Abstract

The image distance transform is a time-consuming global operation. We use optical correlation for the fast computation of binary image distance transforms. A joint transform correlator is proposed to implement the algorithm that accommodates the morphological threshold-decomposition concept. The proposed optical processor is programmable for computing Euclidean, city-block, and chessboard distance transforms in real time. Skeletonization by use of the distance transform and the rotation invariance of the Euclidean distance transform are demonstrated.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  26. F. Y. Shih, O. R. Mitchell, “Threshold decomposition of gray-scale morphology into binary morphology,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 31–42 (1989).
    [CrossRef]
  27. P. Garcia-Martinez, D. Mas, J. Garcia, C. Ferreira, “Nonlinear morphological correlation: optoelectronic implementation,” Appl. Opt. 37, 2112–2118 (1998).
    [CrossRef]
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  30. L. Guibert, Y. Petillot, J.-L. de Bougrenet de la Tocnaye, “Real-time demonstration of an on-board nonlinear joint transform correlator system,” Opt. Eng. 36, 820–824 (1997).
    [CrossRef]
  31. T. J. Grycewicz, “Experimental demonstration of a binary single-lens joint transform correlator using chirp-modulated inputs,” Opt. Eng. 36, 814–819 (1997).
    [CrossRef]
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    [CrossRef]
  34. H.-H. Chang, H. Yan, “Skeletonization of binary digital patterns using a fast Euclidean distance transform,” Opt. Eng. 35, 1003–1008 (1996).
    [CrossRef]
  35. C. Arcelli, G. Sannid di Baja, “A width-independent fast thinning algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 463–474 (1985).
    [CrossRef] [PubMed]

1999 (2)

S. Zhang, M. A. Karim, “Euclidean distance transform by stack filters,” IEEE Signal Process. Lett. 6, 253–256 (1999).
[CrossRef]

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

1998 (2)

P. Garcia-Martinez, D. Mas, J. Garcia, C. Ferreira, “Nonlinear morphological correlation: optoelectronic implementation,” Appl. Opt. 37, 2112–2118 (1998).
[CrossRef]

W. Guan, S. Ma, “A list-processing approach to compute Voronoi diagrams and the Euclidean distance transform,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 757–761 (1998).
[CrossRef]

1997 (2)

L. Guibert, Y. Petillot, J.-L. de Bougrenet de la Tocnaye, “Real-time demonstration of an on-board nonlinear joint transform correlator system,” Opt. Eng. 36, 820–824 (1997).
[CrossRef]

T. J. Grycewicz, “Experimental demonstration of a binary single-lens joint transform correlator using chirp-modulated inputs,” Opt. Eng. 36, 814–819 (1997).
[CrossRef]

1996 (1)

H.-H. Chang, H. Yan, “Skeletonization of binary digital patterns using a fast Euclidean distance transform,” Opt. Eng. 35, 1003–1008 (1996).
[CrossRef]

1995 (2)

H. Breu, J. Gil, D. Kirkpatrick, M. Werman, “Linear time Euclidean distance transform algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 529–533 (1995).
[CrossRef]

M. Gedziorowski, J. Garcia, “Programmable optical–digital processor for rank order and morphological filtering,” Opt. Commun. 119, 207–217 (1995).
[CrossRef]

1994 (3)

Z. Yao, M. Wu, G. Ji, G. Huang, Y. Yan, “New optoelectronic morphological scheme for multiobject recognition,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

L. Liu, “Morphological hit-or-miss transform for binary and gray-tone image processing and its optical implementation,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

J. Li, J. Hu, Y. Wang, “Experimental investigation of a real-time nonlinear joint transform correlator,” Opt. Eng. 33, 3302–3306 (1994).
[CrossRef]

1993 (3)

1992 (4)

M. Fukui, K. Kitayama, “Image logic algebra and its optical implementation,” Appl. Opt. 31, 581–591 (1992).
[CrossRef] [PubMed]

F. Y. Shih, O. R. Mitchell, “A mathematical morphology approach to Euclidean distance transformation,” IEEE Trans. Image Process. 1, 197–204 (1992).
[CrossRef] [PubMed]

A. Fedor, M. O. Freemen, “Optical multiscale morphological processor using a complex-valued kernel,” Appl. Opt. 31, 4042–4050 (1992).
[CrossRef] [PubMed]

D. W. Paglieroni, “A unified distance transform algorithm and architecture,” Mach. Vision Appl. 5, 47–55 (1992).
[CrossRef]

1991 (1)

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

1990 (2)

1989 (3)

Y. Li, A. Kostrzewski, D. H. Kim, G. Eichmann, “Compact parallel real-time programmable optical morphological image processor,” Opt. Lett. 14, 981–983 (1989).
[CrossRef] [PubMed]

B. J. H. Verwer, P. W. Verbeek, S. T. Dekker, “An efficient uniform cost algorithm applied to distance transforms,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 425–429 (1989).
[CrossRef]

F. Y. Shih, O. R. Mitchell, “Threshold decomposition of gray-scale morphology into binary morphology,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 31–42 (1989).
[CrossRef]

1987 (2)

1986 (2)

S. H. Lee, “Optical implementation of digital algorithms,” Opt. Eng. 25, 69–75 (1986).
[CrossRef]

G. Borgefors, “Distance transformations in digital images,” Comput. Vision Graph. Image Proc. 34, 344–371 (1986).
[CrossRef]

1985 (2)

N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical implementation of the Hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
[CrossRef] [PubMed]

C. Arcelli, G. Sannid di Baja, “A width-independent fast thinning algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 463–474 (1985).
[CrossRef] [PubMed]

1966 (1)

Allebach, J. P.

Arcelli, C.

C. Arcelli, G. Sannid di Baja, “A width-independent fast thinning algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 463–474 (1985).
[CrossRef] [PubMed]

Baukens, V.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Borgefors, G.

G. Borgefors, “Distance transformations in digital images,” Comput. Vision Graph. Image Proc. 34, 344–371 (1986).
[CrossRef]

Breu, H.

H. Breu, J. Gil, D. Kirkpatrick, M. Werman, “Linear time Euclidean distance transform algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 529–533 (1995).
[CrossRef]

Buczynski, R.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Cai, J.

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Casasent, D.

R. Schaefer, D. Casasent, “Optical implementation of gray scale morphology,” in Nonlinear Image Processing III, E. R. Dougherty, T. Astula, C. G. Boncelet, eds., Proc. SPIE1658, 287–296 (1992).
[CrossRef]

Chang, H.-H.

H.-H. Chang, H. Yan, “Skeletonization of binary digital patterns using a fast Euclidean distance transform,” Opt. Eng. 35, 1003–1008 (1996).
[CrossRef]

Chavel, P.

Chin, G.

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

David, A. J.

de Bougrenet de la Tocnaye, J.-L.

L. Guibert, Y. Petillot, J.-L. de Bougrenet de la Tocnaye, “Real-time demonstration of an on-board nonlinear joint transform correlator system,” Opt. Eng. 36, 820–824 (1997).
[CrossRef]

Debaes, N.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Dekker, S. T.

B. J. H. Verwer, P. W. Verbeek, S. T. Dekker, “An efficient uniform cost algorithm applied to distance transforms,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 425–429 (1989).
[CrossRef]

Eichmann, G.

Farhat, N. H.

Fedor, A.

Ferreira, C.

Freemen, M. O.

Fukui, M.

Garcia, J.

Garcia-Martinez, P.

Gedziorowski, M.

M. Gedziorowski, J. Garcia, “Programmable optical–digital processor for rank order and morphological filtering,” Opt. Commun. 119, 207–217 (1995).
[CrossRef]

Gil, J.

H. Breu, J. Gil, D. Kirkpatrick, M. Werman, “Linear time Euclidean distance transform algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 529–533 (1995).
[CrossRef]

Glaser, I.

Goodman, J. W.

Goulet, A.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Gregory, D. A.

Grycewicz, T. J.

T. J. Grycewicz, “Experimental demonstration of a binary single-lens joint transform correlator using chirp-modulated inputs,” Opt. Eng. 36, 814–819 (1997).
[CrossRef]

Guan, W.

W. Guan, S. Ma, “A list-processing approach to compute Voronoi diagrams and the Euclidean distance transform,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 757–761 (1998).
[CrossRef]

Guibert, L.

L. Guibert, Y. Petillot, J.-L. de Bougrenet de la Tocnaye, “Real-time demonstration of an on-board nonlinear joint transform correlator system,” Opt. Eng. 36, 820–824 (1997).
[CrossRef]

Heremans, P.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Hu, J.

J. Li, J. Hu, Y. Wang, “Experimental investigation of a real-time nonlinear joint transform correlator,” Opt. Eng. 33, 3302–3306 (1994).
[CrossRef]

Huang, G.

Z. Yao, M. Wu, G. Ji, G. Huang, Y. Yan, “New optoelectronic morphological scheme for multiobject recognition,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

Huang, K.-S.

Ittycheriah, A. P.

Jenkins, B. K.

Ji, G.

Z. Yao, M. Wu, G. Ji, G. Huang, Y. Yan, “New optoelectronic morphological scheme for multiobject recognition,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

Jutamulia, S.

Karim, M. A.

S. Zhang, M. A. Karim, “Euclidean distance transform by stack filters,” IEEE Signal Process. Lett. 6, 253–256 (1999).
[CrossRef]

Kim, D. H.

Kirk, A.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Kirkpatrick, D.

H. Breu, J. Gil, D. Kirkpatrick, M. Werman, “Linear time Euclidean distance transform algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 529–533 (1995).
[CrossRef]

Kitayama, K.

Kostrzewski, A.

Krile, T. F.

Lee, S. H.

S. H. Lee, “Optical implementation of digital algorithms,” Opt. Eng. 25, 69–75 (1986).
[CrossRef]

Li, J.

J. Li, J. Hu, Y. Wang, “Experimental investigation of a real-time nonlinear joint transform correlator,” Opt. Eng. 33, 3302–3306 (1994).
[CrossRef]

Li, Y.

Lim, S. L.

Lin, T. W.

Liu, L.

L. Liu, “Morphological hit-or-miss transform for binary and gray-tone image processing and its optical implementation,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

Z. Zhu, L. Liu, “Optical cellular continuous-logic array for gray-scale image processing,” Appl. Opt. 32, 3676–3683 (1993).
[CrossRef] [PubMed]

Ma, S.

W. Guan, S. Ma, “A list-processing approach to compute Voronoi diagrams and the Euclidean distance transform,” IEEE Trans. Pattern Anal. Mach. Intell. 20, 757–761 (1998).
[CrossRef]

Maragos, P.

P. Maragos, “Morphological correlation and mean absolute error,” in ICASSP-89: 1989 International Conference on Acoustic, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1989), Vol. 3, pp. 1568–1571.

P. Maragos, “Optimal morphological approaches to image matching and object detection,” in Proceedings of the Second International Conference on Computer Vision, Tampa, Fla., 5–8 December 1988, pp. 695–699.

Mas, D.

Mitchell, O. R.

F. Y. Shih, O. R. Mitchell, “A mathematical morphology approach to Euclidean distance transformation,” IEEE Trans. Image Process. 1, 197–204 (1992).
[CrossRef] [PubMed]

F. Y. Shih, O. R. Mitchell, “Threshold decomposition of gray-scale morphology into binary morphology,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 31–42 (1989).
[CrossRef]

Ochoa, E.

Paek, E.

Paglieroni, D. W.

D. W. Paglieroni, “A unified distance transform algorithm and architecture,” Mach. Vision Appl. 5, 47–55 (1992).
[CrossRef]

Petillot, Y.

L. Guibert, Y. Petillot, J.-L. de Bougrenet de la Tocnaye, “Real-time demonstration of an on-board nonlinear joint transform correlator system,” Opt. Eng. 36, 820–824 (1997).
[CrossRef]

Prata, A.

Psaltis, D.

Saleh, B. E. A.

Sannid di Baja, G.

C. Arcelli, G. Sannid di Baja, “A width-independent fast thinning algorithm,” IEEE Trans. Pattern Anal. Mach. Intell. 7, 463–474 (1985).
[CrossRef] [PubMed]

Sawchuk, A. A.

Schaefer, R.

R. Schaefer, D. Casasent, “Optical implementation of gray scale morphology,” in Nonlinear Image Processing III, E. R. Dougherty, T. Astula, C. G. Boncelet, eds., Proc. SPIE1658, 287–296 (1992).
[CrossRef]

Shih, F. Y.

F. Y. Shih, O. R. Mitchell, “A mathematical morphology approach to Euclidean distance transformation,” IEEE Trans. Image Process. 1, 197–204 (1992).
[CrossRef] [PubMed]

F. Y. Shih, O. R. Mitchell, “Threshold decomposition of gray-scale morphology into binary morphology,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 31–42 (1989).
[CrossRef]

Sweeney, D. W.

Szoplik, T.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

J. Garcia, T. Szoplik, C. Ferreira, “Optoelectronic morphological image processor,” Opt. Lett. 18, 1952–1954 (1993).
[CrossRef] [PubMed]

Thienpont, H.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Verbeek, P. W.

B. J. H. Verwer, P. W. Verbeek, S. T. Dekker, “An efficient uniform cost algorithm applied to distance transforms,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 425–429 (1989).
[CrossRef]

Veretennicoff, I.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Verwer, B. J. H.

B. J. H. Verwer, P. W. Verbeek, S. T. Dekker, “An efficient uniform cost algorithm applied to distance transforms,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 425–429 (1989).
[CrossRef]

Vouncks, R.

R. Buczynski, V. Baukens, T. Szoplik, A. Goulet, N. Debaes, A. Kirk, P. Heremans, R. Vouncks, I. Veretennicoff, H. Thienpont, “Fast optical thresholding with an array of optoelectronic transceiver elements,” IEEE Trans. Photon. Technol. Lett. 11, 367–369 (1999).
[CrossRef]

Walkup, J. F.

Wang, C.-H.

Wang, J.-M.

Wang, Y.

J. Li, J. Hu, Y. Wang, “Experimental investigation of a real-time nonlinear joint transform correlator,” Opt. Eng. 33, 3302–3306 (1994).
[CrossRef]

Weaver, C. S.

Weber, A. G.

Werman, M.

H. Breu, J. Gil, D. Kirkpatrick, M. Werman, “Linear time Euclidean distance transform algorithms,” IEEE Trans. Pattern Anal. Mach. Intell. 17, 529–533 (1995).
[CrossRef]

Wu, M.

Z. Yao, M. Wu, G. Ji, G. Huang, Y. Yan, “New optoelectronic morphological scheme for multiobject recognition,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Yan, H.

H.-H. Chang, H. Yan, “Skeletonization of binary digital patterns using a fast Euclidean distance transform,” Opt. Eng. 35, 1003–1008 (1996).
[CrossRef]

Yan, Y.

Z. Yao, M. Wu, G. Ji, G. Huang, Y. Yan, “New optoelectronic morphological scheme for multiobject recognition,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

Yao, Z.

Z. Yao, M. Wu, G. Ji, G. Huang, Y. Yan, “New optoelectronic morphological scheme for multiobject recognition,” Opt. Eng. 33, 3447–3455 (1994).
[CrossRef]

Yu, F. T. S.

Zhang, S.

S. Zhang, M. A. Karim, “Euclidean distance transform by stack filters,” IEEE Signal Process. Lett. 6, 253–256 (1999).
[CrossRef]

Zhou, S.

M. Wu, S. Zhou, J. Cai, G. Chin, “Optical binary image algebra processor,” Opt. Commun. 86, 454–460 (1991).
[CrossRef]

Zhu, Z.

Appl. Opt. (11)

Z. Zhu, L. Liu, “Optical cellular continuous-logic array for gray-scale image processing,” Appl. Opt. 32, 3676–3683 (1993).
[CrossRef] [PubMed]

K.-S. Huang, A. A. Sawchuk, B. K. Jenkins, P. Chavel, J.-M. Wang, A. G. Weber, C.-H. Wang, I. Glaser, “Digital optical cellular image processor (DOCIP): experimental implementation,” Appl. Opt. 32, 166–173 (1993).
[CrossRef] [PubMed]

M. Fukui, K. Kitayama, “Image logic algebra and its optical implementation,” Appl. Opt. 31, 581–591 (1992).
[CrossRef] [PubMed]

E. Ochoa, J. P. Allebach, D. W. Sweeney, “Optical median filtering using threshold decomposition,” Appl. Opt. 26, 252–260 (1987).
[CrossRef] [PubMed]

N. H. Farhat, D. Psaltis, A. Prata, E. Paek, “Optical implementation of the Hopfield model,” Appl. Opt. 24, 1469–1475 (1985).
[CrossRef] [PubMed]

A. P. Ittycheriah, J. F. Walkup, T. F. Krile, S. L. Lim, “Outer product processor using polarization encoding,” Appl. Opt. 29, 275–283 (1990).
[CrossRef] [PubMed]

A. J. David, B. E. A. Saleh, “Optical implementation of the Hopfield algorithm using correlation,” Appl. Opt. 29, 1063–1064 (1990).
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Figures (8)

Fig. 1
Fig. 1

Negative-distance SE’s for (a) a Euclidean DT, (b) a city-block DT, and (c) a chessboard DT.

Fig. 2
Fig. 2

Positive-distance SE’s for (a) a Euclidean DT, (b) a city-block DT, and (c) a chessboard DT.

Fig. 3
Fig. 3

JTC-based optical correlator for processing DT’s.

Fig. 4
Fig. 4

Binary input image (111 × 85 pixels) of an aircraft for processing in the optical correlator.

Fig. 5
Fig. 5

Distance SE images used in the simulation calculations: (a) Euclidean DT, (b) city-block DT, (c) chessboard DT.

Fig. 6
Fig. 6

Distance images obtained from Fig. 5: (a) Euclidean DT, (b) city-block DT, (c) chessboard DT.

Fig. 7
Fig. 7

Skeleton image obtained from the DT shown in Fig. 6(c).

Fig. 8
Fig. 8

Demonstration of the rotation invariance of the Euclidean DT: (a) input image, (b) Euclidean DT, (c) city-block DT, (d) chessboard DT.

Equations (22)

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dep, q=x-s2+y-t21/2.
d4p, q=|x-s|+|y-t|.
d8p, q=Max|x-s|, |y-t|.
DTf=MinqB dp, qif pB0otherwise,
DTf=FΦgk,
Fx, y=Mfx, y.
Mmaxi,j|ki, j|=N.
DTf=FΦgk+N+N=FΦgh+N,
Fnx, y=1if Fx, yn0if Fx, y<n,
hnx, y=1if hx, yn0if hx, y<n,
F1=F2=ΛFn=Λ=FN=f.
FΦgh=n=1N fΦbhn-N,
DTf=n=1N fΦbhn-N+N=n=1N fΦbhn.
DTf=n=1N T|hn|fhn,
DTf=n=1Nfcb hnc=N-n=1N fcb hn,
DTf=N-n=1N T1fchn,
rx, y=fcx+d, y+hnx-d, y.
Ru, ν=|FCu, ν|2+|Hnu, ν|2+FC*u, νHnu, νexp-jud+FCu, νHn*u, νexpjud,
rx, y=fcx, yfcx, y+hnx, yhnx, y+fcx, yhnx, yδx-2d+hnx, yfcx, yδx+2d.
N=L-12/21/2.
N=L-1.
N=L-1/2.

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