Abstract
This chapter discusses a problem defined in a 3-dimensional regular grid. Such a grid is commonly used in 3D image analysis. We may also assume that a general 3D space (e.g., for a robot) is regularly subdivided into cubes of uniform size. The chapter considers shortest paths in such a cuboidal world.
When I was a Boy Scout, we played a game when new Scouts joined the troop. We lined up chairs in a pattern, creating an obstacle course through which the new Scouts, blindfolded, were supposed to manoeuvre. The Scoutmaster gave them a few moments to study the pattern before our adventure began. But as soon as the victims were blindfolded, the rest of us quietly removed the chairs.—I think life is like this game.
Pierce Vincent Eckhart
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Notes
- 1.
Published by Reinhard Klette and Thomas Bülow in 2000; see [32].
- 2.
Published between 2000 and 2002 in papers by Thomas Bülow and Reinhard Klette; see [11].
- 3.
See, for example, [17], pages 595–601.
- 4.
Note that we can classify a simple cube-curve in linear time to be first-class or not, by using the original rubberband algorithm: the curve is first-class iff option (O 2) does not occur at all.
- 5.
See, for example, [12, page 49].
- 6.
See [12, page 49].
- 7.
Formulated as an open problem on [33, page 406].
- 8.
This leads to two new open problems (smallest simple cube-curve without end-angle, and smallest simple cube-curve where none of the MLP vertices is a grid point). See Problem 9.7 at the end of this chapter. We consider that the second problem (i.e., all MLP vertices not at a grid point) is more difficult to solve.
- 9.
Note that it is impossible that four consecutive vertices of ρ are identical.
- 10.
Otherwise, we update e by removing a sufficiently small segment(s) from its endpoint. This is another way to handle the degenerate case (see Sect. 3.4) of the used RBA.
- 11.
For Line 2, see Lemma 6 in [40].
- 12.
Two digits are used only for displaying coordinates. Obviously, in the calculations it is necessary to use higher precision.
- 13.
- 14.
- 15.
Theorem by E. Galois; see also B.L. van der Waerden’s famous example p(x)=x 5−x−1.
- 16.
See Theorem 9 in [5], saying that the ESP problem is in general not solvable by radicals over the field of rationals.
- 17.
- 18.
- 19.
Named after the Norwegian mathematician Niels Henrik Abel (1802–1829).
- 20.
- 21.
See, for example, [36].
- 22.
See [36], page 441.
- 23.
We used GAP, see [22].
- 24.
For each a∈ℤ19, there exists a number j∈{0,1,…,18} such that 2j≡amod 19.
- 25.
The DSS technique represents a digital curve, assumed to be a sequence of grid points, by subsequent digital straight segments (DSSs) of maximum length; the MLP technique considers a digital curve as a sequence of grid cells, calculating a minimum-length polygon (MLP) in the union of these cells.
- 26.
Not to be confused with a 2D image segmentation algorithm of the same name [44].
References
Amarunnishad, T.M., Das, P.P.: Estimation of length for digitized straight lines in three dimensions. Pattern Recognit. Lett. 11, 207–213 (1990)
Asano, T., Asano, T., Guibas, L., Hershberger, J., Imai, H.: Visibility of disjoint polygons. Algorithmica 1, 49–63 (1986)
Asano, T., Kawamura, Y., Klette, R., Obokata, K.: Minimum-length polygons in approximation sausages. In: Proc. Int. Workshop Visual Form. LNCS, vol. 2059, pp. 103–112. Springer, Berlin (2004)
Bailey, D.: An efficient Euclidean distance transform. In: Proc. Int. Workshop Combinatorial Image Analysis. LNCS, vol. 3322, pp. 394–408. Springer, Berlin (2004)
Bajaj, C.: The algebraic complexity of shortest paths in polyhedral spaces. In: Proc. Allerton Conf. Commum. Control Comput., pp. 510–517 (1985)
Bajaj, C.: The algebraic degree of geometric optimization problems. Discrete Comput. Geom. 3, 177–191 (1988)
Berlekamp, E.R.: Factoring polynomials over large finite fields. Math. Comp. 24, 713–735 (1970)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge, UK (2004)
Bülow, T., Klette, R.: Rubber band algorithm for estimating the length of digitized space-curves. In: Proc. Intern. Conf. Pattern Recognition, vol. 3, pp. 551–555 (2000)
Bülow, T., Klette, R.: Approximation of 3D shortest polygons in simple cube curves. In: Proc. Digital and Image Geometry. LNCS, vol. 2243, pp. 281–294. Springer, Berlin (2001)
Bülow, T., Klette, R.: Digital curves in 3D space and a linear-time length estimation algorithm. IEEE Trans. Pattern Anal. Mach. Intell. 24, 962–970 (2002)
Burden, R.L., Faires, J.D.: Numerical Analysis, 7th edn. Brooks Cole, Pacific Grove (2000)
Chattopadhyay, S., Das, P.P.: Estimation of the original length of a straight line segment from its digitization in three dimensions. Pattern Recognit. 25, 787–798 (1992)
Choi, J., Sellen, J., Yap, C.-K.: Approximate Euclidean shortest path in 3-space. In: ACM Conf. Computational Geometry, pp. 41–48. ACM Press, New York (1994)
Choi, J., Sellen, J., Yap, C.-K.: Precision-sensitive Euclidean shortest path in 3-space. In: Proc. Annu. ACM Sympos. Computational Geometry, pp. 350–359 (1995)
Coeurjolly, D., Debled-Rennesson, I., Teytaud, O.: Segmentation and length estimation of 3D discrete curves In: Proc. Digital and Image Geometry. LNCS, vol. 2243, pp. 299–317. Springer, Berlin (2001)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press, Cambridge (2001)
Dorst, L., Smeulders, A.W.M.: Length estimators for digitized contours. Comput. Vis. Graph. Image Process. 40, 311–333 (1987)
Dror, M., Efrat, A., Lubiw, A., Mitchell, J.: Touring a sequence of polygons. In: Proc. STOC, pp. 473–482 (2003)
Ellis, T.J., Proffitt, D., Rosen, D., Rutkowski, W.: Measurement of the lengths of digitized curved lines. Comput. Graph. Image Process. 10, 333–347 (1979)
Ficarra, E., Benini, L., Macii, E., Zuccheri, G.: Automated DNA fragments recognition and sizing through AFM image processing. IEEE Trans. Inf. Technol. Biomed. 9, 508–517 (2005)
GAP—Groups, Algorithms, Programming—a system for computational discrete algebra. www-gap.mcs.st-and.ac.uk/gap.html (2011). Accessed July 2011
Ghosh, S.K., Mount, D.M.: An output sensitive algorithm for computing visibility graphs. SIAM J. Comput. 20, 888–910 (1991)
Herstein, I.N.: Topics in Algebra, 2nd edn. Wiley, New York (1975)
Hochbaum, D.S. (ed.): Approximation Algorithms for NP-Hard Problems. PWS Pub. Co., Boston (1997)
Jonas, A., Kiryati, N.: Length estimation in 3-D using cube quantization. In: Proc. Vision Geometry. SPIE, vol. 2356, pp. 220–230 (1994)
Jonas, A., Kiryati, N.: Length estimation in 3-D using cube quantization. J. Math. Imaging Vis. 8, 215–238 (1998)
Kapoor, S., Maheshwari, S.N.: Efficient algorithms for Euclidean shortest path and visibility problems with polygonal. In: Proc. Annu. ACM Sympos. on Computational Geometry, pp. 172–182 (1988)
Karavelas, M.I., Guibas, L.J.: Static and kinetic geometric spanners with applications. In: Proc. ACM–SIAM Symp. Discrete Algorithms, pp. 168–176 (2001)
Kiryati, N., Kubler, O.: On chain code probabilities and length estimators for digitized three-dimensional curves. Pattern Recognit. 28, 361–372 (1995)
Kiryati, N., Szekely, G.: Estimating shortest paths and minimal distances on digitized three-dimensional surfaces. Pattern Recognit. 26, 1623–1637 (1993)
Klette, R., Bülow, T.: Critical edges in simple cube-curves. In: Proc. Discrete Geometry Computational Imaging. LNCS, vol. 1953, pp. 467–478. Springer, Berlin (2000)
Klette, R., Rosenfeld, A.: Digital Geometry. Morgan Kaufmann, San Francisco (2004)
Klette, R., Yip, B.: The length of digital curves. Mach. Graph. Vis. 9, 673–703 (2000)
Klette, R., Kovalevsky, V., Yip, B.: Length estimation of digital curves. In: Proc. Vision Geometry. SPIE, vol. 3811, pp. 117–129 (1999)
Knuth, D.E.: The Art of Computer Programming, vol. 2, 3rd edn. Addison-Wesley, Reading (1997)
Lee, D.T.: Proximity and reachability in the plane. Ph.D. thesis, University of Illinois at Urbana–Champaign, Urbana (1978)
Li, F., Klette, R.: Minimum-length polygon of a simple cube-curve in 3D space. In: Proc. Int. Workshop Combinatorial Image Analysis. LNCS, vol. 3322, pp. 502–511. Springer, Berlin (2004)
Li, F., Klette, R.: The class of simple cube-curves whose MLPs cannot have vertices at grid points. In: Proc. Discrete Geometry Computational Imaging. LNCS, vol. 3429, pp. 183–194. Springer, Berlin (2005)
Li, F., Klette, R.: Minimum-length polygons of first-class simple cube-curves. In: Proc. Computer Analysis Images Patterns. LNCS, vol. 3691, pp. 321–329. Springer, Berlin (2005)
Li, F., Klette, R.: Shortest paths in a cuboidal world. In: Proc. Int. Workshop Combinatorial Image Analysis. LNCS, vol. 4040, pp. 415–429. Springer, Berlin (2006)
Li, F., Klette, R.: Analysis of the rubberband algorithm. Image Vis. Comput. 25(10), 1588–1598 (2007)
Li, F., Pan, X.: An approximation algorithm for computing minimum-length polygons in 3D images. In: Proc. The 10th Asian Conference on Computer Vision (ACCV 2010). LNCS, vol. 6495, pp. 641–652. Springer, Berlin (2011)
Luo, H., Eleftheriadis, A.: Rubberband: an improved graph search algorithm for interactive object segmentation. In: Proc. Int. Conf. Image Processing, vol. 1, pp. 101–104 (2002)
Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier, Amsterdam (2000)
Mitchell, J.S.B., Sharir, M.: New results on shortest paths in three dimensions. In: Proc. SCG, pp. 124–133 (2004)
Montanari, U.: A note on minimal length polygonal approximations to a digitalized contour. Commun. ACM 13, 41–47 (1970)
Noakes, L., Kozera, R., Klette, R.: Length estimation for curves with different samplings. In: Digital and Image Geometry. LNCS, vol. 2243, pp. 334–346. Springer, Berlin (2001)
Overmars, M.H., Welzl, E.: New methods for constructing visibility graphs. In: Proc. Annu. ACM Sympos. on Computational Geometry, pp. 164–171 (1988)
Papadimitriou, C.H.: An algorithm for shortest path motion in three dimensions. Inf. Process. Lett. 20, 259–263 (1985)
Roberts, A.W., Varberg, V.D.: Convex Functions. Academic Press, New York (1973)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Sharir, M., Schorr, A.: On shortest paths in polyhedral spaces. SIAM J. Comput. 15, 193–215 (1986)
Sklansky, J., Kibler, D.F.: A theory of nonuniformly digitized binary pictures. IEEE Trans. Syst. Man Cybern. 6, 637–647 (1976)
Sloboda, F., Zaťko, B., Klette, R.: On the topology of grid continua. In: Proc. Vision Geometry. SPIE, vol. 3454, pp. 52–63 (1998)
Sloboda, F., Zaťko, B., Stoer, J.: On approximation of planar one-dimensional grid continua. In: Klette, R., Rosenfeld, A., Sloboda, F. (eds.) Advances in Digital and Computational Geometry, pp. 113–160. Springer, Singapore (1998)
Sun, C., Pallottino, S.: Circular shortest path on regular grids. CMIS Report 01/76, CSIRO Math. Information Sciences, Australia (2001)
Sunday, D.: Algorithm 15: convex hull of a 2D simple polygonal path. www.softsurfer.com/Archive/algorithm_0203/ (2011). Accessed July 2011
Talbot, M.: A dynamical programming solution for shortest path itineraries in robotics. Electr. J. Undergrad. Math. 9, 21–35 (2004)
Welzl, E.: Constructing the visibility graph for n line segments in \({\mathcal{O}}(n^{2})\) time. Inf. Process. Lett. 20, 167–171 (1985)
Wolber, R., Stäb, F., Max, H., Wehmeyer, A., Hadshiew, I., Wenck, H., Rippke, F., Wittern, K.: Alpha-Glucosylrutin: Ein hochwirksams Flavonoid zum Schutz vor oxidativem Stress. J. Dtsch. Dermatol. Ges. 2, 580–587 (2004)
Wolfram Mathworld. Good Prime. mathworld.wolfram.com/GoodPrime.html (2011). Accessed July 2011
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Li, F., Klette, R. (2011). Paths in Cube-Curves. In: Euclidean Shortest Paths. Springer, London. https://doi.org/10.1007/978-1-4471-2256-2_9
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