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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Electrical capacitance tomography (ECT) attempts to reconstruct the permittivity distribution of the cross-section of measurement objects from the capacitance measurement data, in which reconstruction algorithms play a crucial role in real applications. Based on the robust principal component analysis (RPCA) method, a dynamic reconstruction model that utilizes the multiple measurement vectors is presented in this paper, in which the evolution process of a dynamic object is considered as a sequence of images with different temporal sparse deviations from a common background. An objective functional that simultaneously considers the temporal constraint and the spatial constraint is proposed, where the images are reconstructed by a batching pattern. An iteration scheme that integrates the advantages of the alternating direction iteration optimization (ADIO) method and the forward-backward splitting (FBS) technique is developed for solving the proposed objective functional. Numerical simulations are implemented to validate the feasibility of the proposed algorithm.

Acquiring the spatial distribution information of materials is vital for improving the system efficiency and reducing pollution emission in chemical reactors or multiphase flow units. ECT is a noninvasive imaging technique, which is used to acquire spatial distribution information from inaccessible objects in order to monitor industrial processes. Owing to its distinct advantages such as the non-intrusive sensing, radiation-free nature, high temporal resolution, affordable measuring device and easy implementation, ECT is proven to be useful in industrial process monitoring, multiphase flow measurements, the visualization of combustion flames in porous media and the identification of two-phase flow patterns [

ECT technology attempts to reconstruct the permittivity distribution of the cross-section via an appropriate reconstruction algorithm from the capacitance measurement data, where reconstructing high-quality images plays a crucial role in real applications. Due to the ill-posed nature of the inverse problem, the ‘soft-field’ effect and the underdetermined problem in ECT image reconstruction, achieving high-accuracy reconstruction of a dynamic object is challenging. The key issue for improving the reconstruction quality has attracted intensive attention, and thus various algorithms, which can be approximately divided into two categories, static and dynamic reconstruction algorithms, had been developed for ECT image reconstruction. Common static reconstruction algorithms include the linear back-projection (LBP) method [

The above-mentioned algorithms have played an important role in promoting the development of ECT technology and found numerous successful applications. It is worth mentioning that static reconstruction algorithms are often used to image a dynamic object [

Based on the RPCA method, a dynamic reconstruction model that utilizes the multiple measurement vectors is presented in this paper, where the evolution process of a dynamic object is regarded as a sequence of 2-D images with different temporal sparse deviations from a common background. An objective functional that simultaneously considers the temporal constraint and the spatial constraint is proposed, in which the images are reconstructed in a batching pattern. An iteration scheme that integrates the merits of the ADIO method and the FBS technique is developed for solving the established objective functional. Numerical simulations are implemented to validate the feasibility of the proposed algorithm.

The rest of this paper is organized as follows: based on the RPCA method, a reconstruction model that utilizes the multiple measurement vectors is proposed in Section 2. The original image reconstruction model is formulated into an optimization problem, and a new objective functional is established in Section 3. In Section 4, an iteration scheme that integrates the advantages of the ADIO method and the FBS algorithm is developed for solving the proposed objective functional. In Section 5, numerical simulations are implemented to evaluate the feasibility of the proposed algorithm, and a concise discussion on the numerical results is provided. Finally, conclusions are presented in Section 6.

The ECT image reconstruction process involves two key phases: the forward problem and the inverse problem. The forward problem solves the capacitance values from a given permittivity distribution. It is worth mentioning that the forward problem is a well-posed problem, and it can be easily solved by numerical methods such as the finite element method or the finite difference method. The relationship between capacitance and the permittivity distribution can be formulated by [

The inverse problem attempts to estimate the permittivity distribution from the given capacitance data. In real applications, the static linearization image reconstruction model can be simplified as [_{i}_{j}_{i}_{i}

In the static reconstruction model, the solution merely reflects an instantaneous measurement without any considerations of temporal dynamics of the underlying process, whereas in the case of the dynamic reconstruction model it reflects a sequence of temporally successive measurement, such that the temporal correlations of a dynamic object of interest should be imposed. In other word, the dependence of the capacitance measurement on the evolution of the permittivity distribution is explicitly considered in the dynamical reconstruction model. If the evolution of the permittivity distribution does not follow any dynamics, the dynamical reconstruction model reduces to the static reconstruction model. Obviously, the static reconstruction model is a special case of the dynamic reconstruction model.

The PCA method is an efficient data processing technique, which have enjoyed wide popularity in various fields. Unfortunately, the performances of the PCA technique suffer from the outliers in the data matrix, and thus different approaches had been developed for improving the PCA method. The RPCA method provides a new insight for modern data analysis approaches, which alleviates the deficiencies of the PCA method by applying the ℓ_{1}-regularization and the nuclear norm on the matrix entries. Therefore, the RPCA method is robust to grossly corrupted observations of the underlying low-rank matrix. In a word, the RPCA method tries to recover principal component _{*} defines the nuclear norm for a matrix, and it can be specified as
_{k}_{1} represents the ℓ_{1}-norm for a matrix, which can be defined as
_{1} is introduced to satisfy the sparsity assumption of matrix

Studies indicate that the evolution process of a dynamic object can be regarded as a sequence of images with different temporal sparse deviations from a common background. Motivated by this observation, the following low-rank and sparse decomposition of _{1} is the low-rank matrix component for modeling the ‘background components’ of _{1} is assumed to resemble each other rather than to be constant in time, which can be described as a low-rank matrix in mathematics under the RPCA framework [_{2} is the sparse matrix component for modeling the sparse deviation from the background _{1}. Submitting

_{F}_{1} > 0, _{2} > 0, _{3} > 0 and _{4} > 0 are the regularization parameters; Ω represents a stabilizing functional;

_{1}‖_{*}, ‖_{2}‖_{1}, Ω(_{1}‖_{*} and ‖_{2}‖_{1} are used to model the background component and the sparse deviation from the background of a dynamic process, respectively. It is worth mentioning that

The design of function Ω is vital for

Following the discussions presented in previous sections, an objective functional for ECT image reconstruction can be specified by:

Several desirable properties can be found from

In traditional reconstruction models, only single measurement data is used to independently implement image reconstruction; in

In

In _{1}-norm are used to model the background component and the sparse component of a dynamic evolution process, respectively.

The measurement noises and the model approximation distortions are simultaneously considered in

ECT image reconstruction process is a typical ill-posed problem, methods that ensure the numerical stability while improving the quality of an inversion solution should be employed. In

The unknown variables in

Developing an efficient algorithm to solve

_{1} and _{2}, and directly solving the equation is challenging. In the ADIO method, different unknown variables can be alternately solved [

According to the above discussions,

The FBS technique is originally proposed for solving the following optimization problem [

According to the corresponding deductions, the resulting FBS algorithm can be formulated as:
_{δμJ}

In the case of
^{T}

Following the above discussions, an iteration scheme can be developed for solving

Step 1. Specify the algorithmic parameters and the initial values.

Step 2. Update variable

Step 3. Update variable _{1} by solving

Step 4. Update variable _{2} by solving

Step 5. Loop to Step 2 until a predetermined iteration stopping criterion is satisfied.

Additionally, it can be known in advance that the inversion solution belongs to the range [Θ_{1}, Θ_{2} ], therefore a projected operator is introduced to the iteration scheme:

According to the above discussions, the proposed reconstruction technique can be concisely called as the multiple measurement vectors dynamic reconstruction (MMVDR) algorithm. In this section, dynamic reconstruction cases are implemented to evaluate the feasibility of the MMVDR algorithm, and the reconstruction quality is compared with the projected Landweber iteration (PLI) method. The initial values are computed by the standard Tikhonov regularization method. All algorithms are implemented using the MATLAB 7.0 software on a PC with a Pentium IV 2.4 G Hz CPU, and 4 G bytes memory. The image error is used to evaluate the reconstruction quality, which is defined as [_{True} and _{Estimated} represent the true and estimated permittivity distributions, respectively. A 12 electrodes square ECT sensor, which is present in

In the section, the image reconstructions with three measurement vectors (

_{1} = 1.5, _{2} = 0.05, _{3} = 0.07, _{4} = 0.001, and the number of iterations is 130.

The images reconstructed by the PLI method are presented in

The images reconstructed by the MMVDR algorithm are illustrated in

Meanwhile, it can be found that in the MMVDR algorithm, the images are reconstructed by a batching pattern, and thus the temporal correlations of a dynamic object can be better considered, which differs from other vector-based reconstruction algorithms. In addition, it can be found that the MMVDR method is an iterative algorithm, and it is hard to achieve the online reconstruction presently. In the future, more investigations on improving reconstruction speed should be implemented.

The image errors are listed in

In this section, the image reconstructions with four measurement vectors (

Algorithmic parameters for the PLI method are listed in

The images reconstructed by the PLI method are presented in

In this section, the noise-contaminated capacitance data is used to evaluate the robustness of the MMVDR algorithm. In this case, two measurement vectors (

Where _{True} stands for the true capacitance data; _{Contaminated} is the noise-contaminated capacitance data; _{Contraminated} = _{True} + _{1} · _{1} represents the standard deviation and

In the MMVDR algorithm, the algorithmic parameters are the same as Section 5.1.

ECT is considered a promising visualization measurement technology, in which reconstructing high quality images is highly desirable for real applications. In this paper, based on the RPCA technique, a dynamic reconstruction model that utilizes the multiple measurement vectors is presented, in which the evolution process of a dynamic object is considered as a sequence of images with different temporal sparse deviations from a common background. An objective functional that simultaneously considers the temporal constraint and the spatial constraint is proposed, where the images are reconstructed in a batching pattern. An iteration scheme that integrates the advantages of the ADIO method and the FBS technique is developed for solving the proposed objective functional. Numerical simulation results indicate that the proposed algorithm can ensure a stable numerical solution. For the cases simulated in this paper, the quality of the images reconstructed by the proposed algorithm is improved, which indicates that the proposed algorithm is successful in solving ECT inverse problems. As a result, a promising algorithm is introduced for ECT image reconstruction.

Applications indicate that each algorithm may show different numerical performances to different reconstruction tasks. In practice, the selection of a reconstruction algorithm depends mainly on the measurement requirements and the prior information of a specific reconstruction task. Our work provides an alternative approach for solving ECT inverse problems, which needs to be validated by more cases in the future. At the same time, more investigations on the improvement of the reconstruction speed should be undertaken.

The authors wish to thank the National Natural Science Foundation of China (Nos. 51206048, 61072005 and 51006106) and the Program for the Changjiang Scholars and Innovative Research Team in University (No.IRT0952) for supporting this research.

Layout of ECT sensor.

Dynamic reconstruction objects.

Reconstructed images by the PLI algorithm.

Reconstructed images by the MMVDR algorithm.

Dynamic reconstruction objects.

Reconstructed images by the PLI algorithm.

Reconstructed images by the MMVDR algorithm.

Dynamic reconstruction objects.

Reconstructed images by the MMVDR algorithm under the noise level of 0%.

Reconstructed images by the MMVDR algorithm under the noise level of 9%.

Reconstructed images by the MMVDR algorithm under the noise level of 24%.

Reconstructed images by the MMVDR algorithm under the noise level of 33%.

Algorithmic parameters for the PLI algorithm.

Relaxation factor | 1 | 1 | 1 |

Number of iteration | 480 | 401 | 432 |

Image errors (%).

PLI | 12.61 | 18.03 | 18.82 |

MMVDR | 9.08 | 13.92 | 15.11 |

Algorithmic parameters for the PLI algorithm.

Relaxation factor | 1 | 1 | 1 | 1 |

Number of iteration | 535 | 500 | 111 | 748 |

Image errors (%).

PLI | 15.77 | 15.09 | 15.22 | 18.39 |

MMVDR | 10.37 | 11.55 | 6.68 | 13.40 |

Image errors (%).

0 | 10.89 | 7.36 |

9% | 11.03 | 7.87 |

24% | 12.63 | 10.06 |

33% | 13.81 | 10.20 |