OSA's Digital Library

Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 1, Iss. 7 — Jul. 17, 2006
« Show journal navigation

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

Mehdi DaneshPanah and Bahram Javidi  »View Author Affiliations


Optics Express, Vol. 14, Issue 12, pp. 5143-5153 (2006)
http://dx.doi.org/10.1364/OE.14.005143


View Full Text Article

Acrobat PDF (338 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

Image Segmentation is a useful operation in image processing [1

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

, 2

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

]. According to Haralick and Shapiro [3

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

], the objective of segmentation is to break down the input image into homogenous partitions with respect to some characteristics such as gray tone or texture. In deed, the characteristics mentioned above are primitive and many problems require much more sophisticated measures of homogeneity [4

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

]. In this paper we address segmentation of 3D complex holographic images in a statistical framework.

Holographic images can be recorded digitally with optoelectronic sensor arrays for future reconstruction, recognition or compression tasks [5

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

19

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

]. There are several approaches to obtain a three dimensional (3D) digital hologram from an object, including off-axis, phase shift and Single Exposure On-Line (SEOL) digital holography. SEOL holography uses a 3D coherent Mach-Zehnder interferometer (see Fig. 1) to record a single on-line Fresnel interference pattern of the reference and object beams on a solid state imaging sensor such as a charged coupled device [12

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

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]

]. 3D microorganism imaging and recognition using SEOL was reported recently in [14

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

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]

]. Single exposure method has some advantages over its multiple exposure counterpart. Particularly, it eliminates the need for multiple interferogram recordings with phase shifts in the reference beam [17

17. T. Zhang and I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998). [CrossRef]

]. This property is important in dynamic imaging of objects [14

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

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]

], since the scene may change during the phase shift recordings.

Off-axis holography also eliminates the need for multiple recordings [18

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

, 19

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

]. However, this method imposes some limitations. Particularly, only a fraction of the space-bandwidth product of the photo sensor is utilized, which results in inefficient usage of the imaging device and reduced quality of visualization. Also, in off-axis holography, the reconstructed image size is a function of the angle between the object and reference beams, thus its application is restricted to objects with limited dimensions.

Fig. 1. Illustrates a SEOL holographic microscopy setup imaging a Sphacelaria alga.

Clearly, for holographic images, bivariate joint magnitude and phase distribution provides a more accurate model, comparing to independent univariate Gaussian analysis. This is achieved by providing a mechanism to employ the correlation between each pixel’s magnitude and phase content for better segmentation. Thus, this method can be used in segmentation of any 3D holographic image with jointly distributed magnitude and phase.

2. Background

Based on the work of Kass [24

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

], region snake was introduced in [20

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]

] as a reformulation of 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

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]

23

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.

]. Region snake develops a statistically optimal segmentation scheme for images with strong additive or multiplicative noise (such as synthetic aperture radar images [23

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.

]), where classic snake active contours exhibit poor performance due to their local processing nature.

Thus, the contribution of statistical region snake was to change the strategy of contour evolution from local pixel processing to region based criterion. A classic difficulty in the original snake active contours is that the snake can easily get trapped in local minimums of image surface if the snake boundary is to be driven by the gradient of local neighboring pixels to the snake contour [24

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

]. The condition gets worse in case the image surface is jagged. This major drawback requires the snake contour to be initialized close to original boundaries of the target for successful convergence. Although some other methods [25

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

,26

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

] have been proposed to overcome the locality issue of snake active contours, more or less, they are specialized for certain type of images and may fail to address a wide range of problems.

Employing the statistical information of regions for segmentation was introduced originally in [27

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

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

]. However, in [20

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]

] statistical region snake uses the concept of non-overlapping properties of target image and background noise [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

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]

] and develops a statistical framework to estimate the target support. This approach enables region snake to enter the scope of Synthetic Aperture Radar (SAR) images [23

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.

], where speckle noise fails most of the classic edge based approaches as well as degrading the performance of more sophisticated methods like watersheds due to over segmentation [31

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]

]. Hence, region snake has inherited the free-shape target detection ability of snake active contours, while broadening its scope of applicability to more sophisticated image models by introducing noise tolerance.

3. Bivariate jointly distributed region snake

In the context of snake active contours, a snake is essentially a closed contour which evolves to minimize a certain criterion known as snake energy [24

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

]. The contour divides the image into inner and outer separate regions which will be denoted by Ωt and Ωb. Throughout this paper, script t and b will be used for target and background, respectively.

Statistical region snake’s primary objective is to find the best Ωt 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 Np 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={si |i∈[1

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

,N]}, where N is the total number of complex pixels. As mentioned before, reconstructed holographic images are complex, thus each pixel value si is a complex number that we denote its magnitude and phase by αi and φi , respectively. Let w={wi |i∈[1

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

,N]} be a binary window that determines the support of the target such that wi =1 for the target pixels and wi =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

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]

]. The latter can be formulated in the compact model of si =a iwi +b i[1

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

-wi ].

3.1. Bivariate Gaussian distribution model

In [22

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]

] several optimal criterion laws have been derived for situations in which the statistical behavior of target and background are from exponential family (i.e. Gaussian, Gamma and etc). However, as mentioned earlier, in digitally reconstructed holographic images, one deals with complex pixels with distinct magnitude and phase, which are jointly distributed in spatial domain. Thus single variable approach can not be applied to capture the correlation of magnitude and phase of each pixel. However, it is rational to assume that the target and background regions have independent distributions.

Considering the one dimensional, non-overlapping image model, and by assuming a bivariate Gaussian distribution for amplitude (α) and phase (φ) random variables inside the target region, one can write the target’s joint probability distribution function as following:

ft(αi,φi)=1σφtΦ(φiμφtσφt)×1σα|φtΦ(αiμα|φtσα|φt),
(1)

where αi =|si | and φi =∠si are the magnitude and phase of pixel si, respectively. Also, function Φ(x)=(2π)-1/2 exp(-x 2/2) is the standard normal probability distribution function.

Let the parameter vector for this distribution be:

Θt={μαt,μφt,σαt,σφt,ρt},
(2)

where 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 Θb and joint probability density function fb (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:

μαφu=μαu+ρuσαu(φμφu)σφu,σαφu2=σα2(1ρu2),
(3)

for u∈{t,b}. For generality, we assume parameter vector Θ={Θt, Θb} a priori unknown.

3.2. Maximum likelihood hypothesis testing

The problem of segmentation is analogous to estimating the most likely binary window (w), best representing the target support. Considering a hypothesis testing approach with hypothesis Hw , one needs to maximize the a posteriori conditional probability P[Hw |s]. Assuming the general case of equiprobable hypotheses, the Bayes rule states that:

P[Hw|s]=P(Hw).P[s|Hw]P(s).
(4)

Thus maximizing the a posteriori probability P[Hw |s] is equivalent to maximizing the conditional probability P[s|Hw ] which corresponds to the likelihood of the hypothesis Hw .

Since the bivariate Gaussian probability density functions of the target and background are assumed to be independent, the likelihood function for the hypothesis Hw can be written as multiplication of corresponding joint probability distribution functions inside and outside of binary window (w) as:

P(s|Hw,Θ)=i=1Nft(αi,φi).wi×i=1Nfb(αi,φi).(1wi),
(5)

Using the above mathematical likelihood model, by substituting (1) in (5) one can get:

P(sHw,Θ)=(12πσφtσαφt)Nt(w)exp[i=1N(φiμφt)2.wi2(σφt)2]×exp[i=1N(αiμαφt)2.wi2(σαφt)2]
×(12πσφbσαφb)Nb(w)exp[i=1N(φiμφb)2.(1wi)2(σφb)2]×exp[i=1N(αiμαφb)2.(1wi)2(σαφb)2],
(6)

where Nt (w), (w)Nb (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:

log{P(s|Hw,Θ)}=Nt(w)log(2πσφtσαt1ρt2)12(σφt)2i=1N(φiμφt)2.wi
12(σα|φt)2i=1N(αiμαtρtσαt(φiμφt)σφt)2.wi
Nb(w)log(2πσφbσαb1ρb2)12(σφb)2i=1N(φμφb)2.(1wi)
12(σα|φb)2i=1N(αiμαbρtσαb(φiμφb)σφt)2.(1wi).
(7)

So far Eq. (7) indicates the log-likelihood function of hypothesis Hw 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.

In order to derive the maximum likelihood estimation of unknown parameters one has to take partial derivatives of the likelihood (or log-likelihood) function with respect to elements of Θ, and set each to zero to eventually find the estimated parameter vector denoted by Θ^ ={Θ^ t, Θ^ b}. The resulting sample mean, variance and correlation are as following:

μ̂αu=1Nu(w)iΩuαi,μ̂φu=1Nu(w)iΩuφi,
(σ̂αu)2=1Nu(w)iΩu(αiμ̂αu)2,(σ̂φu)2=1Nu(w)iΩu(φiμ̂αu)2,
ρ̂u=1Nu(w)σ̂αuσ̂φuiΩu(αiμ̂αu)(φiμ̂αu),
(8)

where u denotes either of the target or background regions.

3.3. Optimization criterion

Substituting the maximum likelihood estimates of Eq. (8) in (7), one can rename, simplify and rewrite the log-likelihood function in Eq. (7) as:

F(s|Hw,Θ̂)=log{P(s|Hw,Θ̂)}=N2log(2π)Nt(w)log(σ̂φtσ̂αt1ρ̂t2)
Nb(w)log(σ̂φbσ̂αb1ρ̂b2)N.
(9)

Since the first and last terms in Eq. (9) are independent of the choice of the binary window function (w), it is obvious that maximization of (9) is analogous to minimization of the following criterion with respect to w:

J(s|Hw,Θ̂)=Nt(w)log(σ̂φtσ̂αt1ρ̂t2)+Nb(w)log(σ̂φbσ̂αb1ρ̂b2).
(10)

Thus the hypothesis Hw 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

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

], Eq. (10) can be interpreted as the snake energy since its minimization guides the snake contour to enclose the target.

4. Stochastic optimization algorithm

The optimization criterion derived in Eq. (10) presents a hyper-plane spanned over 2Np dimensions (with Np 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

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]

22

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]

].

Lets approximate the closed snake contour by an 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:

1. Randomly select a node pi, i∈[1…l] from the current snake node set.

2. Consider a random direction displacement with length d (d: constant) for the selected node, rebuild the binary window with the new node set and denote it as wk+1.

3. Compute J(wk+1, s) by using the slightly deformed new window. If it happens to be J(wk+1, s)<J(w k, s) let w k+1=wk+1 (i.e. Accept the deformation). Otherwise discard the deformation and start the steps over.

This three step simple algorithm will force the cost function to decrease, but it is clear that it does not guarantee global optimization since the initial snake nodes are set manually. The process terminates when there’s no more progress in minimization of 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

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.

For the sake of clarity, the mathematical manipulation for single variable situation is described here. Nevertheless, generalizing this procedure for the bivariate case is a trivial task. In the following, superscripts denote the iteration number and subscripts indicate the region initial. Using the above assumptions we wish to find σ^ak+1 and σbk+1.

In the following, it has been shown that one can use statistics from iteration k and the pixel values in region Ωd to find σa,bk+1:

(σ̂ak+1)2=1NΩak+ΩdsiΩak+Ωd(si(μ̂ak+Δμ))2=1NΩak+ΩdsiΩak+Ωd[(siμ̂ak)2+(Δμ)2+2(Δμ)(siμ̂ak)]
=1NΩak+Ωd{(NΩak+Ωd.(Δμ)2)+siΩak[(siμ̂ak)2+2(Δμ)(siμ̂ak)]+siΩd[(siμ̂ak)2+2(Δμ)(siμ̂ak)]}
=1NΩak+Ωd{(NΩak+Ωd.(Δμ)2)+NΩak.(σ̂ak)2+siΩd(siμ̂ak)2+2(Δμ)siΩd(siμ̂ak)},
(11)

where

Δμ=μ̂ak+1μ̂ak=μ̂ak.NΩak+μ̂d.NΩdNΩak+Ωdμ̂ak.
(12)

By having Ωd added to Ωa, it should be subtracted from Ωb. Following the same vein in Eq. (11), the following result for the background region is evident:

σ̂bk+1=1NΩbkΩd{(NΩbkΩd.(Δμ)2)+NΩbk.σ̂bksiΩd(siμ̂bk)22(Δμ)siΩd(siμ̂bk)},
(13)

with

Δμ=μ̂bk+1μ̂bk=μ̂bk.NΩbkμ̂d.N̂ΩdNΩakΩdμ̂bk.
(14)

It is also a trivial task to derive expression for a situation in which the selected node is moved in a way that Ωd 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 kth parameter vector Θ^ k and pixel values in region Ωd.

4.2. Adaptive node selection and direction inertia

In every iteration of the original stochastic method, a node is selected and moved in a uniform random fashion. However, for most arbitrary initializations of the snake contour, to enclose the target appropriately, some of the nodes need to be moved more often and some of the movement directions are more significant for an individual node. This is particularly true for the cases in which the snake has to creep the structure of an intricate target such as the sphacelaria alga (Fig. 4). Therefore, a uniform node selection/movement scheme is far from optimal and slows down the convergence substantially.

Since we have no prior knowledge of the target shape, we have no information about the nodes or directions which are more significant. However, while the snake evolves, the significant nodes can be identified by watching the number of successful movements of each node relative to total successful deformations. Here we exploit this fact to present the following adaptive node selection scheme. Let all nodes be selected equiprobable at first iteration; that is:

P1(pi)=Np1fori{1Np}.
(16)

For every successful deformation of snake, update the selection probabilities vector P k by the following rule:

Pk(pi)=Numberofsuccessfuldeformationsofnodepiinthelast(k1)iterationsTotalnumberofsuccessfuldeformationsinthelast(k1)iterations.
(17)

Practically, the above can be implemented by initiating equiprobable directions for all nodes. Each time a successful deformation happens in discrete direction j of node p(αpj ), the direction probability vector Q k should be updated using the following rule:

Qk(αpj)=Numberofsuccessfuldeformationsofnodepindirectionjinthelast(k1)iterationsTotalnumberofsuccessfuldeformationsofnodepinthelast(k1)iterations.
(18)

The reader should note that all directions still have the chance of being selected. So even if a node has a strong inertia toward one direction and some other direction happens to become a better choice for that specific node, the algorithm will adapt and push the inertia toward the best direction gradually. Utilizing this schema, the number of successful modifications in a run increases; thereby the snake can segment the target faster.

5. Experimental results

In this section some experimental results on application of bivariate jointly distributed region snakes are presented. Digitally reconstructed images of several microorganisms from SEOL holographic microscopy [see Fig. 1(a)] experiment are being used [14

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]

]. As discussed earlier, bivariate region snake incorporates both the magnitude and phase information simultaneously. But since the holographic images are complex, only the magnitude images are shown here.

The first image in Fig. 3(a) shows a 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 1500th iteration.

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.

This time, the image is intentionally reconstructed out of focus from a SEOL hologram, so it appears blurred without well defined edges to demonstrate the robustness of the proposed algorithm for holographic images. However, bivariate region snake shows promising results in its segmentation.

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.

6. Conclusion

A novel bivariate jointly distributed region snake in a statistical framework has been proposed for segmentation of digitally reconstructed SEOL holographic images. The advantage of holographic images is that they provide both phase and magnitude information from the scene. This additional information aids in a more reliable segmentation. The proposed method extends the region snakes concept [20

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]

23

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.

] to images with complex pixel values by exploiting the joint correlated magnitude-phase distribution and eventually offers an optimization criterion for the snake-based segmentation. Moreover, the stochastic algorithm which is normally used to carry out the optimization process of region snakes is modified with some boosting techniques to increase the robustness, convergence rate and to ensure proper segmentation. We believed that this technique can be used for any 3D digital holographic image with complex pixel values. Future work may include (but not limited to) estimation of a reliable initial snake contour through other boundary detection schemes. Experimental results using biological samples that we presented indicate the segmentation of 3D holographic images.

Acknowledgments

We wish to thank Inkyu Moon for his assistance with optical experiments.

References and Links

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 threedimensional 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]

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  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]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

Supplementary Material


» Media 1: MOV (1100 KB)     
» Media 2: MOV (2405 KB)     

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited