This section reports the numerical validations of DMCICT by reconstructing 3D chemiluminescence distribution of three different phantoms. Figure 3 shows the schematic experimental setup for 3D tomography measurements. All experimental instruments are assembled on an optical platform. The cylinder is made of quartz with uniform refractive index of 1.4, and has a size of 131 mm (outer diameter) × 92 mm (inner diameter) × 42 mm (height). The object coordinate system *O*-*XYZ* is defined as follows: the center of the cylinder bottom surface is defined as the origin *O*; the directions parallel with and normal to the cylinder vertical axis are respectively defined as the *X* and *Y* directions; the cylinder vertical axis is defined as the *Z* direction. Based on the measurement system setup, numerical tomographic reconstructions using DMCICT and CICT on different phantoms are performed prior to any practical experiments.

Three phantoms of different patterns are generated to verify the universality of the DMCICT in Fig. 4. Figure 4(a) shows a simulated flame phantom consisting of a funnel-shaped object and a spindle object, Fig. 5(b) shows a turbulent spray phantom consisted of two branches, and Fig. 5(c) presents a combination of spherical, conical and prismatic shapes. All phantoms have uniform thickness of 2 voxels (i.e., 1 mm). The intensity of phantom voxels is artificially set as unity and the intensity of all other voxels is set as 0. Such intensity distribution simulates a recognizable pattern to help evaluate the quality of tomographic reconstruction. Same as Fig. 2(a), the measurement domain in Fig. 4 is also discretized into 184 voxel (*X*) × 184 voxel (*Y*) × 70 voxel (*Z*). To perform simulation reconstructions, a total of 8 simulated projections are respectively recorded from the azimuth angles of 0°, 30°, 50°, 80°, 100°, 140°, 160° and 180°, and the inclination angle *φ* is set to 0° for all projections. To ensure all phantoms can be entirely recorded, the pixel resolution of projections is set to be 800 pixels × 300 pixels (with the pixel size of 5.6 µm), and the imaging magnification is set to 15. Besides, the focal length and *f* number of virtual lens used are 35 mm and 2.8, respectively. To simulate practical experimental uncertainty, 1% of Gaussian noise and 0.1° angle error is added during the imaging process in accordance with our past effort [34]. During the ART iteration, the iteration number is set to 30 with a relaxation factor of 0.7 to ensure convergence for DMCICT.

For variation and comparison purposes, numerical reconstructions by DMCICT and CICT methods are performed for all phantoms. Figure 6(a) shows the calculating reconstruction time for phantom P1, P2 & P3 by both methods. Based on the order of processes, the total calculating time is divided into two parts: 1) the time cost to establish and store the immense mapping relationship matrix, and 2) the time consumed in following ART iterations. As shown in Fig. 5(a), the total calculation time of DMCICT is reduced by more than 50% compared to that of CICT for all phantoms. Such observation reflects that DMCICT has significantly improved calculation efficiency by optimizing the imaging model (explained in Fig. 1). After the tomographic calculation, Fig. 5(b) compares the quality of the reconstruction results by both algorithms. To quantitatively evaluate the reconstructions, a correlation coefficient *Corr*3 is employed similar to our past tomography approaches [33, 38], which can be calculated using Eq. (3):

$$\begin{array}{c}Corr3({V}_{\text{phan}},{V}_{\text{recon}})=\\ \frac{{\sum }_{\text{j}}\left\{\left[{V}_{\text{phan}}\left(j\right)-\stackrel{-}{{V}_{\text{phan}}}\right]\times \left[{V}_{\text{recon}}\left(j\right)-\stackrel{-}{{V}_{\text{recon}}}\right]\right\}}{\sqrt{{\sum }_{\text{j}}{\left[{V}_{\text{phan}}\left(j\right)-\stackrel{-}{{V}_{\text{phan}}}\right]}^{2}}\times \sqrt{{\sum }_{\text{j}}{\left[{V}_{\text{recon}}\left(j\right)-\stackrel{-}{{V}_{\text{recon}}}\right]}^{2}}}\end{array}$$

3

where *V*phan and *V*recon are the 3D signal intensity matrices of the original and reconstructed phantoms, respectively. The overline above a matrix indicates the average of all non-zero values in matrix. Eq. (3) indicates that the closer *Corr*3 is to 1, the more similar the reconstruction to the original phantom. As shown in Fig. 5(b), for all studied phantoms, the *Corr*3 values by DMCICT are around 0.9, noticeably higher than those by CICT (~ 0.8). Such observation quantitatively validates the accuracy of DMCICT. Compared to CICT, the DMCICT method is believed to possess the capability of quicker convergence and higher accuracy within limited iterations (i.e., 30 in this work, also an often-used number in practical tomography applications). However, it is necessary to clarify that such comparison does not negate the validity of CICT. Under sufficient number of iterations (e.g., 500 in our previous work [34]), the CICT reconstructions also achieved sufficient accuracy (i.e., *Corr*3 close to 0.9). Nevertheless, due to significantly enhanced calculation speed in contrast to CICT, the DMCICT method is managed to obtain precise reconstructions more effectively. Hence, the DMCICT is especially preferred when a dynamic target is reconstructed, usually involving hundreds or thousands of frames sequentially recorded.

To evaluate the robustness of DMCICT on practical reconstruction situations, its performance on different voxel sizes is investigated. Besides previously tested discretization (184 voxel × 184 voxel × 70 voxel, namely dim184), the measurement domain are respectively divided into: 92 voxel × 92 voxel × 35 voxel (dim92) and 276 voxel × 276 voxel× 105 voxel (dim276). Figure 6 compares the reconstruction time and accuracy of P3 by DMCICT and CICT under different voxel sizes. As shown in Fig. 6(a), the reconstruction calculation time is significantly reduced when DMCICT is used instead of CICT for all tested voxels sizes. Moreover, the proportion of calculation time saved by DMCICT rises as the voxel size becomes smaller (i.e., the nominal resolution increases). Specially, under dim276 (voxel size: 0.33 mm), the calculating time for DMCICT is reduced to nearly one third of that for CICT. Figure 6(b) shows the reconstruction accuracy of both methods maintains at approximately same level, though *Corr*3 gradually reduces with smaller voxel size. The method discrepancies in both computational cost and accuracy indicate that DMCICT would be more beneficial for practical tomographic reconstructions desired for higher spatial resolutions.

Figure 7 further evaluates the capacity of DMCICT in withstanding imaging noise. As stated in the simulation setup, the imaging noise is represented by the Gaussian noise and the angle error, set as 1% and 0.1° in previous calculation. Hence, Fig. 7(a) compares the DMCICT reconstructions of all phantoms with Gaussian noise *E*gaussian = 1%, 2% and 3% (angle error *E*angle = 0.1°), while Fig. 8(b) compares the reconstructions with *E*angle = 0.1°, 0.2° and 0.3° (*E*gaussian = 1%). As can be observed in Fig. 8, neither Gaussian noise nor angle error has prominent impact on the reconstruction by DMCICT. Specifically, the value of *Corr*3 maintains above 0.8 under most cases, indicating that the DMCICT method possesses reliable tolerance on imaging noise. The only exception may be the deteriorated reconstruction of P2 with *E*gaussian = 3%. This is possibly due to relatively lower signal-to-noise ratio of P2, since P2 is generated from a practical flame reconstruction while others are consisted of basic geometric shapes.