# A Novel Evolution Strategy of Level Set Method for the Segmentation of Overlapping Cervical Cells

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## Abstract

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## 1. Introduction

- (1)
- We designed a morphological scaling-based topology filter (MSTF) to filter out the false positive fragments caused by improper allocation of the initial contour points of touching cells (i.e., misallocation). In MSTF, we constructed the signed distance function (SDF) as the LSF of the initialized cytoplasm contour based on the linear time Euclidean distance transform (LTEDT) algorithm [62], which is denoted by LTEDT-SDF.
- (2)
- We theoretically derived a new mathematical toolbox about vector calculus for evolution of the LSF as the supplementary for the codimension two-level set method (CTLSM) [63], aiming to keep the initialized contours of the nonoverlapping region fixed. Our proposed evolution method of partially fixed contour is called the 2D codimension two-object level set method (DCTLSM), which can alleviate the accuracy loss of a MSTF.
- (3)
- We proposed a novel evolution strategy of LSF inspired by the watershed method [64]. In this strategy, we provided an effective guidance mechanism for attracting and repelling the LSF to converge towards its actual cell boundary.
- (4)
- We used the dataset published by the First Overlapping Cervical Cytology Image Segmentation Challenge held in conjunction with the IEEE International Symposium on Biomedical Imaging (ISBI-2014 challenge) to evaluate our proposed method. The experimental results showed that cellular clumps consisting of two to 10 cells under an overlap ratio less than 0.2 can be accurately segmented. Furthermore, the segmentation of cellular clumps consisting of two to four cells can be effectively segmented with an overlap ratio less than 0.5. By qualitive and quantitative comparisons, our method outperformed the other segmentation methods.

## 2. Methodology

#### 2.1. Cellular Component Segmentation

#### 2.2. Touching Cell Spliting

#### 2.2.1. Morphological Scaling-Based Topology Filter

#### 2.2.2. 2D Codimension Two-Object Level Set Method

#### 2.3. Overlapping Cell Segmentation

#### 2.3.1. Cutting Line Detection

#### 2.3.2. Contour Scanning Strategy for Segmentation

## 3. Experiments

#### 3.1. Image Datasets

#### 3.2. Evaluation Metrics

## 4. Results and Discussion

#### 4.1. The Determination of Morphological Scaling Threshold for MSTF and DCTLSM

#### 4.2. Quantitative Evaluation of Our Segmentation Results

#### 4.2.1. Quantitative Comparison with Baseline Method

#### 4.2.2. Quantitative Comparison with The-State-of-The-Art Methods

#### 4.2.3. Computational Complexity

#### 4.3. Qualitative Evaluation of Our Segmentation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The architecture of the proposed method takes the contour initialization as the input to perform touching cell splitting (pink arrow) and overlapping cell segmentation (purple arrow). MSTF: morphological scaling-based topology filter and DCTLSM: 2D codimension two-object level set method. The “Fig.” and “Eq.” denote “Figure” and “Equation” respectively.

**Figure 2.**Illustration of the algorithm flow of cellular component segmentation. (

**a**) Cervical cytology image; (

**b**) Super-pixel edge map; (

**c**) Clean edge map; (

**d**) Convex hull of cellular clump; (

**e**) Segmentation of cellular clumps; (

**f**) Nuclei detection and segmentation; (

**g**) Initialization of cytoplasm in the cellular clump.

**Figure 3.**Process of MSTF and DCTLSM. (

**a**) Misallocated contour fragments; (

**b**) Contour shrunk by MSTF; (

**c**) Contour expanded by MSTF; (

**d**) False negative area; (

**e**) Fixed contour ${\varphi}_{0}$ and active contour ϕ

_{1}; (

**f**) Contour shrunk by DCTLSM; (

**g**) Contour expanded by DCTLSM. (

**h**) Refined segmentation by Canny mountain map. TPR: true positive rate, FPR: false positive rate and FNR: false negative rate.

**Figure 4.**Explanation of MSTF and DCTLSM. (

**a**) Linear time Euclidean distance transform-signed distance function (LTEDT-SDF) of the initial contour; (

**b**) LTEDT-SDF of the contour shrunk by MSTF; (

**c**) Canny mountain map; (

**d**) Illustration of the morphological shrinking process of the MSTF; (

**e**) Illustration of the morphological expanding process of the MSTF; (

**f**) Illustration of the additional term ${\delta}_{0}{\varphi}_{1}$.

**Figure 5.**Process of the segmentation of overlapping cells. (

**a**) Construction of the union contour; (

**b**) Detection of the critical state contour; (

**c**) Discriminant point; (

**d**) Cutting line; (

**e**) Contour scanning result; (

**f**) Contour shrunk by the DCTLSM; (

**g**) Contour expanded by the DCTLSM; (

**h**) Refined segmentation by the Canny mountain map.

**Figure 6.**Explanation of the contour scanning strategy. (

**a**) LTEDT−SDF of the union contour; (

**b**) Topology discriminant bridge; (

**c**) Detection of the discriminant point; (

**d**) Velocity field for the evolution of the LSF.

**Figure 7.**Selection of the morphological scaling threshold. (

**a**) Numeric outlier detection of the pixel−based average false positive growth rate ($\Delta \mathrm{FP}$); (

**b**) Validity comparison of the MSTF and DCTLSM with the $\Delta \mathrm{FP}$ and pixel-based average false negative growth rate ($\Delta \mathrm{FN}$) at threshold (TH) = 15.

**Figure 8.**Illustration of quantitative comparisons in terms of mean the Dice coefficient ($\mathrm{DC}$), object−based false negative rate (${\mathrm{FN}}_{\mathrm{O}}$), mean true positive rate (${\mathrm{TP}}_{\mathrm{P}}$), and mean false positive rate (${\mathrm{FP}}_{\mathrm{P}}$) against the baseline method (reproduced from [33], Institute of Electrical and Electronics Engineers, 2015) on the (

**a**) train-45 and (

**b**) test-90 datasets (using the mean and standard deviation results of each measure) at TH = 15. This is a visualization of Table 3.

**Figure 9.**Mean Dice coefficient (

**a**), mean true positive rate (

**b**), mean false positive rate (

**c**) and object−based false negative rate (

**d**) as a function of number of cells and overlap ratio.

**Figure 10.**Running time (in minutes) of the (

**a**) segmentation of the cellular component (initialization); (

**b**) segmentation of the cytoplasm by the baseline method and (

**c**) segmentation of the cytoplasm by our proposed method.

**Figure 11.**Qualitative results of (

**a**) Lu (reproduced from [33], Institute of Electrical and Electronics Engineers, 2015) and (

**b**) our proposed method compared with (

**c**) the ground truth.

**Table 1.**Results (mean) on the train-45 dataset in terms of the pixel-based average false positive growth rate ($\Delta \mathrm{FP}$) and pixel-based average false negative growth rate ($\Delta \mathrm{FN}$) for the selection and verification of a suitable morphological scaling threshold as a function of the number of cells. TH: threshold and MSTF: morphological scaling-based topology filter.

TH | 2 Cells | 3 Cells | 4 Cells | 5 Cells | 6 Cells | 7 Cells | 8 Cells | 9 Cells | 10 Cells |
---|---|---|---|---|---|---|---|---|---|

ISBI-14 Train-45 Dataset (MSTF) | |||||||||

2 | ΔFP^{P} = −0.38ΔFN ^{P} = +1.19 | ΔFP^{P} = −0.05ΔFN ^{P} = +0.67 | ΔFP^{P} = −0.09ΔFN ^{P} = +0.42 | ΔFP^{P} = −0.21ΔFN ^{P} = +0.66 | ΔFP^{P} = −0.06ΔFN ^{P} = +0.46 | ΔFP^{P} = −0.15ΔFN ^{P} = +0.39 | ΔFP^{P} = −0.21ΔFN ^{P} = +0.75 | ΔFP^{P} = −0.19ΔFN ^{P} = +0.61 | ΔFP^{P} = −0.14ΔFN ^{P} = +0.98 |

5 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.06ΔFN ^{P} = +0.70 | ΔFP^{P} = −0.09ΔFN ^{P} = +0.43 | ΔFP^{P} = −0.26ΔFN ^{P} = +0.70 | ΔFP^{P} = −0.06ΔFN ^{P} = +0.47 | ΔFP^{P} = −0.17ΔFN ^{P} = +0.43 | ΔFP^{P} = −0.24ΔFN ^{P} = +0.78 | ΔFP^{P} = −0.24ΔFN ^{P} = +0.63 | ΔFP^{P} = −0.14ΔFN ^{P} = +1.02 |

10 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.06ΔFN ^{P} = +0.77 | ΔFP^{P} = −0.13ΔFN ^{P} = +0.47 | ΔFP^{P} = −0.36ΔFN ^{P} = +0.84 | ΔFP^{P} = −0.08ΔFN ^{P} = +0.56 | ΔFP^{P} = −0.21ΔFN ^{P} = +0.48 | ΔFP^{P} = −0.29ΔFN ^{P} = +0.86 | ΔFP^{P} = −0.28ΔFN ^{P} = +0.70 | ΔFP^{P} = −0.16ΔFN ^{P} = +1.10 |

15 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.07ΔFN ^{P} = +0.90 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.39ΔFN ^{P} = +1.00 | ΔFP^{P} = −0.11ΔFN ^{P} = +0.74 | ΔFP^{P} = −0.27ΔFN ^{P} = +0.56 | ΔFP^{P} = −0.40ΔFN ^{P} = +0.97 | ΔFP^{P} = −0.34ΔFN ^{P} = +0.82 | ΔFP^{P} = −0.19ΔFN ^{P} = +1.34 |

20 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.08ΔFN ^{P} = +1.11 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.39ΔFN ^{P} = +1.11 | ΔFP^{P} = −0.19ΔFN ^{P} = +0.91 | ΔFP^{P} = −0.33ΔFN ^{P} = +0.64 | ΔFP^{P} = −0.47ΔFN ^{P} = +1.09 | ΔFP^{P} = −0.38ΔFN ^{P} = +0.95 | ΔFP^{P} = −0.27ΔFN ^{P} = +1.73 |

25 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.09ΔFN ^{P} = +1.32 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.40ΔFN ^{P} = +1.19 | ΔFP^{P} = −0.28ΔFN ^{P} = +1.11 | ΔFP^{P} = −0.56ΔFN ^{P} = +0.72 | ΔFP^{P} = −0.53ΔFN ^{P} = +1.21 | ΔFP^{P} = −0.43ΔFN ^{P} = +1.14 | ΔFP^{P} = −0.39ΔFN ^{P} = +2.13 |

30 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.14ΔFN ^{P} = +1.86 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.40ΔFN ^{P} = +1.19 | ΔFP^{P} = −0.35ΔFN ^{P} = +1.27 | ΔFP^{P} = −0.56ΔFN ^{P} = +0.76 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.30 | ΔFP^{P} = −0.44ΔFN ^{P} = +1.32 | ΔFP^{P} = −0.72ΔFN ^{P} = +2.36 |

35 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.22ΔFN ^{P} = +2.49 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.40ΔFN ^{P} = +1.19 | ΔFP^{P} = −0.39ΔFN ^{P} = +1.43 | ΔFP^{P} = −0.56ΔFN ^{P} = +0.85 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.37 | ΔFP^{P} = −0.44ΔFN ^{P} = +1.32 | ΔFP^{P} = −0.72ΔFN ^{P} = +2.36 |

50 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.70ΔFN ^{P} = +2.99 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.40ΔFN ^{P} = +1.19 | ΔFP^{P} = −0.56ΔFN ^{P} = +1.65 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.01 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.37 | ΔFP^{P} = −0.44ΔFN ^{P} = +1.32 | ΔFP^{P} = −0.72ΔFN ^{P} = +2.36 |

70 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.70ΔFN ^{P} = +2.99 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.40ΔFN ^{P} = +1.19 | ΔFP^{P} = −0.56ΔFN ^{P} = +1.65 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.01 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.37 | ΔFP^{P} = −0.44ΔFN ^{P} = +1.32 | ΔFP^{P} = −0.72ΔFN ^{P} = +2.36 |

100 | ΔFP^{P} = −0.84ΔFN ^{P} = +1.22 | ΔFP^{P} = −0.70ΔFN ^{P} = +2.99 | ΔFP^{P} = −0.70ΔFN ^{P} = +0.57 | ΔFP^{P} = −0.40ΔFN ^{P} = +1.19 | ΔFP^{P} = −0.56ΔFN ^{P} = +1.65 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.01 | ΔFP^{P} = −0.57ΔFN ^{P} = +1.37 | ΔFP^{P} = −0.44ΔFN ^{P} = +1.32 | ΔFP^{P} = −0.72ΔFN ^{P} = +2.36 |

ISBI-14 Train-45 Dataset (DCTLSM) | |||||||||

15 | ΔFP^{P} = −0.54ΔFN ^{P} = −0.08 | ΔFP^{P} = −0.06ΔFN ^{P} = −0.04 | ΔFP^{P} = −0.67ΔFN ^{P} = −0.03 | ΔFP^{P} = −0.30ΔFN ^{P} = −0.02 | ΔFP^{P} = +0.00ΔFN ^{P} = −0.02 | ΔFP^{P} = −0.14ΔFN ^{P} = +0.02 | ΔFP^{P} = −0.18ΔFN ^{P} = −0.00 | ΔFP^{P} = −0.23ΔFN ^{P} = −0.01 | ΔFP^{P} = −0.10ΔFN ^{P} = +0.05 |

**Table 2.**Results (mean) on the test-90 dataset in terms of the $\Delta \mathrm{FP}$ and $\Delta \mathrm{FN}$ for the verification of a suitable morphological scaling threshold as a function of the number of cells at TH = 15.

TH | 2 Cells | 3 Cells | 4 Cells | 5 Cells | 6 Cells | 7 Cells | 8 Cells | 9 Cells | 10 Cells |
---|---|---|---|---|---|---|---|---|---|

ISBI-14 Test-90 Dataset (MSTF) | |||||||||

15 | ΔFP_{P} = −0.74ΔFN _{P} = +0.83 | ΔFP_{P} = 0.00ΔFN _{P} = 0.00 | ΔFP_{P} = −0.36ΔFN _{P} = +0.97 | ΔFP_{P} = 0.00ΔFN _{P} = 0.00 | ΔFP_{P} = −0.25ΔFN _{P} = +0.64 | ΔFP_{P} = −0.31ΔFN _{P} = +0.72 | ΔFP_{P} = −0.04ΔFN _{P} = +0.90 | ΔFP_{P} = −0.37ΔFN _{P} = +0.91 | ΔFP_{P} = −0.14ΔFN _{P} = +0.79 |

ISBI-14 Test-90 Dataset (DCTLSM) | |||||||||

15 | ΔFP_{P} = −0.74ΔFN _{P} = −0.02 | ΔFP_{P} = 0.00ΔFN _{P} = 0.00 | ΔFP_{P} = −0.36ΔFN _{P} = +0.02 | ΔFP_{P} = 0.00ΔFN _{P} = 0.00 | ΔFP_{P} = −0.23ΔFN _{P} = −0.06 | ΔFP_{P} = −0.21ΔFN _{P} = −0.02 | ΔFP_{P} = −0.01ΔFN _{P} = +0.04 | ΔFP_{P} = −0.32ΔFN _{P} = −0.02 | ΔFP_{P} = −0.10ΔFN _{P} = −0.06 |

**Table 3.**Quantitative comparison against the baseline method [33] on the train-45 and test-90 datasets (using the mean and standard deviation results of each measure), as a function of the Dice coefficient ($\mathrm{DC}$) thresholds at TH = 15. Object-based false negative rate (${\mathrm{FN}}_{\mathrm{O}}$), mean true positive rate (${\mathrm{TP}}_{\mathrm{P}}$) and mean false positive rate (${\mathrm{FP}}_{\mathrm{P}}$).

Methods | DC > 0.6 | DC > 0.7 | DC > 0.8 | DC > 0.9 |
---|---|---|---|---|

ISBI-14 Train-45 Dataset | ||||

Lu [33] | DC = 0.905(0.078), FN_{O} = 0.126(0.181)TP _{P} = 0.917(0.087), FP_{P} = 0.006(0.009) | DC = 0.912(0.066), FN_{O} = 0.148(0.206)TP _{P} = 0.920(0.080), FP_{P} = 0.005(0.007) | DC = 0.924(0.049), FN_{O} = 0.211(0.241)TP _{P} = 0.927(0.066), FP_{P} = 0.004(0.005) | DC = 0.951(0.027), FN_{O} = 0.441(0.309)TP _{P} = 0.943(0.043), FP_{P} = 0.002(0.003) |

Ours | DC = 0.917(0.068), FN_{O} = 0.078(0.098)TP _{P} = 0.930(0.069), FP_{P} = 0.005(0.007) | DC = 0.923(0.059), FN_{O} = 0.104(0.126)TP _{P} = 0.931(0.065), FP_{P} = 0.004(0.006) | DC = 0.935(0.045), FN_{O} = 0.200(0.205)TP _{P} = 0.936(0.056), FP_{P} = 0.004(0.005) | DC = 0.960(0.023), FN_{O} = 0.456(0.317)TP _{P} = 0.947(0.039), FP_{P} = 0.001(0.002) |

ISBI-14 Test-90 Dataset | ||||

Lu [33] | DC = 0.871(0.102), FN_{O} = 0.254(0.268)TP _{P} = 0.892(0.106), FP_{P} = 0.005(0.007) | DC = 0.891(0.082), FN_{O} = 0.317(0.284)TP _{P} = 0.895(0.102), FP_{P} = 0.003(0.006) | DC = 0.920(0.058), FN_{O} = 0.439(0.304)TP _{P} = 0.913(0.082), FP_{P} = 0.002(0.003) | DC = 0.956(0.029), FN_{O} = 0.630(0.301)TP _{P} = 0.947(0.045), FP_{P} = 0.001(0.002) |

Ours | DC = 0.882(0.095), FN_{O} = 0.178(0.177)TP _{P} = 0.906(0.091), FP_{P} = 0.005(0.007) | DC = 0.904(0.073), FN_{O} = 0.281(0.226)TP _{P} = 0.911(0.085), FP_{P} = 0.003(0.005) | DC = 0.931(0.051), FN_{O} = 0.450(0.269)TP _{P} = 0.925(0.071), FP_{P} = 0.002(0.004) | DC = 0.961(0.027), FN_{O} = 0.663(0.292)TP _{P} = 0.949(0.044), FP_{P} = 0.001(0.001) |

**Table 4.**Results (standard deviation) of our cytoplasm segmentation and the state-of-the-art methods on the train-45 dataset.

Methods | FN_{O} | TP_{P} | FP_{P} | DC |
---|---|---|---|---|

ISBI-14 Train-45 Dataset | ||||

Ushizima [23] | 0.267(0.278) | 0.841(0.130) | 0.002(0.003) | 0.872(0.082) |

Nosrati [30] | 0.111(0.166) | 0.875(0.086) | 0.004(0.004) | 0.871(0.075) |

Tareef [28] | 0.296(0.277) | 0.948(0.059) | 0.005(0.007) | 0.914(0.075) |

Lu [33] | 0.148(0.206) | 0.920(0.080) | 0.005(0.007) | 0.912(0.066) |

Lee [26] | 0.137(0.194) | 0.882(0.097) | 0.002(0.003) | 0.897(0.075) |

Ours | 0.104(0.126) | 0.931(0.065) | 0.004(0.006) | 0.923(0.059) |

**Table 5.**Results (standard deviation) of our cytoplasm segmentation and the state-of-the-art methods on the test-90 dataset.

Methods | FN_{O} | TP_{P} | FP_{P} | DC |
---|---|---|---|---|

ISBI-14 Test-90 Dataset | ||||

Ushizima [23] | 0.174(0.210) | 0.826(0.130) | 0.001(0.002) | 0.867(0.083) |

Nosrati [30] | 0.140(0.170) | 0.900(0.090) | 0.005(0.004) | 0.870(0.080) |

Nosrati [31] | 0.110(0.170) | 0.930(0.090) | 0.005(0.004) | 0.880(0.080) |

Lu [33] | 0.317(0.284) | 0.895(0.102) | 0.003(0.006) | 0.891(0.082) |

Tareef [52] | 0.163(0.223) | 0.939(0.064) | 0.005(0.005) | 0.888(0.076) |

Tareef [53] | 0.274(0.277) | 0.907(0.088) | 0.004(0.005) | 0.889(0.073) |

Tareef [50] | 0.222(0.240) | 0.945(0.071) | 0.005(0.005) | 0.897(0.077) |

Huang [51] | 0.100(0.130) | 0.940(0.090) | 0.004(0.003) | 0.890(0.070) |

Ours | 0.281(0.226) | 0.911(0.085) | 0.003(0.005) | 0.904(0.073) |

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## Share and Cite

**MDPI and ACS Style**

Liu, G.; Ding, Q.; Luo, H.; Ju, M.; Jin, T.; He, M.; Dong, G.
A Novel Evolution Strategy of Level Set Method for the Segmentation of Overlapping Cervical Cells. *Appl. Sci.* **2021**, *11*, 443.
https://doi.org/10.3390/app11010443

**AMA Style**

Liu G, Ding Q, Luo H, Ju M, Jin T, He M, Dong G.
A Novel Evolution Strategy of Level Set Method for the Segmentation of Overlapping Cervical Cells. *Applied Sciences*. 2021; 11(1):443.
https://doi.org/10.3390/app11010443

**Chicago/Turabian Style**

Liu, Guangqi, Qinghai Ding, Haibo Luo, Moran Ju, Tianming Jin, Miao He, and Gang Dong.
2021. "A Novel Evolution Strategy of Level Set Method for the Segmentation of Overlapping Cervical Cells" *Applied Sciences* 11, no. 1: 443.
https://doi.org/10.3390/app11010443