Abstract
For 3D images composed of successive scanner slices (e.g. medical imaging, confocal microscopy or computed tomography), the sampling step may vary according to the axes, and specially according to the depth which can take values lower or higher than 1. Hence, the sampling grid turns out to be parallelepipedic. In this paper, 3D anisotropic local distance operators are introduced. The problem of coefficient optimization is addressed for arbitrary mask size. Lower and upper bounds of scaling factors used for integer approximation are given. This allows, first, to derive analytically the maximal normalized error with respect to Euclidean distance, in any 3D anisotropic lattice, and second, to compute optimal chamfer coefficients. As far as large images or volumes are concerned, 3D anisotropic operators are adapted to the measurement of distances between objects sampled on non-cubic grids as well as for quantitative comparison between grey level images.
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Borgefors, G.: Applications using distance transforms. In: Arcelli, C., Cordella, L.P., Sanniti di Baja., G. (eds.) Aspects of Visual Form Processing, pp. 83–108. World Scientific, Singapore (1994)
Borgefors, G.: Distance transformations in arbitrary dimensions. Computer Vision, Graphics and Image Processing 27, 312–345 (1984)
Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics and Image Processing 34, 344–371 (1986)
Cuisenaire, O., Macq, B.: Fast Euclidean distance transformation by propagation using multiple neighborhood. Computer Vision and Image Understanding 76, 163–172 (1999)
Maurer Jr., 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 Pattern Analysis and Machine Intelligence 25(2), 265–270 (2003)
Svensson, S., Borgefors, G.: Digital distance transforms in 3D images using information from neighbourhoods up to 5x5x5. Computer Vision and Image Understanding 88, 24–53 (2002)
Fouard, C., Malandain, G.: 3-D chamfer distances and norms in anisotropic grids. Image and Vision Computing 23, 143–158 (2005)
Fouard, C., Strand, R., Borgefors, G.: Weighted distance transforms generalize to modules and their computation on point lattices. Pattern Recognition 40, 2453–2474 (2007)
Hulin, J., Thiel, E.: Chordal axis on weighted distance transforms. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 271–282. Springer, Heidelberg (2006)
Verwer, B.: Local distances for distance transformations in two and three dimensions. Pattern Recognition Letters 12, 671–682 (1991)
Coquin, D., Bolon, P.: Discrete distance operator on rectangular grids. Pattern Recognition Letters 16, 911–923 (1995)
Remy, E., Thiel, E.: Optimizing 3D chamfer mask with norm constraints. In: Proceedings of International Workshop on Combinatorial Image Analysis, Caen, France, pp. 39–56 (2000)
Strand, R.: Weighted distances based on neighbourhood sequences. Pattern Recognition Letters 28(15) (2007)
Borgefors, G.: On digital distance transforms in three dimensions. Computer Vision and Image Understanding 64, 368–376 (1996)
Borgefors, G.: Weighted digital distance transforms in four dimensions. Discrete Applied Mathematics 125, 161–176 (2003)
Kiselman, C.: Regularity properties of distance transformations in image analysis. Computer Vision and Image Understanding 64, 390–398 (1996)
Sintorn, I.M., Borgefors, G.: Weighted distance transforms in rectangular grids. In: 11th International Conference on Image Analysis and Processing, Palermo, Italy, pp. 322–326 (2001)
Svensson, S., Borgefors, G.: Distance transforms in 3D using four different weights. Pattern Recognition Letters 23, 1407–1418 (2002)
Sintorn, I.M., Borgefors, G.: Weighted distance transforms for images using elongated voxel grids. In: Proc. 10th Discret Geometry for Computer Imagery, Bordeaux, France, pp. 244–254 (2002)
Chehadeh, Y., Coquin, D., Bolon, P.: A generalization to cubic and non cubic local distance operators on parallelepipedic grids. In: Proc. 5th Discret Geometry for Computer Imagery, pp. 27–36 (1995)
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Coquin, D., Bolon, P. (2009). Lower and Upper Bounds for Scaling Factors Used for Integer Approximation of 3D Anisotropic Chamfer Distance Operator. In: Brlek, S., Reutenauer, C., Provençal, X. (eds) Discrete Geometry for Computer Imagery. DGCI 2009. Lecture Notes in Computer Science, vol 5810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04397-0_39
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DOI: https://doi.org/10.1007/978-3-642-04397-0_39
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