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Deconvolution in confocal microscopy with total variation regularization

N. Dey, L. Blanc-Féraud, J. Zerubia INRIA/I3S, Projet ARIANA, BP93, 06902 Sophia Antipolis, France
C. Zimmer, J.-C. Olivo-Marin Institut Pasteur, LAIQ, 25-28 rue du Dr. Roux, 75015 Paris, France
Z. Kam Weizmann Institute of Science, Rehovot, Israel 76100

email: Nicolas.Dey@sophia.inria.fr

Introduction

The confocal laser scanning microscope (CLSM) is an optical fluorescence microscope associated to a laser that scans the specimen in 3D and uses a pinhole to reject most out-of-focus light. This is a powerful and increasingly popular microscopy technique for 3D imaging of biological specimens. Nevertheless, the quality of confocal microscopy images suffers from two basic physical limitations. First, out-of-focus blur due to the diffraction-limited nature of optical microscopy remains substantial, even though it is reduced compared to widefield microscopy. Second, the confocal pinhole drastically reduces the amount of light detected by the photomultiplier, leading to Poisson noise [5]. The images produced by CLSM can therefore benefit from postprocessing by deconvolution methods designed to reduce blur and/or noise. Several deconvolution methods have been proposed for confocal microscopy, such as Tikhonov-Miller inverse filter [9], the Carrington [9] and Richardson-Lucy (RL) algorithms [6,4]. An important drawback of RL deconvolution, however, is that it amplifies noise after a few iterations. This sensitivity to noise can be avoided with the help of regularization constraints, leading to much improved results. Conchello et al. [1] and van Kempen et al. [8] have presented a RL algorithm using energy-based regularization applied to biological images. Conchello's regularization term leaves oscillations introduced by RL iterations in homogeneous areas, whereas Tikhonov-Miller regularization [8] results in over-smoothed edges. Here we propose to use RL algorithm with a regularization term based on total variation, which preserves the edges in the image and smoothes homogeneous areas.

Image formation model

We model the blur by using a standard model [7] of the optical system point spread function (PSF) and we simulate the image degradation by introducing Poisson noise. Thus, if $ o$ is the undegraded image, we model the observed image $ i$ as: $ i=N_{Poiss}(o*h)$ where $ h$ is the PSF. A more detailed description of the PSF was given in [3].


Proposed deconvolution method

A common approach to image restoration uses a probabilistic framework: given an observed degraded image $ i$, what is the image $ o$ that maximizes the probability of observing image $ i$? This probability $ p(o\mid i)$ obeys Bayes' rule: $ p(o\mid i)=p(i\mid o).p(o)/p(i)$. The likelihood probability $ p(i\mid o)$ is a Poisson probability. By maximizing this likelihood probability, which is equivalent to minimize $ -\log p(i\mid o)$, we obtain the well-known Richardson-Lucy (RL) iterative algorithm. Standard RL algorithm applied on noisy images rarely converges to a suitable solution, as it tends to amplify the noise after several iterations. Here, we propose to rather minimize the a posteriori probability $ p(o\mid i)\sim p(i\mid o).p(o)$ where the probability $ p(o)$ define a model on the object to reconstruct and acts as a regularization term. We define this model such that the associated minimization $ -\log p(o\mid i)$ leads to the minimization of the likelihood plus the total variation (TV): $ -\log
p(i\mid o)+\lambda\sum_{{\bf x}}\left\vert\nabla o({\bf x})\right\vert$. This regularization avoids the noise amplification by smoothing homogeneous areas while preserving edges in the images. Eq. [*] gives the regularized RL algorithm, where $ \lambda$ is the regularization parameter ( $ \lambda\sim10^{-2}$ to $ 10^{-3}$):

$\displaystyle o_{n+1}({\bf x})=\left\{ \left[\frac{i({\bf x})}{\left(o_{n}*h\ri...
...frac{\nabla o_{n}({\bf x})}{\left\vert\nabla o_{n}({\bf x})\right\vert}\right)}$ (1)

Results

We present results on synthetic test images (see Fig. [*]). The synthetic images are different geometrical objects, as represented on Fig. [*] (a)-(c). Qualitatively, the proposed deconvolution method greatly improves the results compared to standard RL: in contrast to standard RL, our method preserves the edges and avoids intensity oscillations. Quantitatively, we use Csiszár I-divergence [2] and MSE criterions to compare original data and final estimate. We obtain a great improvement compared to standard RL (from $ 1.$ for standard RL to $ .5$ to $ .7$ for regularized RL with TV).

Figure: Deconvolution of synthetic test images using RL with TV regularization. (a)-(c): synthetic test images. (d)-(f): blurred and noisy images degraded as defined in section [*]. (h)-(j): images deconvolved with standard RL. (k)-(m): images deconvolved with RL using TV regularization.
1.!\includegraphics{figures/f92_Robjects}

(a) (b) (c)


1.!\includegraphics{figures/f92_Rblurnoise}

(d) (e) (f)


1.!\includegraphics{figures/f92_R_RL}

(h) (i) (j)


1.!\includegraphics{figures/f92_R_RLregul}

(k) (l) (m)


Conclusion

We have presented a new deconvolution method for confocal microscopy images. The method is based on the Richardson-Lucy algorithm, regularized using total variation. This yields good deconvolution results for synthetic images. During the talk, we will present some results on real images too. The next step is to improve the point spread function model, to take into account optical aberations.

Bibliography

1
J.-A. Conchello and J.G. McNally (1996). ``Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy'', Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing III, pp. 199-208.

2
I. Csiszár (1991). ``Why least squares and maximum entropy?'', The Annals of Statistics 19, 2032-2066.

3
N. Dey, L. Blanc-Féraud, C. Zimmer, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia (2004). ``A deconvolution method for confocal microscopy with total variation regularization'', Proceedings of ISBI'2004.

4
L.B. Lucy (1974). ``An iterative technique for rectification of observed distributions'', The Astronomical Journal 79, no. 6, 745-765.

5
J.B. Pawley (1996). Handbook of biological confocal microscopy, 2nd ed., Plenum Press, New York.

6
W. H. Richardson (1972). ``Bayesian-based iterative method of image restoration'', Journal of Optical Society of America 62, 55-59.

7
H.T.M. van der Voort (1989). Three dimensional image formation and processing in confocal microscopy, Ph.D. thesis, Amsterdam University.

8
G.M.P. van Kempen and L.J. van Vliet (2000). ``Background estimation in non linear image restoration'', Journal of Optical Society of America A 17, no. 3, 425-433.

9
G.M.P. van Kempen, L.J. van Vliet, P.J. Verveer, and H.T.M van der Voort (1997). ``A quantitative comparison of image restoration methods for confocal microscopy'', Journal of Microscopy 185, 354-365.


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