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Time-Dependent Image Restoration of Low-SNR Live-Cell Ca^{2} Fluorescence Microscopy Data

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Synthetic Data

#### 2.2. Live-Cell Fluorescence Microscopy

## 3. Discussion

## 4. Materials and Methods

#### 4.1. Mathematical Formulation

#### 4.2. Experiments: Imaging Data and Evaluation

#### 4.2.1. Dataset 1: Synthetic Image Data

#### 4.2.2. Dataset 2: Genetically Encoded Ca${}^{2+}$ Indicator for Optimal Imaging (GECO) Tagged to Lysosomal TPC2 in Jurkat T-Cells

#### 4.2.3. Dataset 3: Free Cytosolic Ca${}^{2+}$ Concentration Imaging in Jurkat T-Cells

#### 4.2.4. Dataset 4: Confocal Ca${}^{2+}$ Imaging in Astrocytes In Situ

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

ER | static entropy deconvolution |

GECO | genetically encoded Ca${}^{2+}$ indicator for optimal imaging |

LR | Lucy–Richardson |

PSF | Point spread function |

ROI | Region of interest |

SNR | Signal-to-noise ratio |

SSIM | Structural similarity index |

TD ER | time-dependent entropy deconvolution |

TPC | two pore channel |

## Appendix A. Algorithm Details

#### Appendix A.1. Minimization of Cost Functional (uid13)

Algorithm A1: Deconvolution. |

#### Appendix A.2. Practical Notes

## References

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**Figure 1.**Example of synthetic data processed with the different deconvolution methods. (

**a**) Sample frame of synthetic data without any added noise and before applying the PSF. The yellow box indicates the region of interest pictured in panels (

**b**–

**d**), which show input noisy images for various noise levels as well as image restoration results. The parameters for the entropy deconvolution are $\lambda =0.1$ and (TD ER: $\lambda =0.1,{\lambda}_{T}=0.1$) and $\epsilon =0.001$. LR: Lucy–Richardson deconvolution; ER: entropy regularization-based deconvolution (static); TD ER: time-dependent entropy regularization-based deconvolution.

**Figure 2.**(

**a**) The SSIM of the different image restoration methods, averaged over all time frames of 25 different, randomly generated synthetic datasets, such as the one in Figure 1a, normalized with respect to the SSIM between the noisy and the original image as a function of the input Gaussian noise. Thus, a value larger than one indicates image quality improvement compared to the noisy input image. The measurement points correspond to the average values obtained for three different Poisson noise levels, and the error bars indicate the influence of varying the Poisson noise levels in terms of the standard deviation of the respective different simulations. TD ER SSIM values are significantly higher than ER values and ER SSIM values significantly higher than LR values, except for the smallest Gaussian noise level (p < 0.05; paired, one-sided Wilcoxon signed-rank test with Bonferroni correction; based on the SSIM values of the random synthetic time series, with the values averaged over Poisson noise levels). (

**b**) The results of the Gaussian noise estimation as a function of the input Gaussian noise for the different deconvolution methods as well as for the original noisy image, where the latter is pictured in greys and represents a plausibility check of the applied noise estimation approach. TD ER values are significantly lower than ER values and ER values significantly lower than LR values for all Gaussian noise levels.

**Figure 3.**Comparison of the different deconvolution methods for TPC2-R.GECO.1.2 images captured with different exposure times. (

**a**,

**e**,

**i**,

**m**): raw data, captured at 100 ms, 150 ms, 200 ms and 400 ms; (

**b**,

**f**,

**j**,

**n**): images deconvolved using MATLAB’s Lucy–Richardson (LR) algorithm; (

**c**,

**g**,

**k**,

**o**): images deconvolved using the static entropy algorithm (ER); (

**d**,

**h**,

**l**,

**p**): images deconvolved with the time-dependent entropy algorithm (TD ER). Parameters for the entropy algorithms are $\lambda =2.0$ and (TD ER: $\lambda =2.0,{\lambda}_{T}=2.0$) and $\epsilon =0.001$.

**Figure 4.**Comparison of the different deconvolution methods for a time series of dataset 2, captured at 100 ms exposure time. (

**a**) Raw image. (

**b**) Deconvolved with the MATLAB Lucy–Richardson algorithm. (

**c**) Deconvolved by ER with $\lambda =2.0$. (

**d**) Deconvolved with the proposed TD ER with ($\lambda =2.0,{\lambda}_{T}=2.0$). Each panel includes a zoomed-in region of interest indicated in yellow. (

**e**–

**h**) The intensity profile plotted along the blue line in the frames above. All entropy-based algorithms here use $\epsilon =0.001$.

**Figure 5.**Panel (

**a**): Deconvolution results for [Ca${}^{2+}{]}_{\mathrm{i}}$ imaging and frames using Fluo-4 (upper row) and FuraRed (lower row) as the indicator dye. From left to right: raw image, LR, ER and TD ER result. Entropy-based deconvolution parameters were $\lambda =0.4$ (TD ER: $\lambda =0.4,{\lambda}_{T}=0.4$) and $\epsilon =0.001$. Panels (

**b**,

**c**) show the estimated background noise remaining in the deconvolved images, normalized to the background noise of the raw image for the different deconvolution methods.

**Figure 6.**The ratio of the deconvolution results of the two channels from Figure 5 after postprocessing according to [2]. (

**a**) Fluo-4/FuraRed ratio of raw images, (

**b**) ratio of LR results, (

**c**) ratio of ER results and (

**d**) ratio of TD ER results. Panels (

**e**–

**h**) show the intensity profile plotted along the blue line in the frames above.

**Figure 7.**Deconvolution results for a jGCaMP7b-expressing astrocyte in a mouse brain slice. (

**a**) Raw image, (

**b**) LR result, (

**c**) ER result and (

**d**) TD ER result. Entropy parameters here are $\lambda =0.05$ and ($\lambda =0.05,{\lambda}_{T}=0.05$) and $\epsilon =0.001$. Panels (

**e**–

**h**) show the intensity profile plotted along the blue line in the frames above. Panel (

**i**) shows the amount of background noise remaining in the image after the application of the different deconvolution algorithms, normalized to the noise level of the original data.

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## Share and Cite

**MDPI and ACS Style**

Woelk, L.-M.; Kannabiran , S.A.; Brock , V.J.; Gee , C.E.; Lohr , C.; Guse , A.H.; Diercks , B.-P.; Werner, R.
Time-Dependent Image Restoration of Low-SNR Live-Cell Ca^{2} Fluorescence Microscopy Data. *Int. J. Mol. Sci.* **2021**, *22*, 11792.
https://doi.org/10.3390/ijms222111792

**AMA Style**

Woelk L-M, Kannabiran SA, Brock VJ, Gee CE, Lohr C, Guse AH, Diercks B-P, Werner R.
Time-Dependent Image Restoration of Low-SNR Live-Cell Ca^{2} Fluorescence Microscopy Data. *International Journal of Molecular Sciences*. 2021; 22(21):11792.
https://doi.org/10.3390/ijms222111792

**Chicago/Turabian Style**

Woelk, Lena-Marie, Sukanya A. Kannabiran , Valerie J. Brock , Christine E. Gee , Christian Lohr , Andreas H. Guse , Björn-Philipp Diercks , and René Werner.
2021. "Time-Dependent Image Restoration of Low-SNR Live-Cell Ca^{2} Fluorescence Microscopy Data" *International Journal of Molecular Sciences* 22, no. 21: 11792.
https://doi.org/10.3390/ijms222111792