• Christoph Blankenburg University of Applied Sciences Darmstadt, Germany CNRS, CRAN, UMR 7039, 54516 Vand{\oe}uvre-l\`es-Nancy, France Universit{\'e} de Lorraine, CRAN, UMR 7039, 2 avenue de la For{\^e}t de Haye
  • Christian Daul CNRS, CRAN, UMR 7039, 54516 Vand{\oe}uvre-l\`es-Nancy, France Universit{\'e} de Lorraine, CRAN, UMR 7039, 2 avenue de la For{\^e}t de Haye
  • Joachim Ohser University of Applied Sciences Darmstadt



discrete geometry, integral of torsion, skeleton line


Curvature and torsion of three-dimensional curves are important quantities in fields like material science or biomedical engineering. Torsion has an exact definition in the continuous domain. However, in the discrete case most of the existing torsion evaluation methods lead to inaccurate values, especially for low resolution data. In this contribution we use the discrete points of space curves to determine the Fourier series coefficients which allow for representing the underlying continuous curve with Cesàro’s mean. This representation of the curve suits for the estimation of curvature and torsion values with their classical continuous definition. In comparison with the literature, one major advantage of this approach is that no a priori knowledge about the shape of the cyclic curve parts approximating the discrete curves is required. Synthetic data, i.e. curves with known curvature and torsion, are used to quantify the inherent algorithm accuracy for torsion and curvature estimation. The algorithm is also tested on tomographic data of fiber structures and open foams, where discrete curves are extracted from the pore spaces.


bibitem[{Adams emph{et~al.}(2008)Adams, Okagbare, Feng, McCarley, Murphy, and


Adams AA, Okagbare P, Feng J, McCarley RL, Murphy MC, Soper SA (2008).

newblock Capture and enumeration of circulating tumor cells from peripheral

blood using microfluids.

newblock J Am Chem Soc 130:8633--41.

bibitem[{Bae emph{et~al.}(2013)Bae, Erdonmez, Halloran, and Chiang}]{bae13}

Bae CJ, Erdonmez CK, Halloran JW, Chiang YM (2013).

newblock {Design of battery electrodes with dual-scale porosity to minimize

tortuosity and maximize performance}.

newblock Adv Mat 25:1254--8.


Baerentzen A (2001).

newblock On the implementation of the fast marching methods for 3d lattices.

newblock Tech. rep., Techn. Univ. of Denmark, Dept. Math. Modelling.

bibitem[{Blankenburg emph{et~al.}(2016)Blankenburg, Rack, Daul, and


Blankenburg C, Rack A, Daul C, Ohser J (2016).

newblock Curvature and torsion estimates of particle traces through porous

media observed by fast in-situ microtomography.

newblock J Microsc :submitted.

bibitem[{Carroll emph{et~al.}(2014)Carroll, Hankins, Kose, and


Carroll D, Hankins E, Kose E, Sterling I (2014).

newblock {A survey of the differential geometry of discrete curves}.

newblock The Mathematical Intelligencer 36:28--35.

bibitem[{Chen-Wiegart emph{et~al.}(2014)Chen-Wiegart, DeMike, Erdonmez,

Thornton, Scott A.~Barnett, and Wang}]{chenwiegart14}

Chen-Wiegart YK, DeMike R, Erdonmez C, Thornton K, Scott A.~Barnett SA, Wang J


newblock {Tortuosity characterization of 3D microstructure at nano-scale for

energy storage and conversion materials}.

newblock Journal of Power Sources 249:349--56.

bibitem[{Coeurjolly emph{et~al.}(2001)Coeurjolly, Serge, and


Coeurjolly D, Serge M, Laure T (2001).

newblock Discrete curvature based on osculating circles estimation.

newblock Lecture Notes in Comp Sci 2059:303--12.

bibitem[{Coeurjolly and Svensson(2003)}]{coeurjolly03}

Coeurjolly D, Svensson S (2003).

newblock Destimation of curvature along curves with application to fibres in

{D} images of paper.

newblock Lecture Notes in Comp Sci 2749:247--54.

bibitem[{Coleman and Pritchett(1990)}]{coleman90}

Coleman SY, Pritchett CJ (1990).

newblock Random rotations in simulation with computer 3{D} reconstruction.

newblock Acta Stereol 9:207--18.

bibitem[{Crenshaw emph{et~al.}(2000)Crenshaw, Ciampaglio, and


Crenshaw HC, Ciampaglio CN, McHenry M (2000).

newblock Analysis of the three-dimensional trajectories of organisms:

estimates of the velocity, curvature and torsion from positional information.

newblock J Experimental Biol 203:961--82.

bibitem[{Dharmasiri emph{et~al.}(2009)Dharmasiri, Balamurugan, McCarley,

Spivak, and Soper}]{dharmasiri09}

Dharmasiri U, Balamurugan S, McCarley RL, Spivak D, Soper SA (2009).

newblock Highly efficient capture and enumeration of low abundance prostate

cancer cells using prostate-specific membrane antigen aptamers immobilized to

a polymeric microfluidic device.

newblock Electrophoresis 30:3289--300.


Fullman RL (1953).

newblock Measurement of approximately cylindrical particles in opaque samples.

newblock Trans Metall AIME 197:1267--8.

bibitem[{Gaiselmann emph{et~al.}(2013)Gaiselmann, Manke, Lehnert, and


Gaiselmann G, Manke I, Lehnert W, Schmidt V (2013).

newblock Extraction of curved fibers from 3{D} data.

newblock Image Anal Stereol 32:57--63.

bibitem[{Gommes emph{et~al.}(2009)Gommes, Bons, Blacher, Dunsmuir, and


Gommes CJ, Bons AJ, Blacher S, Dunsmuir JH, Tsou AH (2009).

newblock {Practical methods for measuring the tortuosity of porous materials

from binary or gray-tone tomographic reconstructions}.

newblock AIChE 55:2000--12.

bibitem[{Grisan emph{et~al.}(2003)Grisan, Foracchia, and Ruggeri}]{grisan03}

Grisan E, Foracchia M, Ruggeri A (2003).

newblock {A novel method for the automatic evaluation of retinal vessel


newblock IEEE EMBS .

bibitem[{Kehtarnavaz and de~Figueiredo(1988)}]{kehtarnavaz88}

Kehtarnavaz ND, de~Figueiredo JP (1988).

newblock A 3{D} contour segmentatio scheme based on curvature and torsion.

newblock IEEE Trans Pattern Anal Mach Intell 10:707--13.

bibitem[{Lewinger emph{et~al.}(2005)Lewinger, Gomes, Lopes, and


Lewinger T, Gomes JD, Lopes H, Craizner M (2005).

newblock Curvature and torsion estimators based on parametric curve fitting.

newblock Computers Graphics 29:641--55.

bibitem[{Medina emph{et~al.}(2004)Medina, Wahle, Olszewski, and


Medina R, Wahle A, Olszewski ME, Sonka M (2004).

newblock Curvature and torsion estimation for coronary-artery motion analysis.

newblock SPIE Medical Imaging 5369:504--15.

bibitem[{Meng emph{et~al.}(2008)Meng, Costa, Geyer, Viana, Reiter,

M{"u}ller, and Weninger}]{meng08}

Meng S, Costa LdF, Geyer SH, Viana MP, Reiter C, M{"u}ller GB, Weninger W


newblock Three-dimensional description and mathematical characterization of

the parasellar internal carotid artery in human infants.

newblock J Anatomy 212:636--44.

bibitem[{Moeslang emph{et~al.}(2009)Moeslang, Pieritz, Boller, and


Moeslang A, Pieritz RA, Boller B, Ferrero C (2009).

newblock Gas bubble network formation in irradiated beryllium pebbles

monitored by {X}-ray microtomography.

newblock J Nucl Mat 386--388:1052--5.


Mokhtarian F (1997).

newblock A theory of multiscale, torsion-based shape representation for space


newblock Comp Vision Image Understanding 68:1--19.

bibitem[{Nguyen and Debled-Bennesson(2008)}]{nguyen08}

Nguyen TP, Debled-Bennesson I (2008).

newblock Curvature and torsion estimators from 3{D} curves.

newblock Adv Visual Comp Lecture Notes in Computer Sci 5358:688--99.

bibitem[{Nguyen and Debled-Bennesson(2009)}]{nguyen09}

Nguyen TP, Debled-Bennesson I (2009).

newblock On local properties of digital curves.

newblock I J Shape Modeling 14:105--25.

bibitem[{Ohser emph{et~al.}(2012)Ohser, Ferrero, Wirjadi, Kuznetsova,

D{"u}ll, and Rack}]{ohser12}

Ohser J, Ferrero C, Wirjadi O, Kuznetsova A, D{"u}ll J, Rack A (2012).

newblock Estimation of the probability of finite percolation in porous

microstructures from tomographic images.

newblock Int J Mat Res 103:184--91.

bibitem[{Ohser emph{et~al.}(2009)Ohser, Nagel, and Schladitz}]{ohser09}

Ohser J, Nagel W, Schladitz K (2009).

newblock Miles formulae for {B}oolean models observed on lattices.

newblock Image Anal Stereol 28:77--92.

bibitem[{Ohser and Schladitz(2009)}]{ohser09a}

Ohser J, Schladitz K (2009).

newblock 3{D} Images of Materials Structures -- Processing and Analysis.

newblock Weinheim, Berlin: Wiley VCH.

bibitem[{Pao emph{et~al.}(1992)Pao, Lu, and L.}]{pao92}

Pao YC, Lu JT, L. RE (1992).

newblock Bending and twisting of an in vivo coronary artery at a bifurcation.

newblock J Biomechanics 25:287--95.

bibitem[{Patasius emph{et~al.}(2007)Patasius, Marozas, Lukosevicius, and


Patasius M, Marozas V, Lukosevicius A, Jegelevicius D (2007).

newblock {Model based investigation of retinal vessel tortuosity as a function

of blood pressure: preliminary results}.

newblock In: {Engineering in Medicine and Biology Society, 2007. EMBS 2007.

th Annual International Conference of the IEEE}.

bibitem[{Peker and Helvaci(2008)}]{peker08}

Peker SM, Helvaci SS (2008).

newblock Solid-Liquid Two-Phase Flow.

newblock Amsterdam: Elsevier.

bibitem[{Peyrega and Jeulin(2013)}]{peyrega13}

Peyrega C, Jeulin D (2013).

newblock Estimation of tortuosity and reconstruction of geodesic paths in


newblock Image Anal Streol 32:27--43.

bibitem[{Pieritz emph{et~al.}(2011)Pieritz, Reimann, and Ferrero}]{pieritz11}

Pieritz RA, Reimann J, Ferrero C (2011).

newblock 3{D} tomography analysis of the inner structure of pebbles and pebble


newblock Adv Eng Mat 13:145–155.

bibitem[{Puentes emph{et~al.}(1998)Puentes, Garreau, Lebreton, and


Puentes J, Garreau M, Lebreton H, Roux C (1998).

newblock Understanding coronary artery movement: a knowledge-based approach.

newblock Artif Intell Medicine 13:207--37.


Rhines FN (1977).

newblock Microstructure properties in materials.

newblock Metall Trans 8A:127--33.


Scagliarini A (2011).

newblock Geometric properties of particle trajectories in turbulent flows.

newblock J Turbulence 12:1--12.


Schneider R (1993).

newblock Convex Bodies: The Brunn-Minkowski Theory.

newblock Cambridge: Encyclopedia of Mathematics and Its Application Vol. {bf

}, Cambridge University Press.


Sethian JA (1999).

newblock Level Sets Methods: Evolving Interfaces in Computational Geometry,

Fluid Mechanics, Computer Vision and Materials Science.

newblock Cambridge: Cambridge University Press.


Soille P (1999).

newblock Morphological Image Analysis.

newblock Berlin, Heidelberg: Springer-Verlag.


Spivak M (1979).

newblock A Comprehensive Introduction to Differential Geometry, vol.~II.

newblock Houston: Publish or Perish, Inc.

bibitem[{Strandmark emph{et~al.}(2013)Strandmark, Ul{'e}n, Kahl, and


Strandmark P, Ul{'e}n J, Kahl F, Grady L (2013).

newblock Shortest paths with curvature and torsion.

newblock In: IEEE International Conference on Computer Vision, DOI


bibitem[{Wood emph{et~al.}(2006)Wood, Zhao, Zambanini, Jackson, Gedroyc,

Thom, Hughes, and Xu}]{wood06}

Wood NB, Zhao SZ, Zambanini A, Jackson M, Gedroyc W, Thom AA, Hughes AD, Xu XY


newblock Curvature and tortuosity of the superficial femoral artery: a

possible risk factor for peripheral arterial disease.

newblock J Appl Physiology 101:1412--8.

bibitem[{Zhao emph{et~al.}(2012)Zhao, Wang, and Xiang}]{zhao12}

Zhao D, Wang H, Xiang Y (2012).

newblock Asymptotic behaviors of the stress fields in the vicinity of

dislocations and dislocation segments.

newblock Phil Mag 92:2351--74.

bibitem[{Zhenga and Qib(2011)}]{zhenga11}

Zhenga S, Qib Y (2011).

newblock Motion estimation of 3{D} coronary vessel skeletons from {X}-ray

angiographic sequences.

newblock Computerized Medical Imaging Graphics 35:353--64.


Zygmund A (1988).

newblock Trigonometric series (2nd ed.).

newblock Cambridge: Cambridge University Press.




How to Cite

Blankenburg, C., Daul, C., & Ohser, J. (2016). ESTIMATING TORSION OF DIGITAL CURVES USING 3D IMAGE ANALYSIS. Image Analysis and Stereology, 35(2), 81–91.



Original Research Paper

Most read articles by the same author(s)

1 2 > >>