Abstract

Morphological transformations are typically performed on binary images by convolution with a binary kernel, which is followed by a threshold. We present an alternate approach that uses a complex-valued kernel with odd symmetry to perform these morphological operations. The complex-valued kernel increases the information-processing ability of the processor with no increase in system complexity. One advantage is that the processor operates on all constant regions of a gray-level image in parallel. A scale–space representation of this processor is obtained by varying the size of the kernel continuously through a range of scales. By using redundant information in the scale representation, this system is found to be robust in the presence of noise and spatial nonuniformities in the image. An optical system to perform morphological filtering based on this system is presented.

© 1992 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Serra, Image Analysis and Mathematical Morphology (Academic, New York, 1982).
  2. K. S. O’Neill, W. Rhodes, “Morphological transformations by hybrid optical–electronic methods,” in Hybrid Image Processing, D. Casasent, A. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 41–44 (1986).
  3. D. Casasent, E. Botha, “Optical symbolic substitution for morphological transformations,” Appl. Opt. 27, 3806–3810 (1988).
  4. Y. Li, A. Kostrzewski, D. Kim, G. Eichmann, “Compact parallel real-time programmable optical morphological image processor,” Opt. Lett. 14, 981–983 (1989).
  5. M. Freeman, B. E. A. Saleh, “Centroid scale–space maps,” J. Opt. Soc. Am. A 8, 1474–1487 (1991).
  6. A. P. Witkin, “Scalespace filtering,” in Proceedings of the Eighth International Joint Conference on Artificial Intelligence, A. Bundy, ed. (Kaufmann, Los Angeles, Calif., 1983), pp. 1019–1022.
  7. A. Rosenfeld, A. Kak, Digital Picture Processing (Academic, New York, 1982).
  8. P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).
  9. J. Hereford, W. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  11. D. A. Jared, K. M. Johnson, G. Moddel, “Joint transform correlator using an amorphous silicon ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 76, 97–102 (1990).
  12. B. Javidi, C. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).
  13. B. Javidi, S. Odeh, “Multiple object identification by bipolar joint transform correlation,” Opt. Eng. 27, 295–300 (1988).
  14. T. Hudson, D. Gregory, “Joint transform correlation using an optically addressed ferroelectric LC spatial light modulator,” Appl. Opt. 29, 1064–1066 (1990).
  15. F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).
  16. C. W. Burckhardt, “A simplification of Lee’s method of generating holograms by computer,” Appl. Opt. 9, 1949 (1970).
  17. W. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
  18. A. Fedor, M. O. Freeman, “User’s guide to the computer-generated hologram facility at the Optoelectronic Computing Systems Center—version 2.0,” OCS Tech. Rep. 90-20B (Opto-electronic Computing Systems Center, University of Colorado, Boulder, Colo., 1991).
  19. H. Bartelt, S. K. Case, “High-efficiency hybrid computer-generated holograms,” Appl. Opt. 21, 2886–2890 (1982).
  20. G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).
  21. A. Fedor, “Centroid scale–space maps: applications and implementation,” OCS Tech. Rep. 90-23 (Optoelectronic Computing Systems Center, University of Colorado, Boulder, Colo., 1990).
  22. T. Drabik, M. Handschy, “Silicon VLSI/ferroelectric liquid crystal technology for micropower optoelectronic computing devices,” Appl. Opt. 29, 5220–5223 (1990).
  23. D. A. Jared, K. M. Johnson, “Optically addressed thresholding VLSI/liquid crystal spatial light modulators,” Opt. Lett. 16, 967–969 (1991).

1991 (2)

1990 (3)

1989 (2)

G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).

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

1988 (4)

B. Javidi, S. Odeh, “Multiple object identification by bipolar joint transform correlation,” Opt. Eng. 27, 295–300 (1988).

J. Hereford, W. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

D. Casasent, E. Botha, “Optical symbolic substitution for morphological transformations,” Appl. Opt. 27, 3806–3810 (1988).

B. Javidi, C. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).

1987 (1)

P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).

1984 (1)

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).

1982 (1)

1970 (2)

Bartelt, H.

Botha, E.

Burckhardt, C. W.

Casasent, D.

Case, S. K.

Drabik, T.

Eichmann, G.

Fedor, A.

A. Fedor, “Centroid scale–space maps: applications and implementation,” OCS Tech. Rep. 90-23 (Optoelectronic Computing Systems Center, University of Colorado, Boulder, Colo., 1990).

A. Fedor, M. O. Freeman, “User’s guide to the computer-generated hologram facility at the Optoelectronic Computing Systems Center—version 2.0,” OCS Tech. Rep. 90-20B (Opto-electronic Computing Systems Center, University of Colorado, Boulder, Colo., 1991).

Freeman, M.

Freeman, M. O.

A. Fedor, M. O. Freeman, “User’s guide to the computer-generated hologram facility at the Optoelectronic Computing Systems Center—version 2.0,” OCS Tech. Rep. 90-20B (Opto-electronic Computing Systems Center, University of Colorado, Boulder, Colo., 1991).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gregory, D.

Handschy, M.

Hereford, J.

J. Hereford, W. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

Hudson, T.

Jared, D. A.

D. A. Jared, K. M. Johnson, “Optically addressed thresholding VLSI/liquid crystal spatial light modulators,” Opt. Lett. 16, 967–969 (1991).

D. A. Jared, K. M. Johnson, G. Moddel, “Joint transform correlator using an amorphous silicon ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 76, 97–102 (1990).

Javidi, B.

B. Javidi, S. Odeh, “Multiple object identification by bipolar joint transform correlation,” Opt. Eng. 27, 295–300 (1988).

B. Javidi, C. Kuo, “Joint transform image correlation using a binary spatial light modulator at the Fourier plane,” Appl. Opt. 27, 663–665 (1988).

Johnson, K. M.

D. A. Jared, K. M. Johnson, “Optically addressed thresholding VLSI/liquid crystal spatial light modulators,” Opt. Lett. 16, 967–969 (1991).

D. A. Jared, K. M. Johnson, G. Moddel, “Joint transform correlator using an amorphous silicon ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 76, 97–102 (1990).

G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).

Kak, A.

A. Rosenfeld, A. Kak, Digital Picture Processing (Academic, New York, 1982).

Kim, D.

Kostrzewski, A.

Kuo, C.

Lee, W.

Li, W.

G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).

Li, Y.

Lu, X. J.

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).

Maragos, P.

P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).

Moddel, G.

D. A. Jared, K. M. Johnson, G. Moddel, “Joint transform correlator using an amorphous silicon ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 76, 97–102 (1990).

G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).

O’Neill, K. S.

K. S. O’Neill, W. Rhodes, “Morphological transformations by hybrid optical–electronic methods,” in Hybrid Image Processing, D. Casasent, A. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 41–44 (1986).

Odeh, S.

B. Javidi, S. Odeh, “Multiple object identification by bipolar joint transform correlation,” Opt. Eng. 27, 295–300 (1988).

Pagano-Stauffer, L.

G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).

Rhodes, W.

J. Hereford, W. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

K. S. O’Neill, W. Rhodes, “Morphological transformations by hybrid optical–electronic methods,” in Hybrid Image Processing, D. Casasent, A. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 41–44 (1986).

Rice, R.

G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).

Rosenfeld, A.

A. Rosenfeld, A. Kak, Digital Picture Processing (Academic, New York, 1982).

Saleh, B. E. A.

Serra, J.

J. Serra, Image Analysis and Mathematical Morphology (Academic, New York, 1982).

Witkin, A. P.

A. P. Witkin, “Scalespace filtering,” in Proceedings of the Eighth International Joint Conference on Artificial Intelligence, A. Bundy, ed. (Kaufmann, Los Angeles, Calif., 1983), pp. 1019–1022.

Yu, F. T. S.

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).

Appl. Opt. (7)

Appl. Phys. Lett. (1)

G. Moddel, K. M. Johnson, W. Li, R. Rice, L. Pagano-Stauffer, “High speed binary optically addressing spatial light modulators,” Appl. Phys. Lett. 55, 537–539 (1989).

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

F. T. S. Yu, X. J. Lu, “A real-time programmable joint transform correlator,” Opt. Commun. 52, 10–16 (1984).

D. A. Jared, K. M. Johnson, G. Moddel, “Joint transform correlator using an amorphous silicon ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 76, 97–102 (1990).

Opt. Eng. (3)

B. Javidi, S. Odeh, “Multiple object identification by bipolar joint transform correlation,” Opt. Eng. 27, 295–300 (1988).

P. Maragos, “Tutorial on advances in morphological image processing and analysis,” Opt. Eng. 26, 623–632 (1987).

J. Hereford, W. Rhodes, “Nonlinear optical image filtering by time-sequential threshold decomposition,” Opt. Eng. 27, 274–279 (1988).

Opt. Lett. (2)

Other (7)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. Serra, Image Analysis and Mathematical Morphology (Academic, New York, 1982).

K. S. O’Neill, W. Rhodes, “Morphological transformations by hybrid optical–electronic methods,” in Hybrid Image Processing, D. Casasent, A. Tescher, eds., Proc. Soc. Photo-Opt. Instrum. Eng.638, 41–44 (1986).

A. P. Witkin, “Scalespace filtering,” in Proceedings of the Eighth International Joint Conference on Artificial Intelligence, A. Bundy, ed. (Kaufmann, Los Angeles, Calif., 1983), pp. 1019–1022.

A. Rosenfeld, A. Kak, Digital Picture Processing (Academic, New York, 1982).

A. Fedor, M. O. Freeman, “User’s guide to the computer-generated hologram facility at the Optoelectronic Computing Systems Center—version 2.0,” OCS Tech. Rep. 90-20B (Opto-electronic Computing Systems Center, University of Colorado, Boulder, Colo., 1991).

A. Fedor, “Centroid scale–space maps: applications and implementation,” OCS Tech. Rep. 90-23 (Optoelectronic Computing Systems Center, University of Colorado, Boulder, Colo., 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Erosion and dilation obtained with the absolute-intensity approach: (a) a two-dimensional input image, (b) structuring element B, (c) the corresponding erosion of the image by B, (d) a one-dimensional cross section of the image, (e) a one-dimensional cross section of B, and (f) the corresponding convolution with the thresholds for erosion Te and dilation Td indicated.

Fig. 2
Fig. 2

(a) Centroid kernel, (b) the correlation with a slice of the image shown in Fig. 1(a), (c) the thresholded output, (d) the 2-D version of the output.

Fig. 3
Fig. 3

Simple centroid scale–space map. Zeros occur when the kernel fits completely within a constant region or when the kernel is centered on the centroid of the object.

Fig. 4
Fig. 4

Morphological transformations with a centroid scale–space map: (a) original image; (b)–(g) smallest six scales of the centroid scale–space map; (h), (i) erosion and dilation operations; (j) edge detection; (k) medial-axis skeleton; (l), (m) opening operations.

Fig. 5
Fig. 5

Method for forming the skeleton of an object. The center of each maximum disk (which touches the boundary of the object at two or more points) is kept.

Fig. 6
Fig. 6

Skeleton for a multilevel image computed from a centroid scale–space map: (a) clown image, (b) skeleton of the clown.

Fig. 7
Fig. 7

(a) Noisy input image, (b)–(d) correlation output for several scales, (e)–(g) output thresholded at zero for several scales, (h) erosion of the clean image obtained by reconstructing the zero regions in the scale–space map, (i) erosion reconstructed by raising the zero-crossing detector threshold applied to (b).

Fig. 8
Fig. 8

Schematic of the scale–space morphological processor. SLM, spatial light modulator.

Fig. 9
Fig. 9

Output from the hybrid processor, with a rectangle as input: (a) after correlation, (b) the zero regions obtained after a threshold, (c) computer-simulated output.

Fig. 10
Fig. 10

(a) FLIR image of a truck, (b) hybrid output after correlation, (c) the thresholded output, (c) a computer-simulated output.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

X B = { a B ˇ a X } ,
X B = { a B ˇ a X Ø ,
X B = ( X B ) B ,
X B = ( X B ) B ,
X B = B a X B a .
X B = T d ( X * B ) .
X B = T e ( X * B ) .
c ( x , y ; r ) = ( x + i y ) w ( x , y ; r ) ,
w ( x , y ; r ) = circ [ ( x 2 + y 2 ) 1 / 2 / r ] .
z ( x , y ; r ) = ZEROS [ c ( x , y ; r ) f ( x , y ) ] .
- 1 - 1 - 1 - 1 8 - 1 - 1 - 1 - 1 .
z ( x , y ; r ) = i { a = ( x , y ) B a ( r ) X i } ,
X i B ( r ) = X i z ( x , y ; r ) .
X B ( r ) = X ¯ z ( x , y ; r ) ¯ .
X B ( r ) = X z ( x , y ; r ) ¯ .
[ X B ( r ) ] - [ X B ( r ) ] = [ X ¯ B ( r ) ] ¯ - [ X B ( r ) ] = [ X ¯ B ( r ) ] [ X B ( r ) ] ¯ = z ( x , y ; r ) ¯ ,
S ( X ; r ) = X p ( x , y ; r ) .
X = r > 0 { MD a ( r ) a S ( X ; r ) } ,
X ^ = r k { MD a ( r ) a S ( X ) } .
B ( r ) = k < r { B a ( k ) B a ( k ) B ( r ) } .
X ^ = r k { B ( k ) MD a ( r ) a S ( X ) } = B a ( k ) X B a ( k ) = X B ( k ) .
P ( u , v ) = F ( u , v ) exp ( i π u d ) + G ( u , v ) exp ( - i π u d ) 2 ,

Metrics