Abstract
The time necessary to follow a path defined on a grey scale image is defined as the sum of the image values along the path. The geodesic time associated with two points of the image is nothing but the smallest amount of time necessary to link these two points. Starting from this notion, we define a new geodesic metric on the image plane: the generalized geodesic distance. The generalized geodesic distance betweeen two points is the length of the shortest path(s) linking these points in a minimum amount of time. This distance is used for defining a propagation function. Applications to shape description and interpolation from contour data are provided.
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References
Beucher, S. (1990), Segmentation d’images et morphologie mathématique, PhD thesis, Ecole des Mines de Paris.
Danielsson, P.-E. (1980), ‘Euclidean distance mapping’, Computer Graphics and Image Processing 14, 227–248.
Jeulin, D., Vincent, L. & Serpe, G. (1992), ‘Propagation algorithms on graphs for physical appli-cations’, Journal of Visual Communication and Image Representation 3(2), 161–181.
Kimmel, R. & Bruckstein, A. (1993), Sub¨Cpixel distance maps and weighted distance transforms, in ‘Geometric methods in computer vision II’, Vol. SPIE-2031, pp. 259–268.
Lantuéjoul, C. & Beucher, S. (1980), ‘Geodesic distance and image analysis’, Mikroskopie 37, 138–142.
Lantuéjoul, C. & Beucher, S. (1981), ‘On the use of the geodesic metric in image analysis’, Journal of Microscopy 121, 39–49.
Lantuéjoul, C. & Maisonneuve, F. (1984), ‘Geodesic methods in image analysis’, Pattern Recognition 17(2), 177–187.
Maisonneuve, F. & Lantuéjoul, C. (1984), ‘Geodesic convexity’, ACM Transactions on Graphics 3(2), 169–174.
Maisonneuve, F. & Schmitt, M. (1989), ‘An efficient algorithm to compute the hexagonal and dodecagonal propagation function’, ACM Transactions on Graphics 8(2), 515–520.
Meyer, F. (1992), Integrals and gradients of images, in P. Gader, E. Dougherty & J. Serra, eds, ‘Image Algebra and Morphological Image Processing III’, Vol. Proc. SPIE 1769, pp. 200–211.
Osher, S. & Sethian, J. (1988), ‘Fronts propagating with curvature speed: algorithms based on Hamilton¨CJacobi formulations’, Journal of Computational Physics 79, 12–49.
Piper, J. & Granum, E. (1987), ‘Computing distance transformations in convex and non-convex domains’, Pattern Recognition 20(6), 599–615.
Rutovitz, D. (1968), Data structures for operations on digital images, in G. Cheng, R. Ledley, D. Kollock & A. Rosenfeld, eds, ‘Pictorial pattern recognition’, pp. 105–133.
Sapiro, G., Kimmel, R., Shaked, D., Kimia, B. & Bruckstein, A. (1993), ‘Implementing continuous-scale morphology via curve evolution’, Pattern Recognition 26(9), 1363–1372.
Schmitt, M. (1989), ‘Geodesic arcs in non-Euclidean metrics, application to the propagation function’, Revue d’intelligence artificielle 3(2), 43–76.
Soille, P. (1991), ‘Spatial distributions from contour lines: an efficient methodology based on distance transformations’, Journal of Visual Communication and Image Representation 2(2), 138–150.
Soille, P. (1992), Morphologie mathématique: du relief à la dimensionalité—Algorithmes et méthodes—, PhD thesis, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
Soille, P. (1993a), Generalized geodesy via geodesic time, Submitted to Pattern Recognition Letter.
Soille, P. (1993b), The geodesic time function: principles and applications, in ‘The 1993 Interna-tional Conference on Confocal Microscopy & 3D Image Processing’, Sydney, p. 107.
Usson, Y. & Montanvert, A. (1993), Reconstruction tridimensionnelle à partir des distances dis-cr¨¨tes, in ‘Colloque géométrie discrète en imagerie: fondements et applications’, Strasbourg,pp. 20–21.
Verbeek, P. & Verwer, B. (1990), ‘Shading from shape, the eikonal equation solved by grey-weighted distance transform’, Pattern Recognition Letters 11, 681–690.
Verwer, B., Verbeek, P. & Dekker, S. (1989), ‘An efficient cost algorithm applied to distance transforms’, IEEE Transactions on Pattern Analysis and Machine Intelligence 11(4), 425429.
Vincent, L. (1991a), Efficient computation of various types of skeletons, in ‘Medical Imaging V’, Vol. Proc. SPIE 1445.
Vincent, L. (1991b), Exact Euclidean distance function by chain propagations, in ‘Proc. IEEE Computer Vision and Pattern Recognition’, pp. 520–525.
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© 1994 Springer Science+Business Media Dordrecht
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Soille, P. (1994). Generalized Geodesic Distances Applied to Interpolation and Shape Description. In: Serra, J., Soille, P. (eds) Mathematical Morphology and Its Applications to Image Processing. Computational Imaging and Vision, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1040-2_25
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DOI: https://doi.org/10.1007/978-94-011-1040-2_25
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4453-0
Online ISBN: 978-94-011-1040-2
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