## Segmentation of 3D holographic images using bivariate jointly distributed region snake

Optics Express, Vol. 14, Issue 12, pp. 5143-5153 (2006)

http://dx.doi.org/10.1364/OE.14.005143

Acrobat PDF (338 KB)

### 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

## 1. Introduction

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]

5. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” App. Phys. Lett. **11**, 77–79 (1967). [CrossRef]

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]

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]

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]

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]

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]

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]

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]

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]

## 2. Background

*Kass*[24

24. M. Kass, A. Witkin, and D. Terzopoulus, “Snakes: Active contour models,” Int. J. Comput. Vis. **1**, 321–331 (1987). [CrossRef]

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]

*active contours*in a statistical framework to derive an optimization criterion in analogy with conventional

*snake energy*in snake active contours terminology. In addition, an accompanying stochastic algorithm is proposed to carry out the aforementioned optimization [20

**21**, 1845–1847 (1996). [CrossRef] [PubMed]

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]

28. R. Deriche, “Using Canny’s criteria to derive a recursively implemented optimal edge detector,” Int. J. Comp.Vis. **1**, 167–187 (1987). [CrossRef]

**21**, 1845–1847 (1996). [CrossRef] [PubMed]

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]

## 3. Bivariate jointly distributed region snake

*snake energy*[24

24. M. Kass, A. Witkin, and D. Terzopoulus, “Snakes: Active contour models,” Int. J. Comput. Vis. **1**, 321–331 (1987). [CrossRef]

_{t}and Ω

_{b}. Throughout this paper, script

*t*and

*b*will be used for target and background, respectively.

_{t}matching the original target support. For this purpose, we use hypothesis testing from statistical decision theory. We consider a common assumption in which the probability distribution of pixels’ complex amplitudes of the object are statistically independent of that of background, while no prior knowledge of the distribution parameters is available. In addition, the snake contour can be modeled with a polygon of

*N*

_{p}nodes, while the number of nodes is arbitrary and depends on the desired resolution. Also, for the sake of simplicity, one-dimensional image model is used as

**s**={

*s*

_{i}|

*i*∈[1,

*N*]}, where

*N*is the total number of complex pixels. As mentioned before, reconstructed holographic images are complex, thus each pixel value

*s*

_{i}is a complex number that we denote its magnitude and phase by

*α*

_{i}and

*φ*

_{i}, respectively. Let

**w**={

*w*

_{i}|

*i*∈[1,

*N*]} be a binary window that determines the support of the target such that

*w*

_{i}=1 for the target pixels and

*w*

_{i}=0 elsewhere. Now, the image can be represented as the addition of disjoint target complex pixels (

**a**) inside

**w**and background complex pixels (

**b**) outside the window

**w**[29

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]

*s*

_{i}=

**a**

*iw*

_{i}+

**b**

_{i}[1-

*w*

_{i}].

### 3.1. Bivariate Gaussian distribution model

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]

*α*) and phase (

*φ*) random variables inside the target region, one can write the target’s joint probability distribution function as following:

*α*

_{i}=|

*s*

_{i}| and

*φ*

_{i}=

*∠s*are the magnitude and phase of pixel

_{i}*s*, respectively. Also, function Φ(

_{i}*x*)=(2

*π*)

^{-1/2}exp(-

*x*

^{2}/2) is the standard normal probability distribution function.

*µ, σ*and

*ρ*are marginal mean, standard deviation and correlation coefficients of magnitude (

*α*) and phase (

*φ*) random variables of the target respectively. In the same vein, the background is assumed to have another bivariate Gaussian distribution with a different, independent parameter set Θ

_{b}and joint probability density function

*f*

_{b}(

*α, φ*). Since the separation of two random variables in Eq. (1) is made possible by conditioning

*α*on

*φ*, the corresponding conditional mean and variances can be used for

*α*as follows:

*u*∈{

*t,b*}. For generality, we assume parameter vector Θ={Θ

_{t}, Θ

_{b}}

*a priori*unknown.

### 3.2. Maximum likelihood hypothesis testing

**w**), best representing the target support. Considering a hypothesis testing approach with hypothesis

*H*

_{w}, one needs to maximize the

*a posteriori*conditional probability

*P*[

*H*

_{w}|

**s**]. Assuming the general case of equiprobable hypotheses, the Bayes rule states that:

*a posteriori*probability

*P*[

*H*

_{w}|

**s**] is equivalent to maximizing the conditional probability

*P*[

**s**|

*H*

_{w}] which corresponds to the likelihood of the hypothesis

*H*

_{w}.

*H*

_{w}can be written as multiplication of corresponding joint probability distribution functions inside and outside of binary window (

**w**) as:

*N*

_{t}(w), (w)

*N*

_{b}(W) are the number of hypothetical target and background pixels according to current

**w**. Taking the natural log of the above expression and incorporating conditional mean and variance of (3) yields:

*H*

_{w}with parameter vector of Θ, which is a priori unknown. In order to proceed, one has to estimate unknown parameters. This could be done in several ways including the maximum likelihood method.

_{t},

_{b}}. The resulting sample mean, variance and correlation are as following:

*u*denotes either of the target or background regions.

### 3.3. Optimization criterion

**w**), it is obvious that maximization of (9) is analogous to minimization of the following criterion with respect to

**w**:

*H*

_{w}which minimizes the above criterion, is the optimal binary window and thus the optimal segmentation of the target in a complex image. The reader should note that Eq. (10) depends on parameter estimations over jointly distributed magnitude and phase random variables inside and outside the window function. Moreover, to show the analogy with energy based snakes terminology [24

**1**, 321–331 (1987). [CrossRef]

*snake energy*since its minimization guides the snake contour to enclose the target.

## 4. Stochastic optimization algorithm

*2N*

_{p}dimensions (with

*N*

_{p}denoting the number of snake polygon nodes) and is obviously non-linear with respect to

**w**. Minimization of such a criterion can be accomplished by several stochastic nonlinear optimization methods. In this paper, we use a simple, yet effective method, to carry out the minimization of the criterion in Eq. (10) [20

**21**, 1845–1847 (1996). [CrossRef] [PubMed]

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]

*l*node (

*l*: constant) polygon. Considering this polygonal representation, the following stochastic algorithm will iteratively tend to find a better binary window (

**w**) causing a decrease in the optimization criterion. This algorithm carries out the following steps:

*p*

_{i}

*, i*∈[1…

*l*] from the current snake node set.

*d*(

*d*: constant) for the selected node, rebuild the binary window with the new node set and denote it as

*J*(

*J*(

*J*(

**w**

_{k},

**s**) let

**w**

_{k+1}=

*J*(

**w,s**). It is apparent that this algorithm needs a proper initialization, for that its convergence is dependent on that. Nevertheless, due to statistical region based strategy, this approach is still less sensitive to initialization comparing to snake active contours approach. This fact has been illustrated in the results section.

### 4.1. Parameter estimation over large images

**w**

_{k}(see Fig. 2). We show that it is possible to evaluate

*J*(

*k*th iteration along with pixel values inside the aforementioned quadrilateral region.

_{d}connects additively to Ω

_{a}(i.e. increases the area of Ω

_{a}). Now, let the combination of Ω

_{a}+Ω

_{d}define the new window function

*J*(

_{d}pixel information effectively as follows.

*k*and the pixel values in region Ω

_{d}to find

_{d}added to Ω

_{a}, it should be subtracted from Ω

_{b}. Following the same vein in Eq. (11), the following result for the background region is evident:

_{d}is added to the background region and subtracted from the target region. As it has been shown in Eqs. (11,13), the new statistical parameters can be derived as a function of

*k*th parameter vector

^{k}and pixel values in region Ω

_{d}.

### 4.2. Adaptive node selection and direction inertia

*sphacelaria*alga (Fig. 4). Therefore, a uniform node selection/movement scheme is far from optimal and slows down the convergence substantially.

**P**

_{k}by the following rule:

*memory*for its movement direction. The role of this memory is to set the probability of each movement direction according to its effectiveness in past trials. This means for every node the most effective direction in minimizing

*J*(

*) is assigned the highest chance for selection the next time the algorithm visits that node.*

**w,s***j*of node

*p*(

**Q**

_{k}should be updated using the following rule:

## 5. Experimental results

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]

*diatom alga*over which the snake is initialized with 4 nodes. Although the initial contour is completely different from target boundaries, the bivariate region snake is able to capture the object after approximately 1500 iterations, ending up with 24 nodes [Fig. 3(b)]. As it can be seen in Fig. 3(c), the optimization trace obtains a reasonable slope and shows very slight progress after the 1500

^{th}iteration.

*sphacelaria*alga has been illustrated (Fig. 4). This alga has a branch-like structure. The initialization captures a small portion of the living organism and through the iterations, the snake creeps to capture its whole body.

## 6. Conclusion

## Acknowledgments

## References and Links

1. | A. K. Jain, |

2. | W. K. Pratt, |

3. | R. M. Haralick and L. G. Shapiro, “Image segmentation techniques,” Computer Vision, Graphics, and Image Processing |

4. | R. O. Duda, P. E. Hart, and D. G. Stork, |

5. | J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” App. Phys. Lett. |

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. |

7. | U. Schnars and W. P. O. Juptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. |

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 threedimensional object reconstruction and recognition,” Appl. Opt. |

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. |

12. | B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. |

13. | D. Kim and B. Javidi, “Distortion-tolerant 3-D object recognition by using single exposure on-axis digital holography,” Opt. Express |

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 |

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 |

16. | I. Moon and B. Javidi, “Shape-tolerant three-dimensional recognition of biological microorganisms using digital holography,” Opt. Express |

17. | T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. |

18. | T. Kreis, ed., |

19. | H. J. W. Goodman, |

20. | O. Germain and P. Refregier, “Optimal snake-based segmentation of a random luminance target on a spatially disjoint background,” Opt. Lett. |

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. |

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 |

23. | O. Germain and P. Refregier, “Edge detection and location in SAR images: Contribution of statistical deformable models,” in |

24. | M. Kass, A. Witkin, and D. Terzopoulus, “Snakes: Active contour models,” Int. J. Comput. Vis. |

25. | C. Xu and J. L. Prince, “Snakes, shapes, and gradient vector flow,” IEEE Trans. Image Process. |

26. | L. D. Cohen, “On active contour models and balloons,” CVGIP: Image Understanding |

27. | C. Kervrann and F. Heitz, “A hierarchical statistical framework for the segmentation of deformable objects in image sequences,” in Proceedings of |

28. | R. Deriche, “Using Canny’s criteria to derive a recursively implemented optimal edge detector,” Int. J. Comp.Vis. |

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. |

30. | B. Javidi and J. Wang, “Optimum distortion invariant filters for detecting a noisy distorted target in background noise,” J. Opt. Soc. Am. A |

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 |

**OCIS Codes**

(100.6890) Image processing : Three-dimensional image processing

(170.3880) Medical optics and biotechnology : Medical and biological imaging

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 1, 2006

Revised Manuscript: May 12, 2006

Manuscript Accepted: May 16, 2006

Published: June 12, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

Mehdi DaneshPanah and Bahram Javidi, "Segmentation of 3D holographic images using bivariate jointly distributed region snake," Opt. Express **14**, 5143-5153 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-12-5143

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### References

- A. K. Jain, Fundamentals of digital image processing, (Prentice Hall, 1989).
- W. K. Pratt, Digital Image Processing, (Wiley, 2001). [CrossRef]
- R. M. Haralick and L. G. Shapiro, "Image segmentation techniques," Computer Vision, Graphics, and Image Processing 29, 100-132 (1985). [CrossRef]
- R. O. Duda, P. E. Hart, and D. G. Stork, Pattern classification, 2nd ed. (Wiley Interscience, New York, 2000).
- J. W. Goodman, and R. W. Lawrence, "Digital image formation from electronically detected holograms," App. Phys. Lett. 11, 77-79 (1967). [CrossRef]
- 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]
- 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]
- T. Nomura, A. Okazaki, M. Kameda, Y. Morimoto, and B. Javidi, "Image reconstruction from compressed encrypted digital hologram," Opt. Eng. 44 (2005). [CrossRef]
- 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]
- T. J. Naughton, A. E. Shortt, and B. Javidi, "Nonuniform quantization compression of digital holograms," Opt. Lett. (2006) (submitted).
- 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]
- 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]
- 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, 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]
- 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]
- T. Zhang and I. Yamaguchi, "Three-dimensional microscopy with phase-shifting digital holography," Opt. Lett. 23,1221-1223 (1998). [CrossRef]
- 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).
- 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]
- 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]
- 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]
- 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.
- M. Kass, A. Witkin, and D. Terzopoulus, "Snakes: Active contour models," Int. J. Comput. Vis. 1, 321-331 (1987). [CrossRef]
- C. Xu, and J. L. Prince, "Snakes, shapes, and gradient vector flow," IEEE Trans. Image Process. 7, 359-369 (1998). [CrossRef]
- L. D. Cohen, "On active contour models and balloons," CVGIP: Image Understanding 53, 211-218 (1991). [CrossRef]
- 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.
- R. Deriche, "Using Canny's criteria to derive a recursively implemented optimal edge detector," Int. J. Comp.Vis. 1, 167-187 (1987). [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]
- 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]
- 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]

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