IMPROVED ESTIMATION OF FIBER LENGTH FROM 3-DIMENSIONAL IMAGES
DOI:
https://doi.org/10.5566/ias.v32.p45-55Keywords:
Image analysis, integral of mean curvature, Crofton formula, systematic errorAbstract
A new method is presented for estimating the specific fiber length from 3D images of macroscopically homogeneous fiber systems. The method is based on a discrete version of the Crofton formula, where local knowledge from 3x3x3-pixel configurations of the image data is exploited. It is shown that the relative error resulting from the discretization of the outer integral of the Crofton formula amonts at most 1.2%. An algorithmic implementation of the method is simple and the runtime as well as the amount of memory space are low. The estimation is significantly improved by considering 3x3x3-pixel configurations instead of 2x2x2, as already studied in literature.References
bibitem[{Baddeley and Averback(1983)}]{baddeley83}
Baddeley A, Averback P (1983).
newblock Stereology of tubular structures.
newblock J Microsc 131:323--–340.
bibitem[{Federer(1959)}]{federer59}
Federer H (1959).
newblock Curvature measures.
newblock Trans Amer Math Soc 93:418--91.
bibitem[{Guderlei emph{et~al.}(2007)Guderlei, Klenk, Mayer, Schmidt, and
Spodarev}]{guderlei07}
Guderlei R, Klenk S, Mayer J, Schmidt V, Spodarev E (2007).
newblock Algorithms for the computation of {M}inkowski functionals of
deterministic and random polyconvex sets.
newblock Image Vision Computing 25:464--74.
bibitem[{Hadwiger(1957)}]{hadwiger57}
Hadwiger H (1957).
newblock Vorlesungen {"u}ber Inhalt, Oberfl{"a}che und
Iso-pe-ri-me-trie.
newblock Berlin: Springer-Verlag.
bibitem[{Kampf(2012)}]{kampf12}
Kampf J (2012).
newblock A limitation of the estimation of intrinsic volumes via pixel
configuration counts :preprint.
bibitem[{Kiderlen and Rataj(2006)}]{kiderlen06a}
Kiderlen M, Rataj J (2006).
newblock On infinitesimal increase of volumes of morphological transforms.
newblock Mathematika 53:103--27.
bibitem[{Klenk emph{et~al.}(2006)Klenk, Schmidt, and Spodarev}]{klenk06}
Klenk S, Schmidt V, Spodarev E (2006).
newblock A new algorithmic approach to the computation of {M}inkowski
functionals of germ-grain models.
newblock Comp Geom Th Appl 34:127--48.
bibitem[{Klette emph{et~al.}(1999)Klette, Kovalevski, and Yip}]{klette99}
Klette R, Kovalevski V, Yip P (1999).
newblock Length estimation of digital curves.
newblock In: Vision Geometry VIII. 117--29.
bibitem[{Klette and Rosenfeld(2004)}]{klette04}
Klette R, Rosenfeld A (2004).
newblock Digital Geometry.
newblock Amsterdam: Morgan & Kaufman Publ.
bibitem[{Lang emph{et~al.}(2001)Lang, Ohser, and Hilfer}]{lang01}
Lang C, Ohser J, Hilfer R (2001).
newblock On the analysis of spatial binary images.
newblock J Microsc 203:303--13.
bibitem[{Lindblad(2005)}]{lindblad05}
Lindblad J (2005).
newblock Surface area estimation of digitized 3{D} objects using weighted
local configurations.
newblock Image Vision Comp 23:111--22.
bibitem[{Lindblad and Nystr{"o}m(2002)}]{lindblad02}
Lindblad J, Nystr{"o}m I (2002).
newblock Surface area estimation of digitized 3{D} objects using local
computations.
newblock In: 10th International Conference on Discrete Geometry for Computer
Imagery, vol. 2301 of emph{LNCS}. DGCI, Bordeaux, France, Berlin,
Heidelberg, New York: Springer.
bibitem[{Louis emph{et~al.}(2011)Louis, Riplinger, Spiess, and
Spodarev}]{louis11}
Louis AK, Riplinger M, Spiess M, Spodarev E (2011).
newblock Inversion algorithms for the spherical {R}adon and cosine transform.
newblock Inverse Problems 27:035015.
bibitem[{Mecke and Nagel(1980)}]{mecke80}
Mecke J, Nagel W (1980).
newblock {S}tation{"a}re r{"a}umliche {F}aserprozesse und ihre
{S}chnittzahlrosen.
newblock Elektron Informationsverarb Kyb 16:475--83.
bibitem[{Meschenmoser and Spodarev(2012)}]{meschenmoser12}
Meschenmoser D, Spodarev E (2012).
newblock On the computation of intrinsic volumes :preprint.
bibitem[{Mrkvi{v c}ka and Rataj(2008)}]{mrkvicka08}
Mrkvi{v c}ka T, Rataj J (2008).
newblock On the estimation of intrinsic volume densities of stationary random
closed sets.
newblock Stochastic Proc Appl 118:213--31.
bibitem[{Mrkvi{v c}ka and Rataj(2009)}]{mrkvicka09}
Mrkvi{v c}ka T, Rataj J (2009).
newblock On the estimation of intrinsic volume densities of stationary random
closed sets via parallel sets in the plane.
newblock Kybernetika 45:931--45.
bibitem[{Mullikin and Verbeek(1993)}]{mullikin93}
Mullikin J, Verbeek P (1993).
newblock Surface estimation of digital planes.
newblock Bioimaging 1:6--16.
bibitem[{Nagel emph{et~al.}(2000)Nagel, Ohser, and Pischang}]{nagel00}
Nagel W, Ohser J, Pischang K (2000).
newblock An integral-geometric approach for the {E}uler-{P}oincar{'e}
characteristic of spatial images.
newblock J Microsc 198:54--62.
bibitem[{Ohser and M{"u}cklich(2000)}]{ohser00}
Ohser J, M{"u}cklich F (2000).
newblock Statistical Analysis of Microstructures in Materials Science.
newblock Chichester, New York: J Wiley & Sons.
bibitem[{Ohser emph{et~al.}(2002)Ohser, Nagel, and Schladitz}]{ohser02}
Ohser J, Nagel W, Schladitz K (2002).
newblock The {E}uler number of discretized sets -- on the choice of adjacency
in homogeneous lattices.
newblock In: Mecke KR, Stoyan D, eds., Morphology of Condensed Matter, Lecture
Notes in Physics. Berlin: Springer.
bibitem[{Ohser emph{et~al.}(2003)Ohser, Nagel, and Schladitz}]{ohser03}
Ohser J, Nagel W, Schladitz K (2003).
newblock The {E}uler number of discretised sets -- surprising results in three
dimensions.
newblock Image Anal Stereol 22:11--9.
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[{Santal{'o}(1976)}]{santalo76}
Santal{'o} LA (1976).
newblock Integral Geometry and Geometric Probability.
newblock Reading, Mass: Addison-Wesley.
bibitem[{Schladitz emph{et~al.}(2006{natexlab{a}})Schladitz, Ohser, and
Nagel}]{schladitz06a}
Schladitz K, Ohser J, Nagel W (2006{natexlab{a}}).
newblock Measurement of intrinsic volumes of sets observed on lattices.
newblock In: Kuba A, Nyul LG, Palagyi K, eds., 13th International Conference
on Discrete Geometry for Computer Imagery, LNCS. DGCI, Szeged, Hungary,
Berlin, Heidelberg, New York: Springer.
bibitem[{Schladitz emph{et~al.}(2006{natexlab{b}})Schladitz, Peters,
Reinel-Bitzer, Wiegmann, and Ohser}]{schladitz06}
Schladitz K, Peters S, Reinel-Bitzer D, Wiegmann A, Ohser J
(2006{natexlab{b}}).
newblock Design of aoustic trim based on geometric modeling and flow
simulation for non-woven.
newblock Comp Materials Sci 38:56--66.
bibitem[{Schmidt and Spodarev(2005)}]{schmidt05}
Schmidt V, Spodarev E (2005).
newblock Joint estimators for the specific intrinsic volumes of stationary
random sets.
newblock Stochastic Processes and their Applications 115:959--81.
bibitem[{Schneider(1993)}]{schneider93}
Schneider R (1993).
newblock Convex Bodies: The Brunn-Minkowski Theory.
newblock Cambridge: Encyclopedia of Mathematics and Its Application Vol. {bf
}, Cambridge University Press.
bibitem[{Svane(2012)}]{svane12}
Svane AM (2012).
newblock Local digital estimators of intrinsic volumes for {B}oolean models
and in the design based setting :preprint.
bibitem[{Vogel emph{et~al.}(2010)Vogel, Weller, and Schl{"u}ter}]{vogel10}
Vogel HJ, Weller U, Schl{"u}ter S (2010).
newblock Quantification of soil structure based on {M}inkowski functions.
newblock Computers Geosciences 36:1236--46.
bibitem[{Weitkamp emph{et~al.}(2011)Weitkamp, Haas, Wegrzynek, and
Rack}]{weitkamp11}
Weitkamp T, Haas D, Wegrzynek D, Rack A (2011).
newblock Ankaphase: software for single-distance phase-retrieval from inline
{X}-ray phase contrast radiographs.
newblock J Synchrotron Radiation 18:617--29.
bibitem[{Windreich emph{et~al.}(2003)Windreich, Kiryati, and
Lohmann}]{windreich03}
Windreich G, Kiryati N, Lohmann G (2003).
newblock Surface area estimation in practice.
newblock In: Nystr{"o}m I, di~Baja GS, Svensson S, eds., 11th International
Conference on Discrete Geometry for Computer Imagery, vol. 2886 of
emph{LNCS}. DGCI, Naples, Italy, Berlin, Heidelberg, New York: Springer.
bibitem[{Ziegel and Kiderlen(2010)}]{ziegel10}
Ziegel J, Kiderlen M (2010).
newblock Estimation of the surface area and surface area measure of
three-dimensional sets from digitizations.
newblock Image Vision Computing 28:64--77.
