Laser absorption tomography based on unstructured meshing

Laser absorption tomography is an imaging technique used to monitor gas flow and combustion processes. The measurement plane is typically discretized using square pixels in which the flow field is assumed to be uniform. This approach can introduce large discretization errors compared to a finite element mesh with linear interpolation, especially when the domain has an irregular and non-rectangular boundary. In this work, we assess the influence of triangular mesh and quadrilateral mesh schemes on reconstruction accuracy. Our theoretical analysis and numerical studies demonstrate that linear interpolation yields superior performance compared to the conventional uniform-pixel method, and the error can be reduced from 0.250 to 0.097 and 0.160 to 0.105 for triangular and quadrilateral meshes, respectively. Non-body-fitted mesh can increase error by 70 % –99 % . The experiment also proves the superiority of body-fitted meshes with linear interpolation. Moreover, triangular mesh exhibits better noise immunity than quadrilateral mesh. The reconstruction quality appears to be insensitive to the mesh quality for the quadrilateral mesh.


Introduction
Laser absorption tomography (LAT) is a powerful gas flow and combustion diagnostic that is widely used to measure time-resolved two-dimensional (2D) and three-dimensional (3D) distributions of target species and temperature in reacting flows [1,2], and can potentially measure pressure [3,4].Light from one or more lasers is directed through the measurement constrained by limited optical access to the flow field, extreme conditions, the size of the transmitter and receiver optics, and the cost of high-speed, high-power lasers as well as fast and reliable detection and data acquisition systems.For example, representative 2D and 3D LAT setups consist of 32 [13] and 96 paths [14], respectively.The gas is represented using a finite basis, and a large number of basis functions are required to capture flow structures of interest, i.e. with considerably more basis functions than projections.As a result, LAT almost always yields a rank-deficient inverse problem, such that an infinite set of gas distributions can fully satisfy any measurement vector [1].However, even given a large number of independent projections, the resulting inversion necessarily amplifies small perturbations to the data, so reconstructions are highly sensitive to noise, resulting in a so-called discrete illposed inverse problem.Either way, information in addition to the measurement must be incorporated into the reconstruction algorithm to produce a physically-plausible estimate of the gas state; this step is called regularization.Regularization is applied either implicitly (e.g. by truncating an iterative solver [15]) or explicitly (e.g. using a statistical prior [3]).Ultimately, reconstruction accuracy depends on the arrangement of projections, selection of absorption features, the temporal and spectral resolution of the laser-detector system, level of noise, and veracity of added information.Crucially, there are numerous implicit modes of regularization that are often ignored by LAT practitioners to the detriment of reconstruction accuracy.In particular, this work examines the influence of the discretization scheme.
Most reported LAT experiments feature a rectangular or elliptical measurement domain that is discretized using square pixels [1,2].The flow field is assumed to be uniform inside each pixel, which we call the uniform-pixel basis (UPB).Such a discretization scheme features the advantages of easy implementation and uniform pixel resolution.However, in reality, the region of interest (ROI) is usually confined and the boundary of ROI may be a circular or complicated geometry.Typically, additional pixels are used outside the ROI so that the reconstruction can be applied to an overall domain that is rectangular.Of course, in reality, the physical quantities within a single pixel are typically non-uniform.Thus, such a strategy suffers from large discretization errors.To overcome the aforementioned limitations, the finite element basis (FEB) was introduced into LAT, where triangular discretization was adopted [3,16].Grauer et al [16] investigated the selection of meshing density, interpolation strategy, and incorporation of a priori information via the Bayesian model.Different from the UPB, whose variables are stored at pixels, the FEB stores the variables at the nodes and assumes a polynomial distribution within each element.Although FEB has been demonstrated in several tomographic demonstrations, e.g.optical tissue [17] and bioluminescence tomography [18], previous works exclusively feature triangular element meshes.Recently, Liu et al introduced the FEB into flame emission tomography [19].However, there is no systematic assessment of the meshing schemes in the tomographic system, nor any mathematical interpretation of mesh performance.With the development of meshing technology, various meshing schemes have been proposed for computational fluid dynamics (CFD), e.g.triangular, quadrilateral, and hybrid meshes.These schemes feature easy body-fitness of complex geometries and local refinement of the mesh density.Thus, this work aims to investigate the fidelity and robustness of FEB with various meshing schemes in the context of LAT.

Principle of LAT
Before introducing our numerical study, we briefly summarize the fundamentals of LAT.Monochromatic light is emitted by a laser and attenuated by the gas according to the Beer-Lambert law, which is rearranged to obtain an absorbance (α v ) [20]: In this expression, I v,0 and I v,L are the intensity of light at the transmitter and receiver, respectively; v is the wavenumber of light; k v is the local spectral absorption coefficient, related to the gas composition and thermodynamic state by a spectroscopic model such as HITRAN [21]; and r is an indicator function that converts the progress variable l to a 2D (or 3D) position along a beam of length L. The spatially-resolved gas state is represented using the FEB: where φ j is the jth orthonormal basis function, of n total functions, and f j is the corresponding coefficient.Equation ( 1) is discretized accordingly: where p i is the ith LOS measurement of m total measurements, and A i,j is the sensitivity of the ith projection to the gas state in the jth basis function: where r i and L i are specific to the ith beam.In this summary, p is α v and f is k v .Absorbance measurements and unknown gas parameters are arranged vector-wise, i.e. p = {p i } and f = {f j }, resulting in the m × n matrix system, Af = p, and reconstruction consists in estimating f given the measurement vector p.This problem is necessarily ill-posed, as described above, and we employ Tikhonov regularization [22] to generate smooth distributions due to its ease of implementation and demonstrated performance in LAT applications.

Numerical setup
Our numerical studies were coded in MATLAB R2020.The phantoms within the circular ROI are assumed to be consistent with a multivariate normal distribution.To obtain the To make the number of variables comparable for both meshing schemes, triangular mesh, and quadrilateral mesh hold a similar number of nodes; while the amount of elements for triangular mesh is around twice that for quadrilateral mesh.More detailed parameters for the meshes are listed in table 1.
For FEB, the interpolation within the element was adopted and the number of variables equals the number of nodes.The shape functions for linear interpolation are plotted in figures 1(c) and (d); while for constant interpolation, the shape function is a flat plane with the coefficient of 1/3 for triangular element and 1/4 for quadrilateral element.The local spectral absorption coefficient within the element is a weighted summation of the values at the corresponding nodes and the summation of the weighted coefficients is equal to one.For UPB, each element has one quantity and the number of variables equals the number of elements.Note that UPB and FEB with constant interpolation both treat each element as one value; the difference is the total number of the variables and the method to obtain the local spectral absorption coefficient.These different distribution assumptions within each element can be integrated into the reconstruction model when calculating the absorption along the path.

Numerical results
Firstly, FEB with constant and linear interpolation are compared to UPB with two different mesh densities for both mesh schemes.We will explain the results in detail for Phantom 1.The associated reconstructions are plotted in figure 2. The criterion applied is a relative error based on the two-norm (E) [15], which is defined as: where x phan and x rec are the values of phantoms and reconstructions under the benchmark mesh, respectively.The relative errors are labeled in figure 2 as well.As can be seen from the figure, all methods can recover the major features of Phantom 1.For both low-density mesh schemes, the relative errors of FEB with linear interpolation are the smallest; followed by FEB with constant interpolation and UPB.By replacing UPB with FEB and applying linear interpolation, the reconstruction error can be decreased from 0.250 to 0.097 for triangular mesh and from 0.160 to 0.105 for quadrilateral mesh.The results for FEB with linear interpolation  are the smoothest ones compared with others; the results for FEB with constant interpolation feature jagged edges; while those for UPB are noisy.For FEB with linear interpolation, both meshes can obtain similar reconstructions.However, the results for UPB with triangular mesh contain large noise.
The main reason is that there are much more variables compared other cases, leading to more severe ill-posedness.
For high-density meshes, although smoothness regularization, i.e.Tikhonov regularization, is applied, the reconstruction results are much worse than those for low-density meshes, resulting from too much variables.
In most previous demonstrations in the literature, the irregular ROI is supplemented into a rectangle and the rectangular meshes are applied [5].Such mesh is called non-bodyfitted mesh, i.e.Mesh 3, Mesh 4, Mesh 7 and Mesh 8 in this work.With the assistance of ICEM, it is simple and convenient to generate the so-called body-fitted mesh, i.e.Mesh 1, Mesh 2, Mesh 5 and Mesh 6 in this work.Figure 3 shows the reconstructions for Phantom 1 by non-body-fitted mesh for all tested cases.The counterparts with body-fitted mesh are shown in figure 2. The reconstruction relative errors are added in the figure as well.As can be seen, non-body-fitted meshes can still recover the major features of Phantom 1 and similar conclusion to body-fitted meshes can be derived as: FEB with linear interpolation is best, followed by FEB with constant interpolation and UPB; the further increase of mesh density cannot improve the reconstruction quality for UPB.However, the relative errors for the non-body-fitted meshes are much larger than those for body-fitted mesh.For example, non-body-fitted mesh can increase error by 70%-99% for FEB with linear interpolation.Additionally, the body-fitted mesh can retrieve clearer structures around the edge, while the reconstructions near the edges for non-body-fitted meshes are blurry, due to the supplemented pixels.The results indicate that the body-fitted mesh is superior to the non-body-fitted mesh.
To study the behavior of triangular mesh and quadrilateral mesh under noisy conditions, uniform white noises are added to the projections.Figure 4 plots the reconstruction relative errors as a function of noise levels for tested phantoms.Error bars represent the standard deviation of 50 cases with the same noise levels for Phantom 1.For the body-fitted triangular mesh with linear interpolation, the condition number of the weight matrix is 15; while that for the body-fitted quadrilateral mesh is 22.The larger condition number of the quadrilateral mesh is caused by the fact that the opposite edges of the quadrilateral element should be as parallel as possible, which increases the linear dependence of the tomographic system  and the condition number.As indicated by the figure, bodyfitted triangular mesh shows better reconstruction performance than body-fitted quadrilateral mesh and holds better noise immunity.For both meshes, the errors and error bars increase with the increase of the noise levels.
For CFD, mesh quality is much more important than mesh topology.However, no evidence has indicated the same conclusion for tomographic reconstructions.Thus, the impact of mesh quality was investigated.The quality is defined as [23]: where |J| is the determinant of the Jacobian matrix at each node of the elements.The optimal average quality of Mesh 1 is 0.92; while that of Mesh 2 is 0.76.Note that the definitions of the mesh quality for the two mesh types are different, and it is meaningless to compare the mesh quality between the mesh types.To obtain a new mesh with lower average quality, the inner nodes of the optimal mesh were randomly moved to a new position while the nodes on the edge keep unchanged.Figure 5 plots the reconstruction relative errors for various mesh qualities ranging from 0.60 to 0.92 for triangular mesh and 0.60 to 0.76 for quadrilateral mesh.The error bars indicate the standard deviation of 200 phantoms without noise.
As can be seen, the errors are smaller than 0.12 for all average mesh qualities.Moreover, with the increase of the average mesh quality, the error gradually decreases for triangular mesh, suggesting a better average mesh quality can improve the reconstruction quality.However, within the tested cases, the improvement is quite limited for quadrilateral mesh.The phenomenon indicates the reconstruction quality is insensitive to the mesh quality for quadrilateral mesh.

Experimental setup and data acquisition
To further compare triangular mesh and quadrilateral mesh structures and evaluate the benefits of the FEB, this section continues the analysis of the experimental data from Nasir and Sander [24].Full experimental details are provided in the reference; briefly, the distribution of ammonia in diesel engine exhaust was reconstructed using absorption spectroscopic tomography.Unlike the absorption spectroscopic tomography techniques based on water vapor and CO 2 , the experiment used a quantum cascade laser (Alpes Lasers HHL-470) to detect absorption of ammonia at wavelengths between 10.39 and 10.40 µm.The detector used was a mercury cadmium telluride detector (Vigo Systems PVI-3TE-10.6).The laser and detector were arranged on a U-shaped bracket, which could be translated using a displacement stage or rotated around the detection region using a rotation stage.With this setup, a total of 30 projections were obtained, spaced at 6 • intervals, and each projection direction consisted of 71 rays with a spacing of 5 mm between them.For the original data processing, the detection region was discretized into a 71 × 71 square mesh, and the ammonia spatial distribution was reconstructed using the inverse Radon transform.The reconstruction system consisted of 2130 known measurements and 5041 unknown variables.Figure 6 illustrates the distribution of ammonia obtained under stationary assumptions with an ambient temperature of approximately 464 K, over a duration of 5 ms.According to the reconstruction results, the estimated diameter of the ammonia distribution is approximately 280 mm, which aligns with the diameter of the tube in the experiment.From the figure, it can be observed that the reconstructed region is slightly larger than the actual region where ammonia is present.Additionally, the reconstruction exhibits noticeable noise and some degree of artifacts, which are inherent characteristics of non-body-fitted meshes.Moreover, a crescent-shaped local maximum region can be observed in the lower right part of the figure, indicating the presence of swirling airflow.

Experimental results
Various meshes were utilized with the data from the previous publication [24] to explore the connection between reconstruction quality and mesh density.As shown in figure 7, the number of nodes for triangular mesh was set to 50, 100, 250, and 500, while for quadrilateral mesh, the number of nodes was set to 52, 105, 260, and 481.It is important to note that it is not possible to precisely control the node number to be exactly the same for both types of meshes while ensuring mesh quality.However, efforts were made to maintain them at the same order of magnitude.
In this section, FEB with linear interpolation was used, since numerical results as shown above suggest that it will be superior.The reconstruction results are shown in figure 8.The labeled 'Residual' in the figure represents the ratio of the two-norm of the difference between the projection and reprojection to the two-norm of the projection.From the perspective of the residual, it can be observed that the quadrilateral mesh converges more easily than the triangular mesh.Additionally, as the number of mesh nodes increases, the residual tends to decrease.However, once the number of mesh nodes reaches the order of 500, the residual remains essentially unchanged.
Comparing the reconstruction results in this section with the results obtained using the inverse Radon transform in section 4.1, it can be observed that the all methods yield similar structures.However, the FEB method produces smoother results and exhibits less noise, even though under low mesh density.This indicates that the FEB method can achieve better results with fewer variables.However, when the number of mesh nodes is too small (less than 100), the FEB reconstruction results may contain varying degrees of sharp edges.But as the number of mesh nodes increases to around 250, this phenomenon gradually disappears.Furthermore, as the number of mesh nodes increases, the differences between triangular mesh and quadrilateral mesh become smaller.
Table 2 lists the time cost for reconstruction under triangular mesh and quadrilateral mesh.It can be observed that as the mesh density increases, the time cost for model construction and reconstruction also increases.However, the time cost for reconstruction is almost negligible compared to the time cost for model construction.Moreover, the time cost for constructing models using quadrilateral mesh is consistently lower than that using triangular mesh.This is because calculating the chord length within each mesh element is necessary during model construction, and under the same order of magnitude of mesh node number, the number of elements in triangular mesh is much larger than that in quadrilateral mesh, resulting in longer computation time.However, there is no significant difference in the reconstruction time between triangular mesh and quadrilateral mesh; they are generally within the same order of magnitude.

Conclusion
In summary, this work assessed the FEB with various mesh schemes for LAT, i.e. body-fitted triangular mesh, body-fitted quadrilateral mesh, non-body-fitted triangular mesh, and nonbody-fitted quadrilateral mesh.An extensive numerical study was designed and conducted.200 representative phantoms consistent with multivariate normal distributions were produced and applied to simulate different experiment occasions.
The results show that the FEB with linear interpolation outperforms the constant interpolation and classical UPB and the error can be reduced from 0.250 to 0.097 and 0.160 to 0.105 for triangular and quadrilateral meshes, respectively.FEB with linear interpolation can get better reconstruction under low mesh density than UPB under high mesh density.The bodyfitted mesh can reconstruct more accurate information than non-body-fitted mesh around the edge of non-rectangular ROI.Triangular mesh has better noise immunity than quadrilateral mesh.Better mesh quality can reduce the reconstruction error; however, the improvement is limited for quadrilateral mesh.The mesh schemes were validated via experiment, which also shows the superiority of body-fitted mesh.Although this work is discussed under the context of 2D LAT, the conclusions can be easily extended to 3D tomography by adopting tetrahedral mesh and hexahedron mesh.This work is also valuable for other tomographic modalities, e.g.background-oriented Schlieren tomography [25].

Figure 1 .
Figure 1.(a): Sample phantom; (b) ray configuration; (c) and (d): shape function for triangular mesh and quadrilateral mesh, respectively; (e)-(l): triangular mesh and quadrilateral mesh for circle and square, respectively, referred to as Mesh 1 to Mesh 8.

Figure 2 .
Figure 2. Reconstructions for Phantom 1: (a) UPB with low-density triangular mesh; (b) FEB with constant interpolation and low-density triangular mesh; (c) FEB with linear interpolation and low-density triangular mesh; (d) UPB with high-density triangular mesh; (e)-(h) counterparts for quadrilateral mesh.

Figure 3 .
Figure 3. Reconstructions for Phantom 1 by non-body-fitted mesh: (a) UPB with low-density triangular mesh; (b) FEB with constant interpolation and low-density triangular mesh; (c) FEB with linear interpolation and low-density triangular mesh; (d) UPB with high-density triangular mesh; (e)-(h) counterparts for quadrilateral mesh.

Figure 4 .
Figure 4. Reconstruction relative errors as a function of noise level for body-fitted triangular mesh and quadrilateral mesh with linear interpolation.

Figure 5 .
Figure 5. Reconstruction relative errors as a function of mesh quality.

Figure 6 .
Figure 6.Distribution of ammonia based on inverse Radon transformation (the temperature is around 464 K, Reproduced from [24], with permission from Springer Nature.)

Table 2 .
Time cost for triangular mesh and quadrilateral mesh.