Comparison of genetic algorithm and algebraic reconstruction for X-ray tomography in bubbling fluidized beds

Highlights

• A novel method, AGA, is applied to tomographic reconstruction for fluidized bed.

• AGA performs better than SART in low resolution.

• There is not clear trend for SART when the resolution increases.

• AGA is less sensitive to noise than SART.

• For AGA, varying results were obtained in higher spatial resolutions.

Abstract

The performance of two tomographic reconstruction algorithms, Simultaneous Algebraic Reconstruction Technique (SART) and Adaptive Genetic Algorithm (AGA), is evaluated based on synthetic data mimicking X-ray computed tomography of a bubbling fluidized bed. The simulations are based on a high speed X-ray tomography system, consisting of 3 X-ray sources and 32 detectors for each source. The comparison between SART and AGA is made for image resolutions ranging from 20 × 20 to 50 × 50 pixels, for the cases of 2 phantoms (artificial voids) and 3 phantoms in a 23 cm diameter column. The influence of noise on the reconstructions for both algorithms is also considered. It is found that AGA provides better reconstructions than SART at low resolutions. At high resolutions, the reconstruction quality is comparable, but the calculation times for AGA are much longer. AGA is better at finding the phantom position as it is less sensitive to measurement noise.

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.