Abstract

In this paper, we describe the bivariate jointly distributed region snake method in segmentation of microorganisms in Single Exposure On-Line (SEOL) holographic microscopy images. 3D images of the microorganisms are digitally reconstructed and numerically focused from any arbitrary depth from a single recorded digital hologram without mechanical scanning. Living organisms are non-rigid and they vary in shape and size. Moreover, they often do not exhibit clear edges in digitally reconstructed SEOL holographic images. Thus, conventional segmentation techniques based on the edge map may fail to segment these images. However, SEOL holographic microscopy provides both magnitude and phase information of the sample specimen, which could be helpful in the segmentation process. In this paper, we present a statistical framework based on the joint probability distribution of magnitude and phase information of SEOL holographic microscopy images and maximum likelihood estimation of image probability density function parameters. An optimization criterion is computed by maximizing the likelihood function of the target support hypothesis. In addition, a simple stochastic algorithm has been adapted for carrying out the optimization, while several boosting techniques have been employed to enhance its performance. Finally, the proposed method is applied for segmentation of biological microorganisms in SEOL holographic images and the experimental results are presented.

© 2006 Optical Society of America

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  1. A. K. Jain, Fundamentals of digital image processing, (Prentice Hall, 1989).
  2. W. K. Pratt, Digital Image Processing, (Wiley, 2001).
    [CrossRef]
  3. R. M. Haralick and L. G. Shapiro, "Image segmentation techniques," Computer Vision, Graphics, and Image Processing 29, 100-132 (1985).
    [CrossRef]
  4. R. O. Duda, P. E. Hart, and D. G. Stork, Pattern classification, 2nd ed. (Wiley Interscience, New York, 2000).
  5. J. W. Goodman, and R. W. Lawrence, "Digital image formation from electronically detected holograms," App. Phys. Lett. 11, 77-79 (1967).
    [CrossRef]
  6. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses" Appl. Opt. 13, 2693-2703 (1974).
    [CrossRef]
  7. U. Schnars and W. P. O. Juptner, "Direct recording of holograms by a CCD target and numerical reconstruction," Appl. Opt. 33, 179-181 (1994).
    [CrossRef] [PubMed]
  8. T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005).
    [CrossRef]
  9. T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three-dimensional object reconstruction and recognition," Appl. Opt. 41, 4124-4132 (2002).
    [CrossRef] [PubMed]
  10. T. J. Naughton, A. E. Shortt, and B. Javidi, "Nonuniform quantization compression of digital holograms," Opt. Lett. (2006) (submitted).
  11. O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
    [CrossRef] [PubMed]
  12. B. Javidi and D. Kim, "Three-dimensional-object recognition by use of single-exposure on-axis digital holography," Opt. Lett. 30, 236-238 (2005).
    [CrossRef] [PubMed]
  13. D. Kim and B. Javidi, "Distortion-tolerant 3-D object recognition by using single exposure on-axis digital holography," Opt. Express 12, 5539-5548 (2005).
    [CrossRef]
  14. 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]
  15. 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]
  16. I. Moon and B. Javidi, "Shape-tolerant three-dimensional recognition of biological microorganisms using digital holography," Opt. Express 13, 9612-9622 (2005).
    [CrossRef] [PubMed]
  17. T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23,1221-1223 (1998).
    [CrossRef]
  18. T. Kreis, ed., Handbook of Holographic Interferometry, (Wiley, VCH, 2005).
  19. H. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, New York, 1996).
  20. O. Germain and P. Refregier, "Optimal snake-based segmentation of a random luminance target on a spatially disjoint background," Opt. Lett. 21, 1845-1847 (1996).
    [CrossRef] [PubMed]
  21. C. Chesnaud, V. Page, and P. Refregier, "Improvement in robustness of the statistically independent region snake-based segmentation method of target-shape tracking," Opt. Lett. 23, 488-490 (1998).
    [CrossRef]
  22. C. Chesnaud, P. Refregier and V. Boulet, "Statistical region snake-based segmentation adapted to different physical noise models," IEEE Trans. on Pattern Analysis and Machine Intelligence 21, 1145-1157 (1999).
    [CrossRef]
  23. O. Germain, and P. Refregier, "Edge detection and location in SAR images: Contribution of statistical deformable models," in Image Recognition and Classification: Algorithms, Systems, and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), Chap. 4.
  24. M. Kass, A. Witkin, and D. Terzopoulus, "Snakes: Active contour models," Int. J. Comput. Vis. 1, 321-331 (1987).
    [CrossRef]
  25. C. Xu, and J. L. Prince, "Snakes, shapes, and gradient vector flow," IEEE Trans. Image Process. 7, 359-369 (1998).
    [CrossRef]
  26. L. D. Cohen, "On active contour models and balloons," CVGIP: Image Understanding 53, 211-218 (1991).
    [CrossRef]
  27. C. Kervrann, and F. Heitz, "A hierarchical statistical framework for the segmentation of deformable objects in image sequences," in Proceedings of IEEE Conf. on Computer Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, Seattle, 1994), pp. 724-728.
  28. R. Deriche, "Using Canny's criteria to derive a recursively implemented optimal edge detector," Int. J. Comp.Vis. 1, 167-187 (1987).
    [CrossRef]
  29. B. Javidi and J. Wang, "Limitations of the classic definition of the signal-to-noise ratio in matched filter based optical pattern recognition," Appl. Opt. 31, 6826-6829 (1992).
    [CrossRef] [PubMed]
  30. B. Javidi and J. Wang, "Optimum distortion invariant filters for detecting a noisy distorted target in background noise," J. Opt. Soc. Am. A 12, 2604-2614 (1995).
    [CrossRef]
  31. L. Vincent, and P. Soille, "Watersheds in digital spaces: an efficient algorithm based on immersion simulations," IEEE Trans. on Pattern Analysis and Machine Intelligence 13, 583-598 (1991).
    [CrossRef]

2006 (2)

2005 (5)

2002 (2)

1999 (1)

C. Chesnaud, P. Refregier and V. Boulet, "Statistical region snake-based segmentation adapted to different physical noise models," IEEE Trans. on Pattern Analysis and Machine Intelligence 21, 1145-1157 (1999).
[CrossRef]

1998 (3)

1996 (1)

1995 (1)

1994 (1)

1992 (1)

1991 (2)

L. D. Cohen, "On active contour models and balloons," CVGIP: Image Understanding 53, 211-218 (1991).
[CrossRef]

L. Vincent, and P. Soille, "Watersheds in digital spaces: an efficient algorithm based on immersion simulations," IEEE Trans. on Pattern Analysis and Machine Intelligence 13, 583-598 (1991).
[CrossRef]

1987 (2)

R. Deriche, "Using Canny's criteria to derive a recursively implemented optimal edge detector," Int. J. Comp.Vis. 1, 167-187 (1987).
[CrossRef]

M. Kass, A. Witkin, and D. Terzopoulus, "Snakes: Active contour models," Int. J. Comput. Vis. 1, 321-331 (1987).
[CrossRef]

1985 (1)

R. M. Haralick and L. G. Shapiro, "Image segmentation techniques," Computer Vision, Graphics, and Image Processing 29, 100-132 (1985).
[CrossRef]

1974 (1)

1967 (1)

J. W. Goodman, and R. W. Lawrence, "Digital image formation from electronically detected holograms," App. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Bertaux, N.

Boulet, V.

C. Chesnaud, P. Refregier and V. Boulet, "Statistical region snake-based segmentation adapted to different physical noise models," IEEE Trans. on Pattern Analysis and Machine Intelligence 21, 1145-1157 (1999).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Carapezza, E.

Chesnaud, C.

C. Chesnaud, P. Refregier and V. Boulet, "Statistical region snake-based segmentation adapted to different physical noise models," IEEE Trans. on Pattern Analysis and Machine Intelligence 21, 1145-1157 (1999).
[CrossRef]

C. Chesnaud, V. Page, and P. Refregier, "Improvement in robustness of the statistically independent region snake-based segmentation method of target-shape tracking," Opt. Lett. 23, 488-490 (1998).
[CrossRef]

Cohen, L. D.

L. D. Cohen, "On active contour models and balloons," CVGIP: Image Understanding 53, 211-218 (1991).
[CrossRef]

Daneshpanah, M.

Deriche, R.

R. Deriche, "Using Canny's criteria to derive a recursively implemented optimal edge detector," Int. J. Comp.Vis. 1, 167-187 (1987).
[CrossRef]

Frauel, Y.

Gallagher, J. E.

Germain, O.

Goodman, J. W.

J. W. Goodman, and R. W. Lawrence, "Digital image formation from electronically detected holograms," App. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Haralick, R. M.

R. M. Haralick and L. G. Shapiro, "Image segmentation techniques," Computer Vision, Graphics, and Image Processing 29, 100-132 (1985).
[CrossRef]

Herriott, D. R.

Javidi, B.

T. J. Naughton, A. E. Shortt, and B. Javidi, "Nonuniform quantization compression of digital holograms," Opt. Lett. (2006) (submitted).

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]

I. Moon and B. Javidi, "Shape-tolerant three-dimensional recognition of biological microorganisms using digital holography," Opt. Express 13, 9612-9622 (2005).
[CrossRef] [PubMed]

D. Kim and B. Javidi, "Distortion-tolerant 3-D object recognition by using single exposure on-axis digital holography," Opt. Express 12, 5539-5548 (2005).
[CrossRef]

B. Javidi and D. Kim, "Three-dimensional-object recognition by use of single-exposure on-axis digital holography," Opt. Lett. 30, 236-238 (2005).
[CrossRef] [PubMed]

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]

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005).
[CrossRef]

O. Matoba, T. J. Naughton, Y. Frauel, N. Bertaux, and B. Javidi, "Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram," Appl. Opt. 41, 6187-6192 (2002).
[CrossRef] [PubMed]

T. J. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three-dimensional object reconstruction and recognition," Appl. Opt. 41, 4124-4132 (2002).
[CrossRef] [PubMed]

B. Javidi and J. Wang, "Optimum distortion invariant filters for detecting a noisy distorted target in background noise," J. Opt. Soc. Am. A 12, 2604-2614 (1995).
[CrossRef]

B. Javidi and J. Wang, "Limitations of the classic definition of the signal-to-noise ratio in matched filter based optical pattern recognition," Appl. Opt. 31, 6826-6829 (1992).
[CrossRef] [PubMed]

Juptner, W. P. O.

Kameda, M.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005).
[CrossRef]

Kass, M.

M. Kass, A. Witkin, and D. Terzopoulus, "Snakes: Active contour models," Int. J. Comput. Vis. 1, 321-331 (1987).
[CrossRef]

Kim, D.

Lawrence, R. W.

J. W. Goodman, and R. W. Lawrence, "Digital image formation from electronically detected holograms," App. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Matoba, O.

Moon, I.

Morimoto, Y.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005).
[CrossRef]

Naughton, T. J.

Nomura, T.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005).
[CrossRef]

Okazaki, A.

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005).
[CrossRef]

Page, V.

Prince, J. L.

C. Xu, and J. L. Prince, "Snakes, shapes, and gradient vector flow," IEEE Trans. Image Process. 7, 359-369 (1998).
[CrossRef]

Refregier, P.

Rosenfeld, D. P.

Schnars, U.

Shapiro, L. G.

R. M. Haralick and L. G. Shapiro, "Image segmentation techniques," Computer Vision, Graphics, and Image Processing 29, 100-132 (1985).
[CrossRef]

Shortt, A. E.

T. J. Naughton, A. E. Shortt, and B. Javidi, "Nonuniform quantization compression of digital holograms," Opt. Lett. (2006) (submitted).

Soille, P.

L. Vincent, and P. Soille, "Watersheds in digital spaces: an efficient algorithm based on immersion simulations," IEEE Trans. on Pattern Analysis and Machine Intelligence 13, 583-598 (1991).
[CrossRef]

Tajahuerce, E.

Terzopoulus, D.

M. Kass, A. Witkin, and D. Terzopoulus, "Snakes: Active contour models," Int. J. Comput. Vis. 1, 321-331 (1987).
[CrossRef]

Vincent, L.

L. Vincent, and P. Soille, "Watersheds in digital spaces: an efficient algorithm based on immersion simulations," IEEE Trans. on Pattern Analysis and Machine Intelligence 13, 583-598 (1991).
[CrossRef]

Wang, J.

White, A. D.

Witkin, A.

M. Kass, A. Witkin, and D. Terzopoulus, "Snakes: Active contour models," Int. J. Comput. Vis. 1, 321-331 (1987).
[CrossRef]

Xu, C.

C. Xu, and J. L. Prince, "Snakes, shapes, and gradient vector flow," IEEE Trans. Image Process. 7, 359-369 (1998).
[CrossRef]

Yamaguchi, I.

Yeom, S.

Zhang, T.

App. Phys. Lett. (1)

J. W. Goodman, and R. W. Lawrence, "Digital image formation from electronically detected holograms," App. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Appl. Opt. (5)

Computer Vision, Graphics, and Image Processing (1)

R. M. Haralick and L. G. Shapiro, "Image segmentation techniques," Computer Vision, Graphics, and Image Processing 29, 100-132 (1985).
[CrossRef]

CVGIP: Image Understanding (1)

L. D. Cohen, "On active contour models and balloons," CVGIP: Image Understanding 53, 211-218 (1991).
[CrossRef]

IEEE Trans. Image Process. (1)

C. Xu, and J. L. Prince, "Snakes, shapes, and gradient vector flow," IEEE Trans. Image Process. 7, 359-369 (1998).
[CrossRef]

IEEE Trans. on Pattern Analysis and Machine Intelligence (2)

C. Chesnaud, P. Refregier and V. Boulet, "Statistical region snake-based segmentation adapted to different physical noise models," IEEE Trans. on Pattern Analysis and Machine Intelligence 21, 1145-1157 (1999).
[CrossRef]

L. Vincent, and P. Soille, "Watersheds in digital spaces: an efficient algorithm based on immersion simulations," IEEE Trans. on Pattern Analysis and Machine Intelligence 13, 583-598 (1991).
[CrossRef]

Int. J. Comp.Vis. (1)

R. Deriche, "Using Canny's criteria to derive a recursively implemented optimal edge detector," Int. J. Comp.Vis. 1, 167-187 (1987).
[CrossRef]

Int. J. Comput. Vis. (1)

M. Kass, A. Witkin, and D. Terzopoulus, "Snakes: Active contour models," Int. J. Comput. Vis. 1, 321-331 (1987).
[CrossRef]

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

Opt. Eng. (1)

T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Other (7)

O. Germain, and P. Refregier, "Edge detection and location in SAR images: Contribution of statistical deformable models," in Image Recognition and Classification: Algorithms, Systems, and Applications, B. Javidi, ed. (Marcel Dekker, New York, 2002), Chap. 4.

C. Kervrann, and F. Heitz, "A hierarchical statistical framework for the segmentation of deformable objects in image sequences," in Proceedings of IEEE Conf. on Computer Vision and Pattern Recognition, (Institute of Electrical and Electronics Engineers, Seattle, 1994), pp. 724-728.

T. Kreis, ed., Handbook of Holographic Interferometry, (Wiley, VCH, 2005).

H. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill, New York, 1996).

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern classification, 2nd ed. (Wiley Interscience, New York, 2000).

A. K. Jain, Fundamentals of digital image processing, (Prentice Hall, 1989).

W. K. Pratt, Digital Image Processing, (Wiley, 2001).
[CrossRef]

Supplementary Material (2)

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Figures (4)

Fig. 1.
Fig. 1.

Illustrates a SEOL holographic microscopy setup imaging a Sphacelaria alga.

Fig. 2.
Fig. 2.

(a) Illustrates the hypothetical status of the snake contour in kth iteration. (b) Shows the resulting quadrilateral region resulting form deformation of a single node.

Fig. 3.
Fig. 3.

(a) (533 KB) Movie of snake evolution on a diatom alga with 4 point contour initialization. (b) Final segmentation with bivariate region snake after ~1500 iterations. (c) Optimization trace during the experiment.

Fig. 4.
Fig. 4.

(a) (2.17 MB) Movie of the evolution of snake on an out-of-focus sphacelaria alga reconstructed from a SEOL hologram and the 5 point snake initialization. (b) Final segmented microorganism. (c) Optimization curve during the evolution process.

Equations (26)

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

f t ( α i , φ i ) = 1 σ φ t Φ ( φ i μ φ t σ φ t ) × 1 σ α | φ t Φ ( α i μ α | φ t σ α | φ t ) ,
Θ t = { μ α t , μ φ t , σ α t , σ φ t , ρ t } ,
μ α φ u = μ α u + ρ u σ α u ( φ μ φ u ) σ φ u , σ α φ u 2 = σ α 2 ( 1 ρ u 2 ) ,
P [ H w | s ] = P ( H w ) . P [ s | H w ] P ( s ) .
P ( s | H w , Θ ) = i = 1 N f t ( α i , φ i ) . w i × i = 1 N f b ( α i , φ i ) . ( 1 w i ) ,
P ( s H w , Θ ) = ( 1 2 π σ φ t σ α φ t ) N t ( w ) exp [ i = 1 N ( φ i μ φ t ) 2 . w i 2 ( σ φ t ) 2 ] × exp [ i = 1 N ( α i μ α φ t ) 2 . w i 2 ( σ α φ t ) 2 ]
× ( 1 2 π σ φ b σ α φ b ) N b ( w ) exp [ i = 1 N ( φ i μ φ b ) 2 . ( 1 w i ) 2 ( σ φ b ) 2 ] × exp [ i = 1 N ( α i μ α φ b ) 2 . ( 1 w i ) 2 ( σ α φ b ) 2 ] ,
log { P ( s | H w , Θ ) } = N t ( w ) log ( 2 π σ φ t σ α t 1 ρ t 2 ) 1 2 ( σ φ t ) 2 i = 1 N ( φ i μ φ t ) 2 . w i
1 2 ( σ α | φ t ) 2 i = 1 N ( α i μ α t ρ t σ α t ( φ i μ φ t ) σ φ t ) 2 . w i
N b ( w ) log ( 2 π σ φ b σ α b 1 ρ b 2 ) 1 2 ( σ φ b ) 2 i = 1 N ( φ μ φ b ) 2 . ( 1 w i )
1 2 ( σ α | φ b ) 2 i = 1 N ( α i μ α b ρ t σ α b ( φ i μ φ b ) σ φ t ) 2 . ( 1 w i ) .
μ ̂ α u = 1 N u ( w ) i Ω u α i , μ ̂ φ u = 1 N u ( w ) i Ω u φ i ,
( σ ̂ α u ) 2 = 1 N u ( w ) i Ω u ( α i μ ̂ α u ) 2 , ( σ ̂ φ u ) 2 = 1 N u ( w ) i Ω u ( φ i μ ̂ α u ) 2 ,
ρ ̂ u = 1 N u ( w ) σ ̂ α u σ ̂ φ u i Ω u ( α i μ ̂ α u ) ( φ i μ ̂ α u ) ,
F ( s | H w , Θ ̂ ) = log { P ( s | H w , Θ ̂ ) } = N 2 log ( 2 π ) N t ( w ) log ( σ ̂ φ t σ ̂ α t 1 ρ ̂ t 2 )
N b ( w ) log ( σ ̂ φ b σ ̂ α b 1 ρ ̂ b 2 ) N .
J ( s | H w , Θ ̂ ) = N t ( w ) log ( σ ̂ φ t σ ̂ α t 1 ρ ̂ t 2 ) + N b ( w ) log ( σ ̂ φ b σ ̂ α b 1 ρ ̂ b 2 ) .
( σ ̂ a k + 1 ) 2 = 1 N Ω a k + Ω d s i Ω a k + Ω d ( s i ( μ ̂ a k + Δ μ ) ) 2 = 1 N Ω a k + Ω d s i Ω a k + Ω d [ ( s i μ ̂ a k ) 2 + ( Δ μ ) 2 + 2 ( Δ μ ) ( s i μ ̂ a k ) ]
= 1 N Ω a k + Ω d { ( N Ω a k + Ω d . ( Δ μ ) 2 ) + s i Ω a k [ ( s i μ ̂ a k ) 2 + 2 ( Δ μ ) ( s i μ ̂ a k ) ] + s i Ω d [ ( s i μ ̂ a k ) 2 + 2 ( Δ μ ) ( s i μ ̂ a k ) ] }
= 1 N Ω a k + Ω d { ( N Ω a k + Ω d . ( Δ μ ) 2 ) + N Ω a k . ( σ ̂ a k ) 2 + s i Ω d ( s i μ ̂ a k ) 2 + 2 ( Δ μ ) s i Ω d ( s i μ ̂ a k ) } ,
Δ μ = μ ̂ a k + 1 μ ̂ a k = μ ̂ a k . N Ω a k + μ ̂ d . N Ω d N Ω a k + Ω d μ ̂ a k .
σ ̂ b k + 1 = 1 N Ω b k Ω d { ( N Ω b k Ω d . ( Δ μ ) 2 ) + N Ω b k . σ ̂ b k s i Ω d ( s i μ ̂ b k ) 2 2 ( Δ μ ) s i Ω d ( s i μ ̂ b k ) } ,
Δ μ = μ ̂ b k + 1 μ ̂ b k = μ ̂ b k . N Ω b k μ ̂ d . N ̂ Ω d N Ω a k Ω d μ ̂ b k .
P 1 ( p i ) = N p 1 for i { 1 N p } .
P k ( p i ) = Number of successful deformations of node p i in the last ( k 1 ) iterations Total number of successful deformations in the last ( k 1 ) iterations .
Q k ( α p j ) = Number of successful deformations of node p in direction j in the last ( k 1 ) iterations Total number of successful deformations of node p in the last ( k 1 ) iterations .

Metrics