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Flame emission tomography based on finite element basis and adjustable mask

Hecong Liu, Qian Wang, Fan Peng, Zhao Qin, and Weiwei Cai

Open Access Open Access

Abstract

Flame emission tomography is a promising tool for combustion diagnostics. Up to now, this technique relies on the assumption that the distribution within a voxel is uniform. Such an assumption suffers from a large gradient between adjacent voxels and the loss of information within the voxel. Hence, this work aims to develop a method based on the finite element basis to address the above shortcomings. From the reconstruction side, an adjustable mask is developed to suppress the artifacts and speed up reconstruction. Both simulative and experimental studies have been designed and conducted. The simulative results show that the finite element basis can decrease the reconstruction errors (defined as normalized 2-norm difference) from 0.616 to 0.241 and 0.386 to 0.203 by the algebraic reconstruction technique for two representative flame phantoms. The adjustable mask can decrease the reconstruction error for both bases, especially under low voxel resolution. The experimental results show that the predicted projection error of the finite element basis with the adjustable mask is decreased by 57% compared with that of the uniform voxel basis with the fixed mask when the voxel resolution is 16 × 16 × 20. Furthermore, the finite element basis can generate better reconstructions with fewer voxels. Both simulative and experimental studies suggested the superiority of the finite element basis with the adjustable mask.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Combustion diagnostic is an important topic both for scientific research and industrial applications. Up to now, various techniques have been developed and adopted for this purpose, e.g., direct photography of the flame. Due to the surged advances in computational power and sensor technology, optical tomography has been a promising tool for combustion diagnostics due to its non-intrusiveness and the capability for three-dimensional (3D) measurements. Among the various tomographic modalities, flame emission tomography is one of the most prevalent as no light source (typically a laser) is required. Flame emission tomography was first proposed by Hertz et al. in 1988 [1]; in their work, the spatial distribution of CH* emission of a Bunsen flame was reconstructed. Later on, various groups around the world contributed to the development of this technique. For example, Floyd et al. realized an instantaneous 3D measurement of a turbulent opposed jet flame [2] and a matrix burner flame [3]; Ma et al. [4] measured the 3D curvature at 5 kHz; Hossain et al. [5] and Anikin et al. [6] introduced fiber bundles to significantly reduce the number of cameras required; Mohri et al. [7] found that equiangular distribution can get a better reconstruction when the view number is not larger than 12.

Apart from the above developments, another focus is the imaging model of flame emission tomography. For instance, Wang et al. [8] introduced lens imaging into the imaging model where the blur circle was calculated according to the camera aperture. Wan et al. [9] proposed a blurry-spot model, which includes the inhomogeneous distribution of the blur circle, i.e., the outer region of the spot receives less radiation. However, the aforementioned models are all under the assumption that the distribution of the emission within a voxel is uniform, which is referred to as the uniform voxel basis in this work. Such an assumption makes the implementation of the flame models much easier at the cost of several shortcomings, e.g., the large gradient between adjacent voxels and the loss of information within one voxel, leading to certain discretization errors. The so-called finite element basis has been introduced into tomographic communities, such as optical tissue tomography [10], chemical species tomography [11], and bioluminescence tomography [12]. Compared with other tomographic modalities, flame emission tomography has its characteristic, e.g., the light is emitted from the high-temperature flame itself and no external light source is used, which leaves questions whether the finite element basis can be used. In the finite element basis, the interpolation function is necessary to estimate the distribution within each voxel. As the flame is a continuous object, the interpolation will make the reconstruction more smooth and closer to the flame. Therefore, the shortcomings of the uniform voxel basis can be overcome. From the reconstruction side, how to restrict the reconstruction artifacts is an important topic. In a traditional way, a fixed mask is applied to identify the reconstruction outline and accelerate the reconstruction speed [13,14]. However, the fixed mask will introduce large errors under the low voxel resolution. In this work, the finite element basis is proposed for flame emission tomography; an adjustable mask is introduced into flame emission tomography. Both simulative and experimental studies are designed and conducted to demonstrate the effectiveness of the finite element basis and adjustable mask.

2. Theory

2.1 Imaging model

A detailed description of flame emission tomography can be found in other publications [8,15]. Here, only a summary is provided. Figure 1 shows the general imaging process for flame emission tomography. As shown, the volume of interest (VOI) which covers the flame is discretized into numerous cubic voxels, and the voxel at the lower right corner is zoomed for a better display. Each voxel contains eight nodes as labeled in the figure. Within each voxel, numerous locations are randomly selected, where the photons are assumed to be emitted. The emitted photons pass through the flame and the collecting lens and finally hit on the sensor. The radiation intensities at these locations are referred to as f(x,y,z). As shown in the figure, the image of a point (green) in the object space is a blurred circle (purple) due to the off-focus effect.

 figure: Fig. 1.

Fig. 1. Illustration of the imaging process for flame tomography.

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For the uniform voxel basis, the intensity within one voxel is uniform, i.e., f(x,y,z) is a constant within one voxel. The so-called voxel spread function (VSF) can be calculated as the percentage of the number of photons within a specific voxel received by the pixels, and the mathematical expression can be identified as:

$${\boldsymbol{VS}}{{\boldsymbol F}_{i,j}} = {n_{i,j}}/{N_j}\textrm{ for the uniform voxel basis,}$$
where ${N_j}$ is the total number of the photons in j-th voxel; ${n_{i,j}}$ is the number of the photons hit on the i-th pixel, and ${\boldsymbol{VS}}{{\boldsymbol F}_{i,j}}$ represents the impact of the j-th voxel on the i-th pixel.

The key point of the finite element basis is that the variables are stored at the nodes and that the intensities within each voxel are interpolated from the intensities at the nodes. In this work, the tri-linear interpolation is employed as:

$${\boldsymbol f}({x,y,z} )= \mathop \sum \limits_{q = 1}^8 {\alpha _q} \times {{\boldsymbol f}_q},$$
where ${{\boldsymbol f}_q}$ is the intensity at the q-th node; ${\alpha _q}$ is the weight coefficient of the q-th node, which can be calculated from the distance between the node and the interpolated location. As shown in Fig. 1, the green point is positioned in the interpolated location. Take Node 1 for example, ${\alpha _1}$ can be expressed as:
$${\alpha _1} = ({1 - {l_1}/d} )\times ({1 - {l_2}/d} )\times ({1 - {l_3}/d} ),$$
where d is the length of the cubic voxel; ${l_1}$ is the distance between Node 1 and the green point along the x-axis direction; ${l_2}$ is the distance between Node 1 and the green point along the y-axis direction; ${l_3}$ is the distance between Node 1 and the green point along the z-axis direction. For the different interpolated locations, i.e., the random positions where photons are emitted, the l-values are different. The interpolation is only implemented in the reconstructions but not to projections. Note that the sum of weight coefficients for one voxel should be equal to unity. Therefore, the VSF for the finite element basis can be expressed as:
$${\boldsymbol{VS}}{{\boldsymbol F}_{i,j}} = \mathop \sum \nolimits_{m = 1}^M {\alpha _m} \times {n_{i,m}}/{N_m}\textrm{ for the finite element basis,}$$
where ${\boldsymbol{VS}}{{\boldsymbol F}_{i,j}}\; $ represents the impact of the j-th node on the i-th pixel; ${\alpha _m}\; $ is the tri-linear coefficient and can be obtained from (3); ${N_m}$ is the total number of the photons while ${n_{i,m}}$ is the number of photons received by the i-th pixel. M is the total number of the voxels around the j-th node and is defined as:
$$M = \left\{ \begin{array}{ll} 1&{if\;j - th\;node\;at\;the\;corner{}} \\ 2&{if\;j - th\;node\;on\;the\;edge {}} \\ 4&{if\;j - th\;node\;on\;the\;surface} \\ 8&{otherwise}\end{array}\right. .$$

Repeating all the voxels one by one, the VSFs for both bases can be obtained and the imaging model can be expressed in a uniform form as a system of linear equations:

$${{\boldsymbol p}_i} = \mathop \sum \nolimits_{j = 1}^J {{\boldsymbol A}_{i,j}} \times {{\boldsymbol f}_j} = {{\boldsymbol A}_i}{\boldsymbol f,}$$
where ${{\boldsymbol p}_i}$ is the intensity of the i-th pixel recorded by the camera sensor; ${{\boldsymbol f}_j}$ is the flame emission at the j-th voxel or node; J is the total number of variables; ${{\boldsymbol A}_i}$ is the i-th row of A. A is the weight matrix and is another form of VSFs, i.e., ${{\boldsymbol A}_{i,j}} = {\boldsymbol{VS}}{{\boldsymbol F}_{i,j}}$.

2.2 Reconstruction technique

Equation (6) is usually ill-posed and can be solved by several well-established algorithms. In this work, the algebraic reconstruction technique (ART) [15] and Maximum Likelihood Expectation Maximization (MLEM) [16] are adopted due to their ease of implementation and well performance in the tomographic community. The iterative formulation of ART can be expressed as:

$${\boldsymbol f}_j^{k + 1} = {\boldsymbol f}_j^k + {\lambda _{ART}} \times \frac{{{{\boldsymbol p}_i} - {{\boldsymbol A}_i}{{\boldsymbol f}^k}}}{{{{\|\boldsymbol A}_i}\|_2}} \times {{\boldsymbol A}_{i,j}}s.t.{\boldsymbol f} \ge 0,$$
where ${\boldsymbol f}_j^k$ is the reconstructed value of the j-th variable after the k-th iteration; and ${\lambda _{ART}}$ is the relaxation factor which can control the convergence and the speed. The typical value of the relaxation factor varies from zero to two [17] and is set as one in the following work.

The iterative formulation of MLEM can be expressed as:

$${\boldsymbol f}_j^{k + 1} = {\boldsymbol f}_j^k \times \frac{{\mathop \sum \nolimits_i \left( {{{\boldsymbol A}_{i,j}} \times \frac{{{{\boldsymbol p}_i}}}{{{{\boldsymbol A}_i}{{\boldsymbol f}^k}}}} \right)}}{{\mathop \sum \nolimits_i {{\boldsymbol A}_{i,j}}}}s.t.{\boldsymbol f} \ge 0,$$

The iteration process can be summarized as follows: set the initial guess of f and employ (7) or (8) to update the unknown variables, the initial guess for (7) is zero while that for (8) is one; in each iteration, if ${{\boldsymbol f}_j}$ becomes less than zero, apply the non-negative constraint, i.e., set ${{\boldsymbol f}_j}$ as zero. The iteration was terminated when the relative difference between consecutive iterations is smaller than 10−5 or the maximum iterative number is achieved, i.e, 50, in this work.

To effectively suppress the artifacts, an adjustable mask is employed. The main difference between the adjustable mask and the traditional fixed mask is how to identify the flame outline. With the traditional fixed mask, if the intensity of the pixel (${{\boldsymbol p}_i}$) is zero, the corresponding unknown variables (${{\boldsymbol f}_j}$) are set as zero [14]. This method can introduce large errors, especially under low voxel resolution. The basic principle is that a flame is a continuous object; when discretizing the flame, discretization errors would be introduced. For example, for a specific voxel around the flame surface, only part of voxel contains flame emission. This voxel may correspond to several pixels (e.g., ten pixels), and only one of these pixel does not receive any signal. In this case, setting the voxel value to be zero is unreasonable and will introduce a large reconstruction error as most of the periphery voxels has such issue. With our adjustable mask, the aforementioned shortcoming can be overcome by selecting a proper flame outline and setting a proper threshold value. More specifically, if a variable (${{\boldsymbol f}_j}$) corresponds to multiple pixels (${{\boldsymbol p}_i}$), of which the intensity is zero, the variable (${{\boldsymbol f}_j}$) is constrained as zero and would not be updated in the following iterations. Otherwise, the variable (${{\boldsymbol f}_j}$) is continuously updated in each iteration. The mask can be mathematically expressed as:

$${{\boldsymbol f}_j} = 0,\;if\;{Q_j} > {Q_c}$$
where ${Q_j}$ is the number of pixels whose intensity is zero; ${Q_c}$ is a threshold value that is a function of magnification and voxel resolution, and cannot be expressed explicitly. For a specific experiment, one simple method to determine ${Q_c}$ can be summarized as follows: selecting several potential values of ${Q_c}$; reconstructing and re-projecting to obtain predicted projection; comparing the measured projection and the predicted projection to find the best value of ${Q_c}$. Note that for a fixed mask, ${Q_c}$ equals zero.

3. Simulation and results

3.1 Simulative setup

In the simulative studies, phantoms, which are regarded as the ground truth, are usually adopted and serve as the input of the simulative program to generate artificial projections. The program was coded by Intel Visual Fortran 2020 integrated with Visual Studio 2019. The output of the program is the reconstruction of the phantom. By imposing different simulation conditions, e.g., noise level and discretization, the reconstruction will deviate from the ground truth to some extent. Therefore, the impacts of the added factors can be quantified and the robustness of the imaging model and the bases can be evaluated. Figures 3(a) and 3(f) present two phantoms used in this work and referred to as Phantom 1 and Phantom 2, respectively hereafter. The phantoms are adopted from the previous flame tomographic experiments [18]. Phantom 1 and 2 are both diffusion flames with different swirl strengths. Phantom 1 is a high swirl flame; while Phantom 2 is a low swirl one and behaves like a jet flame. The phantoms have a dimensionality of ${\mathrm{40}}\;{\mathrm{mm}} \times {\mathrm{40}}\;{\mathrm{mm}} \times {\mathrm{60}}\;{\mathrm{mm}}{ }$ and a voxel resolution of ${\mathrm{80}} \times {\mathrm{80}} \times {\mathrm{120}}$, leading to the spatial resolution of 2 voxels/mm. The simulative study assumes that a camera captures the phantoms from different angles. The camera sensor has a pixel resolution of ${\mathrm{1024}} \times {\mathrm{1024}}$, and the size of the pixel is ${\mathrm{20}}\;{\mathrm{\mu}}\mathrm{m} \times {\mathrm{20}}\;{\mathrm{\mu}}\mathrm{m}$. The focal length of the camera lens is 55 mm. The previous study has shown that nine views have the best performance for a single camera endoscopic tomographic system [19]. Hence, nine perspectives are assumed to be imaged onto one camera sensor in this work. Each perspective includes ${\mathrm{341}} \times {\mathrm{341}}$ pixels and the angle difference between every two adjacent views is ${\mathrm{20}}^\circ $. The magnification of each perspective is 0.11 and the resolution of the camera is 5.68 pixels/mm.

The VOI is down-sampled by four times, i.e., the voxel resolution of the reconstructions is reduced to ${\mathrm{20}} \times {\mathrm{20}} \times {\mathrm{30}}$. The 10% uniform random noise is added to the simulated projections. To make the results comparable, the reconstructions are then interpolated into the same voxel resolution as the phantom based on the uniform voxel basis or finite element basis. The criteria applied to assess the performance of the two bases and two masks are the reconstruction error (${{E}_{{{recon}}}}$) and re-projection error (${{E}_{{{re}} - {{proj}}}}$) based on the 2-norm, which are commonly applied in flame emission tomography. The mathematical formulas can be expressed as:

$$\left\{ {\begin{array}{c} {{E_{recon}} = {{\|\boldsymbol f}^{re{\boldsymbol c}}} - {{\boldsymbol f}^{phan}}\|_2/{\|{\boldsymbol f}^{phan}}\|_2}\\ {{E_{re - proj}} = {{\|\boldsymbol p}^{re - proj}} - {{\boldsymbol p}^{true}}\|_2/{{\|\boldsymbol p}^{true}}\|_2} \end{array}} \right.,$$
where $\boldsymbol{f}^{\boldsymbol{p} \boldsymbol{h}\boldsymbol{a} \boldsymbol{n}}$ and $\boldsymbol{f}^{\boldsymbol{r} \boldsymbol{e}\boldsymbol{c}}$ are the value of phantoms and reconstructions, respectively; $\boldsymbol{p}^{\boldsymbol{r} \boldsymbol{e}-\boldsymbol{p} \boldsymbol{r} \boldsymbol{o} \boldsymbol{j}}$ and $\boldsymbol{p}^{\boldsymbol{t} \boldsymbol{r}\boldsymbol{u} \boldsymbol{e}}$ are the predicted projection and true projection, respectively. In the simulative study, the true projection can be generated by the simulation; in the experimental study, the true projection is captured by the imaging system.

3.2 Comparison between bases and masks

Two methods are used to assess the bases and the reconstructions. The first method is an indirect method, which is called as re-projection method. The key point of this method is to randomly select eight projections to reconstruct the phantom, re-project to the remaining perspective to obtain the predicted projection, and lastly compare the predicted projection and the simulated projection. The re-projection method is commonly used in flame emission tomography [18,20]. Take Phantom 1 for example, eight projections (expect the sixth projection) are used to reconstruct and the reconstructions are then projected to the sixth perspective. Figure 2 shows the simulated projection and the corresponding predicted projections for both bases with both masks. Panels (a)-(e) are the predicted projections for ART; while Panels (f)-(i) are those for MLEM. The re-projection errors are labeled in the figure as well. As can be seen, both bases with both masks can predict the projection correctly. However, the flame tip for both bases is a little underestimated. The underestimation is more obvious for the finite element basis. As the voxel resolution is limited, it is impossible to estimate a perfect flame outline. What we can do is to choose a proper threshold value to decrease the error. Of course, we can choose another threshold value to get a better estimation of the flame tip. If so, the flame bottom will be overestimated, leading to the increase in the error. The results of the finite element basis are much smoother than those of the uniform voxel basis. Besides, the adjustable mask consistently shows better performance than the fixed mask. For both algorithms, the uniform voxel basis with both masks has moderate errors. The finite element basis with the adjustable mask consistently produces the smallest errors, while the finite element basis with the fixed mask has the largest errors. Note that the adjustable mask can improve the performance of the finite element basis while the fixed mask can degrade it. The possible reason is that the fixed mask will filter out more voxels/nodes than the adjustable mask. Therefore, the area of the predicted projection will be smaller than the simulated projection, leading to a larger error. This phenomenon for the finite element basis is more obvious than that for the uniform voxel basis because that there are more variables for the finite element basis and the percentage of the filtered-out nodes is much larger than that for the uniform voxel basis. The above discussions also indicate that both algorithms have similar reconstruction capability and the performance of the bases is independent of the algorithms.

 figure: Fig. 2.

Fig. 2. (a): Simulated projection for the sixth perspective; (b)-(c): predicted projections of the uniform voxel basis with fixed and adjustable masks for ART, respectively; (d)-(e): predicted projections of the finite element basis with fixed and adjustable masks for ART, respectively; (f)-(i): counterparts for MLEM.

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The second method is the direct comparison between the phantoms and reconstructions. Within this method, all nine projections are used to reconstruct the phantom. With different bases, the resulting weight matrices will be different. The weight matrix for finite element basis has a dimensionality of $13671 \times 1048576$ and 0.38% nonzero elements; while that for uniform voxel basis has a dimensionality of $12000 \times 1048576$ and 0.10% nonzero elements. For the simplicity of comparison, only the adjustable mask is used since the re-projection method indicates that the adjustable mask is better than the fixed mask for both bases. Figures 3(b)-(e) show the 3D renderings of the reconstructions for Phantom 1; while Figs. 3(g)-(j) are the counterparts for Phantom 2. Figures 3(b)-(c) and Figs. 3(g)-(h) are reconstructed by ART; while Figs. 3(d)-(e) and Figs. 3(i)-(j) are reconstructed by MLEM. The renderings are observed from a perspective that is different from those of the projections. Reconstruction errors are labeled in the figure as well. As can be seen, both methods can successfully reveal the major structural features of both phantoms. The finite element basis has a smoother structure compared with the uniform voxel basis. Besides, the reconstruction error of the finite element basis is much smaller than that of the uniform voxel basis.

 figure: Fig. 3.

Fig. 3. 3D renderings for (a): Phantom 1, (b): Reconstruction 1 with the uniform voxel basis by ART, (c): Reconstruction 1 with the finite element basis by ART, (d): Reconstruction 1 with the uniform voxel basis by MLEM, (e): Reconstruction 1 with the finite element basis by MLEM, and (f): Phantom 2, (g): Reconstruction 2 with the uniform voxel basis by ART, (h): Reconstruction 2 with the finite element basis by ART; (i): Reconstruction 2 with the uniform voxel basis by MLEM, (j): Reconstruction 2 with the finite element basis by MLEM.

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For better comparison, Fig. 4 shows the representative horizontal slices for Phantom 1 and corresponding reconstructions with the uniform voxel basis and finite element basis by both reconstruction algorithms, respectively. The layer indices are labeled above every slice. As can be seen, the slices with the finite element basis are quite similar to that of Phantom 1; while the slices with the uniform voxel basis only capture the outlines due to limited voxel resolution. The reconstructions also confirmed that the finite element basis with the adjustable mask has better performance than the uniform voxel basis with the adjustable mask. Moreover, both algorithms have similar performance, which verified the conclusion in the re-projection method that the performance of the bases is independent of the algorithms.

 figure: Fig. 4.

Fig. 4. Representative slices of (a)-(c): Phantom 1, (d)-(f): Reconstruction 1 with the uniform voxel basis by ART, (g)-(i): Reconstruction 1 with the uniform voxel basis by MLEM, (j)-(l): Reconstruction 1 with the finite element basis by ART, (m)-(o): Reconstruction 1 with the finite element basis by MLEM.

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4. Experiment and results

4.1 Experiment setup

Besides simulative studies, a proof-of-concept experiment was also designed and conducted based on a lab-scale flame. The photo of the experimental setup is shown in Fig. 5(a), and the corresponding schematic is shown in Fig. 5(b). As can be seen, a single-camera endoscopic flame tomographic system is established, which consists of one high-speed monochromatic camera, one customized nine-to-one fiber bundle, and a computer. A detailed description of the system can be found in [18]. The high-speed camera worked at a frame rate of 1 kHz and an exposure time of 0.125 ms. The burner used here is a commercial Bunsen burner that was fed with butane and air. Besides, jet air was imposed on the flame to generate the different flame structures. The calibration plate, the background in darkness, and the flame emission were captured in sequence. Before reconstruction, the imaging system was calibrated based on Zhang’s model [21] and the data pre-processing was applied, including background subtraction and thresholding. Thereafter, the calibration data and flame projections were input into the same reconstruction program as used in the simulative studies.

 figure: Fig. 5.

Fig. 5. (a): Photo and (b): schematic of the experimental setup.

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4.2 Experimental validation on sample flame

The VOI in the experiment is $32 \;\textrm{mm} \times 32 \; \textrm{mm} \times 40 \; \textrm{mm}$. To evaluate the fidelity and robustness of the finite element basis and adjustable mask, reconstructions were performed with four different voxel resolutions by ART, from $16 \times 16 \times 20$ to $48 \times 48 \times 60$, corresponding to a voxel size from 2 mm to 0.66 mm. As the ground truth is hard to obtain, the assessment is based on the comparison between the captured images and the predicted projections, which has been detailed in the simulative section. The re-projection error of the eighth predicted projection at the instant of 1 ms with different voxel resolutions is plotted in Fig. 6. As can be seen, the re-projection errors are larger than 0.25 for the four cases. With the increase of the voxel resolution, the errors tend to be smaller for all methods. For both frameworks, the adjustable mask has better performance than the fixed mask. The finite element basis with the adjustable mask is the best one among the four cases. Besides, when the voxel resolution is up to $24 \times 24 \times 30$, the errors of the predicted projections of both bases with the adjustable mask are 0.310 and 0.476, respectively. The further increase of the voxel resolution cannot significantly improve the quality. However, for the fixed mask, the qualities of the predicted projections for both bases become better with the increase of the voxel resolution in this work but are always worse than those for the adjustable mask. Moreover, as shown in Fig. 6, with the increase of the voxel resolution, the difference between the finite element basis and uniform voxel basis diminishes. In principle, if the voxel resolution keeps increasing, the reconstruction quality of the uniform voxel basis can approach that of the finite element basis. However, we did not test at a larger voxel resolution simply because a larger memory is needed which is beyond our computation capability. The re-projection error of the finite element basis with the adjustable mask is decreased by 57% when the voxel resolution is $16 \times 16 \times 20$ compared with the uniform voxel basis with the fixed mask. Besides, the finite element basis with the adjustable mask can obtain a similar reconstruction quality under a lower voxel resolution, e.g., the re-projection error for the finite element basis with the adjustable mask under the voxel resolution of $16 \times 16 \times 20$ is 0.380; while those for the uniform voxel basis with the adjustable mask under the voxel resolution of $40 \times 40 \times 50$ is 0.377.

 figure: Fig. 6.

Fig. 6. Reconstruction error as a function of voxel resolution for both bases with both masks (Projections are captured at the instant of 1 ms).

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Figure 7 shows the captured image and the corresponding predicted projections for the eighth perspective at the instant of 1 ms. Two voxel resolutions are illustrated, i.e., $24 \times 24 \times 30$ and $48 \times 48 \times 60$, for both bases with both masks, respectively. The re-projection errors are labeled in the figure. As can be seen, the predicted projections (except the case shown in Fig. 7(d)) are all similar to the captured image, validating the effectiveness of the reconstruction algorithm. Besides, the predicted projection of the adjustable mask is better than those of the fixed mask. The finite element basis with the adjustable mask has the best performance; while the finite element basis with the fixed mask is the worst. In detail, the area of the predicted projections with the fixed mask is smaller than that with the adjustable mask. For an extreme example, the predicted projection of the finite element basis with the fixed mask under the voxel resolution of $24 \times 24 \times 30$ only has two discrete points as shown in Fig. 7(d). The reason behind the unsuccessful prediction and smaller area of the fixed mask is that the fixed mask will filter out more voxels/nodes than the adjustable mask. The phenomenon is more obvious under the lower resolution since the percentage of the filtered-out voxels/nodes is larger.

 figure: Fig. 7.

Fig. 7. Captured image and predicted projections for View 8 at the instant of 1 ms. (a) Captured image (b)-(c): predicted projections with a voxel resolution of $24 \times 24 \times 30$ for the uniform voxel basis with fixed and adjustable mask, respectively; (d)-(e): predicted projections with a voxel resolution of $24 \times 24 \times 30$ for the finite element basis with fixed and adjustable mask, respectively; (f)-(i): counterparts for the resolution of $48 \times 48 \times 60$.

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Figure 8 shows sample reconstructions by the uniform voxel basis and finite element basis under the voxel resolution of $48 \times 48 \times 60$ with both masks when nine projections are used. As can be seen, while all reconstructions have similar structures, Figs. 8(a) and 8(c) show a spindlier structure compared with Figs. 8(b) and 8(d), which is consistent with the predicted projections in Fig. 7.

 figure: Fig. 8.

Fig. 8. (a)-(b): Reconstructions for the uniform voxel basis with the fixed mask and adjustable mask at 1 ms, respectively; (c)-(d): counterparts for the finite element basis at 1 ms.

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Table 1 lists the time cost under different voxel resolutions for the uniform voxel basis and finite element basis, respectively. As can be seen from the table, with the increase of the voxel resolution, the time cost increases. Compared with the uniform voxel basis under the same voxel resolution, the finite element basis needs 70%∼100% more cost time. However, considering that the finite element basis can reveal a similar structure under the lower voxel resolution as the uniform voxel basis under the higher voxel resolution, the time cost of the finite element basis is much smaller than the uniform voxel basis, e.g., 55.34 s for the finite element basis under the voxel resolution of $16\; \times 16\; \times 20$ and 428.52 s for uniform voxel basis under the voxel resolution of $40\; \times 40\; \times 50$.

Tables Icon

Table 1. Time cost for both bases under different voxel resolutions.

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5. Conclusion

In summary, this work integrated the finite element basis into flame emission tomography. The adjustable mask is adopted to suppress artifacts. Both simulative and experimental studies have been implemented. In the simulative study, both the reconstructions and the corresponding predicted projections suggested that the finite element basis with the adjustable mask can achieve better reconstruction quality. In the experimental study, the flames are reconstructed with different voxel resolutions. The results indicated that based on the same discretization scheme, both uniform voxel basis and finite element basis can reveal the major features of the flame. However, the finite element basis with adjustable mask outperforms the uniform voxel basis consistently; the finite element basis can obtain a similar reconstruction quality under a lower voxel resolution, which is more cost-effective in terms of time. Although this work focuses on the cubic mesh and flame emission tomography, the results are applicable and referable for other meshes, i.e., prism mesh, or other tomographic modalities, i.e., volumetric laser-induced fluorescence.

Funding

Foundation of Science and Technology on Combustion and Explosion Laboratory (6142603200508); National Natural Science Foundation of China (51976122, 52061135108); National Major Science and Technology Projects of China (2017-III-0007-0033).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)
Fig. 1.
Fig. 1. Illustration of the imaging process for flame tomography.

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Fig. 2.
Fig. 2. (a): Simulated projection for the sixth perspective; (b)-(c): predicted projections of the uniform voxel basis with fixed and adjustable masks for ART, respectively; (d)-(e): predicted projections of the finite element basis with fixed and adjustable masks for ART, respectively; (f)-(i): counterparts for MLEM.

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Fig. 3.
Fig. 3. 3D renderings for (a): Phantom 1, (b): Reconstruction 1 with the uniform voxel basis by ART, (c): Reconstruction 1 with the finite element basis by ART, (d): Reconstruction 1 with the uniform voxel basis by MLEM, (e): Reconstruction 1 with the finite element basis by MLEM, and (f): Phantom 2, (g): Reconstruction 2 with the uniform voxel basis by ART, (h): Reconstruction 2 with the finite element basis by ART; (i): Reconstruction 2 with the uniform voxel basis by MLEM, (j): Reconstruction 2 with the finite element basis by MLEM.

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Fig. 4.
Fig. 4. Representative slices of (a)-(c): Phantom 1, (d)-(f): Reconstruction 1 with the uniform voxel basis by ART, (g)-(i): Reconstruction 1 with the uniform voxel basis by MLEM, (j)-(l): Reconstruction 1 with the finite element basis by ART, (m)-(o): Reconstruction 1 with the finite element basis by MLEM.

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Fig. 5.
Fig. 5. (a): Photo and (b): schematic of the experimental setup.

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Fig. 6.
Fig. 6. Reconstruction error as a function of voxel resolution for both bases with both masks (Projections are captured at the instant of 1 ms).

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Fig. 7.
Fig. 7. Captured image and predicted projections for View 8 at the instant of 1 ms. (a) Captured image (b)-(c): predicted projections with a voxel resolution of $24 \times 24 \times 30$ for the uniform voxel basis with fixed and adjustable mask, respectively; (d)-(e): predicted projections with a voxel resolution of $24 \times 24 \times 30$ for the finite element basis with fixed and adjustable mask, respectively; (f)-(i): counterparts for the resolution of $48 \times 48 \times 60$.

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Fig. 8.
Fig. 8. (a)-(b): Reconstructions for the uniform voxel basis with the fixed mask and adjustable mask at 1 ms, respectively; (c)-(d): counterparts for the finite element basis at 1 ms.

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Tables (1)

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Table 1. Time cost for both bases under different voxel resolutions.

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Equations (10)

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V S F i , j = n i , j / N j  for the uniform voxel basis,
f ( x , y , z ) = ∑ q = 1 8 ⁡ α q × f q ,
α 1 = ( 1 − l 1 / d ) × ( 1 − l 2 / d ) × ( 1 − l 3 / d ) ,
V S F i , j = ∑ m = 1 M ⁡ α m × n i , m / N m  for the finite element basis,
M = { 1 i f j − t h n o d e a t t h e c o r n e r 2 i f j − t h n o d e o n t h e e d g e 4 i f j − t h n o d e o n t h e s u r f a c e 8 o t h e r w i s e .
p i = ∑ j = 1 J ⁡ A i , j × f j = A i f ,
f j k + 1 = f j k + λ A R T × p i − A i f k ‖ A i ‖ 2 × A i , j s . t . f ≥ 0 ,
f j k + 1 = f j k × ∑ i ⁡ ( A i , j × p i A i f k ) ∑ i ⁡ A i , j s . t . f ≥ 0 ,
f j = 0 , i f Q j > Q c
{ E r e c o n = ‖ f r e c − f p h a n ‖ 2 / ‖ f p h a n ‖ 2 E r e − p r o j = ‖ p r e − p r o j − p t r u e ‖ 2 / ‖ p t r u e ‖ 2 ,
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