# A Comparison of Different Counting Methods for a Holographic Particle Counter: Designs, Validations and Results

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## Abstract

**:**

## 1. Introduction

## 2. Fringe Patterns and Its Features

#### 2.1. Information Content of Fringe Patterns

#### 2.2. Features to Extract

#### 2.3. Intensity Dependence of Fringe Patterns

## 3. Methods

#### 3.1. Customized Hough Transform

#### 3.1.1. Working Principle

#### 3.1.2. Image Preprocessing

#### 3.1.3. Parameterization

#### 3.2. Blob Detection

#### 3.2.1. Blob Extraction Using Template Matching

#### 3.2.2. Blob Segmentation

#### 3.2.3. Blob Labeling and Counting

#### 3.3. Deep Convolutional Neural Network (DCNN)

#### 3.3.1. Working Principle

#### 3.3.2. Training

#### 3.3.3. Data Processing and Evaluation

## 4. Results

#### 4.1. Customized HT

#### 4.2. Blob Detection

#### 4.3. DCNN

#### 4.4. Comparison of Detection Performance

#### 4.4.1. Details on Customized HT

#### 4.4.2. Details on Blob Detection

#### 4.4.3. Details on DCNN

#### 4.5. Comparison of Computational Speed

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CPC | Condensation Particle Counter |

PC | Particle Counter |

HPC | Holographic Particle Counter |

HPCs | Holographic Particle Counters |

CNM | Condensation Nucleus Magnifier |

OPCs | Optical Particle Counters |

PN | Particle Number |

HPIV | Holography Particle Image Velocimetry |

HT | Hough Transform |

CHT | Circular Hough Transform |

PIU | Particle Imaging Unit |

APM | Aerosol Particle Model |

ASM | Angular Spectrum Method |

FZP | Fresnel Zone Plate |

FZPs | Fresnel Zone Plates |

DIH | Digital Inline Holography |

3D | Three-Dimensional |

2D | Two-Dimensional |

SNR | Signal to Noise Ratio |

DNN | Deep Neural Network |

DCNN | Deep Convolutional Neural Network |

LoG | Laplacian of Gaussian |

DoF | Depth of Field |

ReLu | Rectified Linear Unit |

AI | Artificial Intelligence |

RT | Real-Time |

GPU | Graphics Processing Unit |

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**Figure 1.**(

**a**) in-line holographic principle where a single particle creates a fringe pattern at the camera plane (particles are single neclei in droplets); (

**b**) schematic of the in-line holographic counting unit, subsequently called Particle Imaging Unit (PIU) cf. [1].

**Figure 2.**(

**a**) example of a typical detection plane at low particle number concentration with overlapping fringe patterns and patterns of different extent as a result of the ${z}_{prt}$-location in the sampling channel; (

**b**) the normalized intensity histogram.

**Figure 3.**Fresnel Zone Plate (FZP) to estimate the radius ${R}_{n}$ of fringes and the size of fringe patterns; $\Delta dr$ is the distance between two successive zone centers of the same parity (even or odd) and may be interpreted as the smallest detail to preserve when lowpass filtering fringe patterns; $d{r}_{n}$ is the width of nth zone.

**Figure 4.**U-Net architecture example for 32 × 32 pixels in the lowest resolution. Each blue box corresponds to a multi-channel feature map. The number of channels is denoted on top of the box. The $xy$-size is provided at the lower left edge of the box. White boxes represent copied feature maps. The arrows denote the different operations.

**Figure 5.**Detection result of the customized HT (selected image section) (

**a**) Gaussian filtered fringe patterns (${\sigma}_{lp}=2.62$) where all detected fringes are highlighted with circles; (

**b**) the original fringe patterns. Its determined centroids equal the actual position of the particles in the $xy$- plane. There is one missing hit (orange).

**Figure 6.**Detection result of blob detection (selected image section). (

**a**) histogram and the optimal threshold ${k}_{opt}$ of the whole image, obtained by maximum entropy thresholding. Equation (8) needs to be confined to a lower threshold limit set to ${L}_{1}=0.5$ and an upper limit of ${L}_{2}=0.84$ which is the peak of the mainlobe; (

**b**) fringe patterns overlaid with the corresponding blobs that result from a threshold at ${k}_{opt}=0.74$. Four hits are missing (orange).

**Figure 7.**Comparison of the monitored particle number concentration to the counting rates obtained by the PIU. (

**a**) customized HT; (

**b**) blob detection with maximum entropy thresholding; (

**c**) DCNN based on a U-Net.

**Figure 8.**Zoomed segments of measurement samples from Figure 7b that suffer strong background fluctuations: (

**1**) zero-particle frame; (

**2**) particle number concentration of ${C}_{N}=194$ #/cm${}^{3}$; while the customized HT outputs correct hits (only True Positives), the blob detection in both scenarios fails (also False Positives) because of a misinterpreted intensity threshold in the histogram.

Number of Particles | Precision | Accuracy |
---|---|---|

53 | 0.55 | 0.98 |

88 | 0.45 | 0.91 |

103 | 0.36 | 0.87 |

155 | 0.35 | 0.74 |

180 | 0.25 | 0.69 |

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**MDPI and ACS Style**

Brunnhofer, G.; Hinterleitner, I.; Bergmann, A.; Kraft, M.
A Comparison of Different Counting Methods for a Holographic Particle Counter: Designs, Validations and Results. *Sensors* **2020**, *20*, 3006.
https://doi.org/10.3390/s20103006

**AMA Style**

Brunnhofer G, Hinterleitner I, Bergmann A, Kraft M.
A Comparison of Different Counting Methods for a Holographic Particle Counter: Designs, Validations and Results. *Sensors*. 2020; 20(10):3006.
https://doi.org/10.3390/s20103006

**Chicago/Turabian Style**

Brunnhofer, Georg, Isabella Hinterleitner, Alexander Bergmann, and Martin Kraft.
2020. "A Comparison of Different Counting Methods for a Holographic Particle Counter: Designs, Validations and Results" *Sensors* 20, no. 10: 3006.
https://doi.org/10.3390/s20103006