Abstract

In this paper, we address the identification of biological microorganisms using microscopic integral imaging (II). II senses multiview directional information of 3D objects illuminated by incoherent light. A micro-lenslet array generates a set of elemental images by projecting a 3D scene onto a detector array. In computational reconstruction of II, 3D volumetric scenes are numerically reconstructed by means of a geometrical ray projection method. The identification of the biological samples is performed using the 3D volume of the reconstructed object. In one approach, the multivariate statistical distribution of the reference sample is measured in 3D space and compared with an unknown input sample by means of statistical discriminant functions. The multivariate empirical cumulative density of the 3D volume image is determined for classification. On the other approach, the graph matching technique is applied to 3D volumetric images with Gabor feature extraction. The reference morphology is identified in unknown input samples using 3D grids. Experimental results are presented for the identification of sphacelaria alga and tribonema aequale alga. We present experimental results for both 3D and 2D imaging. To the best of our knowledge, this is the first report on 3D identification of microorganisms using II.

© 2006 Optical Society of America

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  1. B. Javidi, I. Moon, S. Yeom, and E. Carapezza, "Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography," Opt. Express 13, 4492-4506 (2005).
    [CrossRef] [PubMed]
  2. B. Javidi, S. Yeom, I. Moon, and M. Daneshpanah, "Real-time automated 3D sensing, detection, and recognition of dynamic biological micro-organic events," Opt. Express 14, 3806-3829 (2006).
    [CrossRef] [PubMed]
  3. S. Yeom, I Moon, and B. Javidi, "Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms," Proceedings of IEEE 94, 550-566 (2006).
    [CrossRef]
  4. T. Kreis, ed., Handbook of holographic interferometry, (Wiley, VCH, 2005).
  5. A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
    [CrossRef]
  6. H. Kwon and N. M. Nasrabadi, "Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery," IEEE Trans. Geosci. Remote Sens. 43, 388-397 (2005).
    [CrossRef]
  7. P. Refregier, Noise theory and application to physics, (Springer, 2003).
  8. F. Sadjadi, ed., Selected papers on automatic target recognition, (SPIE-CDROM, 1999).
  9. <bok>. B. Javidi and F. Okano eds, Three dimensional television, video, and display technologies, (Springer, Berlin, 2002).</bok>
  10. T. Okoshi, "Three-dimensional displays," Proceedings of IEEE 68, 548-564 (1980).
    [CrossRef]
  11. M. G. Lippmann, "Epreuves reversibles donnant la sensation du relief," J. Phys. 7, 821-825 (1908).
  12. R. Martínez-Cuenca, G. Saavedra, M. Martínez-Corral and B. Javidi, "Enhanced depth of field integral imaging with sensor resolution constraints," Opt. Express 12, 5237-5242 (2004).
    [CrossRef] [PubMed]
  13. J. Jang and B. Javidi, "Three-dimensional integral imaging of micro-objects," Opt. Lett. 29, 1230-1232 (2004).
    [CrossRef] [PubMed]
  14. H. Arimoto and B. Javidi, "Integral three-dimensional imaging with digital reconstruction", Opt. Lett. 26, 157-159 (2001).
    [CrossRef]
  15. S. Hong, J. Jang, and B. Javidi, "Three-dimensional volulmetric object reconstruction using computational integral imaging," Opt. Express 12, 483-491 (2004),
    [CrossRef] [PubMed]
  16. A. Stern and B. Javidi, "3D image sensing and reconstruction with time-division multiplexed computational integral imaging (CII)," Appl. Opt. 42, 7036-7042 (2003).
    [CrossRef] [PubMed]
  17. S. Kishk and B. Javidi, "Improved resolution 3D object sensing and recognition using time multiplexed computational integral imaging," Opt. Express 11, 3528-3541 (2003).
    [CrossRef] [PubMed]
  18. S. Yeom, B. Javidi, and E. Watson, "Photon counting passive 3D image sensing for automatic target recognition," Opt. Express 13, 9310-9330 (2005), http://www.opticsexpress.org/abstract.cfm?id=86216.
    [CrossRef] [PubMed]
  19. A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
    [CrossRef]
  20. S.-K. Treskatis, V. Orgeldinger, H. wolf, and E. D. Gilles, "Morphological characterization of filamentous microorganisms in submerged cultures by on-line digital image analysis and pattern recognition," Biotechnol. Bioeng. 53, 191-201 (1997).
    [CrossRef] [PubMed]
  21. J. M. S. Cabral, M. Mota, and J. Tramper eds., Multiphase bioreactor design: chap2 image analysis and multiphase bioreactor, (London, Taylor & Francis, 2001).
    [CrossRef]
  22. M. Hollander and D. A. Wolfe, Nonparametric statistical methods, (Wiley, 1999).
  23. A. C. Rencher, Methods of multivariate analysis, (Wiley, 2002).
    [CrossRef]
  24. J. G. Daugman, "Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters," J. Opt. Soc. Am. 2, 1160-1169 (1985).
    [CrossRef]
  25. T. S. Lee, "Image representation using 2D Gabor wavelets," IEEE Trans. Pattern. Anal. Mach. Intell. 18, 959-971 (1996).
    [CrossRef]
  26. M. Lades, J. C. Vorbruggen, J. Buhmann, J. Lange, C. v.d. Malsburg, R. P. Wurtz, and W. Konen, "Distortion invariant object recognition in the dynamic link architecture," IEEE Trans. Comput. 42, 300-311 (1993).
    [CrossRef]
  27. Y. Frauel, O. Matoba, E. Tajahuerce, and B. Javidi, "Comparison of passive ranging integral imaging and active imaging digital holography for 3D object recognition," Appl. Opt. 43, 452-462 (2004).
    [CrossRef] [PubMed]
  28. C. E. Lunneborg, Data analysis by resampling: concepts and applications, (Duxbury Press, 1999).
  29. R. C. Gonzalez and R. E. Woods, Digital imaging processing, (Prentice Hall, 2002).

2006 (2)

B. Javidi, S. Yeom, I. Moon, and M. Daneshpanah, "Real-time automated 3D sensing, detection, and recognition of dynamic biological micro-organic events," Opt. Express 14, 3806-3829 (2006).
[CrossRef] [PubMed]

S. Yeom, I Moon, and B. Javidi, "Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms," Proceedings of IEEE 94, 550-566 (2006).
[CrossRef]

2005 (3)

2004 (6)

2003 (2)

2001 (1)

1997 (1)

S.-K. Treskatis, V. Orgeldinger, H. wolf, and E. D. Gilles, "Morphological characterization of filamentous microorganisms in submerged cultures by on-line digital image analysis and pattern recognition," Biotechnol. Bioeng. 53, 191-201 (1997).
[CrossRef] [PubMed]

1996 (1)

T. S. Lee, "Image representation using 2D Gabor wavelets," IEEE Trans. Pattern. Anal. Mach. Intell. 18, 959-971 (1996).
[CrossRef]

1993 (1)

M. Lades, J. C. Vorbruggen, J. Buhmann, J. Lange, C. v.d. Malsburg, R. P. Wurtz, and W. Konen, "Distortion invariant object recognition in the dynamic link architecture," IEEE Trans. Comput. 42, 300-311 (1993).
[CrossRef]

1985 (1)

J. G. Daugman, "Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters," J. Opt. Soc. Am. 2, 1160-1169 (1985).
[CrossRef]

1980 (1)

T. Okoshi, "Three-dimensional displays," Proceedings of IEEE 68, 548-564 (1980).
[CrossRef]

1908 (1)

M. G. Lippmann, "Epreuves reversibles donnant la sensation du relief," J. Phys. 7, 821-825 (1908).

Amaral, A. L.

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

Arimoto, H.

Buhmann, J.

M. Lades, J. C. Vorbruggen, J. Buhmann, J. Lange, C. v.d. Malsburg, R. P. Wurtz, and W. Konen, "Distortion invariant object recognition in the dynamic link architecture," IEEE Trans. Comput. 42, 300-311 (1993).
[CrossRef]

Carapezza, E.

da Motta, M.

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

Daneshpanah, M.

Daugman, J. G.

J. G. Daugman, "Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters," J. Opt. Soc. Am. 2, 1160-1169 (1985).
[CrossRef]

Ferreira, E. C.

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

Frauel, Y.

Hong, S.

Jang, J.

Javidi, B.

B. Javidi, S. Yeom, I. Moon, and M. Daneshpanah, "Real-time automated 3D sensing, detection, and recognition of dynamic biological micro-organic events," Opt. Express 14, 3806-3829 (2006).
[CrossRef] [PubMed]

S. Yeom, I Moon, and B. Javidi, "Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms," Proceedings of IEEE 94, 550-566 (2006).
[CrossRef]

B. Javidi, I. Moon, S. Yeom, and E. Carapezza, "Three-dimensional imaging and recognition of microorganism using single-exposure on-line (SEOL) digital holography," Opt. Express 13, 4492-4506 (2005).
[CrossRef] [PubMed]

S. Yeom, B. Javidi, and E. Watson, "Photon counting passive 3D image sensing for automatic target recognition," Opt. Express 13, 9310-9330 (2005), http://www.opticsexpress.org/abstract.cfm?id=86216.
[CrossRef] [PubMed]

J. Jang and B. Javidi, "Three-dimensional integral imaging of micro-objects," Opt. Lett. 29, 1230-1232 (2004).
[CrossRef] [PubMed]

S. Hong, J. Jang, and B. Javidi, "Three-dimensional volulmetric object reconstruction using computational integral imaging," Opt. Express 12, 483-491 (2004),
[CrossRef] [PubMed]

Y. Frauel, O. Matoba, E. Tajahuerce, and B. Javidi, "Comparison of passive ranging integral imaging and active imaging digital holography for 3D object recognition," Appl. Opt. 43, 452-462 (2004).
[CrossRef] [PubMed]

R. Martínez-Cuenca, G. Saavedra, M. Martínez-Corral and B. Javidi, "Enhanced depth of field integral imaging with sensor resolution constraints," Opt. Express 12, 5237-5242 (2004).
[CrossRef] [PubMed]

S. Kishk and B. Javidi, "Improved resolution 3D object sensing and recognition using time multiplexed computational integral imaging," Opt. Express 11, 3528-3541 (2003).
[CrossRef] [PubMed]

A. Stern and B. Javidi, "3D image sensing and reconstruction with time-division multiplexed computational integral imaging (CII)," Appl. Opt. 42, 7036-7042 (2003).
[CrossRef] [PubMed]

H. Arimoto and B. Javidi, "Integral three-dimensional imaging with digital reconstruction", Opt. Lett. 26, 157-159 (2001).
[CrossRef]

Kishk, S.

Kwon, H.

H. Kwon and N. M. Nasrabadi, "Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery," IEEE Trans. Geosci. Remote Sens. 43, 388-397 (2005).
[CrossRef]

Lades, M.

M. Lades, J. C. Vorbruggen, J. Buhmann, J. Lange, C. v.d. Malsburg, R. P. Wurtz, and W. Konen, "Distortion invariant object recognition in the dynamic link architecture," IEEE Trans. Comput. 42, 300-311 (1993).
[CrossRef]

Lange, J.

M. Lades, J. C. Vorbruggen, J. Buhmann, J. Lange, C. v.d. Malsburg, R. P. Wurtz, and W. Konen, "Distortion invariant object recognition in the dynamic link architecture," IEEE Trans. Comput. 42, 300-311 (1993).
[CrossRef]

Lee, T. S.

T. S. Lee, "Image representation using 2D Gabor wavelets," IEEE Trans. Pattern. Anal. Mach. Intell. 18, 959-971 (1996).
[CrossRef]

Lippmann, M. G.

M. G. Lippmann, "Epreuves reversibles donnant la sensation du relief," J. Phys. 7, 821-825 (1908).

Mahalanobis, A.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef]

Martínez-Corral, M.

Martínez-Cuenca, R.

Matoba, O.

Moda, M.

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

Moon, I

S. Yeom, I Moon, and B. Javidi, "Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms," Proceedings of IEEE 94, 550-566 (2006).
[CrossRef]

Moon, I.

Muise, R. R.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef]

Nasrabadi, N. M.

H. Kwon and N. M. Nasrabadi, "Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery," IEEE Trans. Geosci. Remote Sens. 43, 388-397 (2005).
[CrossRef]

Nevel, A. V.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef]

Okoshi, T.

T. Okoshi, "Three-dimensional displays," Proceedings of IEEE 68, 548-564 (1980).
[CrossRef]

Orgeldinger, V.

S.-K. Treskatis, V. Orgeldinger, H. wolf, and E. D. Gilles, "Morphological characterization of filamentous microorganisms in submerged cultures by on-line digital image analysis and pattern recognition," Biotechnol. Bioeng. 53, 191-201 (1997).
[CrossRef] [PubMed]

Pons, M. N.

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

Roche, N.

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

Saavedra, G.

Stanfill, S. R.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef]

Stern, A.

Tajahuerce, E.

Treskatis, S.-K.

S.-K. Treskatis, V. Orgeldinger, H. wolf, and E. D. Gilles, "Morphological characterization of filamentous microorganisms in submerged cultures by on-line digital image analysis and pattern recognition," Biotechnol. Bioeng. 53, 191-201 (1997).
[CrossRef] [PubMed]

Vivier, H.

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

Vorbruggen, J. C.

M. Lades, J. C. Vorbruggen, J. Buhmann, J. Lange, C. v.d. Malsburg, R. P. Wurtz, and W. Konen, "Distortion invariant object recognition in the dynamic link architecture," IEEE Trans. Comput. 42, 300-311 (1993).
[CrossRef]

Watson, E.

Yeom, S.

Appl. Opt. (2)

Biotechnol. Bioeng. (1)

S.-K. Treskatis, V. Orgeldinger, H. wolf, and E. D. Gilles, "Morphological characterization of filamentous microorganisms in submerged cultures by on-line digital image analysis and pattern recognition," Biotechnol. Bioeng. 53, 191-201 (1997).
[CrossRef] [PubMed]

Environmentrics (1)

A. L. Amaral, M. da Motta, M. N. Pons, H. Vivier, N. Roche, M. Moda, and E. C. Ferreira, "Survey of protozoa and metazoa populations in wastewater treatment plants by image analysis and discriminant analysis," Environmentrics 15, 381-390 (2004).
[CrossRef]

IEEE Trans. Aerosp. Electron. Syst. (1)

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, "Design and application of quadratic correlation filters for target detection," IEEE Trans. Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef]

IEEE Trans. Comput. (1)

M. Lades, J. C. Vorbruggen, J. Buhmann, J. Lange, C. v.d. Malsburg, R. P. Wurtz, and W. Konen, "Distortion invariant object recognition in the dynamic link architecture," IEEE Trans. Comput. 42, 300-311 (1993).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

H. Kwon and N. M. Nasrabadi, "Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery," IEEE Trans. Geosci. Remote Sens. 43, 388-397 (2005).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell. (1)

T. S. Lee, "Image representation using 2D Gabor wavelets," IEEE Trans. Pattern. Anal. Mach. Intell. 18, 959-971 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. G. Daugman, "Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensional visual cortical filters," J. Opt. Soc. Am. 2, 1160-1169 (1985).
[CrossRef]

J. Phys. (1)

M. G. Lippmann, "Epreuves reversibles donnant la sensation du relief," J. Phys. 7, 821-825 (1908).

Opt. Express (6)

Opt. Lett. (2)

Proceedings of IEEE (2)

T. Okoshi, "Three-dimensional displays," Proceedings of IEEE 68, 548-564 (1980).
[CrossRef]

S. Yeom, I Moon, and B. Javidi, "Real-time 3D sensing, visualization and recognition of dynamic biological micro-organisms," Proceedings of IEEE 94, 550-566 (2006).
[CrossRef]

Other (9)

T. Kreis, ed., Handbook of holographic interferometry, (Wiley, VCH, 2005).

P. Refregier, Noise theory and application to physics, (Springer, 2003).

F. Sadjadi, ed., Selected papers on automatic target recognition, (SPIE-CDROM, 1999).

<bok>. B. Javidi and F. Okano eds, Three dimensional television, video, and display technologies, (Springer, Berlin, 2002).</bok>

C. E. Lunneborg, Data analysis by resampling: concepts and applications, (Duxbury Press, 1999).

R. C. Gonzalez and R. E. Woods, Digital imaging processing, (Prentice Hall, 2002).

J. M. S. Cabral, M. Mota, and J. Tramper eds., Multiphase bioreactor design: chap2 image analysis and multiphase bioreactor, (London, Taylor & Francis, 2001).
[CrossRef]

M. Hollander and D. A. Wolfe, Nonparametric statistical methods, (Wiley, 1999).

A. C. Rencher, Methods of multivariate analysis, (Wiley, 2002).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Block diagrams for volumetric 3D recognition of biological microorganisms using (a) the multivariate statistical approach and (b) the morphology-based approach.

Fig. 2.
Fig. 2.

A schematic setup of II for microorganism recording.

Fig. 3.
Fig. 3.

Sections of elemental images for (a) sphacelaria alga and (b) tribonema aequale alga.

Fig. 4.
Fig. 4.

Reconstructed images at depth d=330 µm for (a) sphacelaria alga and (b) tribonema aequale alga.

Fig. 5.
Fig. 5.

The statistical distribution of the criterion discriminant function [see Eq. (3)] generated from the multi-dimensional data sets.

Fig. 6.
Fig. 6.

The mean-square-distance (MSD) between the criterion discriminant function and the actual discriminant function [see Eq. (2)] generated from the multiple section images. Null hypothesis is the true training class. The data is obtained from 3D II volume images. (a) null hypothesis and (b) non-training true class and false class.

Fig. 7.
Fig. 7.

Results for one 2D image. The MSD between the criterion discriminant function and the actual discriminant function [see Eq. (2)] generated from one section image. Null hypothesis is the training true class. The data is obtained from 2D reconstructed image. (a) null hypothesis, (b) non-training true class, and (c) false class.

Fig. 8.
Fig. 8.

An example of RGM results. (a) A reference graph in the center image of the reference set and (b) the input graphs detected in the center image of the fifth input set.

Fig. 9.
Fig. 9.

Maximum similarity function for 10 input image sets of sphacelaria alga. Image set #5 is the referenced true class, others (1–4 and 6–10) are non-training true class image sets.

Fig. 10.
Fig. 10.

Number of detections and the maximum similarity function for 10 input image sets of sphacelaria alga. Image set #5 is the referenced true class, others (1–4 and 6–10) are nontraining true class image sets.

Equations (13)

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

F X 1 , , X p ( x 1 , , x p ) = P ( X 1 x 1 , , X p x p ) = # { X 1 ( n ) x 1 , , X p ( n ) x p } n × p ,
Λ ̂ ( x 1 , , x p ) = F ref ( x 1 , , x p ) F ref ( x 1 , , x p ) + F ref ( x 1 , , x p ) ,
Λ ( x 1 , , x p ) = F ref ( x 1 , , x p ) F ref ( x 1 , , x p ) + F input ( x 1 , , x p ) ,
MSD = d ̂ = E { [ Λ ( x 1 , , x p ) Λ ̂ ( x 1 , , x p ) ] 2 } ,
g uv ( x ) = k uv 2 σ 2 exp ( k uv 2 x 2 2 σ 2 ) [ exp ( j k uv · x ) exp ( σ 2 2 ) ] ,
y uv = g uv * o ,
v ( x i ) = [ v = 1 V y 1 v ( x i ) v = 1 V y uv ( x i ) ] t .
Ω r = { o r ( d j r ) ; j = 1 , , D } and Ω s = { o s ( d j s ) ; j = 1 , , D } ,
x k ( p r , θ r ) = A θ r ( x k o x c o ) + p r , k = 1 , , K grid ,
A θ = [ cos θ sin θ sin θ cos θ ] ,
Γ RS ( p s , θ s ) = 1 D × K grid j = 1 D k = 1 K grid v r [ p r , θ r ; d j r ] , v s [ x k ( p s , θ s ; d j s ] v r [ x k ( p r , θ r ; d j r ) ] | | | | v s [ x k ( p s , θ s ; d j s ) ] ,
Γ RS ( p s , θ ̂ s ) > α Γ ,
θ ̂ s = max θ s Γ RS ( p s , θ s ) .

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