# Dynamic Reconstruction Algorithm of Three-Dimensional Temperature Field Measurement by Acoustic Tomography

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Principle of Acoustic Pyrometry and the Static Model

_{i}represents the time of flight of the wave along the ith sound path; L

_{i}is the ith sound wave ray transmission path; (x,y,z) is the location of the unit, v

_{j}(x,y,z) is the sound speed of the jth imaging unit; f

_{j}(x,y,z) is the slowness of the jth pixel units (i.e., the reciprocal of velocity), n

_{i}is the measurement noise. An equation set is obtained after one measurement cycle. Formula (2) can be simplified as a static reconstruction model in the form of a matrix equation:

## 3. Modeling and Solving of the Dynamic Reconstruction Model

#### 3.1. The Establishment of the Dynamic Reconstruction Model

_{k}represents the slowness variable at time k; g(·) describes the dynamic development information expressed by a series of partial differential equations in the temperature field measurement; h(·) is a measurement equation; y

_{k}represents the TOF value at moment k; w

_{k}and u

_{k}represents the uncertainty in the dynamics equation and measurement equation respectively; and the subscript k is the index of the discrete time. Formulas (5) and (6) can be approximated to linear equations in order to realize a rapid reconstruction

_{k}is the state transition matrix at time k; A

_{k}is measurement operator. If B

_{k}= I, I is a unit matrix. Formula (7) can be regarded as a pure random walk evolution model which is usually adopted in practice when no better dynamic model is known [43].

_{1}and λ

_{2}are non-negative regularization factors, ${\Vert D{F}_{k}\Vert}^{2}$ and D are the regularization term and regularization matrix respectively which function to balance the accuracy and stability of the solution.

_{i}is the pixel sets that are adjacent to the boundary of ith space pixel; Φ

_{i}is the pixel sets that are adjacent to the vertex of ith space pixel. The relationship between a space pixel and its adjacent space pixel can be classified into four conditions (shown in Figure 1), because the distribution of three-dimensional temperature field is continuous in the measured area. It is shown in Figure 1a when the space pixels are located on the eight corners; it is shown in Figure 1b when the space pixel is located in the middle of the side boundary; it is shown in Figure 1c when the space pixel is located in the center of the plane; it is shown in Figure 1d when the space pixel is located in the center of the cubic; it is shown in Figure 1e when the space pixel are adjacent to the boundary of the ith space pixel; it is shown in Figure 1f when the space pixels are adjacent to the vertex of the ith space pixel. The N × N regularization matrix D is obtained in turn in which p is the sum of the pixel sets that are adjacent to the ith space pixel and N is the sum of the space pixels.

#### 3.2. Extension of Objective Function by Using Robust Estimation

_{1}and λ

_{2}is a problem of choosing multiple regularization parameters which is commonly solved using a L-hyperplane [54]. The L hyper plane is considered as the multi-dimensional extension of a typical L curve method that is a curve representing the residual norm and constrained norm with a proper scale. Intuitively, the “generalized corner” of a L-hyperplane should be the approximate balance point between the regularization error and the disturbance error. The main disadvantage of the L-hyperplane method is the high computation cost of estimating the maximum of the Gaussian curvature for a large number of regularization parameters. Moreover, positioning the maximum of the Gaussian curvature by regular optimization techniques is limited by the fact that there are multiple extrema in the Gaussian curvature function.

_{1},b

_{2}) represent the coordinates of the origin. The MDF ν(

**λ**) is the distance from the origin to the point on the L-hyperplane.

**λ**and F

^{*}is as follow

^{*}can usually be found by any optimization technique, but many optimization algorithms require calculating the high-order derivative of z(λ) and x

_{i}(λ) with respect to λ

_{i}and these derivatives are obtained by solving a linear system whose size is the same as that of the original problem and are calculated in sequence from $\frac{\partial {F}^{*}(\lambda )}{\partial {\lambda}_{i}}$. For ease of calculation, a fixed-point algorithm for λ

^{*}is obtained using the basic characteristics of the MDF.

^{−4}.

#### 3.3. Solving of the Objective Function

_{0}is given to find a local minimum F* of the objective function $Z(F)$ by a classical nonlinear programming algorithm.

_{0}to satisfy $P({F}_{0},{F}^{*})$ ≤ 0 starting from any point in the neighborhood of F*. For this purpose, any classical nonlinear programming algorithm is used to minimize $P({F}_{0},{F}^{*})$ in the tunneling phase. The function value of the minimization series is not greater than that of previous minimizations by carrying out these two phases alternatively, that is, the function value in the local minimization is decreasing. It is obvious that the design of the tunneling function is very important in the application. These are some commonly used tunneling functions and modified tunneling functions [42,59].

## 4. Temperature Field Reconstruction with Noiseless Measurement Signals

_{k}is the time delay of the kth acoustic ray, L

_{k}is the sound wave path of the kth sound ray.

^{−3}. The images of the reconstruction temperature fields show that the five algorithms can reconstruct images confirming the model temperature fields.

_{M}(j) is the temperature value of the model temperature field, T

_{Ma}is the average temperature of the model temperature field, T

_{Mmax}is the maximum of the model temperature field, T

_{R}

_{(j)}is the temperature value of reconstructed temperature field, T

_{Ra}is the average temperature of the reconstructed temperature field, T

_{Rmax}is the maximum of the reconstructed temperature field.

^{16}and it is seriously ill-posed.

## 5. Temperature Field Reconstruction Using Measurement Signals with Noise

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The adjacent relation between space pixels. (

**a**) The space pixels are located on the eight corners; (

**b**) The space pixel is located in the middle of the side boundary; (

**c**) The space pixel is located in the center of the plane; (

**d**) The space pixel is located in the center of the cubic; (

**e**) The space pixels are adjacent to the boundary of the ith space pixel; (

**f**) The space pixels are adjacent to the vertex of the ith space pixel.

**Figure 2.**Illustration of the transducer arrangement and the rays. (

**a**) Transducers arrangement; (

**b**) Sound rays between transducers.

Property | Model 1 | Model 2 | Model 3 | Model 4 |
---|---|---|---|---|

relaxation factor | 0.24 | 0.22 | 0.20 | 0.20 |

iteration step | 120 | 120 | 120 | 120 |

Algorithm | Model | E1 | E2 | E3 |
---|---|---|---|---|

LSM | model 1 | 1.02 | 3.30 | 1.88 |

model 2 | 7.21 | 2.45 | 6.67 | |

model 3 | 11.26 | 6.82 | 9.86 | |

model 4 | 26.05 | 11.53 | 10.37 | |

ART | model 1 | 1.02 | 3.30 | 1.88 |

model 2 | 5.41 | 2.02 | 5.52 | |

model 3 | 7.83 | 6.23 | 9.74 | |

model 4 | 16.04 | 10.68 | 10.34 | |

STR | model 1 | 1.02 | 3.30 | 1.88 |

model 2 | 7.23 | 2.53 | 6.69 | |

model 3 | 12.37 | 5.36 | 9.87 | |

model 4 | 26.37 | 11.50 | 10.38 | |

SIRT | model 1 | 1.01 | 3.30 | 1.88 |

model 2 | 4.25 | 1.94 | 5.21 | |

model 3 | 7.04 | 6.11 | 7.81 | |

model 4 | 15.24 | 10.13 | 10.15 | |

DRARE | model 1 | 0.43 | 0.89 | 0.33 |

model 2 | 2.02 | 1.07 | 0.41 | |

model 3 | 4.85 | 2.27 | 2.06 | |

model 4 | 8.36 | 6.15 | 4.21 |

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**MDPI and ACS Style**

Li, Y.; Liu, S.; Inaki, S.H.
Dynamic Reconstruction Algorithm of Three-Dimensional Temperature Field Measurement by Acoustic Tomography. *Sensors* **2017**, *17*, 2084.
https://doi.org/10.3390/s17092084

**AMA Style**

Li Y, Liu S, Inaki SH.
Dynamic Reconstruction Algorithm of Three-Dimensional Temperature Field Measurement by Acoustic Tomography. *Sensors*. 2017; 17(9):2084.
https://doi.org/10.3390/s17092084

**Chicago/Turabian Style**

Li, Yanqiu, Shi Liu, and Schlaberg H. Inaki.
2017. "Dynamic Reconstruction Algorithm of Three-Dimensional Temperature Field Measurement by Acoustic Tomography" *Sensors* 17, no. 9: 2084.
https://doi.org/10.3390/s17092084