STRUCTURE DETECTION WITH SECOND ORDER RIESZ TRANSFORMS
DOI:
https://doi.org/10.5566/ias.1964Keywords:
image analysis, local fiber orientation, tubular structuresAbstract
A frequently applied indicator of tubular structures is based on the eigenvalues of the Hessian matrix of the original image convolved with a Gaussian, whose standard derivation depends on the size of the tubes. Hence the tube size must either be known in advance or a whole scale of standard deviations has to be tested resulting in higher computational costs – a serious obstacle for data with varying tube thickness.
In this paper, we propose to modify the structure indicator by replacing the derivatives of the Gaussian smoothed function by the Riesz transform. We show by various numerical examples that the resulting structure indicator is scale independent. Smoothing with a Gaussian is just necessary to cope with the noise in the image, but is not related to the size of the tubular structures. We apply the novel structure indicator for the fiber orientation analysis of fibrous materials and for the segmentation of leather. The latter one was a special challenging application since all scales are present in the microstructure of leather.
References
bibitem[{Aach emph{et~al.}(2006)Aach, Mota, Stuke, M"uhlich, and
Barth}]{Aach06}
Aach T, Mota C, Stuke I, M"uhlich M, Barth E (2006).
newblock Analysis of superimposed oriented patterns.
newblock Image Processing IEEE Transactions on 15:3690--700.
bibitem[{Altendorf and Jeulin(2009)}]{altendorf09}
Altendorf H, Jeulin D (2009).
newblock 3d directional mathematical morphology for analysis of fiber
orientations.
newblock Image Analysis and Stereology 28:143--53.
bibitem[{Basil-Jones emph{et~al.}(2010)Basil-Jones, Edmonds, Allsop, Cooper,
Holmes, Norris, Cookson, Kirby, and Haverkamp}]{haverkamp2010}
Basil-Jones MM, Edmonds RL, Allsop TF, Cooper SM, Holmes G, Norris GE, Cookson
DJ, Kirby N, Haverkamp RG (2010).
newblock Leather structure determination by small-angle x-ray scattering
(saxs): Cross sections of ovine and bovine leather.
newblock Journal of Agricultural and Food Chemistry 58:5286--91.
bibitem[{Beucher(1994)}]{beu94}
Beucher S (1994).
newblock Watershed, hierarchical segmentation and waterfall algorithm.
newblock In: Serra J, Soille P, eds., Computational Imaging and Vision Vol. 2:
Mathematical Morphology and Its Applications to Image Processing. Dordrecht:
Kluwer.
bibitem[{Bittrich emph{et~al.}(2014)Bittrich, Schladitz, Meyndt, Schulz, and
Godehardt}]{jalca2014}
Bittrich E, Schladitz K, Meyndt R, Schulz H, Godehardt M (2014).
newblock Micro-computed tomography studies for three-dimensional leather
structure analysis.
newblock Journal of the American Leather Chemists Association 109:367--71.
bibitem[{Cloetens emph{et~al.}(2002)Cloetens, Ludwig, Boller, Helfen, Salvo,
Mache, and Schlenker}]{cloetens02:fiber-felt}
Cloetens P, Ludwig W, Boller E, Helfen L, Salvo L, Mache R, Schlenker M (2002).
newblock Quantitative phase contrast tomography using coherent synchrotron
radiation.
newblock Proc SPIE 4503:82--91.
bibitem[{Constantin(2016)}]{Constantin2016}
Constantin A (2016).
newblock Fourier Analysis, Volume 1:Theory.
newblock Cambridge: Cambridge University Press.
bibitem[{Du emph{et~al.}(1995)Du, Parker, and Davis}]{DPD1995}
Du YP, Parker DL, Davis WL (1995).
newblock Vessel enhancement filtering in three-dimensional {MR} angiography.
newblock Journal of Magnetic Resonance Imaging 5:353--9.
bibitem[{Felsberg and Granlund(2004)}]{FG2004}
Felsberg M, Granlund G (2004).
newblock {POI} detection using channel clustering and the 2d energy tensor.
newblock In: Rasmussen CE, B{"u}lthoff HH, Sch{"o}lkopf B, Giese MA, eds.,
Pattern Recognition: 26th DAGM Symposium, T{"u}bingen, Germany, August 30 -
September 1, 2004. Proceedings. Berlin, Heidelberg: Springer Berlin
Heidelberg.
bibitem[{Felsberg and Sommer(2001)}]{FS2001}
Felsberg M, Sommer G (2001).
newblock The monogenic signal.
newblock IEEE Transactions on Signal Processing 49:3136--44.
bibitem[{F{"o}rstner and G{"u}lch(1987)}]{FG1987}
F{"o}rstner W, G{"u}lch E (1987).
newblock A fast operator for detection and precise location of distinct
points, corners and centres of circular features.
newblock In: Proc. ISPRS Intercommission Conference on Fast Processing of
Photogrammetric Data.
bibitem[{Frangi emph{et~al.}(1998)Frangi, Niessen, Vincken, and
Viergever}]{FNVV98}
Frangi AF, Niessen WJ, Vincken KL, Viergever MA (1998).
newblock Multiscale vessel enhancement filtering.
newblock In: Medical Image Computing and Computer-Assisted
Interventation—MICCAI’98. Springer, 130--7.
bibitem[{Godehardt emph{et~al.}(2017)Godehardt, Schladitz, Dietrich, Meyndt,
and Schulz}]{ismm2016}
Godehardt M, Schladitz K, Dietrich S, Meyndt R, Schulz H (2017).
newblock Segmentation of collagen fiber bundles in 3d by waterfall on
orientations.
newblock In: Mathematical Morphology and Its Applications to Signal and Image
Processing - 13th International Symposium, {ISMM} 2017, Fontainebleau, May
-17, 2017. Proceedings.
bibitem[{H"auser emph{et~al.}(2014)H"auser, Heise, and Steidl}]{HHS13}
H"auser S, Heise B, Steidl G (2014).
newblock Linearized {R}iesz transform and quasi-monogenic shearlets.
newblock International Journal of Wavelets Multiresolution and Information
Processing 12:1450027--1 – 1450027--25.
bibitem[{Heieck emph{et~al.}(2017)Heieck, Hermann, Middendorf, and
Schladitz}]{heieck2017}
Heieck F, Hermann F, Middendorf P, Schladitz K (2017).
newblock Influence of the cover factor of 2d biaxial and triaxial braided
carbon composites on their in-plane mechanical properties.
newblock Composite Structures 163:114 -- 122.
bibitem[{K{"o}the(2003)}]{Kot2003}
K{"o}the U (2003).
newblock Edge and junction detection with an improved structure tensor.
newblock In: Joint Pattern Recognition Symposium. Springer.
bibitem[{K{"o}the and Felsberg(2005)}]{KF2005}
K{"o}the U, Felsberg M (2005).
newblock Riesz-transforms versus derivatives: On the relationship between the
boundary tensor and the energy tensor.
newblock In: Kimmel R, Sochen NA, Weickert J, eds., Scale Space and PDE
Methods in Computer Vision: 5th International Conference, Scale-Space 2005.
Springer Berlin Heidelberg.
bibitem[{Krause emph{et~al.}(2010)Krause, Hausherr, Burgeth, Herrmann, and
Krenkel}]{krause10}
Krause M, Hausherr J, Burgeth B, Herrmann C, Krenkel W (2010).
newblock Determination of the fibre orientation in composites using the
structure tensor and local x-ray transform.
newblock J Material Science 45:888--96.
bibitem[{Larkin emph{et~al.}(2001)Larkin, Bone, and Oldfield}]{LBO01}
Larkin KG, Bone DJ, Oldfield MA (2001).
newblock Natural demodulation of two-dimensional fringe patterns. i. general
background of the spiral phase quadrature transform.
newblock Journal of the Optical Society of America A 18:1862--70.
bibitem[{Lindeberg(1999)}]{L1999}
Lindeberg T (1999).
newblock Principles for automatic scale selection.
newblock In: Handbook on Computer Vision and Applications, vol.~2. Academic
Press.
bibitem[{Ohser and Schladitz(2009)}]{ohser-schladitz09book}
Ohser J, Schladitz K (2009).
newblock 3d Images of Materials Structures -- Processing and Analysis.
newblock Weinheim: Wiley VCH.
bibitem[{Prince(1994)}]{P1994}
Prince MR (1994).
newblock Gadolinium-enhanced {MR} aortography.
newblock Radiology 191:155--64.
bibitem[{Redenbach emph{et~al.}(2012)Redenbach, Rack, Schladitz, Wirjadi, and
Godehardt}]{redenbach12:_beyon_imagin}
Redenbach C, Rack A, Schladitz K, Wirjadi O, Godehardt M (2012).
newblock Beyond imaging: on the quantitative analysis of tomographic volume
data.
newblock International Journal of Materials Research 2:217--27.
bibitem[{Rief(2005)}]{rief2005nonlinear}
Rief S (2005).
newblock Nonlinear flow in porous media-numerical solution of the
navier-stokes system with two pressures and application to paper making.
newblock Ph.D. thesis, Technische Universit{"a}t Kaiserslautern.
bibitem[{Riesz(1928)}]{riesz1928fonctions}
Riesz M (1928).
newblock Sur les fonctions conjugu{'e}es.
newblock Mathematische Zeitschrift 27:218--44.
bibitem[{Robb emph{et~al.}(2007)Robb, Wirjadi, and Schladitz}]{robb07}
Robb K, Wirjadi O, Schladitz K (2007).
newblock Fiber orientation estimation from 3d image data: Practical
algorithms, visualization, and interpretation.
newblock In: Proc. 7th Int. Conf. Hybrid Intelligent Systems. Kaiserslautern,
Germany.
bibitem[{Schausberger emph{et~al.}(2010)Schausberger, Heise, Maurer, Bernet,
Ritsch-Marte, and Stifter}]{SHMB10}
Schausberger SE, Heise B, Maurer C, Bernet S, Ritsch-Marte M, Stifter D (2010).
newblock Flexible contrast for low-coherence interference microscopy by
{F}ourier-plane filtering with a spatial light modulator.
newblock Optics letters 35:4154--6.
bibitem[{Sedlazeck(2008)}]{Sed2008}
Sedlazeck A (2008).
newblock Local feature detection by higher order {R}iesz transforms on images.
newblock Diplomarbeit, Christian-Albrechts-Universitat zu Kiel, Germany.
bibitem[{Sliseris emph{et~al.}(2016)Sliseris, Andr{"a}, Kabel, Wirjadi, Dix,
and Plinke}]{Sliseris2016}
Sliseris J, Andr{"a} H, Kabel M, Wirjadi O, Dix B, Plinke B (2016).
newblock Estimation of fiber orientation and fiber bundles of mdf.
newblock Materials and Structures 49:4003--12.
bibitem[{Steidl and Teuber(2009)}]{ST09}
Steidl G, Teuber T (2009).
newblock Anisotropic smoothing using double orientation.
newblock In: Lie A, Lysaker M, Morken K, Tai XC, eds., 2nd International
Conference on Scale Space Methods and Variational Methods in Computer Vision,
SSVM 2009, Voss, Norway, June 1-5, 2009., vol. 5567 of emph{Lecture Notes in
Computer Science}. Springer.
bibitem[{Stein and Weiss(1971)}]{SW1971}
Stein EM, Weiss GL (1971).
newblock Introduction to Fourier Analysis on Euclidean Spaces, vol.~1.
newblock Princeton University Press.
bibitem[{Storath(2011)}]{Sto2011}
Storath M (2011).
newblock Directional multiscale amplitude and phase decomposition by the
monogenic curvelet transform.
newblock SIAM Journal on Imaging Sciences 4:57--78.
bibitem[{Unser and Van De~Ville(2010)}]{UV2010}
Unser M, Van De~Ville D (2010).
newblock Wavelet steerability and the higher-order {R}iesz transform.
newblock Image Processing IEEE Transactions on 19:636--52.
bibitem[{Wells emph{et~al.}(2016)Wells, Holmes, and
Haverkamp}]{haverkamp2016}
Wells HC, Holmes G, Haverkamp RG (2016).
newblock Looseness in bovine leather: microstructural characterization.
newblock Journal of the Science of Food and Agriculture 96:2731--6.
bibitem[{Wirjadi emph{et~al.}(2014)Wirjadi, Godehardt, Schladitz, Wagner,
Rack, Gurka, Nissle, and Noll}]{wirjadi2014:multilayer}
Wirjadi O, Godehardt M, Schladitz K, Wagner B, Rack A, Gurka M, Nissle S, Noll
A (2014).
newblock Characterization of multilayer structures in fiber reinforced polymer
employing synchrotron and laboratory {X}-ray {CT}.
newblock International Journal of Materials Research 105:645--54.
bibitem[{Wirjadi emph{et~al.}(2016)Wirjadi, Schladitz, Easwaran, and
Ohser}]{wirjadi16}
Wirjadi O, Schladitz K, Easwaran P, Ohser J (2016).
newblock Estimating fibre direction distributions of reinforced composites
from tomographic images.
newblock Image Analysis and Stereology 35:167--79.
bibitem[{Wirjadi emph{et~al.}(2009)Wirjadi, Schladitz, Rack, and
Breuel}]{wirjadi2009applications}
Wirjadi O, Schladitz K, Rack A, Breuel T (2009).
newblock Applications of anisotropic image filters for computing 2d and
d-fiber orientations.
newblock In: Stereology and Image Analysis--10th European Congress of ISS.
bibitem[{Wonisch and W{"u}st(2014)}]{wonisch2014}
Wonisch A, W{"u}st A (2014).
newblock More precise part design:accurate simulation of fiber orientation of
glass fiber-reinforced plastics.
newblock Kunststoffe International :80--3.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2019 Image Analysis & Stereology
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
