Optical remote sensing (ORS) combined with the computerized tomography (CT)
technique is a powerful tool to retrieve a two-dimensional concentration map over an area under investigation. Whereas medical CT usually uses a beam number of hundreds of thousands, ORS-CT usually uses a beam number of dozens, thus severely limiting the spatial resolution and the quality of the reconstructed map. The smoothness a priori information is, therefore, crucial for ORS-CT. Algorithms that produce smooth reconstructions include smooth basis function minimization, grid translation and multiple grid (GT-MG), and low third derivative (LTD), among which the LTD algorithm is promising because of the fast speed. However, its theoretical basis must be clarified to better understand the characteristics of its smoothness constraints. Moreover, the computational efficiency and reconstruction quality need to be improved for practical applications. This paper first treated the LTD algorithm as a special case of the Tikhonov regularization that uses the approximation of the third-order derivative as the regularization term. Then, to seek more flexible smoothness constraints, we successfully incorporated the smoothness seminorm used in variational interpolation theory into the reconstruction problem. Thus, the smoothing effects can be well understood according to the close relationship between the variational approach and the spline functions. Furthermore, other algorithms can be formulated by using different seminorms. On the basis of this idea, we propose a new minimum curvature (MC) algorithm by using a seminorm approximating the sum of the squares of the curvature, which reduces the number of linear equations to half that in the LTD algorithm. The MC algorithm was compared with the non-negative least square (NNLS), GT-MG, and LTD algorithms by using multiple test maps. The MC algorithm, compared with the LTD algorithm, shows similar performance in terms of reconstruction quality but requires only approximately 65

Measuring the concentration distribution of atmospheric chemicals over large areas is required in many environmental applications, such as locating hotspots or emission sources of air pollutants (Wu et al., 1999), understanding air pollutant dispersion and airflow patterns, and quantifying emission rates or ventilation efficiency (Samanta and Todd, 2000; Belotti et al., 2003; Arghand et al., 2015). The traditional network method uses multiple point samplers placed at various locations in the region under investigation. This method is intrusive, time-consuming, and limited in its temporal and spatial resolution (Cehlin, 2019). The advanced method is based on the combination of optical remote sensing and computerized tomography techniques (ORS-CT). ORS-CT is a powerful technique for the sensitive mapping of air contaminants throughout kilometer-sized areas in real time (Du et al., 2011). There are two commonly used ORS techniques that use an open-path tunable diode laser (TDL) and open-path Fourier transform infrared spectrometer. The ORS analyzer emits a light beam, targeted at multiple mirrors, which reflect the beam back to the analyzer. For each beam path, the path-integrated concentration (PIC) is obtained. After multiple PICs are collected, a two-dimensional concentration map can be generated through tomographic reconstruction algorithms (Hashmonay et al., 2001). The ORS-CT method provides a better spatial and temporal resolution than the network approach, and it is more sensitive than the range-resolved optical techniques. It is also non-intrusive and suitable for continuous long-term monitoring.

In ORS-CT mapping of atmospheric chemicals, owing to factors including system cost, response time, and beam configuration, the number of beams is only tens, whereas the number of beams in medical CT is hundreds of thousands. The very small beam number poses several challenges in tomographic reconstruction algorithms. In practice, transform methods based on the theory of Radon transformation, using a filtered back projection formula, are not feasible because of noise and artifacts in the reconstructions (Radon, 1986; Herman, 2009). Series expansion methods, which discretize the reconstruction problem before any mathematical analysis, are usually used in ORS-CT. The underlying distribution is represented by a linear combination of a finite set of basis functions (Censor, 1983). The simplest type is the pixel-based approach, which divides an area into multiple grid pixels and assigns a unit value inside each pixel. The path integral is approximated by the summation of the product of the pixel value and the length of the path in that pixel. A system of linear equations can be set up for multiple beams. The inverse problem involves finding the optimal set of pixel concentrations according to criteria including the least square criterion to minimize the summation of the squared errors between the observed and model-predicted PICs, the maximum likelihood (ML) criterion to maximize the probability of the PIC observations, given the distribution of the random variables of the concentrations and observation errors, and the maximum entropy criterion to maximize the entropy of the reconstructed maps, given that the average concentration of the map is known (Herman, 2009). Commonly used pixel-based algorithms are algebraic reconstruction techniques (ART), non-negative least square (NNLS), and expectation–maximization (EM; Tsui et al., 1991; Lawson and Janson, 1995; Todd and Ramachandran, 1994). The NNLS algorithm has similar performance to the ART algorithm but a shorter computation time (Hashmonay et al., 1999). It has been used in the U.S. Environmental Protection Agency Other Test Method (OTM) 10 for the horizontal radial plume mapping of air contaminants (EPA, 2005). The EM algorithm is mainly used for ML-based minimization. These traditional pixel-based algorithms are suitable for rapid CT, but they produce maps with poor spatial resolution, owing to the requirement that the pixel number must not exceed the beam number, or they may have problems of indeterminacy associated with substantially underdetermined systems (Hashmonay, 2012).

To mitigate the problem of indeterminacy and improve the spatial resolution of reconstructions without substantially increasing the system cost, the smooth basis function minimization (SBFM) algorithm has been proposed. This algorithm represents the distribution map by a linear combination of several bivariate Gaussian functions (Drescher et al., 1996; Giuli et al., 1999). Each bivariate Gaussian has six unknown parameters (normalizing coefficient, correlation coefficient, peak locations, and standard deviations) to be determined. The problem requires fitting these parameters to the observed PIC data. This method performs better than the traditional pixel-based algorithms for ORS-CT applications because the patterns of air dispersion are physically smooth in shape (Wu and Chang, 2011). However, the resultant equations defined by the PICs are nonlinear because of the unknown parameters. The search for the set of parameters with the best fit, minimizing the mean squared difference between predicted and measured path integrals, can be performed through an iterative minimization procedure, such as the simplex method or simulated annealing.

The reported methods using simulated annealing to find a global minimization are highly computationally intensive, thereby limiting the SBFM algorithm's practical applications, such as rapid reconstruction in the industrial monitoring of chemical plants. However, an algorithm converging toward a smooth concentration distribution consistent with the path-integrated data has been demonstrated to be a rational choice. To improve the computational speed and append the smoothness a priori information to the inverse problem, the pixel-based low third derivative (LTD) algorithm has been proposed. This algorithm sets the third derivative at each pixel to zero, thus resulting in a new system of linear equations that is overdetermined. The LTD algorithm has been reported to work as well as the SBFM algorithm but is approximately 100 times faster (Price et al., 2001). Another method to produce the smoothness effect is the grid translation (GT) algorithm, which shifts the basis grid by different distances (e.g., one-third or two-thirds of the width of the basis grid) horizontally and vertically while keeping the basis grid fixed (Verkruysse and Todd, 2004). Smoothness is achieved by averaging the reconstruction results after each shifting. An improved version, called the grid translation and multigrid (GT-MG), applies the GT algorithm at different basis grid resolutions (Verkruysse and Todd, 2005). This method has been used with the ML-EM algorithm to improve the reconstruction accuracy, particularly in determining the peak location and value (Cehlin, 2019).

The success of these algorithms demonstrates the need to apply smoothness
restriction to the ORS-CT gas mapping. With the LTD algorithm, a smooth
reconstruction is achieved by simply adding the third-order derivative
constraints. The generated solutions are locally quadratic. To understand the
characteristics of these constraints and apply the method to the specific
application, the theoretical basis of the algorithm must be understood.
However, this basis is not clearly defined in the literature. With the purpose of introducing smoothness constraints, the LTD algorithm can be treated as a special case of the Tikhonov regularization, a well-known technique for solving the ill-posed inverse problem (Tikhonov and Arsenin, 1977; Rudin et al., 1992). The Tikhonov

The interpolation techniques are based on the given sample points, in contrast to tomographic reconstruction, in which only the line integrals are known. However, we have found that the interpolation can be adopted in the reconstruction process to produce a smooth solution by using the smoothness seminorm for interpolation as a smoothness regularization factor for the tomographic reconstruction problem. In view of the variational spline interpolation, the characteristics of algorithms using different seminorms have been well explored in the literature. The LTD algorithm can be considered as one case that minimizes the seminorm consisting of the third-order derivatives (Bini and Capovani, 1986). Other algorithms can also be formulated by using different seminorms. On the basis of this idea, we propose a new minimum curvature (MC) algorithm using a seminorm approximating the integral of the squares of the curvature. This algorithm generates a smooth reconstruction approximating the application of cubic spline interpolation. We compared the algorithm with the NNLS, LTD, and GT-MG algorithms by using multiple test maps. We demonstrated its effectiveness and two main aspects of this method. First, a smoothing effect similar to spline interpolation is achieved during the reconstruction process by using high-resolution grid division, and second, the computational efficiency is markedly better than that of the LTD algorithm through halving the number of linear equations according to the new smoothness seminorm. This approach achieves the same performance but is easier to perform than the GT-MG algorithm, which has complicated operations involving multiple grids and grid translation. More specific algorithms applied for the ORS-CT method for mapping atmospheric chemicals could be further formulated and evaluated similarly.

The area of the test field was

The beam configuration and grid division. The field was divided
into

The wavelength of the laser beam is tuned to the absorption line of the target gas and is transparent to other species. For a general detection of atmospheric pollutants, the laser absorption is in the linear regime, and the attenuation of the laser beam is governed by the Beer–Lambert law. The predicted PIC for one beam is equal to the sum of the multiplication of the pixel concentration and the length of the beam inside the pixel. In general, let us assume that the site is divided into

The LTD algorithm introduces the smoothness information through setting the
third-order derivative of the concentration to zero at each pixel in both

A weight needs to be assigned to each equation, depending on the uncertainty of the observation. Under the assumption that the analyzers have the same
performance, the uncertainty is mainly associated with the path length.
Therefore, equations are assigned weights inversely proportional to the path
length to ensure that different paths have equal influences. Herein, the
lengths of the laser paths are approximately equal to each other. Therefore,
their weights are set to the same value and scaled to be 1. The weights for
the third-derivative prior equations are assigned as the same value of

The LTD algorithm actually constructs a regularized inverse problem. It can be viewed as a special case of the well-known Tikhonov regularization
technique. The Tikhonov

The regularization parameter determines the balance between data fidelity and
regularization terms. Determination of the optimum regularization parameter is an important step in the regularization method. However, the regularization parameter is problem and data dependent. There is no general purpose parameter choice algorithm that will always produce a good parameter. For simplicity, we use the method based on the discrepancy principle (Hamarik
et al., 2012). The regularization parameter

Splines are special types of piecewise polynomials, which have been demonstrated to be very useful in numerical analysis and in many applications in science and engineering problems. They match given values at some points (called knots) and have continuous derivatives up to some order at these points (Champion et al., 2000). Spline interpolation is preferred over polynomial interpolation by fitting low-degree polynomials between each of the pairs of the data points instead of fitting a single high-degree polynomial. Normally, the spline functions can be found by solving a system of linear equations with unknown coefficients of the low-degree polynomials defined by the given boundary conditions.

The variational approach provides a new way of finding the interpolating splines and opens up directions in theoretical developments and new applications (Champion et al., 2000). Variational interpolation was motivated by the minimum curvature property of natural cubic splines, i.e., the interpolated surface minimizes an energy functional that corresponds to the potential energy stored in a bent elastic object. This principle provides flexibility in controlling the behavior of the generated spline. Given an observation

We can see that the minimizing problem in Eq. (

Under the assumption that the unknown concentration distribution is described
by a function

Beam geometry and a

For conventional, pixel-based reconstruction algorithms, the number of pixels
(unknowns) should not exceed the number of beams (equations) to obtain a
well-posed problem. Because only tens of beams are usually used in ORS-CT
applications, the resultant spatial resolution is very coarse. The GT
algorithm is one way to increase the resolution, but it requires several steps
to complete the entire translation because each translation uses a different
grid division, and the reconstruction process must be conducted for each grid
division. In the MC algorithm, we use only one division of high-resolution
grids directly during the reconstruction. The resultant system of linear
equations remains determined because of the smoothness restriction at each
pixel. As shown in Fig. 2,

The NNLS, LTD, and MC algorithms were compared by using multiple test maps.
The results were also compared with those of the GT-MG algorithm. We set up
test conditions similar to those used in Verkruysse and Todd (2005). The
concentration distribution from one source is defined by a bivariate Gaussian
distribution as follows:

The source number varies from 1 to 5. For multiple sources, the resultant concentration distribution is the superposition value due to each source. For each source number, 100 maps were generated by randomly setting the source strength, location, and peak width from the defined ranges or set above.

The concentration filed is discretized with a resolution of

A conventional image quality measure, called nearness, is used to describe the discrepancy between the original maps and the reconstructed maps. Nearness evaluates errors over all grid cells on the map are as follows (Verkruysse and Todd, 2005):

The effectiveness of locating the emission source is evaluated by the peak
location error, which calculates the distance between the true and
reconstructed peak locations.

Exposure error percentage is used to evaluate how well the average concentrations
in the whole field are reconstructed. It can reflect the accuracy of measuring chemical air emissions and emission rates from fugitive sources, such as agricultural sources and landfills, as follows (Verkruysse and Todd, 2004):

For the LTD and MC approaches using high-resolution grids, the kernel matrix

In these tests, the traditional NNLS algorithm uses

Original test maps (first column) and corresponding maps reconstructed with the NNLS (second column), LTD (third column), and MC (fourth column) algorithms, showing

Nearness is the most important measure of accuracy of the reconstructed map.
It represents the reconstruction of peak heights, shapes, and the production
of artifacts. The smaller the nearness value, the better the reconstruction
quality. In Table 1, the LTD, MC, and GT-MG algorithms generally reduce the
nearness values by more than 50

Mean and standard deviation of the nearness.

As shown in Table 2, the LTD and MC algorithms show better performance in peak location error than the NNLS algorithm. They generally improve the accuracy of peak location by 1 to 2

Mean and standard deviation of the peak location error.

One reason for this finding is that when two or more peaks with comparable peak magnitudes are on the map (Fig. 3), the algorithm may not identify the correct location of the highest peak. Therefore, a large error may occur when the highest value on the reconstructed map is located on the wrong peak.

The exposure error of NNLS can be severely affected by the spline interpolation of the reconstruction results. Therefore, a nearest interpolation was used. As shown in Table 3, MC and LTD algorithms show approximately 2 to 5 times better performance than the NNLS algorithm. The exposure error reflects the accuracy of the overall emissions measurement other than the concentration distribution. The performance of the LTD and MC algorithms is very similar, whereas the MC algorithm illustrates slightly better performance than the LTD algorithm. Unlike the trends shown by the NNLS in the nearness and peak location error, its performance in exposure error improves with an increasing source number. A plausible cause of this phenomenon may be that the distribution becomes more uniform with larger numbers of sources. Because the NNLS algorithm uses coarse grid division, it produces concentrations with very low spatial resolution and fits the true distribution better when the distribution becomes more uniform.

Mean and standard deviation of the exposure error.

The computations were run on a computer with an Intel Core
i7-6600U processor of 2.6

Mean and standard derivation of the computation time.

Contour plots of the resolution matrices for the LTD and MC algorithms are shown in Fig. 4a and b. Each row represents the weight strength of all the pixels for the current pixel. The fitness values for the LTD and MC algorithm are 1.4411 and 1.3878, respectively. The MC algorithm shows slightly better performance. The off-diagonal elements are not zeros. The reconstructed concentration at each pixel is a weighted average of the concentrations of the surrounding pixels according to the smoothness regularization. Each row of the averaging kernel matrix can be regarded as smoothing weights. Because the pixels have a 2-D arrangement, we show the 2-D display of the row of the 106th pixel (row and column indices are 4 and 16) in the averaging kernel matrix for the LTD and MC algorithms in Fig. 4c and d, as an example. The dependence on the beam geometry can be seen in both pictures. Because the beam configuration is fixed, the difference between the fitness values is mainly caused by the use of different regularization approaches. The fitness difference between the LTD and MC algorithms is very small, which may indicate that both algorithms have similar smoothness effects. This result coincides with the results from other measures discussed above. The 2-D display of the diagonal elements of the averaging kernel matrix are shown in Fig. 4e and f, which are not of much use in this case.

Contour plot of the averaging kernel matrix for

The derivatives are approximated by the finite differences during the
discretization process. The finite grid length causes discretization error and affects the reconstruction results. We studied the influences of different grid divisions by investigating the changes in the nearness, peak location error, exposure error, and computation time with respect to the pixel number. In total, the following Five different grid divisions were used:

The change in

The nearness, peak location error, and exposure error generally illustrate
decreasing trends with increasing pixel number. The MC algorithm shows a
slightly better performance than the LTD algorithm with increasing pixel
number. The performance improvement becomes slow for both algorithms when the
division is finer than

To understand the characteristics of the smoothness constraints and to seek more flexible smooth reconstruction, we first identified the LTD algorithm as being a special case of Tikhonov regularization. Then, more flexible smoothness constraints were found through the smoothness seminorms, according to variational interpolation theory. The smoothness seminorms were successfully adopted in ORS-CT inverse problems. On the basis of the variational approach, we proposed a new MC algorithm by using a seminorm approximating the sum of the squares of the curvature. The new algorithm improves computational efficiency through reducing the number of linear equations to half that of the LTD algorithm. It is simpler to perform than the GT-MG algorithm by directly using high-resolution grids during the reconstruction.

The MC, LTD, and NNLS algorithms were compared by using multiple test maps.
The new MC algorithm shows a similar performance to the LTD algorithm but only requires approximately 65

These comparisons demonstrate the feasibility of introducing the theory of variational interpolation. On the basis of the seminorms, it is easier to understand the advantages and the drawbacks of different algorithms. Common problems, such as the over-smoothing issue, may be improved by formulating more algorithms suitable for ORS-CT applications. Note that, although the smoothness is very good a priori information for the reconstruction problem, beam configuration and the underlying concentration distribution are also important factors affecting the reconstruction quality. To further improve the reconstruction quality, extra a priori information according to specific application may be added to the inverse problem. For example, statistic information of the underlying distribution or information resulting from the fluid mechanics.

Data and code are available on request by contacting the authors.

KD was responsible for acquiring funding for this research. SL designed the algorithm and conducted the tests. SL and KD were both involved in data analysis. Both authors contributed to writing and editing the paper.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Canada Foundation for Innovation (grant no. 35468) and the Natural Sciences and Engineering Research Council of Canada (grant no. RGPIN-2020-05223).

This paper was edited by Thomas von Clarmann and reviewed by Joern Ungermann and three anonymous referees.