Recently, tomographic particle image velocimetry (TPIV) has been widely employed in the measurement of the flow field around a cylinder, turbulent boundary layer, flame field, and other flow fields since it is highly accurate, multi-point, three-dimensional, and instantaneous. The principle of TPIV is to reconstruct the three-dimensional scattering intensity distribution of particles in the flow field at the adjacent time and combine the three-dimensional cross-correlation algorithm to obtain its instantaneous flow field. 3D particle field reconstruction is the basis of TPIV and the premise of obtaining an accurate 3D flow field. Therefore, it is necessary to develop fast and high-precision 3D particle field reconstruction algorithms. The improvement of the reconstruction algorithm includes two aspects. The first is to improve the reconstruction accuracy, which is the similarity between the reconstructed particle field and the actual particle field, thus affecting the accuracy of the flow field. The second is to shorten the reconstruction time, since the reconstruction process needs to calculate the weight coefficient that is the intensity contribution value of spatial voxels to pixels, and iteration is adopted to update the intensity value of voxels. Thus, the reconstruction process often takes a lot of time, which is the biggest bottleneck in the TPIV application. Therefore, the calculation method of the weight coefficient plays an important role in reconstruction accuracy and reconstruction time.

According to the imaging principle, the line of sight received by a pixel is a spatial volume, so only some of the voxels that contribute to the intensity of a certain pixel can be fully projected into the pixel. The partially projected voxels involve the weight coefficient calculation, which is related to the setting of the camera's internal and external parameters and the spatial volume. The traditional method often employs back projection to calculate the weight coefficient. However, due to a large number of divided spatial voxels and pixels, the order of the weight coefficient is usually large. Additionally, the back projection method not only needs to calculate each line of sight equation but also needs to calculate the number and volume of voxels intersected with the line of sight, thereby resulting in a huge amount of the weight matrix calculation. Therefore, reducing the calculation time of the weight matrix is the key to improving the reconstruction speed. In this paper, the area of voxels projected on the corresponding pixel is calculated as the weight coefficient, and a forward projection method (FPA) is proposed.

Firstly, a multi-view projection imaging simulation program based on the pinhole camera model for particles in 3D space is constructed, and artificial images are generated for analysis and verification. Secondly, FPA is combined with the current mainstream reconstruction algorithms (such as MART, MLOS+MART, and MLOS+SMART) to analyze reconstruction accuracy and time consumption. The results show that when FPA is employed for the reconstruction volume described in this paper, compared with the traditional backward method and the sub-grid method, the number of FPA weight matrix elements is reduced by about three and one orders of magnitude respectively, thus reducing calculation time and computer memory occupation. When the commonly experimental particle concentration p_ppp (particle per pixel) is 0.05, the reconstruction accuracy of this method combined with the current mainstream reconstruction algorithm will be higher than 0.8. In addition, based on the artificial images, the influence of the best camera acquisition angle and the experimental camera noise on the reconstruction results is analyzed, which proves that the reconstruction accuracy still meets the requirements of three-dimensional flow field reconstruction under the experimental noise conditions.

A forward projection weight calculation method (FPA) based on single voxel is proposed in this paper. A particle projection imaging program in 3D space is constructed to verify the correctness of the proposed method. Taking the simulated imaging image as the reconstruction input, it is shown that the matrix elements of FPA combined with the MLOS algorithm can be reduced by about three and one orders of magnitude respectively compared with the traditional backward method and TSM, and the computing time can be reduced by 97% and 85% respectively, greatly reducing the computer memory consumption. Through similarity analysis, the average similarity of the weight matrix calculated by FPA and the traditional backward method is higher than 0.9974, which proves the reliability of FPA. Comparison between the reconstruction results of FPA and those of predecessors in the simplified two-dimensional plane shows that the reconstruction results of the FPA method combined with MART and MLOS+SMART only lose 0.02 reconstruction accuracy. Additionally, FPA together with MART and MLOS+SMART algorithms has good reconstruction results, and the reconstruction accuracy can reach more than 0.8 under the common experimental particle concentration (p_ppp = 0.05). By comparison, MLOS+FPA+MART has higher reconstruction accuracy and faster reconstruction speed, which is suitable for 3D flow field reconstruction. After the experimental noise is added to the imaging process, MLOS+FPA+MART is employed for reconstruction. The results show that the reconstruction accuracy after adding noise is still higher than 0.75, indicating that FPA has good robustness against noise. The analysis of the cross-symmetry camera layout in 3D space shows that the best acquisition angle of CCD2 in the cross-symmetry type is 15°–45° and 10°–40° respectively.