Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access November 28, 2020

Adaptive Mathematical Morphology on Irregularly Sampled Signals in Two Dimensions

  • Teo Asplund EMAIL logo , Cris L. Luengo Hendriks , Matthew J. Thurley and Robin Strand


This paper proposes a way of better approximating continuous, two-dimensional morphology in the discrete domain, by allowing for irregularly sampled input and output signals. We generalize previous work to allow for a greater variety of structuring elements, both flat and non-flat. Experimentally we show improved results over regular, discrete morphology with respect to the approximation of continuous morphology. It is also worth noting that the number of output samples can often be reduced without sacrificing the quality of the approximation, since the morphological operators usually generate output signals with many plateaus, which, intuitively do not need a large number of samples to be correctly represented. Finally, the paper presents some results showing adaptive morphology on irregularly sampled signals.

MSC 2010: 68U10; 94A08; 94A20


[1] T. Asplund, C. L. Luengo Hendriks, M. J. Thurley, and R. Strand. Estimating the gradient for images with missing samples using elliptical structuring elements. Submitted.Search in Google Scholar

[2] T. Asplund, C. L. Luengo Hendriks, M. J. Thurley, and R. Strand. Mathematical morphology on irregularly sampled signals. In Asian Conference on Computer Vision - DGMM4CV Workshop, pages 506–520. Springer, 2016.10.1007/978-3-319-54427-4_37Search in Google Scholar

[3] T. Asplund, C. L. Luengo Hendriks, M. J. Thurley, and R. Strand. Mathematical morphology on irregularly sampled data in one dimension. Mathematical Morphology-Theory and Applications, 2(1):1–24, 2017.10.1515/mathm-2017-0001Search in Google Scholar

[4] T. Asplund, A. Serna, B. Marcotegui, R. Strand, and C. L. L. Hendriks. Mathematical morphology on irregularly sampled data applied to segmentation of 3D point clouds of urban scenes. In International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, pages 375–387. Springer, 2019.10.1007/978-3-030-20867-7_29Search in Google Scholar

[5] M. Breuß, B. Burgeth, and J. Weickert. Anisotropic continuous-scale morphology. In Iberian Conference on Pattern Recognition and Image Analysis, pages 515–522. Springer, 2007.10.1007/978-3-540-72849-8_65Search in Google Scholar

[6] R. W. Brockett and P. Maragos. Evolution equations for continuous-scale morphology. In Acoustics, Speech, and Signal Processing, 1992. ICASSP-92., 1992 IEEE International Conference on, volume 3, pages 125–128. IEEE, 1992.10.1109/ICASSP.1992.226260Search in Google Scholar

[7] R. W. Brockett and P. Maragos. Evolution equations for continuous-scale morphological filtering. Signal Processing, IEEE Transactions on, 42(12):3377–3386, 1994.10.1109/78.340774Search in Google Scholar

[8] S. Calderon and T. Boubekeur. Point morphology. ACM Transactions on Graphics (TOG), 33(4):45, 2014.10.1145/3263365Search in Google Scholar

[9] L. Cammoun, C. A. Castaño-Moraga, E. Muñoz-Moreno, D. Sosa-Cabrera, B. Acar, M. Rodriguez-Florido, A. Brun, H. Knutsson, and J. Thiran. A review of tensors and tensor signal processing. In Tensors in Image Processing and Computer Vision, pages 1–32. Springer, 2009.10.1007/978-1-84882-299-3_1Search in Google Scholar

[10] O. Cuisenaire. Distance transformations: fast algorithms and applications to medical image processing. PhD thesis, Université Catholique de Louvain, Belgique, 1999.Search in Google Scholar

[11] W. Förstner. A feature based correspondence algorithm for image matching. Int. Arch. of Photogrammetry and Remote Sensing, 26:150–166, 1986.Search in Google Scholar

[12] W. Förstner and E. Gülch. A fast operator for detection and precise location of distinct points, corners and centres of circular features. In Proc. ISPRS intercommission conference on fast processing of photogrammetric data, pages 281–305. Interlaken, 1987.Search in Google Scholar

[13] J. Gomes and O. Faugeras. Reconciling distance functions and level sets. Journal of Visual Communication and Image Representation, 11(2):209–223, 2000.10.1006/jvci.1999.0439Search in Google Scholar

[14] C. L. Hendriks, G. M. van Kempen, and L. J. van Vliet. Improving the accuracy of isotropic granulometries. Pattern Recognition Letters, 28(7):865–872, 2007.10.1016/j.patrec.2006.12.001Search in Google Scholar

[15] H. Knutsson. Representing local structure using tensors. In 6th Scandinavian Conference on Image Analysis, Oulu, Finland, pages 244–251. Linköping University Electronic Press, 1989.Search in Google Scholar

[16] H. Knutsson and C.-F. Westin. Normalized and differential convolution. In Computer Vision and Pattern Recognition, 1993. Proceedings CVPR’93., 1993 IEEE Computer Society Conference on, pages 515–523. IEEE, 1993.Search in Google Scholar

[17] A. Landström and M. J. Thurley. Adaptive morphology using tensor-based elliptical structuring elements. Pattern Recognition Letters, 34(12):1416–1422, 2013.10.1016/j.patrec.2013.05.003Search in Google Scholar

[18] H. Nyquist. Certain topics in telegraph transmission theory. Transactions of the AIEE, pages 617–644, 1928. [reprinted in: Proceedings of the IEEE, vol. 90, no. 2, pp. 280-305, February 2002].10.1109/5.989875Search in Google Scholar

[19] S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed: algorithms based on hamilton-jacobi formulations. Journal of computational physics, 79(1):12–49, 1988.10.1016/0021-9991(88)90002-2Search in Google Scholar

[20] G. Sapiro, R. Kimmel, D. Shaked, B. B. Kimia, and A. M. Bruckstein. Implementing continuous-scale morphology via curve evolution. Pattern recognition, 26(9):1363–1372, 1993.10.1016/0031-3203(93)90142-JSearch in Google Scholar

[21] C. E. Shannon. Communication in the presence of noise. Proceedings of the IRE, 37(1):10–21, 1949. [reprinted in: Proceedings of the IEEE, vol. 86, no. 2, pp. 447–457, February 1998].10.1109/JRPROC.1949.232969Search in Google Scholar

[22] S. R. Sternberg. Grayscale morphology. Computer vision, graphics, and image processing, 35(3):333–355, 1986.10.1016/0734-189X(86)90004-6Search in Google Scholar

[23] M. J. Thurley. Three dimensional data analysis for the separation and sizing of rock piles in mining (PhD thesis). Monash University, 2002.Search in Google Scholar

[24] R. van den Boomgaard. Mathematical Morphology: Extensions Towards Computer Vision. PhD thesis, 1992.Search in Google Scholar

[25] R. van den Boomgaard and A. Smeulders. The morphological structure of images: The differential equations of morphological scale-space. IEEE transactions on pattern analysis and machine intelligence, 16(11):1101–1113, 1994.10.1109/34.334389Search in Google Scholar

[26] L. J. Van Vliet, I. T. Young, and G. L. Beckers. A nonlinear Laplace operator as edge detector in noisy images. Computer vision, graphics, and image processing, 45(2):167–195, 1989.10.1016/0734-189X(89)90131-XSearch in Google Scholar

[27] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transactions on image processing, 13(4):600–612, 2004.10.1109/TIP.2003.819861Search in Google Scholar

[28] J. Weickert. Anisotropic diffusion in image processing. Teubner Stuttgart, 1998.Search in Google Scholar

Received: 2019-10-24
Accepted: 2020-11-02
Published Online: 2020-11-28

© 2020 Teo Asplund et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 29.11.2023 from
Scroll to top button