# Froth Image Acquisition and Enhancement on Optical Correction and Retinex Compensation

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## Abstract

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## 1. Introduction

## 2. Design of Uniform Light Distribution

- (1)
- First of all, we need to establish a simple optical system consisting of LED, free surface, and receiving surface, as shown in Figure 2 and Figure 3, where p
_{0}is the vertex of the surface, located in the geometry center shaft between LED and the receiving surface. The incident ray via p_{0}on the surface refracts to t_{0}(the receiving surface). - (2)
- The refraction of the incident ray vector $\overrightarrow{I}$ of p
_{i}, the emergent light vector $\overrightarrow{O}$, and the normal vector $\overrightarrow{N}$ at that meeting point can be expressed as:$${[1\text{}+\text{}{n}^{2}\text{}-\text{}2n(\overrightarrow{O}\text{}\times \text{}\overrightarrow{I})]}^{\frac{1}{2}}\text{}\times \text{}\overrightarrow{N}\text{}-\text{}n\overrightarrow{I}$$_{t}is greater than the p_{o}; $\overrightarrow{O}$ can be expressed using the approximate point coordinates of t as, $\left\{{x}_{t,}{y}_{t,}{z}_{t}\right\}/\sqrt{{x}_{t}^{2}+{y}_{t}^{2}+{z}_{t}^{2}}$ where, z_{t}is the distance between the illuminated area and light.The unit normal vector of point p is,$$\overrightarrow{N}\text{}=\text{}\left(\frac{\partial z}{\partial x},\frac{\partial z}{\partial y},-1\right)/\sqrt{{\left(\frac{\partial z}{\partial x}\right)}^{2}+{\left(\frac{\partial z}{\partial y}\right)}^{2}+1}$$Put the above three vector expressions into refraction formulae:$$\frac{\partial z}{\partial x}\text{}=\text{}\left(n\mathrm{sin}u\mathrm{sin}v\text{}-\text{}\frac{{x}_{i}}{\sqrt{{x}_{i}^{2}+{y}_{i}^{2}+{z}_{i}^{2}}}\right)/\left(\frac{{z}_{i}}{\sqrt{{x}_{i}^{2}+{y}_{i}^{2}+{z}_{i}^{2}}}\text{}-\text{}n\mathrm{sin}u\mathrm{cos}v\right)$$$$\frac{\partial z}{\partial y}\text{}=\text{}\left(n\mathrm{cos}u-\text{}\frac{{y}_{i}}{\sqrt{{x}_{i}^{2}+{y}_{i}^{2}+{z}_{i}^{2}}}\right)/\left(\frac{{z}_{i}}{\sqrt{{x}_{i}^{2}+{y}_{i}^{2}+{z}_{i}^{2}}}\text{}-\text{}n\mathrm{sin}u\mathrm{cos}v\right)$$ - (3)
- We determine the corresponding relationship between the incident light and the emergent light without considering energy loss on the basis of the energy conservation, with the mapping relationship $u\to y,v\to x$:$${{\displaystyle \iint}}^{\text{}}I\left(u,v\right)\mathrm{sin}u\mathrm{dud}v={{\displaystyle \iint}}^{\text{}}E\left({x}_{t},{y}_{t}\right)\mathrm{d}x\mathrm{d}y$$
_{t},y_{t}) is the light intensity of plane illuminated. The point coordinates t can be obtained by (x_{t},y_{t}) = (f(u,v),g(u,v)), put into the expression of the normal vector of N by combining with boundary conditions to solve the flotation surface.

_{min}of luminance in the area is equal to 10.17lx, and the maximal value E

_{max}of luminance in the area is equal to 11.03lx.

## 3. Image Quality Compensation Algorithm Based on Adaptive MSR

_{th}spectral band image at point (x,y); ${R}_{n}\left(x,y\right)$ is the Retinex result of the n

_{th}spectral band at point (x,y); G(x,y) is a Gaussian function; $\mathsf{\sigma}$ is the constant of Gauss around space; P is a scale; and * is the convolution symbol.

_{1}, k

_{2}, b are constant factors used for changed curve, $N{R}_{{n}_{i}}\left(x,y\right)$ is the normalization of ${R}_{{n}_{i}}\left(x,y\right)$, and ${\overline{c}}_{n}$ is the constant surrounded by a normalized Gauss function. It is defined as,

## 4. Experimental Results and Analysis

_{max}and A

_{min}, respectively, are the maximum and the minimum of grey level values of a 3 × 3 window centered by $\phi \left(m,n\right)$. Table 1 gives the quantitative indicators of the three different types of bubble images. Thus, it can be seen that the contrast and SNR are increased significantly by the improved adaptive MSR.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Comparison of images acquired by three kinds of illumination, with the same illumination and different lamps. (

**a**) PHILIPS E27 Fluorescent; (

**b**) PHILIPS MHN-TDW Halide; (

**c**) HY 03-033 Lamp.

**Figure 4.**Flare contrast for ordinary light and the light in this design. (

**a**) Conventional LED’s reflection effect (

**b**) Free curved LED’s reflection effect.

**Figure 5.**Different types of images acquired by using free-curved lens LED light (red-color-encircled parts are shaded parts), and their corresponding Histogram Equalization (HE) results. (

**a**) First type; (

**b**) Second type; (

**c**) Third type; (

**d**) HE on First type; (

**e**) HE on Second type; (

**f**) HE on Third type.

**Figure 6.**Different types of images (see Figure 5) with improved adaptive Multi-Scale Retinex (MSR). (

**a**) First type; (

**b**) Second type; (

**c**) Third type.

**Figure 7.**First type of bubble image segmentation. (

**a**) Image with optical correction; (

**b**) Segmentation result; (

**c**) Adaptive MSR segmentation.

**Figure 8.**Second type of bubble image segmentation. (

**a**) Image with optical correction; (

**b**) Segmentation result; (

**c**) Adaptive MSR segmentation.

**Figure 9.**Third type of bubble image segmentation. (

**a**) Image with optical correction; (

**b**) Segmentation result; (

**c**) Adaptive MSR segmentation.

Image | First Type | Second Type | Third Type | |||
---|---|---|---|---|---|---|

Contrast | SNR | Contrast | SNR | Contrast | SNR | |

Original image | 0.2264 | 7.2647 | 0.1924 | 7.0581 | 0.2075 | 8.1625 |

Enhanced | 0.2743 | 12.6114 | 0.2235 | 14.6851 | 0.2483 | 13.6425 |

Type | Segmentation Directly | Adaptive MSR Segmentation | Manual Segmentation |
---|---|---|---|

First type | 112 | 134 | 138 |

Second type | 87 | 102 | 114 |

Third type | 198 | 235 | 251 |

Type | Segmentation Directly | Adaptive MSR Segmentation | Manual Segmentation |
---|---|---|---|

First type | 134 | 151 | 176 |

Second type | 107 | 124 | 148 |

Third type | 210 | 251 | 287 |

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**MDPI and ACS Style**

Wang, W.; Liu, W.; Lang, F.; Zhang, G.; Gao, T.; Cao, T.; Wang, F.; Liu, S.
Froth Image Acquisition and Enhancement on Optical Correction and Retinex Compensation. *Minerals* **2018**, *8*, 103.
https://doi.org/10.3390/min8030103

**AMA Style**

Wang W, Liu W, Lang F, Zhang G, Gao T, Cao T, Wang F, Liu S.
Froth Image Acquisition and Enhancement on Optical Correction and Retinex Compensation. *Minerals*. 2018; 8(3):103.
https://doi.org/10.3390/min8030103

**Chicago/Turabian Style**

Wang, Weixing, Wei Liu, Fangnian Lang, Guangnan Zhang, Ting Gao, Ting Cao, Fengping Wang, and Sheng Liu.
2018. "Froth Image Acquisition and Enhancement on Optical Correction and Retinex Compensation" *Minerals* 8, no. 3: 103.
https://doi.org/10.3390/min8030103