Learned prior-guided algorithm for flow field visualization in electrical capacitance tomography

Abstract

Electrical capacitance tomography provides great potential advantages in measuring flow field parameters by providing information about spatial-temporal medium distributions, but it is plagued by low-quality reconstructions. In order to overcome this challenge, in this work, the learned prior (LP) that bridges the measurement physics and data-driven modeling paradigms is introduced and coupled with the measurement physics and the domain knowledge into a novel imaging model for reshaping the tomographic reconstruction problem. The LP captures spatial details of imaging objects and guides the search to discover high-quality solutions. A new multi-fidelity deep learning is developed to predict the LP by using deep convolutional encoder-decoder network and multi-fidelity samples, which reduces the difficulty and cost of collecting high-fidelity samples. The established imaging model is solved within the framework of the half-quadratic optimization method. This work transforms the image reconstruction paradigm by fusing the measurement physics and data-driven modeling paradigms. The assessment results have validated that this novel method provides a range of advantages over popular methods, including higher reconstruction quality (RQ) and better robustness.

Introduction

In dynamic industrial processes involving two- or multi-phase flows, such as solar-driven fuel production, metallurgy, chemical reactor, etc., it is essential to reconstruct flow field or gain the time-varying distribution of materials for building digital twin systems for quality control, system safety, and efficiency improvement. However, this task is extremely challenging for both academia and industry. The electrical capacitance tomography, as an imaging-enabled measurement modality, brings a new light to this challenge. Once the scan or measurement is completed, an imaging algorithm, which is a key component of the technology and affects the quality of the reconstructed image, is carried out to visualize the target. Based on these visualizations, subsequent quantitative or qualitative analysis can be carried out to gain deeper insights into monitored objects. In recent decade, the object of its application has been successfully extended to different areas by providing tomograms of rapidly changing objects. For example, this technology has been used to measure media distributions in two-phase flow systems, visualize combustion flames, and more.

Despite the promise of this technology, one of the main technical obstacles is the low quality of tomograms. Especially, the ill-conditioned property and complex noises and imaging targets make this problem more complicated and difficult to solve. As a research hotspot, a large number of algorithms have been developed to address this problem [1], [2], [3], [4], [5], [6], [7], [8]. To deal with the ill-posed property, the iterative reconstruction methods are preferred, which are often reduced into the optimization of a loss function and have the ability to encode image priors or handle complex noises [3], [4], [5], [6], [7], [8].

Depending on the given objective function, these methods consist of two categories, namely iterative reconstruction algorithms that use regularization and those that eliminate the need to use regularization. The former excels at integrating or fusing complex priors and can handle complex noises and imaging tasks. The latter eliminates the use of the regularization term and is less computationally burdensome, but still struggles with low reconstruction quality (RQ) [1], [2].

Regularized iterative reconstruction methods using one regularization term [3], [4], [5], [6], [7], [8] or two or more regularization terms [9], [10], [11], [12], [13], [14], [15], [16] have been studied in depth, but the RQ remains low because they are difficult to capture details of imaging objects using only the priors related to the domain knowledge. Usually, these methods need to solve an optimization problem, and many excellent optimizers have been proposed, such as the split Bregman algorithm [17], the Douglas-Rachford splitting method [18], [19], the block coordinate descent algorithm [20], the fast iteration shrinkage thresholding algorithm [21], the forward backward splitting method [22], [23], the alternating direction method of multipliers [24], [25], [26], the primal-dual algorithm [27], and more. The Bregman method has a significant impact on optimization problems, and more comprehensive discussions and applications can be found in [28], [29], [30], [31], [32]. Some new methods have been developed for treating with optimization problems in the field of image processing [33], [34], [35], [36], [37].

Additionally, some new techniques have been proposed in the literature. For example, the reader can refer to [38], [39], [40], [41], [42]. The usefulness of these algorithms has been confirmed, but they are still struggling with a low RQ. In [43], [44], the non-iterative reconstruction methods that eliminate the iterative process were employed to perform image reconstruction tasks. They have the advantage of fast reconstruction, but the tomograms are usually filled with artifacts.

The past decade has witnessed the deep learning revolution. Deep learning or machine learning has revolutionized the solution of the tomographic reconstruction problem [45], [46], [47], [48], but they still face many challenges [49]. These approaches usually lack a mathematical justification of the results and are not sufficiently interpretable. The self-supervised learning is also used to solve inverse problems in addition to supervised learning techniques. Self-supervised learning intends to mine the supervised information from massive unsupervised samples by devising an appropriate auxiliary task. This constructed supervised information is used to train the network for learning useful and valuable representations for downstream tasks [50], [51]. The deep generative model has been developed to solve inverse problems [52]. In any case, deep learning or machine learning has profoundly influenced and changed this field with far-reaching effects.

It has been demonstrated that traditional regularization algorithms based on physical models are not flexible and effective enough, although they have a good interpretability and mathematical basis [3], [4], [5], [6], [7], [8]. Deep learning-based methods are flexible, but their efficacy is challenged by the generalization performance, interpretability and robustness. Moreover, these methods do not use the imaging physics, and the solution may deviate from the physical meaning of the imaging problem. To overcome the mentioned technical challenges, a novel approach is proposed to integrate and fuse these two types of approaches, with the main purpose of exploiting their strengths and avoiding their weaknesses. Our work is an important step towards revolutionizing the tomographic reconstruction paradigm.

In previous years, the value and usefulness of the hand-crafted domain knowledge-based priors have been verified, but the use of such knowledge alone does not capture spatial details of imaging targets and does not guarantee high-quality reconstructions. This fact has been confirmed by many studies [3], [4], [5], [6], [7], [8], [38], [39], [40], [41], [42]. Moreover, the existing well-studied domain knowledge-based priors emphasize the characteristics of imaging objects, and do not provide a reference image to guide the search to discover high-quality solutions. To address the problem, a promising and efficient way is to add more effective priors, including pushing the boundaries and types of conventional priors and innovating integration methods. The advent of deep learning has brought new light and hope to achieve this goal. Deep learning or machine learning can build the nonlinear mapping from the input to the output and demonstrates an impressive ability to solve complex problems [53], [54], [55], [56], [57], [58], [59], [60]. We introduce the deep learning-based learned prior (LP) by learning from a given dataset. Further, the LP, the domain knowledge and the electrical measurement mechanism are incorporated to create a more flexible and robust optimization-based imaging model, where the maximum correntropy criterion as a data misfit term is used to restrict the adverse effect of interferences or noises and the minimax concave penalty is used to model the LP.

Before using machine learning methods, we recognize the cost and difficulty of collecting high-fidelity training data. Overall, changes in modeling techniques, numerical methods and measurement techniques have led to heterogeneous data types: high-fidelity and low-fidelity data. In our applications, low-fidelity samples are defined as measurements from low precision measurement methods or results from low precision numerical simulations, while high-fidelity samples are defined as ground-truth permittivity distributions. High-fidelity samples are accurate but costly to obtain, while low-fidelity samples are abundant but provide an imprecise approximation. It is worth noting that each type of fidelity data has its own advantages. Thus, single-fidelity models trained using only low-fidelity or high-fidelity data miss out on high accuracy or generalizability, respectively. Multi-fidelity learning combines models with different fidelities to provide more accurate results at a relatively low computational cost.

To reduce the cost of collecting high-fidelity samples, we design a new multi-fidelity deep learning by using a deep convolutional encoder-decoder network (DCEDN) and multi-fidelity samples to predict the LP, and formulate the training task as an optimization problem solved by the split Bregman method. Our multi-fidelity deep learning aims to accurately approximate high-fidelity responses to unknown inputs by building a new model to correct the low-fidelity model with a limited number of high-fidelity samples. The DCEDN uses the encoder-decoder structure and the convolutional operator, which serves as the low-fidelity model. A novel two-stage learning method is proposed to train the DCEDN by following the measurement physics in collecting training samples, leading to the reduced training difficulty and the improved performance of the model. The LP is an image, as opposed to conventional priors, and captures spatial details of imaging objects.

Owing to the solution paradigm shift of the tomographic reconstruction problem, in our study, we have to solve a novel optimization problem. Because our objective function contains a maximum correntropy criterion, a minimax concave penalty and a weight $L_1$ norm, we use the half-quadratic optimization method and further solve the derived sub-problem by the half-quadratic splitting method and the forward-backward splitting method. The LP has two main functions. The first function is that it is integrated into the novel imaging model as a useful prior or reference image. The second function is that it serves as a good starting solution for accelerating the convergence of the algorithm.

Our study overcomes the limitations of existing reconstruction methods by reshaping the imaging paradigm. In the following, we summarize the main contributions.

(1) Our study introduces the LP that bridges the measurement physics and data-driven modeling paradigms. The measurement physics, LP and sparsity-induced prior are fused into a unique imaging model, where the data misfit term is modeled by the maximum correntropy criterion for restricting the adverse effects of interferences or noises and the LP is encoded by the minimax concave penalty. The novel optimization problem not only achieves the complementary strengths of different image priors, but also increases the flexibility and robustness of the imaging method and achieves the paradigm shift of the tomographic reconstruction. Unlike data-driven models, one of the key strengths of our new model is its established theoretical performance guarantees and interpretability.

(2) Our study proposes an effective optimizer for solving the novel optimization model within the framework of the half-quadratic optimization method. The proposed numerical method allows the optimization problem to be split into less difficult sub-problems, which are solved by the half-quadratic splitting technique and the forward-backward splitting method, thus making the solution process less computationally burdensome.

(3) We build a new multi-fidelity deep learning method to infer the LP. Our multi-fidelity deep learning method approximates high-fidelity responses to unknown inputs by correcting the low-fidelity model (the DCEDN) with a small amount of high-fidelity samples, significantly reducing the cost of collecting high-fidelity samples. The training problem is solved within the framework of the split Bregman method, making the training less computationally difficult.

(4) Case studies on challenging reconstructions demonstrate that our new method can achieve the better tomograms and robustness compared to popular imaging techniques. Our findings confirm that the LP is useful, and the coupling of the measurement physics, LP and sparsity-induced prior not only enables the complementary advantages, but also extends the flexibility and adaptability of the model, which is conducive to reducing reconstruction errors. Our study opens up exciting new opportunities for reshaping the tomographic reconstruction paradigm.

This paper proceeds as follows. Section 2 describes the measurement physics and details our novel imaging model. Section 3 expatiates our multi-fidelity deep learning and prediction of the LP. Section 4 presents a new optimizer for solving the novel imaging model. Section 5 provides a qualitative analysis of the novel model. Section 6 quantifies the performance. We conclude the paper in Section 7.