Abstract
We present a method for estimating the surface volume of four-dimensional objects in discrete binary images. A surface volume weight is assigned to each 2 × 2 × 2 × 2 configuration of image elements. The total surface volume of a digital 4D object is given by a summation of the local volume contributions. Optimal volume weights are derived in order to provide an unbiased estimate with minimal variance for randomly oriented digitized planar hypersurfaces. Only 14 out of 64 possible boundary configurations appear on planar hypersurfaces. We use a marching hypercubes tetrahedrization to assign surface volume weights to the non-planar cases. The correctness of the method is verified on four-dimensional balls and cubes digitized in different sizes. The algorithm is appealingly simple; the use of only a local neighbourhood enables efficient implementations in hardware and/or in parallel architectures.
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Bhaniramka, P., Wenger, R., Crawfis, R.: Isosurface construction in any dimension using convex hulls. IEEE Trans. on Vision and Computer Graphics 10(2), 130–141 (2004)
Borgefors, G.: Weighted digital distance transforms in four dimensions. Discrete Applied Mathematics 125(1), 161–176 (2003)
Coeurjolly, D., Flin, F., Teytaud, O., Tougne, L.: Multigrid convergence and surface area estimation. In: Asano, T., Klette, R., Ronse, C. (eds.) Geometry, Morphology, and Computational Imaging. LNCS, vol. 2616, pp. 101–119. Springer, Heidelberg (2003)
Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators. In: Proceedings of the 16th International Conference on Pattern Recognition (ICPR), Quebec, vol. IV, pp. 330–334. IEEE Computer Science, Los Alamitos (2002)
Freeman, H.: Boundary encoding and processing. In: Lipkin, B.S., Rosenfeld, A. (eds.) Picture Processing and Psychopictorics, New York, pp. 241–266. Academic Press, London (1970)
Jonker, P.P.: Skeletons in N dimensions using shape primitives. Pattern Recognition Lett. 23(4), 677–686 (2002)
Kenmochi, Y., Klette, R.: Surface area estimation for digitized regular solids. In: Latecki, L.J., Melter, R.A., Mount, D.M., Wu, A.Y. (eds.) Proc. SPIE Vision Geometry IX, vol. 4117, pp. 100–111 (2000)
Klette, R.: Multigrid convergence of geometric features. In: Bertrand, G., Imiya, A., Klette, R. (eds.) Digital and Image Geometry. LNCS, vol. 2243, pp. 314–333. Springer, Heidelberg (2002)
Klette, R., Sun, H.J.: Digital planar segment based polyhedrization for surface area estimation. In: Arcelli, C., Cordella, L.P., Sanniti di Baja, G. (eds.) IWVF 2001. LNCS, vol. 2059, pp. 356–366. Springer, Heidelberg (2001)
Kulpa, Z.: Area and perimeter measurement of blobs in discrete binary pictures. Computer Graphics and Image Processing 6, 434–454 (1977)
Lindblad, J.: Surface area estimation of digitized planes using weighted local configurations. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 348–357. Springer, Heidelberg (2003)
Lindblad, J.: Surface area estimation of digitized 3D objects using weighted local configurations. Image and Vision Computing 23(2), 111–122 (2005); Special issue on Discrete Geometry for Computer Imagery.
Lorensen, W.E., Cline, H.E.: Marching Cubes: A high resolution 3D surface construction algorithm. In: Proceedings of the 14th ACM SIGGRAPH on Computer Graphics, vol. 21, pp. 163–169 (1987)
Proffit, D., Rosen, D.: Metrication errors and coding efficiency of chain-encoding schemes for the representation of lines and edges. Computer Graphics and Image Processing 10, 318–332 (1979)
Roberts, J.C., Hill, S.: Piecewise linear hypersurfaces using the marching cubes algorithm. In: Erbacher, R.F., Pang, A. (eds.) Proceedings of SPIE Visual Data Exploration and Analysis VI, vol. 3643, pp. 170–181 (1999)
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Lindblad, J. (2005). Surface Volume Estimation of Digitized Hyperplanes Using Weighted Local Configurations. In: Andres, E., Damiand, G., Lienhardt, P. (eds) Discrete Geometry for Computer Imagery. DGCI 2005. Lecture Notes in Computer Science, vol 3429. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31965-8_24
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DOI: https://doi.org/10.1007/978-3-540-31965-8_24
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