Marques, J., Genant, H.K., Lillholm, M., Dam, E.B.: Diagnosis of osteoarthritis and prognosis of tibial cartilage loss by quantification of tibia trabecular bone from MRI. Magn. Reson. Med. 70(2), 568–575 (2013). https://doi.org/10.1002/mrm.24477

    Article  Google Scholar 

  12. Medvedev, N., Popov, V., Henneberger, C., Kraev, I., Rusakov, D.A., Stewart, M.G.: Glia selectively approach synapses on thin dendritic spines. Philos. Trans. R Soc. B Biol. Sci. 369(1654), 20140047 (2014)

    Article  Google Scholar 

  13. Richards, D., Guatimosim, C., Betz, W.: Two endocytic recycling routes selectively fill two vesicle pools in frog motor nerve terminals. Neuron 27(3), 551–559 (2000)

    Article  Google Scholar 

  14. Ripley, B.D.: Tests of randomness for spatial point patterns. J. Roy. Stat. Soc. Ser. B (Methodol.) 41(3), 368–374 (1979). https://doi.org/10.1111/j.2517-6161.1979.tb01091.x

    Article  MATH  Google Scholar 

  15. Ripley, B.D.: Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge (1988)

    Book  Google Scholar 

  16. Ronneberger, O., Fischer, P., Brox, T.: U-net: convolutional networks for biomedical image segmentation. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 234–241. Springer (2015)

  17. Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Heidelberg (2008)

    Book  Google Scholar 

  18. Scientific, T.F.: Amira-avizo software (2020)

  19. Sethian, J.A.: Fast marching methods. SIAM Rev. 41(2), 199–235 (1999)

  20. Si, H.: TetGen, a delaunay-based quality tetrahedral mesh generator. ACM Trans. Math. Softw. 41(2), 1–36 (2015)

    Article  MathSciNet  Google Scholar 

  21. Sporring, J., Waagepetersen, R., Sommer, S.: Generalizations of Ripley’s k-function with application to space curves. In: International Conference on Information Processing in Medical Imaging, pp. 731–742. Springer (2019)

  22. Stephensen, H.J.T., Darkner, S., Sporring, J.: Restoring drifted electron microscope volumes using synaptic vesicles at sub-pixel accuracy. Commun. Biol. 3(1), 1–7 (2020)

    Article  Google Scholar 

  23. Stephensen, H.J.T., Sporring, J.: Rodent neuronal volume annotations and segmentations (2020). https://www.doi.org/10.17894/ucph.33bd30d2-5796-48f4-a0a8-96fcc0ce6af5

  24. Svane, A.M.: Valuations in image analysis. In: Tensor Valuations and Their Applications in Stochastic Geometry and Imaging, pp. 435–454. Springer (2017)

  25. Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Jarrod Millman, K., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C., Polat, İ, Feng, Y., Moore, E.W., Van der Plas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P., Contributors, S., et al.: SciPy 1.0: fundamental algorithms for scientific computing in python. Nat. Methods 17, 261–272 (2020). https://doi.org/10.1038/s41592-019-0686-2

  26. Zhang, D., Lu, G.: Review of shape representation and description techniques. Pattern Recogn. 37(1), 1–19 (2004). https://doi.org/10.1016/j.patcog.2003.07.008

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Copenhagen, Copenhagen, Denmark

    Hans J. T. Stephensen & Jon Sporring

  2. Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark

    Anne Marie Svane

  3. Center for Translational Neuromedicine, University of Copenhagen, Copenhagen, Denmark

    Carlos B. Villanueva & Steven A. Goldman

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  1. Hans J. T. Stephensen
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  2. Anne Marie Svane
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  3. Carlos B. Villanueva
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  4. Steven A. Goldman
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  5. Jon Sporring
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Correspondence to Hans J. T. Stephensen.

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Supported by the Centre for Stochastic Geometry and Advanced Bioimaging and The Center for Quantification of Imaging Data from MAX IV.

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Stephensen, H.J.T., Svane, A.M., Villanueva, C.B. et al. Measuring Shape Relations Using r-Parallel Sets. J Math Imaging Vis 63, 1069–1083 (2021). https://doi.org/10.1007/s10851-021-01041-3

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