Abstract
Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherence-enhancing diffusion methods. Therefore, we propose a method for coherence-enhancing diffusion on the invertible orientation score of a 2D image. In an orientation score, the local orientation is represented by an additional third dimension, ensuring that crossing elongated structures are separated from each other. We consider orientation scores as functions on the Euclidean motion group, and use the group structure to apply left-invariant diffusion equations on orientation scores. We describe how we can calculate regularized left-invariant derivatives, and use the Hessian to estimate three descriptive local features: curvature, deviation from horizontality, and orientation confidence. These local features are used to adapt a nonlinear coherence-enhancing, crossing-preserving, diffusion equation on the orientation score. We propose two explicit finite-difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis. Experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherence-enhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore, the use of deviation from horizontality makes it feasible to reduce the number of sampled orientations while still preserving crossings.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Antoine, J.-P., & Murenzi, R. (1996). Two-dimensional directional wavelets and the scale-angle representation. Signal Processing, 52(3), 241–272.
Antoine, J.-P., Murenzi, R., & Vandergheynst, P. (1999). Directional wavelets revisited: Cauchy wavelets and symmetry detection in patterns. Applied and Computational Harmonic Analysis, 6(3), 314–345.
August, J. (2001). The curve indicator random field. Ph.D. thesis, Yale University.
Candès, E. J., & Donoho, D. L. (1999a). Curvelets—a surprisingly effective nonadaptive representation for objects with edges. In A. Cohen, C. Rabut, & L. L. Schumaker (Eds.), Curve and surface fitting: Saint-Malo 1999. Nashville: Vanderbilt University Press.
Candès, E. J., & Donoho, D. L. (1999b). Ridgelets: the key to high dimensional intermittency? Philosophical Transactions of the Royal Society of London A, 357, 2495–2509.
Chen, J., Sato, Y., & Tamura, S. (2000). Orientation space filtering for multiple orientation line segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(5), 417–429.
Citti, G., & Sarti, A. (2006). A cortical based model of perceptional completion in the roto-translation space. Journal of Mathematical Imaging and Vision, 24(3), 307–326.
Cottet, G.-H., & Germain, L. (1993). Image processing through reaction combined with nonlinear diffusion. Mathematics of Computation, 61, 659–667.
Duits, R. (2005). Perceptual organization in image analysis. Ph.D. thesis, Technische Universiteit Eindhoven. http://www.bmi2.bmt.tue.nl/Image-Analysis/People/RDuits/THESISRDUITS.pdf.
Duits, R., & Burgeth, B. (2007). Scale spaces on Lie groups. In F. Sgallari, A. Murli, & N. Paragios (Eds.), Lecture notes in computer science : Vol. 4485. Scale space and variational methods in computer vision: proceedings of the first international conference, SSVM 2007 (pp. 300–312). Ischia, Italy, May–June 2007. Berlin: Springer.
Duits, R., & Franken, E. M. (2007). Left-invariant stochastic evolution equations on SE(2) and its applications to contour enhancement and contour completion via invertible orientation scores. arXiv: 0711.0951v4, 2007. http://arxiv.org/abs/0711.0951. Also available as CASA report nr. 35, Eindhoven University of Technology.
Duits, R., & Franken, E. M. (2009a, accepted). Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores—part I: Linear left-invariant diffusion equations on SE(2). Quarterly on Applied Mathematics.
Duits, R., & Franken, E. M. (2009b, accepted) Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores—part II: Nonlinear left-invariant diffusions on invertible orientation scores. Quarterly on Applied Mathematics.
Duits, R., & van Almsick, M. A. (2008). The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-Euclidean motion group. AMS Quarterly of Applied Mathematics, 66, 27.
Duits, R., van Almsick, M. A., Duits, M., Franken, E. M., & Florack, L. M. J. (2004). Image processing via shift-twist invariant operations on orientation bundle functions. In N. Zhuralev et al. (Eds.), 7th International conference on pattern recognition and image analysis (PRIA-7-2004) (pp. 193–196). St. Petersburg, October 2004.
Duits, R., Duits, M., van Almsick, M. A., & ter Haar Romeny, B. M. (2007). Invertible orientation scores as an application of generalized wavelet theory. Pattern Recognition and Image Analysis, 17(1), 42–75.
Felsberg, M., Forssén, P.-E., & Scharr, H. (2006). Channel smoothing: Efficient robust smoothing of low-level signal features. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(2), 209–222.
Florack, L. M. J., ter Haar Romeny, B. M., Koenderink, J. J., & Viergever, M. A. (1993). Cartesian differential invariants in scale-space. Journal of Mathematical Imaging and Vision, 3(4), 327–348.
Foolen, J., van Donkelaar, C., Nowlan, N., Murphy, P., Huiskes, R., & Ito, K. (2008). Collagen orientation in periosteum and perichondrium is aligned with preferential directions of tissue growth. Journal of Orthopaedic Research: official publication of the Orthopaedic Research Society.
Franken, E. M. (2008). Enhancement of crossing elongated structures in images. Ph.D. thesis, Eindhoven University of Technology, Department of Biomedical Engineering, Eindhoven, The Netherlands.
Franken, E. M., Rongen, P., van Almsick, M. A., & ter Haar Romeny, B. M. (2006). Detection of electrophysiology catheters in noisy fluoroscopy images. In Lecture notes in computer science : Vol. 4191. Proceedings of the 9th international conference on medical image computing and computer-assisted intervention—MICCAI 2006 (pp. 25–32). Copenhagen, Denmark, 1–6 October 2006. Berlin: Springer.
Franken, E. M., Duits, R., & ter Haar Romeny, B. M. (2007a). Nonlinear diffusion on the 2D Euclidean motion group. In F. Sgallari, A. Murli, & N. Paragios (Eds.), Lecture notes in computer science : Vol. 4485. Scale space and variational methods in computer vision: proceedings of the first international conference, SSVM 2007 (pp. 461–472). Ischia, Italy, May–June 2007. Berlin: Springer.
Franken, E. M., Duits, R., & ter Haar Romeny, B. M. (2007b). Curvature estimation for enhancement of crossing curves. In W. Niessen, C.-F. Westin, & M. Nielsen (Eds.), Proceedings of the 8th IEEE Computer Society workshop on mathematical methods in biomedical image analysis, held in conjunction with the IEEE international conference on computer vision, Rio de Janeiro, Brazil, 14–20 October 2007. Omnipress, Digital proceedings.
Gerschgorin, S. (1931). Über die Abgrenzung der Eigenwerte einer Matrix. Izv. Akad. Nauk. USSR Otd. Fiz.-Mat. Nauk., 7, 749–754.
Granlund, G. H., & Knutsson, H. (1995). Signal processing for computer vision. Dordrecht: Kluwer Academic.
Heitger, F., & von der Heydt, R. (1993). A computational model of neural contour processing. In Proceedings of the 4th international conference on computer vision (pp. 32–40). Berlin, Germany, 20–23 June 1993. Washington: IEEE Computer Society Press.
Kalitzin, S. N., ter Haar Romeny, B. M., & Viergever, M. A. (1997). Invertible orientation bundles on 2D scalar images. In B. M. ter Haar Romeny, L. M. J. Florack, J. Koenderink, & M. Viergever (Eds.), Scale space theory in computer vision (pp. 77–88).
Kalitzin, S. N., ter Haar Romeny, B. M., & Viergever, M. A. (1999). Invertible apertured orientation filters in image analysis. International Journal of Computer Vision, 31(2–3), 145–158.
Manniesing, R., & Niessen, W. J. (2005). Multiscale vessel enhancing diffusion in ct angiography noise filtering. In Lecture notes in computer science : Vol. 3565. Information processing in medical imaging (pp. 138–149). Berlin: Springer.
Manniesing, R., Viergever, M. A., & Niessen, W. J. (2006). Vessel enhancing diffusion: A scale space representation of vessel structures. Medical Image Analysis, 10(6), 815–825.
Mumford, D. (1994). Elastica and computer vision. In C. L. Bajaj (Ed.), Algebraic geometry and its applications (pp. 491–506). New York: Springer.
Nitzberg, M., & Shiota, T. (1992). Nonlinear image filtering with edge and corner enhancement. IEEE Transactions on Pattern Analysis and Machine Intelligence, 14, 826–833.
Rubbens, M. P., Mol, A., Boerboom, R. A., Bank, R. A., Baaijens, F. P. T., & Bouten, C. V. C. (2008, in press). Intermittent straining accelerates the development of tissue properties in engineered heart valve tissue. Tissue Engineering Part A, 14. doi:10.1089/ten.tea.2007.0396.
Scharr, H. (2006). Diffusion-like reconstruction schemes from linear data models. In Lecture notes in computer science : Vol. 4174. Pattern recognition: 28th DAGM symposium (pp. 51–60). Berlin, Germany, 12–14 September 2006. Berlin: Springer.
Shaw, C. S., Jones, D. A., & Wagenmakers, A. J. M. (2008). Network distribution of mitochondria and lipid droplets in human muscle fibres. Histochemistry and Cell Biology, 129(1), 65–72.
Starck, J.-L., Candès, J. E., & Donoho, D. L. (2002). The curvelet transform for image denoising. IEEE Transactions on Image Processing, 11(6), 670–684.
Tschumperlé, D. (2006). Fast anisotropic smoothing of multi-valued images using curvature-preserving pde’s. International Journal of Computer Vision, 68(1), 65.
Tuch, D. S., Weisskoff, R. M., Belliveau, J. W., & Wedeen, V. J. (1999). High angular resolution diffusion imaging of the human brain. In Proc. of the 7th annual meeting of ISMRM, Philadelphia (p. 321).
Unser, M. (1999). Splines: A perfect fit for signal and image processing. IEEE Signal Processing Magazine, 16(6), 22–38.
Van Almsick, M. A. (2007). Context models of lines and contours. Ph.D. thesis, Eindhoven University of Technology, Department of Biomedical Engineering, Eindhoven, The Netherlands.
Van Ginkel, M. (2002). Image analysis using orientation space based on steerable filters. Ph.D. thesis, Technische Universiteit Delft, The Netherlands.
Walters, D. (1987). Selection of image primitives for general-purpose visual processing. Computer Vision, Graphics, and Image Processing, 37(2), 261–298.
Weickert, J. A. (1998). Anisotropic diffusion in image processing. In European Consortium for Mathematics in Industry series. Stuttgart: Teubner.
Weickert, J. A. (1999). Coherence-enhancing diffusion filtering. International Journal of Computer Vision, 31(2–3), 111–127.
Weickert, J., & Scharr, H. (2002). A scheme for coherence-enhancing diffusion filtering with optimized rotation invariance. Journal of Visual Communication and Image Representation, 13(1–2), 103–118.
Welk, M., Weickert, J., & Steidl, G. (2006). From tensor-driven diffusion to anisotropic wavelet shrinkage. In A. Leonardis, H. Bischof, & A. Prinz (Eds.), Lecture notes in computer science : Vol. 3951–3954. Proceedings of the ninth European conference on computer vision (pp. 391–403). Graz, Austria, May 2006. Berlin/Heidelberg: Springer.
Williams, L. R., & Jacobs, D. W. (1997). Stochastic completion fields: a neural model of illusory contour shape and salience. Neural Computing, 9(4), 837–858.
Zweck, J., & Williams, L. R. (2004). Euclidean group invariant computation of stochastic completion fields using shiftable-twistable functions. Journal of Mathematical Imaging and Vision, 21(2), 135–154.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Franken, E., Duits, R. Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores. Int J Comput Vis 85, 253–278 (2009). https://doi.org/10.1007/s11263-009-0213-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-009-0213-5