Abstract
Quantifying shape complexity is useful in several practical problems in addition to being interesting from a theoretical point of view. In this paper, instead of assigning a single global measure of complexity, we propose a distributed coding where to each point on the shape domain a measure of its contribution to complexity is assigned. We define the shape simplicity as the expressibility of the shape via a prototype shape. To keep discussions concrete we focus on a case where the prototype is a rectangle. Nevertheless, the constructions in the paper is valid in higher dimensions where the prototype is a hyper-cuboid. Thanks to the connection between differential operators and mathematical morphology, the proposed construction naturally extends to the case where diamonds serve as the prototypes.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Moser, D., Zechmeister, H.G., Plutzar, C., Sauberer, N., Wrbka, T., Grabherr, G.: Landscape patch shape complexity as an effective measure for plant species richness in rural landscapes. Landscape Ecol. 17(7), 657–669 (2002)
Ngo, T.-T., Collet, C., Mazet, V.: Automatic rectangular building detection from VHR aerial imagery using shadow and image segmentation. In: IEEE International Conference on Image Processing, pp. 1483–1487 (2015)
Maragos, P.: Pattern spectrum and multiscale shape representation. IEEE Trans. Pattern Anal. Mach. Intell. 11(7), 701–716 (1989)
Rosin, P.L., Zunic, J.: Measuring squareness and orientation of shapes. J. Math. Imaging Vis. 39(1), 13–27 (2011)
Arslan, M.F., Tari, S.: Complexity of shapes embedded in \(\mathbb{Z}^n\) with a bias towards squares. IEEE Trans. Image Process. 5(6), 922–937 (2020)
Genctav, A., Tari, S.: Discrepancy: local/global shape characterization with a roundness bias. J. Math. Imaging Vis. 61(1), 160–171 (2019)
Varshney, K.R., Willsky, A.S.: Classification using geometric level sets. J. Mach. Learn. Res. 11, 491–516 (2010)
Fawzi, A., Moosavi-Dezfooli, S., Frossard, P.: The robustness of deep networks: a geometrical perspective. IEEE Signal Process. Mag. 34(6), 50–62 (2017)
Oberman, A.: A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions. Math. Comput. 74(251), 1217–1230 (2005)
Breuß, M., Weickert, J.: Highly accurate PDE-based morphology for general structuring elements. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 758–769. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02256-2_63
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Arslan, M.F., Tari, S. (2021). Local Culprits of Shape Complexity. In: Elmoataz, A., Fadili, J., Quéau, Y., Rabin, J., Simon, L. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2021. Lecture Notes in Computer Science(), vol 12679. Springer, Cham. https://doi.org/10.1007/978-3-030-75549-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-75549-2_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-75548-5
Online ISBN: 978-3-030-75549-2
eBook Packages: Computer ScienceComputer Science (R0)