Comparison of genetic algorithm and algebraic reconstruction for X-ray tomography in bubbling fluidized beds
Graphical abstract
Introduction
Fluidized beds play an important role in gas–solid reactions in the chemical industry. The study of fluidized beds started early [1], but the hydrodynamics are still a challenging field [2]. Advanced experimental techniques are required to help in solving the fundamental problems of gas–solid flow, even though numerical methods have contributed significantly to the current understanding [3], [4]. There is big obstacle for observing fluidized beds directly: the non-transparency of the gas–solid phase. However, tomographic techniques such as X-ray CT (Computed Tomography) and ECT (Electrical Capacitance Tomography), which do not rely on visible light, make it possible to explore the internal motion of fluidized beds [5], [6]. Comparing X-ray CT and ECT, ECT is cheaper and easier to use, but X-ray CT is easier to interpret. Moreover, with X-ray CT a higher spatial resolution can be obtained [7].
Although X-ray CT is well developed for medical application, the application to fluidized bed only started in the 1990s, see [8]. It was first applied to determine the time-averaged gas hold-up in a gas–solid bed [9], [10]. After that, an X-ray CT system was applied to measure the time-averaged local solids distribution in a circulating fluidized bed by [11]. An X-ray imaging system was developed by [12] to achieve 3-D visualization for fluidized beds, which can also provide time-averaged CT imaging for the fluidized beds [13], [14]. All the early applications faced a practical limit of X-ray CT: the scanning time of traditional X-ray CT is too long to record the dynamics of the flow of a fluidized bed [15]. Time averaged data is rather limited when analyzing the dynamic phenomena of fluidized beds.
Time-resolved tomographic measurement of fluidized beds is thus urgently needed. A high speed X-ray tomography system has been developed for measuring the bubbling flow in a fluidized bed by [16]. Three X-ray sources are set up at equal angles around the bed to generate projections simultaneously, so that detector arrays receive the attenuated X-ray signals at the same time. A time resolution of 2500 frames/s can be reached. Meanwhile, another fast X-ray system was built by Bieberle et al. [17] and used with gas liquid flow. The frame rate can be as high as 5000 frames/s [18]. It later was also applied in a fluidized bed measurement [19]. Both of these fast X-ray CT systems have high enough time resolution to study multiphase fluid flow. Mudde and co-workers studied fluidized beds with a diameter up to 25 cm. A maximum pixel size of 5 mm is desired to be able to image bubbles as small as 15 mm.
Although virtually all reconstruction problems in CT are ill-posed, this problem in high-speed X-ray CT is particularly serious. In high speed X-ray CT, the number of data points depends on the number of detectors which is restricted by space and detector size. The traditional CT algorithms such as Filtered Back Projection (FBP) are seriously influenced by the ill-posed problem. It is easy to produce unknown errors when the order of the image matrix is much larger than the order of the raw data matrix [20]. The Simultaneous Algebraic Reconstruction Technique (SART) [21] was firstly applied to solve such a problem. Alternatively, the Genetic Algorithm (GA) [22] is introduced to tomographic image reconstruction [23], [24], [25], [26], [27], [28]. Wu et al. ([25]) applied GA to fluidized beds. They reconstructed, amongst other cases, four bubbles that were simultaneously in the cross section of the bed. These bubbles were all of roughly the same size, i.e. 20% of the bed diameter. They found that GA is superior over filtered linear back projection algorithms, regardless of the number of different viewing angles used for generating the data. The paper also discusses a strategy to refine the reconstruction mesh and showed that high spatial resolution can be reached with GA. Furthermore, they imposed a bonus in the GA algorithm on mutations towards grouped pixels of the same binary value, thereby incorporating the characteristic of a bubbling fluidized bed. This is similar to the so-called one-step-late algorithm that we use in SART (see below).
It is proved in [25] that GA has performed well in reconstructing images from ill-posed and noisy data. However, a comparison between SART and GA has not been made. In the present paper, we compare the performance of SART and GA for the reconstruction of the data from a high speed X-ray tomography system with three sources. Based on this system, the measuring principle and the mathematical theory of the reconstruction algorithms are discussed. We use synthetic data as we then know exactly the input to the reconstructions, which allows a better comparison of the two algorithms. A so-called Adaptive Genetic Algorithm (AGA) and a SART algorithm are employed to do the image reconstruction. The outcomes of the two approaches are compared, and the possibility of improving the reconstructed image quality and accuracy is discussed. Finally, a test is run with real data from our X-ray tomography.
Section snippets
High speed X-ray tomography
The high speed X-ray tomography system developed by Mudde et al. [16] has been applied for bubbling fluidized bed measurements [29], [30]. It has been shown that this system is able to measure the individual bubbles with an equivalent diameter larger than 2 cm in a fluidized bed of 25 cm diameter. The present paper further explores the reconstruction algorithm for this system. The algorithm evaluation is first done via artificial data, which can play a similar role as real data in the
The reconstruction problem
Tomographic reconstruction is an inverse procedure for computing the solids fraction α(x,y) from the measured X-ray intensity. There are three basic steps.
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Convert the X-ray intensity Im to solids path length ls, i.e. the total amount of solids on the measuring line. ls is usually referred to as the ray sum, . In the real measurement, this procedure requires a calibration [31].
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Mesh the computing domain. We use a uniform meshing in this paper. In the reconstruction, we estimate the solids
Results & discussion
To compare the capability and accuracy of SART and AGA under the same conditions, a comparison has been made using the same reconstruction program, with only a different core reconstruction algorithm. We use circular phantoms as the object to be reconstructed, because these resemble the bubble shape in a cross section of the fluidized bed.
A criterion known as Structural Similarity (SSIM) [39] is employed to estimate the quality of the reconstructed images. SSIM is a combination of luminance
Conclusion
The reconstruction of images for X-ray tomography is challenging, since it is an ill-posed problem: the number of grid cells is much higher that the number of data points. In this study, we compared the Simultaneous Algebraic Reconstruction Technique (SART) and an Adaptive Genetic Algorithm (AGA) for this purpose using artificial data for a bubbling fluidized bed. AGA performs much better than SART for low spatial resolution reconstructions (20 × 20 and 30 × 30 pixels). The accuracy of AGA goes down
Acknowledgment
We would like to thank Prof. Yi Cheng and Dr. Changing Wu from Tsinghua University for the helpful information on GA based tomographic reconstruction.
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2018, International Journal of Multiphase FlowCitation Excerpt :Wu et al. (2007) does not have a fast X-ray tomography system, but used artificially generated images of fast X-ray tomography data from gas bubbles in water to claim that GA is superior over conventional techniques such as FBP. Yang et al. (2014) showed that the quality of AGA reconstructed images is better for low resolutions of 20 × 20 and 30 × 30 pixels and for SART 40 × 40 pixels. To improve the reconstructed images, Yang et al. (2015) combined both reconstruction techniques, where the images are pre-processed with AGA and subsequently treated with SART to yield the final reconstructed images.
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