## Cell imaging beyond the diffraction limit using sparse deconvolution spatial light interference microscopy |

Biomedical Optics Express, Vol. 2, Issue 7, pp. 1815-1827 (2011)

http://dx.doi.org/10.1364/BOE.2.001815

Acrobat PDF (1810 KB)

### Abstract

We present an imaging method, dSLIM, that combines a novel deconvolution algorithm with spatial light interference microscopy (SLIM), to achieve 2.3x resolution enhancement with respect to the diffraction limit. By exploiting the sparsity of the phase images, which is prominent in many biological imaging applications, and modeling of the image formation via complex fields, the very fine structures can be recovered which were blurred by the optics. With experiments on SLIM images, we demonstrate that significant improvements in spatial resolution can be obtained by the proposed approach. Moreover, the resolution improvement leads to higher accuracy in monitoring dynamic activity over time. Experiments with primary brain cells, i.e. neurons and glial cells, reveal new subdiffraction structures and motions. This new information can be used for studying vesicle transport in neurons, which may shed light on dynamic cell functioning. Finally, the method is flexible to incorporate a wide range of image models for different applications and can be utilized for all imaging modalities acquiring complex field images.

© 2011 OSA

## 1. Introduction

1. D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science **300**, 82–86 (2003). [CrossRef] [PubMed]

2. F. Zernike, “How I discovered phase contrast,” Science **121**(3141), 345–349 (1955) [CrossRef] [PubMed]

5. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express **19**, 1016–1026 (2011). [CrossRef] [PubMed]

6. S. Van Aert, D. Van Dyck, and A. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—classical criteria and a statistical alternative,” Opt. Express **14**, 3830–3839 (2006). [CrossRef] [PubMed]

7. J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods **19**, 373–385 (1999). [CrossRef] [PubMed]

12. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**, 19462–19478 (2010) [CrossRef] [PubMed]

13. B. Kemper, P. Langehanenberg, and G. Bally, “Digital holographic microscopy: a new method for surface analysis and marker-free dynamic life cell imaging,” Optik Photonik **2**, 41–44 (2007). [CrossRef]

*sparsity*properties of the transform coefficients. This model is especially useful in capturing fine-scale structures, and it successfully reveals the details in the phase components lost due to the instruments optical transfer function. In addition, due to the very low noise floor provided by SLIM (0.3 nm spatially and 0.03 nm temporally [5

5. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express **19**, 1016–1026 (2011). [CrossRef] [PubMed]

**h**and

**H**denote vectors and matrices, respectively, with transposes

**h**

^{T}and

**H**

^{T}. The spatial coordinates within a image are denoted by (

*x,y*), operator * denotes convolution, and

*i*is equal to

## 2. Overview of Spatial Light Interference Microscopy (SLIM)

5. Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express **19**, 1016–1026 (2011). [CrossRef] [PubMed]

*U*(

*x,y*) = |

*U*(

*x,y*)|exp[

*i*Φ(

*x,y*)], is used for illumination, which is decomposed into scattered and unscattered fields after passing the specimen. Let us denote the unscattered light as

*U*

_{0}and the scattered light as

*U*

_{1}(

*x, y*). A liquid crystal phase modulator (LCPM) is used to introduce phase modulations to the unscattered field, such that where

*ϕ*

_{0}is the intentionally added phase delay, and

*ϕ*

_{1}(

*x, y*) is the phase difference between the scattered and unscattered fields caused by the specimen. The unscattered field contains the uniform background of the image field, whereas the scattered light provides information on the structure of the specimen. The recorded intensity is expressed as In traditional phase-contrast microscopy [2

2. F. Zernike, “How I discovered phase contrast,” Science **121**(3141), 345–349 (1955) [CrossRef] [PubMed]

*ϕ*

_{0}is fixed at

*ϕ*

_{1}(

*x, y*) cannot be uniquely retrieved). In contrast, SLIM uses multiple phase delays 0,

*π*, and

*ϕ*

_{1}can be uniquely determined. Specifically,

*ϕ*

_{1}can be extracted from the four recordings using Moreover, the phase associated with the complex field can be calculated by where we define by

## 3. Image formation and deconvolution model

*U*(

*x,y*) =

*|U*(

*x,y*)

*|*exp[

*i*Φ(

*x,y*)] can be uniquely determined using SLIM (using Eqs. (3) and (5)). However, as in all imaging systems, only a degraded version of this field can be observed in practice. Modeling the imaging process as a linear, spatially invariant degradation system, the measured image can be expressed as the convolution of the original complex field with the instrument point spread function (PSF) as where

*h*(

*x,y*) is the PSF of the system, and

*n*(

*x,y*) is the additive signal independent noise. In general, both the magnitude and phase of the complex image function is degraded via the optical transfer function. As in traditional deconvolution [10

10. P. Sarder and A. Nehorai, “Deconvolution methods for 3D fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

*σ*

^{2}is the noise variance, and the functional

*R*(·) is used to regularize and impose constraints on the estimates of the magnitude and phase.

12. Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express **18**, 19462–19478 (2010) [CrossRef] [PubMed]

*Ũ*(

*x,y*)| ≈ |

*U*(

*x,y*)| = const. This assumption makes the problem in Eq. (7) linear, and it is also very useful in avoiding instabilities due to nonlinearity. With this approximation, the problem Eq. (7) becomes For mathematical convenience and clarity, let us denote by

*g*(

*x,y*) the observed field exp[

*i*Φ̃(

*x,y*)], and by

*f*(

*x,y*) the unknown field exp[

*i*Φ(

*x,y*)]. Due to the linearity of the degradation, the problem Eq. (8) can be expressed equivalently in matrix vector form as where

**g**and

**f**are images

*g*(

*x,y*) and

*f*(

*x,y*) in vector forms, respectively, and

**H**is the convolution matrix corresponding to the PSF

*h*(

*x,y*).

**f̂**can be found as (

**H**

^{T}

**H**)

^{−1}

**H**

^{T}

**g**(equivalent to the inverse filter). However, this approach generally leads to noise amplification and ringing artifacts due to heavy suppression of high spatial frequencies. The role of regularization is to impose desired characteristics on the image estimates to avoid this noise amplification and to increase the resolution. The parameter

*β*is used to control the trade-off between the data-fidelity and the smoothness of the estimates.

## 4. Complex field deconvolution using sparsity

### 4.1. Image model

*sparsity*principle, that is, our main assumption is that the phase images can be very accurately represented in some transform domain with sparse coefficients [17

17. S. Gazit, A. Szameit, Y. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express **17**, 23920–23946 (2009). [CrossRef]

*L*linear transforms

**D**

*of the complex image*

_{k}**f**with

*k*= 1,...,

*L*. These transforms are chosen to be high-pass filters, such that their application provide complex images with a large number of coefficients with small values with only a few coefficients containing the most of the signal energy. The selection of the linear transforms that most accurately capture the image characteristics is crucial in the final image quality. We employ a collection of difference operators to capture signal variation at varying scales. The directional contourlets [19

19. A. L. Cunha, J. Zhou, and M. N. Do, “The nonsubsampled contourlet transform: Theory, design, and applications,” IEEE Trans. Image Process. **15**(10), 3089–3101 (2006). [CrossRef] [PubMed]

20. A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Review **51**(1), 34–81 (2009). [CrossRef]

*l*-pseudonorm, and

_{p}*α*are the weighting coefficients. It is known from the compressive sensing and sparse representation literature [15

_{k}15. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory **52**, 489–509 (2006) [CrossRef]

16. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory **52**, 1289–1306 (2006). [CrossRef]

20. A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Review **51**(1), 34–81 (2009). [CrossRef]

*p*≤ 1 enforces sparsity on

**D**

_{k}**f**, with smaller

*p*values increasing the sparsity effect. Hence, the prior in Eq. (12) enforces sparsity in the transform coefficients

**D**

_{k}**f**, which in turn leads to smoothness in the image estimate.

**f**. Instead, we use separate Gaussian priors on each transform coefficient or in a more compact form as where

**A**

*are diagonal matrices with*

_{k}*α*,

_{ki}*i*= 1,...

*N*in the diagonal. Compared to Eq. (12), where a single parameter is assigned to all coefficients of

*k*

^{th}filter output, separate parameters are used for each coefficient. It can be shown that Eq. (14) is equivalent to Eq. (12) in the limit

*p*→ 0 [21], hence it highly enforces sparsity. The model Eq. (14) has the advantage of being convex (as opposed to Eq. (12)), and therefore optimization over Eq. (14) is much easier and more robust compared to

*l*minimization.

_{p}*α*have a special important role in Eq. (13): they represent the local spatial activity at each location, and hence they are a measure of spatial variation in the corresponding filters direction. It is clear that the model Eq. (14) requires a large number of parameters, whose manual selection is not practical. We can, however, estimate them simultaneously with the complex image. For their estimation, we employ an additional level of model and assign uniform priors Notice that this modeling assigns equal probability to all possible values of

_{ki}*α*, hence no prior knowledge is assumed on its value.

_{ki}*compressible*[22

22. J. Romberg, “Imaging via compressive sensing,” IEEE Signal Process. Mag. **25**(2), 14–20 (2008). [CrossRef]

### 4.2. Noise model

*σ*

^{2}the noise variance. An additional level of modeling (as in Eq. (15)) can be incorporated to estimate this parameter as well. However, SLIM provides images with very high SNRs (on the order of 1000 or more), and therefore

*σ*

^{2}is generally very small and can be estimated experimentally from an uniform area of the observed image. In addition, the Gaussian noise assumption becomes an accurate description of the noise in SLIM due to the high SNR (as in fluorescence microscopy [10

10. P. Sarder and A. Nehorai, “Deconvolution methods for 3D fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

### 4.3. Algorithm

**f**and the parameters

*α*using the

_{ki}*maximum a posteriori*(MAP) estimates, that is, This problem is convex in

**f**and

*α*, but not jointly, and therefore we resort to an iterative scheme to estimate the unknowns in an alternating fashion. The optimal estimate of the complex image can be found by taking the derivative of Eq. (18) and setting it equal to zero, which results in

_{ki}*α*can be estimated in a similar way by equating the corresponding derivatives to zero, which results in where

_{ki}*ε*is a small number (e.g., 10

^{−6}) used to avoid numerical instability. It follows from Eq. (20) that the parameters

*α*are functions of the

_{ki}*k*

^{th}filter response at pixel

*i*, and therefore a

*spatially-adaptive*estimation is employed for

**f**in Eq. (19) through their joint estimation. Notice also that matrices

**A**

*are spatial-adaptivity matrices controlling the smoothness applied at each location; when the filter responses at a pixel are very small, the algorithm assumes that the pixel has low spatial variation, and applies a large amount of smoothness at that point. On the other hand, if the filter responses are high, the pixel is likely to be close to an edge and the smoothness amount is lowered to preserve the image structure.*

_{k}**f**using Eq. (19), and the spatial adaptivity matrices

**A**

*using Eq. (20). The block diagram of a single dSLIM iteration is shown in Fig. 2. The filters*

_{k}**D**

*consist of the derivative operators in Eq. (10) and Eq. (11), and the directional contourlets shown in Fig. 1. The estimate of the image Eq. (19) can be computed very efficiently using the conjugate gradient (CG) method. The matrices*

_{k}**H**and

**D**

*do not have to be explicitly constructed during CG iterations; all operations in Eq. (19) can be performed via convolutions in the spatial domain or multiplications in the Fourier domain. Empirically, we found that the algorithm converges rapidly; a few iterations (up to 5–10) is generally enough to provide high-quality results. Hence, the proposed method can be applied to large images very efficiently.*

_{k}*σ*

^{2}) that needs to be set by the user. In our experiments, we empirically estimated its value by taking a rectangular region of the image with uniform values and computing the variance in this region. As mentioned above, this estimate is known to be reliable in images with high SNR [10

10. P. Sarder and A. Nehorai, “Deconvolution methods for 3D fluorescence microscopy images,” IEEE Signal Process. Mag. **23**, 32–45 (2006). [CrossRef]

*i*Φ̃(

*x,y*)] rather than to the measured intensities

*I*(

*x,y,ϕ*

_{0}) in Eq. (3), the quantitative phase information is preserved. In contrast, traditional deconvolution methods [10

**23**, 32–45 (2006). [CrossRef]

## 5. Experiments

*λ*= 530 nm); the field of view is 75

*μ*m × 100

*μ*m with the CCD resolution of 1040 × 1388. In all reported experiments, the specimen is relatively thin such that the whole image is in focus, and the degradation in the image is only due to a planar PSF. The PSF, depicted in Fig. 3(a), is obtained experimentally by imaging a sub-resolution 200nm microbead treated as a point-source. Due to the high SNR provided by SLIM, this PSF closely matches the actual optical transfer function of the imager.

^{−7}–10

^{−6}(for a maximum signal value of 1), which is used as the value of the parameter

*σ*

^{2}. The NAs of the objective and condenser are NA

_{o}= 0.75 and NA

_{c}= 0.55, respectively. The experimentally measured full-width-at-half-maximum (FWHM) of the PSF is 540 nm, which is comparable with the expected Rayleigh limit, calculated as

*μ*m in the SLIM image, whereas it is measured as 0.63

*μ*m in the dSLIM image. The reduction in the length of the particle is approximately 2.3, which is in agreement with the result of the PSF deconvolution experiment (Fig. 3).

## 6. Conclusion

## Acknowledgments

## References and links

1. | D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science |

2. | F. Zernike, “How I discovered phase contrast,” Science |

3. | D. Murphy, “Differential interference contrast (DIC) microscopy and modulation contrast microscopy,” in |

4. | G. Popescu, “Quantitative phase imaging of nanoscale cell structure and dynamics,” in |

5. | Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express |

6. | S. Van Aert, D. Van Dyck, and A. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—classical criteria and a statistical alternative,” Opt. Express |

7. | J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods |

8. | W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques |

9. | F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3D steerable filters,” Opt. Express |

10. | P. Sarder and A. Nehorai, “Deconvolution methods for 3D fluorescence microscopy images,” IEEE Signal Process. Mag. |

11. | Z. Zalevsky and D. Mendlovic, |

12. | Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express |

13. | B. Kemper, P. Langehanenberg, and G. Bally, “Digital holographic microscopy: a new method for surface analysis and marker-free dynamic life cell imaging,” Optik Photonik |

14. | J. P. Haldar, Z. Wang, G. Popescu, and Z. P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” International Conference of the IEEE Engineering in Medicine and Biology Society, Buenos Aires, 2010, pp. 3382–3385. |

15. | E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory |

16. | D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory |

17. | S. Gazit, A. Szameit, Y. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express |

18. | A sparse modeling is also used in [17], but in contrast to our work, sparsity is enforced directly on the intensity image. This modeling is specifically suited for point-like structures, whereas our formulation can model a wide range of structures via the employment of transforms. |

19. | A. L. Cunha, J. Zhou, and M. N. Do, “The nonsubsampled contourlet transform: Theory, design, and applications,” IEEE Trans. Image Process. |

20. | A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Review |

21. | S. D. Babacan, L. Mancera, R. Molina, and A. K. Katsaggelos “Bayesian compressive sensing using non-convex priors,” in EUSIPCO’09, Glasgow, Scotland, Aug. (2009). |

22. | J. Romberg, “Imaging via compressive sensing,” IEEE Signal Process. Mag. |

**OCIS Codes**

(100.1830) Image processing : Deconvolution

(100.5070) Image processing : Phase retrieval

(100.6640) Image processing : Superresolution

(110.0180) Imaging systems : Microscopy

**ToC Category:**

Microscopy

**History**

Original Manuscript: March 28, 2011

Revised Manuscript: May 4, 2011

Manuscript Accepted: May 24, 2011

Published: June 2, 2011

**Citation**

S. Derin Babacan, Zhuo Wang, Minh Do, and Gabriel Popescu, "Cell imaging beyond the diffraction limit using sparse deconvolution spatial light interference microscopy," Biomed. Opt. Express **2**, 1815-1827 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-7-1815

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### References

- D. J. Stephens and V. J. Allan, “Light microscopy techniques for live cell imaging,” Science 300, 82–86 (2003). [CrossRef] [PubMed]
- F. Zernike, “How I discovered phase contrast,” Science 121(3141), 345–349 (1955) [CrossRef] [PubMed]
- D. Murphy, “Differential interference contrast (DIC) microscopy and modulation contrast microscopy,” in Fundamentals of Light Microscopy and Digital Imaging (Wiley-Liss, 2001) pp. 153–168.
- G. Popescu, “Quantitative phase imaging of nanoscale cell structure and dynamics,” in Methods in Cell Biology , B. Jena, Ed. (Elsevier Inc., 2008) vol. 90, pp. 87–115.
- Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19, 1016–1026 (2011). [CrossRef] [PubMed]
- S. Van Aert, D. Van Dyck, and A. den Dekker, “Resolution of coherent and incoherent imaging systems reconsidered—classical criteria and a statistical alternative,” Opt. Express 14, 3830–3839 (2006). [CrossRef] [PubMed]
- J. G. McNally, T. Karpova, J. Cooper, and J. A. Conchello, “Three-dimensional imaging by deconvolution microscopy,” Methods 19, 373–385 (1999). [CrossRef] [PubMed]
- W. Wallace, L. H. Schaefer, and J. R. Swedlow, “A workingperson’s guide to deconvolution in light microscopy,” Biotechniques 31, 1076 (2001).
- F. Aguet, S. Geissbühler, I. Märki, T. Lasser, and M. Unser, “Super-resolution orientation estimation and localization of fluorescent dipoles using 3D steerable filters,” Opt. Express 17, 6829–6848 (2009). [CrossRef] [PubMed]
- P. Sarder and A. Nehorai, “Deconvolution methods for 3D fluorescence microscopy images,” IEEE Signal Process. Mag. 23, 32–45 (2006). [CrossRef]
- Z. Zalevsky and D. Mendlovic, Optical Superresolution (Springer, 2004) vol. 91.
- Y. Cotte, M. F. Toy, N. Pavillon, and C. Depeursinge, “Microscopy image resolution improvement by deconvolution of complex fields,” Opt. Express 18, 19462–19478 (2010) [CrossRef] [PubMed]
- B. Kemper, P. Langehanenberg, and G. Bally, “Digital holographic microscopy: a new method for surface analysis and marker-free dynamic life cell imaging,” Optik Photonik 2, 41–44 (2007). [CrossRef]
- J. P. Haldar, Z. Wang, G. Popescu, and Z. P. Liang, “Label-free high-resolution imaging of live cells with deconvolved spatial light interference microscopy,” International Conference of the IEEE Engineering in Medicine and Biology Society, Buenos Aires , 2010, pp. 3382–3385.
- E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006) [CrossRef]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]
- S. Gazit, A. Szameit, Y. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17, 23920–23946 (2009). [CrossRef]
- A sparse modeling is also used in [17], but in contrast to our work, sparsity is enforced directly on the intensity image. This modeling is specifically suited for point-like structures, whereas our formulation can model a wide range of structures via the employment of transforms.
- A. L. Cunha, J. Zhou, and M. N. Do, “The nonsubsampled contourlet transform: Theory, design, and applications,” IEEE Trans. Image Process. 15(10), 3089–3101 (2006). [CrossRef] [PubMed]
- A. M. Bruckstein, D. L. Donoho, and M. Elad, “From sparse solutions of systems of equations to sparse modeling of signals and images,” SIAM Review 51(1), 34–81 (2009). [CrossRef]
- S. D. Babacan, L. Mancera, R. Molina, and A. K. Katsaggelos “Bayesian compressive sensing using non-convex priors,” in EUSIPCO’09, Glasgow, Scotland, Aug. (2009).
- J. Romberg, “Imaging via compressive sensing,” IEEE Signal Process. Mag. 25(2), 14–20 (2008). [CrossRef]

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