Fuzzy Association Rule Based Froth Surface Behavior Control in Zinc Froth Flotation
Abstract
:1. Introduction
2. Process Description
3. Methods
3.1. Froth Surface Behavior Representation
3.1.1. Bubble Size Distribution
3.1.2. Froth Velocity Distribution
3.2. Rule Base Construction
3.2.1. Association Rule Mining
Algorithm I: Fuzzy AprioriBased Setpoint Association Rule Mining. 
Input: A collection of froth behavior data at reagent adjusting time, 
their corresponding froth behavior data captured after reagent adjustment, 
and XRF grade analysis result of collected froth after reagent adjustment 
Output: 

3.2.2. Fuzzy Inference System
3.3. Reagent Control
3.3.1. Flotation Causal Inference Model Construction
3.3.2. Distance Metric
 (i)
 $d\left(x,y\right)\ge 0$;
 (ii)
 $d\left(x,y\right)=0\iff x=y$;
 (iii)
 $d\left(x,y\right)=d\left(y,x\right)$; and
 (iv)
 $d\left(x,z\right)\le d\left(x,y\right)+d\left(y,z\right)$.
3.3.3. Reagent Control Based on Receding Horizon Optimization
4. Results and Discussion
4.1. Relationship between Visual Features and Process Variables
4.2. Validation of the Fuzzy A Priori Association Rule Mining
4.3. Industrial Experiments
5. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Procedures for Keypoint Detector 


Procedures for Keypoint Description 

1. Define a binary test $\tau $ of image patch $p$ 
$\tau \left(p;x,y\right)=\{\begin{array}{ll}1& p\left(x\right)<p\left(y\right)\\ 0& p\left(x\right)\ge p\left(y\right)\end{array}$ 
Define the feature as a vector of binary tests 
${f}_{n}\left(p\right)={\displaystyle \sum _{1\le i\le n}{2}^{i1}\tau \left(p;{x}_{i},{y}_{i}\right)}$ 
2. For any feature set of $\left({x}_{i},{y}_{i}\right)$, define the $2\times n$ matrix 
$S=\left(\begin{array}{l}{x}_{1},\cdots ,{x}_{n}\\ {y}_{1},\cdots ,{y}_{n}\end{array}\right)$ 
3. Use the patch orientation $\theta $ and the corresponding rotation matrix ${R}_{\theta}$ to construct a “steered” version of $S$ as ${S}_{\theta}={R}_{\theta}S$, thus 
${g}_{n}\left(p,\theta \right)={f}_{n}\left(p\right)\left({x}_{i},{y}_{i}\right)\in {S}_{\theta}$ 
Feature  Bubble size Distribution  Froth Velocity Distribution  

Mu  Sigma  Mu  Sigma  
Grade  −0.47  −0.42  −0.32  0.51 
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Zhang, J.; Tang, Z.; Ai, M.; Gui, W. Fuzzy Association Rule Based Froth Surface Behavior Control in Zinc Froth Flotation. Symmetry 2018, 10, 216. https://doi.org/10.3390/sym10060216
Zhang J, Tang Z, Ai M, Gui W. Fuzzy Association Rule Based Froth Surface Behavior Control in Zinc Froth Flotation. Symmetry. 2018; 10(6):216. https://doi.org/10.3390/sym10060216
Chicago/Turabian StyleZhang, Jin, Zhaohui Tang, Mingxi Ai, and Weihua Gui. 2018. "Fuzzy Association Rule Based Froth Surface Behavior Control in Zinc Froth Flotation" Symmetry 10, no. 6: 216. https://doi.org/10.3390/sym10060216