Multidimensional Characterisation of Time-dependent Image Data: A Case Study for the Peripheral Nervous System in Ageing Mice


  • Matthias Weber Institute of Stochastics Ulm University D-89069 Ulm Germany
  • Thomas Wilhelm Institute of Stochastics Ulm University D-89069 Ulm Germany
  • Volker Schmidt Institute of Stochastics Ulm University D-89069 Ulm Germany



automated segmentation, copula, electron microscopy, parametric modelling, quadriceps nerve, statistical image analysis


Segmentation of µm-resolution image data of irregularly shaped objects poses challenges to existing segmentation algorithms. This is especially true, when imperfections like noise, uneven lightning or traces of sample preparation are present in the image data. In this paper, considering electron micrographs of femoral quadriceps nerve sections of mice, a segmentation method to extract single axons surrounded by myelin sheaths is developed which is able to cope with various imperfections and artefacts. This approach successfully combines established methods like local thresholding and marker-based watershed transform to achieve a reliable segmentation of the given data. Indeed, the resulting segmentation map can be used to quantitatively determine geometrical characteristics of the axons and myelin sheaths. This is exemplified by modelling the joint probability distribution of axon area and myelin sphericity using a parametric copula approach, and by analysing the evolution of the model parameters for image data obtained from mice of different ages.


Akima H (1974). A method of bivariate interpolation and smooth surface fitting based on local procedures. Commun ACM 17:18–20.

Beucher S, Lantuéjoul C (1979). Use of watersheds in contour detection. In: Proceedings of the International Workshop on Image Processing, Real-Time Edge and Motion Detection/Estimation. Rennes, France.

Bickel PJ, Doksum KA (2015). Mathematical Statistics: Basic Ideas and Selected Topics, vol. 1. CRC Press, 2nd ed.

Carlson RE, Fritsch FN (1985). Monotone piecewise bicubic interpolation. SIAM J Numer Anal 22:386–400.

Durante F, Sempi C (2010). Copula therory: An introduction. In: Jaworski P, Durante F, Härdle W, Rychlik T, eds., Copula Theory and Its Application. Springer, 3–31.

Friede R (1986). Relation between myelin sheath thickness, internode geometry, and sheath resistance. Exp Neurol 92:234–47.

Furat O, Leißner T, Bachmann K, Gutzmer J, Peuker U, Schmidt V (2019a). Stochastic modeling of multidimensional particle characteristics using parametric copulas. Microsc Microanal 25:720–34.

Furat O, Prifling B, Westhoff D, Weber M, Schmidt V (2019b). Statistical 3D analysis and modeling of complex particle systems based on tomographic image data. Prakt Metallogr Pr M 56:787–96.

Gonzalez RC, Woods RE (2007). Digital Image Processing. Pearson.

Heumans H, Nacken P, Toet A, Vincent L (1992). Graph morphology. J Vis Commun Image R 3:24–38.

Hoshen J, Kopelman R (1976). Percolation and cluster distribution. i. cluster multiple labeling technique and criticalconcentration algorithm. Phys Rev 14:3438–45.

Joe H (1997). Multivariate Models and Multivariate Dependence Concepts. Chapman & Hall.

Lantuéjoul C (1982). Geodesic segmentation. In: Uhr L, Preston K, eds., Multicomputers and Image Processing: Algorithms and Programs. Academic Press, 111–24.

Neumann M, Machado-Charry E, Zojer K, Schmidt V (2021). On variability and interdependence of local porosity and local tortuosity in porous materials: a case study for sack paper. Methodol Comput Appl 23:613–27.

Roerdink JB, Meijster A (2000). The watershed transform: Definitions, algorithms and parallelization strategies. Fund Inform 41:187–228.

Sklar A (1959). Fonctions de répartition à n dimensions et leurs marges. Publications de lInstitut Statistique de lUniversité de Paris 8:229–31.

Smith K, Blakemore W, Murray J, Patterson R (1982). Internodal myelin volume and axon surface area: A relationship determining myelin thickness? J Neurol Sci 55:231–46.

Soille P (2013). Morphological Image Analysis: Principles and Applications. Springer.

Vincent L (1989). Graphs and mathematical morphology. Signal Process 16:365–88.

von Loeper F, Schaumann P, de Langlard M, Hess R, Bäsmann R, Schmidt V (2020). Probabilistic prediction of solar power supply to distribution networks, using forecasts of global horizontal irradiation. Sol Energy 203:145–56.

Wadell H (1932). Volume, shape, and roundness of rock particles. J Geol 40:443–51.

Weber M, Bäuerle A, Schmidt M, Neumann M, Fändrich M, Ropinski T, Schmidt V (2020). Automatic identification of crossovers in cryo-EM images of murine amyloid protein A fibrils with machine learning. J Microsc Oxford 277:12–22.

Yuan X, Klein D, Kerscher S, West B, J. Weis IK, Martini R (2018). Macrophage depletion ameliorates peripheral neuropathy in aging mice. J Neurosci 38:4610–20.

Zaimi A, Wabartha M, Herman V, Antonsanti PL, Perone CS, Cohen-Adad J (2018). AxonDeepSeg: automatic axon and myelin segmentation from microscopy data using convolutional neural networks. Sci Rep UK 8:1–11.




How to Cite

Weber, M., Wilhelm, T., & Schmidt, V. (2021). Multidimensional Characterisation of Time-dependent Image Data: A Case Study for the Peripheral Nervous System in Ageing Mice. Image Analysis and Stereology, 40(2), 85–94.



Original Research Paper

Most read articles by the same author(s)