Abstract
When fitting archaeological artifacts, one would like to have a representation that simplifies fragments while preserving their complementarity. In this paper, we propose to employ the scale-spaces of mathematical morphology to hierarchically simplify potentially fitting fracture surfaces. We study the masking effect when morphological operations are applied to selected subsets of objects. Since fitting locally depends on the complementarity of fractures only, we introduce ‘Boundary Morphology’ on surfaces rather than volumes. Moreover, demonstrating the Lipschitz nature of the terracotta fractures informs our novel extrusion method to compute both closing and opening operations simultaneously. We also show that in this proposed representation the effects of abrasion and uncertainty are naturally bounded, justifying the morphological approach. This work is an extension of our contribution earlier published in the proceedings of ISMM2019 [10].
References
[1] Attali, D., Boissonnat, J.D., Edelsbrunner, H.: Stability and computation of medial axes - a state-of-the-art report pp. 109–125 (2008)10.1007/b106657_6Search in Google Scholar
[2] van den Boomgaard, R., Dorst, L.: The morphological equivalent of Gaussian scale space. In: Proceedings of Gaussian Scale-Space Theory. pp. 203–220. Kluwer (1997)10.1007/978-94-015-8802-7_15Search in Google Scholar
[3] Bosworth, J.H., Acton, S.T.: Morphological scale-space in image processing. Digital Signal Processing 13(2), 338–367 (2003)10.1016/S1051-2004(02)00033-7Search in Google Scholar
[4] Brown, B., Laken, L., Dutré, P., Gool, L.V., Rusinkiewicz, S., Weyrich, T.: Tools for virtual reassembly of fresco fragments. International Journal of Heritage in the Digital Era 1(2), 313–329 (2012)10.1260/2047-4970.1.2.313Search in Google Scholar
[5] Calderon, S., Boubekeur, T.: Point morphology. ACM Trans. Graph. 33(4), 1–13 (2014)Search in Google Scholar
[6] Chen, M.h., P.-F., Y.: A multiscaling approach based on morphological filtering. IEEE Transactions on PAMI 11(7), 694–700 (1989)10.1109/34.192464Search in Google Scholar
[7] Crane, K., de Goes, F., Desbrun, M., Schröder, P.: Digital geometry processing with discrete exterior calculus. In: ACM SIGGRAPH 2013 courses. SIGGRAPH ’13, ACM, New York, NY, USA (2013)10.1145/2504435.2504442Search in Google Scholar
[8] Dorst, L., van den Boomgaard, R.: The systems theory of contact. In: Sommer, G., Zeevi, Y. (eds.) Algebraic Frames for the Perception-Action Cycle, Lecture Notes in Computer Science, vol. 1888, pp. 22–47. Springer Verlag (2000)10.1007/10722492_2Search in Google Scholar
[9] ElNaghy, H., Dorst, L.: Geometry based faceting of 3D digitized archaeological fragments. In: 2017 IEEE ICCV Workshops. pp. 2934–2942 (2017)10.1109/ICCVW.2017.346Search in Google Scholar
[10] ElNaghy, H., Dorst, L.: Complementarity-preserving fracture morphology for archaeological fragments. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds.) Mathematical Morphology and Its Applications to Signal and Image Processing. pp. 403–414. Springer International Publishing, Cham (2019)10.1007/978-3-030-20867-7_31Search in Google Scholar
[11] Fitzgibbon, A.: Robust registration of 2D and 3D point sets. Image and Vision Computing 21, 1145–1153 (2003)10.1016/j.imavis.2003.09.004Search in Google Scholar
[12] Funkhouser, T., Shin, H., Toler-Franklin, C., Castañeda, A.G., Brown, B., Dobkin, D., Rusinkiewicz, S., Weyrich, T.: Learning how to match fresco fragments. J. Comput. Cult. Herit. 4(2), 1–13 (2011)Search in Google Scholar
[13] Hachenberger, P.: 3D Minkowski sum of polyhedra. In: CGAL User and Reference Manual. CGAL Editorial Board, 4.11.1 edn. (2018)Search in Google Scholar
[14] Huang, Q.X., Flöry, S., Gelfand, N., Hofer, M., Pottmann, H.: Reassembling fractured objects by geometric matching. In: ACM SIGGRAPH Papers. pp. 569–578. SIGGRAPH’06, ACM, New York, NY, USA (2006)10.1145/1141911.1141925Search in Google Scholar
[15] Jackway, P.T., Deriche, M.: Scale-space properties of the multiscale morphological dilation-erosion. IEEE Transactions on PAMI 18(1), 38–51 (1996)10.1109/34.476009Search in Google Scholar
[16] Kazhdan, M., Bolitho, M., Hoppe, H.: Poisson surface reconstruction. In: Proceedings of the Fourth Eurographics Symposium on Geometry Processing. pp. 61–70. SGP ’06, Eurographics Association, Aire-la-Ville, Switzerland, Switzerland (2006)Search in Google Scholar
[17] Maurer, C.R., Qi, R., Raghavan, V.: A linear time algorithm for computing exact Euclidean distance transforms of binary images in arbitrary dimensions. IEEE Transactions on PAMI 25(2), 265–270 (2003)10.1109/TPAMI.2003.1177156Search in Google Scholar
[18] McBride, J.C., Kimia, B.B.: Archaeological fragment reconstruction using curve-matching. In: IEEE CVPR Workshops. vol. 1, pp. 3–3 (2003)10.1109/CVPRW.2003.10008Search in Google Scholar
[19] Nacken, P.F.M.: Chamfer metrics, the medial axis and mathematical morphology. Journal of Mathematical Imaging and Vision 6(2), 235–248 (Jun 1996)10.1007/BF00119841Search in Google Scholar
[20] Palmas, G., Pietroni, N., Cignoni, P., Scopigno, R.: A computer-assisted constraint-based system for assembling fragmented objects. In: 2013 Digital Heritage International Congress (DigitalHeritage). vol. 1, pp. 529–536 (2013)10.1109/DigitalHeritage.2013.6743793Search in Google Scholar
[21] Papaioannou, G., Schreck, T., Andreadis, A., Mavridis, P., Gregor, R., Sipiran, I., Vardis, K.: From reassembly to object completion: A complete systems pipeline. J. Comput. Cult. Herit. 10(2), 1–22 (2017)Search in Google Scholar
[22] Salinas, D., Lafarge, F., Alliez, P.: Structure-aware mesh decimation. In: Computer Graphics Forum. vol. 34, pp. 211–227. The Eurographs Association (2015)10.1111/cgf.12531Search in Google Scholar
[23] Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, Inc., Orlando, FL, USA (1983)Search in Google Scholar
[24] Serra, J.C.: Lipschitz lattices and numerical morphology. SPIE Conf. Image Algebra and Morphological Image Processing II 1568, 1568–12 (1991)Search in Google Scholar
[25] Sethian, J.: Fast marching methods and level set methods for propagating interfaces. Computational Fluid Dynamics, 29th Annual Lecture Series (1998)10.1007/978-3-0348-8724-3_36Search in Google Scholar
[26] Toler-Franklin, C., Brown, B., Weyrich, T., Funkhouser, T., Rusinkiewicz, S.: Multi-feature matching of fresco fragments. ACM Trans. Graph. 29(6), 1–12 (2010)Search in Google Scholar
[27] van den Boomgaard, R., Smeulders, A.: The morphological structure of images: the differential equations of morphological scale-space. IEEE Transactions on Pattern Analysis and Machine Intelligence 16(11), 1101–1113 (Nov 1994)10.1109/34.334389Search in Google Scholar
[28] Wu, Z., Song, S., Khosla, A., Yu, F., Zhang, L., Tang, X., Xiao, J.: 3D shapenets: A deep representation for volumetric shapes. In: IEEE CVPR. pp. 1912–1920 (2015)Search in Google Scholar
© 2020 Hanan ElNaghy et al., published by De Gruyter
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