Abstract
We consider the problem of interpolating a surface based on sparse data such as individual points or level lines. We derive interpolators satisfying a list of desirable properties with an emphasis on preserving the geometry and characteristic features of the contours while ensuring smoothness across level lines. We propose an anisotropic third-order model and an efficient method to adaptively estimate both the surface and the anisotropy. Our experiments show that the approach outperforms AMLE and higher-order total variation methods qualitatively and quantitatively on real-world digital elevation data.
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References
Almansa, A.: échantillonnage, interpolation et détection. applications en imagerie satellitaire. Technical report, ENS Cachan (2002)
Almansa, A., Cao, F., Gousseau, Y., Rouge, B.: Interpolation of digital elevation models using AMLE and related methods. Geoscience and Remote Sensing 40, 314–325 (2002)
Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations of image processing. Arch. Rational Mech. 123, 199–257 (1993)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Clarendon Press (2000)
Carr, J.C., Fright, W.R., Beatson, R.K.: Surface interpolation with radial basis functions for medical imaging. Trans. Med. Imaging 16(1), 96–107 (1997)
Caselles, V., Morel, J.-M., Sbert, C.: An axiomatic approach to image interpolation. Trans. Image Proc. 7(3), 376–386 (1998)
Cressie, N.: Statistics for Spatial Data. Wiley, New York (1993)
Duchon, J.: Interpolation des fonctions de deux variables suivant le principe de la flexion des plaques minces. R.A.I.R.O. Anal. Numér. 10, 5–12 (1976)
Franke, R.: Scattered data interpolation: Test of some methods. Math. Comput. 38, 181–200 (1982)
Gesch, D., Evans, G., Mauck, J., Hutchinson, J., Carswell Jr., W.J.: The national map – elevation. U.S. Geological Survey Fact Sheet 3053 (2009)
Journel, A.G., Huijbregts, C.J.: Mining Geostatistics. Academic (1978)
Masnou, S., Morel, J.: Level lines based disocclusion. In: 5th IEEE Int’l Conf. on Image Processing, Chicago, IL, October 4-7, pp. 259–263 (1998)
Matheron, G.: La théorie des variables régionalisées, et ses applications. Technical Report 5, Les Cahiers du Centre de Morphol. Math. de Fontainebleau (1971)
Meinguet, J.: Surface Spline Interpolation: Basic Theory and Computational Aspects. In: Approximation Theory and Spline Functions, Dordrecht, Holland, pp. 124–142 (1984)
Meyer, T.: Coastal elevation from sparse level curves. Summer project under the guidance of T. Wittman, A. Bertozzi, and A. Chen, UCLA (2011)
Meyers, D., Skinner, S., Sloan, K.: Surfaces from contours. Trans. on Graphics 11(3), 228–258 (1992)
Mitas, L., Mitasova, H.: Spatial Interpolation. Wiley (1999)
Savin, O.: C 1 regularity for infinity harmonic functions in two dimensions. Arch. Ration. Mech. Anal. 176(3), 351–361 (2005)
Soille, P.: Spatial distributions from contour lines: An efficient method based on distance transformations. J. Vis. Commun. Image Represent. 2(2), 138–150 (1991)
Soille, P.: Generalized Geodesic Distances Applied to Interpolation and Shape Description. In: Mathematical Morphology and its Applications to Image Processing. Kluwer (1994)
Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer (1999)
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Lellmann, J., Morel, JM., Schönlieb, CB. (2013). Anisotropic Third-Order Regularization for Sparse Digital Elevation Models. In: Kuijper, A., Bredies, K., Pock, T., Bischof, H. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2013. Lecture Notes in Computer Science, vol 7893. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38267-3_14
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DOI: https://doi.org/10.1007/978-3-642-38267-3_14
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