# Combustion State Recognition of Flame Images Using Radial Chebyshev Moment Invariants Coupled with an IFA-WSVM Model

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[1,2,3]. However, these methods all require the sensors to keep a certain distance from the flame. Optical sensors, such as photosensitive sensors and infrared sensors, have the advantages of quick response and long recognition distance. However, they are sometimes difficult to deploy such as outdoor locations and large open facilities.

## 2. Feature Selection Using Radial Chebyshev Moment Invariants

#### 2.1. Pre-Processing of Candidate Image

#### 2.2. Radial Chebyshev Moments

#### 2.3. Radial Chebyshev Moment Invariants

#### 2.4. Feature Selection

## 3. Improved Firefly Algorithm-Wavelet Support Vector Machine

#### 3.1. Wavelet Support Vector Machine (WSVM)

#### 3.2. Firefly Algorithm

#### 3.3. Improved Firefly Algorithm

#### 3.4. WSVM Parameter Optimization Based on Improved Firefly Algorithm

- Step 1.
- Initialize the improved firefly algorithm, and set up the parameters, including the number of fireflies $K$, initial and final randomization parameters ${\alpha}_{0}$ and ${\alpha}_{\infty}$, maximum iterations ${t}_{\mathrm{max}}$, attractiveness ${\beta}_{0}$, basic attractiveness ${\beta}_{b}$ and light absorption coefficient $\gamma $.
- Step 2.
- Generate the initial locations of fireflies at random as ${x}_{i}\left(i=1,2,\cdots ,K\right)$, every firefly is composed of the WSVM parameters $C$ and $\tau $.
- Step 3.
- Calculate the fitness value of every firefly to determine or update its brightness.
- Step 4.
- Rank the fireflies by their brightness and regard the firefly with minimum brightness as the global-best point.
- Step 5.
- Evaluate the brightness of every current firefly and compare the brightness of any two fireflies $i$ and $j$. If ${I}_{i}>{I}_{j}$, calculate the distance ${r}_{ij}$ to obtain the improved attractiveness $\tilde{\beta}\left({r}_{ij}\right)$. Then, move firefly $i$ towards $j$ in all dimensions using Equation (28). The randomization parameter $\alpha \left(t+1\right)$ is updated by Equation (26).
- Step 6.
- Check termination condition. If the maximum iteration limit is reached, stop the optimization process and output the values of optimal parameters $C$ and $\tau $. Otherwise, return to Step 3.

## 4. Combustion State Recognition Based on IFA-WSVM Model

- Step 1.
- Apply a video camera to collect a series of candidate images. Then, the pre-processed and edge images are obtained by pre-processing the candidate images.
- Step 2.
- Calculate the radial Chebyshev moment invariants of every pre-processed and edge image. Select the region and contour moment invariants to construct the corresponding feature vector ${F}_{i}$. The dataset is composed of all the feature vectors and their labels.
- Step 3.
- Divide the dataset into training set and testing set. The former is employed to construct model, and the latter is used for validating the IFA-WSVM model.
- Step 4.
- Train the WSVM with the feature vectors of training set, and apply IFA to determine the penalty factor $C$ and the dilation parameter $\tau $. When the termination condition is met, the optimal parameters are obtained.
- Step 5.
- Input the training data into the WSVM with optimal parameters once again, rebuilding the IFA-WSVM model. The features vectors of testing set are input into the IFA-WSVM model to predict their belonging labels, achieving the combustion state recognition.

## 5. Case Studies

#### 5.1. Study Setup

#### 5.2. Results and Discussion

#### 5.2.1. Results for the First Case Study and Comparison

#### 5.2.2. Results for the Second Case Study and Comparison

#### 5.2.3. Results for the Third Case Study and Comparison

#### 5.2.4. Results for the Fourth Case Study and Comparison

#### 5.2.5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Candidate images: (

**a**–

**e**) stable burning state; (

**f**–

**j**) extinguishing state; (

**k**–

**o**) final extinguished state; (

**p**–

**t**) non-flame state.

**Figure 4.**Results of pre-processing: (

**a**–

**d**) original images; (

**e**–

**h**) pre-processed images; (

**i**–

**l**) edge images.

**Figure 6.**Pre-processed images of RTS transformation set. (

**a**) Original image; (

**b**) image rotated by −20° and translated with [35, 25]; (

**c**) image rotated by 15° and translated with [−35, −25]; (

**d**) image translated with [−35, −25] and scaled with 0.8; (

**e**) image translated with [35, 25] and scaled with 0.8; (

**f**) image rotated by −20° and scaled with 0.8; (

**g**) image rotated by 20° and scaled with 1.2; (

**h**) image rotated by −15° and scaled with 1.2; (

**i**) image rotated by 15°, translated with [35, 25] and scaled with 1.2

**.**

${\tilde{\mathit{C}}}_{\mathbf{13}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{21}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{23}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{25}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{26}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{32}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{44}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{55}}^{\prime}$ | |
---|---|---|---|---|---|---|---|---|

Stable burning state | 0.05764 | 0.1434 | 0.06644 | 0.07413 | 0.05933 | 0.1205 | 0.07037 | 0.05952 |

Extinguishing state | 0.09043 | 0.1052 | 0.07653 | 0.06037 | 0.04384 | 0.1627 | 0.09681 | 0.04934 |

Final extinguished state | 0.1105 | 0.1751 | 0.05140 | 0.04476 | 0.03403 | 0.06471 | 0.05474 | 0.1346 |

Non-flame state | 0.1022 | 0.1882 | 0.2518 | 0.1948 | 0.2879 | 0.3698 | 0.2118 | 0.2461 |

Coefficient of variation | 25.73% | 24.18% | 84.35% | 73.34% | 114.36% | 74.18% | 65.54% | 74.19% |

${\tilde{\mathit{C}}}_{\mathbf{13}}^{\u2033}$ | ${\tilde{\mathit{C}}}_{\mathbf{21}}^{\u2033}$ | ${\tilde{\mathit{C}}}_{\mathbf{23}}^{\u2033}$ | ${\tilde{\mathit{C}}}_{\mathbf{25}}^{\u2033}$ | ${\tilde{\mathit{C}}}_{\mathbf{26}}^{\u2033}$ | ${\tilde{\mathit{C}}}_{\mathbf{32}}^{\u2033}$ | ${\tilde{\mathit{C}}}_{\mathbf{44}}^{\u2033}$ | ${\tilde{\mathit{C}}}_{\mathbf{55}}^{\u2033}$ | |
---|---|---|---|---|---|---|---|---|

Stable burning state | 0.2231 | 0.2069 | 0.2632 | 0.2318 | 0.2162 | 0.5859 | 0.3427 | 0.3169 |

Extinguishing state | 0.2418 | 0.2789 | 0.3329 | 0.2783 | 0.2824 | 0.4407 | 0.4145 | 0.4081 |

Final extinguished state | 0.5374 | 0.7651 | 0.2971 | 0.2147 | 0.3470 | 0.7293 | 0.4514 | 0.4533 |

Non-flame state | 0.2017 | 0.4388 | 0.3742 | 0.2721 | 0.2329 | 0.5646 | 0.3281 | 0.4908 |

Coefficient of variation | 52.63% | 58.74% | 15.04% | 12.40% | 21.78% | 20.39% | 15.25% | 17.97% |

Parameter | Value | Range | Description |
---|---|---|---|

$K$ | 10 | — | Number of fireflies |

${\alpha}_{0}$ | 0.5 | — | Initial randomization parameters |

${\alpha}_{\infty}$ | ${10}^{-4}$ | — | Final randomization parameters |

${t}_{\mathrm{max}}$ | 50 | — | Maximum number of iterations |

${\beta}_{0}$ | 1 | — | Attractiveness |

${\beta}_{b}$ | 0.2 | — | Basic attractiveness |

$\gamma $ | 1 | — | Light absorption coefficient |

$C$ | — | $[0.01,100]$ | Penalty factor of WSVM |

$\tau $ | — | $[0.01,100]$ | Dilation parameter of WSVM |

Parameter | Region Moments | Region and Contour Moments | ||||
---|---|---|---|---|---|---|

Hu | Zernike | Radial-Chebyshev | Hu | Zernike | Radial-Chebyshev | |

Recognition amount | 1751 | 1808 | 1875 | 1796 | 1844 | 1916 |

Recognition rate | 90.54% | 93.49% | 96.95% | 92.86% | 95.35% | 99.07% |

Average operating time of a single image | 32.19 ms | 103.54 ms | 41.07 ms | 66.16 ms | 205.79 ms | 83.38 ms |

**Table 5.**Radial Chebyshev moment invariants of the images in Figure 6.

${\tilde{\mathit{C}}}_{\mathbf{13}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{21}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{23}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{25}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{26}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{32}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{44}}^{\prime}$ | ${\tilde{\mathit{C}}}_{\mathbf{55}}^{\prime}$ | |
---|---|---|---|---|---|---|---|---|

(a) | 0.05444 | 0.07434 | 0.04488 | 0.03860 | 0.07514 | 0.2510 | 0.08892 | 0.07073 |

(b) | 0.06002 | 0.07957 | 0.04718 | 0.03732 | 0.07636 | 0.2244 | 0.08912 | 0.07578 |

(c) | 0.05684 | 0.07554 | 0.04697 | 0.03652 | 0.07114 | 0.2494 | 0.09563 | 0.06927 |

(d) | 0.05888 | 0.07493 | 0.04831 | 0.03963 | 0.08176 | 0.2232 | 0.09072 | 0.07867 |

(e) | 0.05713 | 0.07609 | 0.04288 | 0.03778 | 0.07301 | 0.2431 | 0.08103 | 0.07241 |

(f) | 0.05563 | 0.07076 | 0.04622 | 0.03576 | 0.07406 | 0.2499 | 0.08882 | 0.07744 |

(g) | 0.05484 | 0.07380 | 0.04410 | 0.03947 | 0.07741 | 0.2507 | 0.09012 | 0.07114 |

(h) | 0.05863 | 0.07114 | 0.04512 | 0.03465 | 0.07286 | 0.2206 | 0.09840 | 0.07812 |

(i) | 0.05261 | 0.07264 | 0.04923 | 0.04097 | 0.08093 | 0.2646 | 0.08392 | 0.08116 |

Coefficient of variation | 4.24% | 3.64% | 4.43% | 5.34% | 4.81% | 6.39% | 5.88% | 5.59% |

Parameter | Region Moments | Region and Contour Moments | ||||
---|---|---|---|---|---|---|

Hu | Zernike | Radial-Chebyshev | Hu | Zernike | Radial-Chebyshev | |

Recognition amount | 376 | 391 | 419 | 397 | 412 | 433 |

Recognition rate | 85.45% | 88.86% | 95.23% | 90.23% | 93.64% | 98.41% |

Average operating time of a single image | 33.39 ms | 100.90 ms | 46.37 ms | 67.84 ms | 198.16 ms | 90.42 ms |

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**MDPI and ACS Style**

Yang, M.; Bian, Y.; Yang, J.; Liu, G.
Combustion State Recognition of Flame Images Using Radial Chebyshev Moment Invariants Coupled with an IFA-WSVM Model. *Appl. Sci.* **2018**, *8*, 2331.
https://doi.org/10.3390/app8112331

**AMA Style**

Yang M, Bian Y, Yang J, Liu G.
Combustion State Recognition of Flame Images Using Radial Chebyshev Moment Invariants Coupled with an IFA-WSVM Model. *Applied Sciences*. 2018; 8(11):2331.
https://doi.org/10.3390/app8112331

**Chicago/Turabian Style**

Yang, Meng, Yongming Bian, Jixiang Yang, and Guangjun Liu.
2018. "Combustion State Recognition of Flame Images Using Radial Chebyshev Moment Invariants Coupled with an IFA-WSVM Model" *Applied Sciences* 8, no. 11: 2331.
https://doi.org/10.3390/app8112331