# Estimation Methods of the Point Spread Function Axial Position: A Comparative Computational Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. PSF Model

#### 2.1. OPD Function Implementation

#### 2.2. Numerical Integration of the Model

#### 2.3. Implementation of the 3D PSF Function

#### 2.4. General Model for a Noisy PSF

## 3. Comparing Experimental and Theoretical 3D PSF

**θ**makes $\mathrm{OF}\left(\mathit{\theta}\right)$ optimum. This is,

#### 3.1. Least Square Error

#### 3.2. Csiszár I-Divergence Minimization

- $I\left(p\right|\left|r\right)\ge 0$
- $I\left(p\right|\left|r\right)=0$ when $p=r$

#### 3.3. Maximum Likelihood

#### 3.4. Implementation of the Estimation Methods

#### 3.5. Lower Bound of the Estimation Error

## 4. Results

#### 4.1. Tests on Simulated Data

#### 4.1.1. Cramer–Rao Lower Bound Analysis

#### 4.1.2. Convergence Test

#### 4.1.3. Exploratory Testing

#### 4.1.4. Extensive Testing of Methods

#### 4.2. Axial Position Estimation of an Experimental PSF

## 5. Discussion

#### 5.1. Gibson and Lanni Model

#### 5.2. Computational Convergence

#### 5.3. General Performance of Methods

#### 5.4. Test with an Experimental PSF

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Cramer–Rao lower bound computed with Equation (20) for the noisy point spread function (PSF) model given by Equation (5) under several acquisition conditions. In black, the theoretical PSF intensity axial profile for reference. (

**a**) CRB versus photon number; (

**b**) CRB versus background; (

**c**) CRV versus pixel number; (

**d**) CRB versus pixel size.

**Figure 3.**Descriptive statistics of the residuals of initial estimates and estimations of the methods. Residuals of initial estimates where computed as the absolute value of the difference between the randomized initial estimate and the ground truth. (

**a**) Without noise; (

**b**) Noise Case I (NCI); (

**c**) NCII; (

**d**) NCIII.

**Figure 4.**Samples of diffracted-limited images obtained from the Gibson and Lanni model [13] are shown in the column marked (

**a**). The same images degraded by different noise conditions are shown in the columns marked (

**b**–

**d**). The last row depicts the samples of sagittal slices. (

**a**) Without noise; (

**b**) NCI; (

**c**) NCII; (

**d**) NCIII.

**Figure 5.**Success measurements of the methods: MIDIV (red), ML (green) and LSQR (blue). A dashed line at $95\%$ has been plotted in all subfigures for reference. Estimation results are presented as the mean point estimates of the success percentages and their confidence intervals. (

**a**) NCI; (

**b**) NCII; (

**c**) NCIII.

**Figure 6.**Iteration measurements of the methods: MIDIV (red), ML (green) and LSQR (blue). Iteration number results are presented as the mean point estimates of the iteration number and their confidence intervals. (

**a**) NCI; (

**b**) NCII; (

**c**) NCIII.

**Figure 7.**Time measurements of the methods: MIDIV (red), ML (green) and LSQR (blue). Results are presented as the mean point estimates of the total computing times per parameter estimation and their confidence intervals. (

**a**) NCI; (

**b**) NCII; (

**c**) NCIII.

**Figure 8.**Accuracy measurements of methods: MIDIV (red), ML (green) and LSQR (blue). The dashed line at 5 $\mathsf{\mu}$m represents the ground truth (GT), the actual value of the parameter from which data were generated. The results of accuracy are presented as the mean point estimates of the set of estimations of ${t}_{s}$ and their confidence intervals. (

**a**) NCI; (

**b**) NCII; (

**c**) NCIII.

**Figure 9.**Precision measurements of the methods: MIDIV (red), ML (green) and LSQR (blue). The dashed line represents the theoretical standard deviation of the initial estimates (${\sigma}_{IEs}$). The Cramer–Rao bound (CRB) is shown in solid line segments for each level of intensity. Precision results are presented as the point estimates of the standard deviations of the ${t}_{s}$ estimations and their confidence intervals. (

**a**) NCI; (

**b**) NCII; (

**c**) NCIII.

**Figure 10.**Optical axis estimation results of the methods: MIDIV (red), ML (green) and LSQR (blue). (

**a**) NCI; (

**b**) NCII; (

**c**) NCIII.

**Figure 11.**Pseudocolor images of $45\times 31$ pixels. Upper left, image of fluorescent nanobead of 0.170 ± 0.005 $\mathsf{\mu}$m in diameter. In the counter clockwise direction: synthetic PSFs obtained by the estimations of the axial ${t}_{s}$ by MIDIV, LSQR and ML, respectively.

**Figure 12.**Intensity profiles along the optical axis of the Gibson and Lanni model. Green profiles, ${t}_{s}=0$ $\mathsf{\mu}$m; blue profiles, ${t}_{s}=3$ $\mathsf{\mu}$m; and red profiles, ${t}_{s}=10$ $\mathsf{\mu}$m. Center image, ${n}_{oil}$ in the design condition; left and right plots, ${n}_{oil}$ in the non-design conditions. (

**a**) ${n}_{oil}=1.510$; (

**b**) ${n}_{oil}=1.515$; (

**c**) ${n}_{oil}=1.520$.

Parameter | Description |
---|---|

NA | Numerical aperture |

M | Total magnification |

${n}_{oi{l}^{*}}$ | Nominal refractive index of the immersion oil |

${t}_{oi{l}^{*}}$ | Nominal working distance of the lens |

${n}_{oil}$ | Refractive index of the immersion oil used |

${n}_{{g}^{*}}$ | Nominal refractive index of the coverslip |

${t}_{{g}^{*}}$ | Nominal thickness of the coverslip |

${n}_{g}$ | Refractive index of the coverslip used |

${t}_{g}$ | Thickness of the coverslip used |

${n}_{s}$ | Refractive index of the medium where the point source is |

${t}_{s}$ | Point-source position measured from the coverslip’s inner side |

${z}_{{d}^{*}}$ | Tube length design condition |

${z}_{d}$ | Tube length in the non-design condition |

${x}_{d},{y}_{d}$ | Point source planar position measured from the optical axis |

$\Delta z$ | System defocus |

**Table 2.**Convergence test. ROC, rate of convergence; MIDIV, I-divergence minimization; LSQR, non-linear least square.

Method | Iterations | ${\mathit{\alpha}}_{\mathbf{ROC}}$ |
---|---|---|

MIDIV | 9 | 1.0 |

ML | 9 | 1.0 |

LSQR | 10 | 1.0 |

**Table 3.**FWHM measurements on experimental PSF and the Gibson and Lanni model after estimating ${t}_{s}$ with the MIDIV, ML and LSQR methods.

Axial ($\mathsf{\mu}$m) | Lateral ($\mathsf{\mu}$m) | |
---|---|---|

Exp. PSF | 1.303 | 0.377 |

MIDIV | 1.301 | 0.387 |

ML | 1.301 | 0.387 |

LSQR | 1.301 | 0.387 |

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**MDPI and ACS Style**

Diaz Zamboni, J.E.; Casco, V.H.
Estimation Methods of the Point Spread Function Axial Position: A Comparative Computational Study. *J. Imaging* **2017**, *3*, 7.
https://doi.org/10.3390/jimaging3010007

**AMA Style**

Diaz Zamboni JE, Casco VH.
Estimation Methods of the Point Spread Function Axial Position: A Comparative Computational Study. *Journal of Imaging*. 2017; 3(1):7.
https://doi.org/10.3390/jimaging3010007

**Chicago/Turabian Style**

Diaz Zamboni, Javier Eduardo, and Víctor Hugo Casco.
2017. "Estimation Methods of the Point Spread Function Axial Position: A Comparative Computational Study" *Journal of Imaging* 3, no. 1: 7.
https://doi.org/10.3390/jimaging3010007