Shadowgraph Tomography of a High Pressure GDI Spray

An isooctane spray from a high-pressure multi-hole GDI injector (Bosch HDEV6) was characterized by means of optical extinction tomography, relying on parallel illumination by a focused-shadowgraph setup. The tests were carried out in air at ambient conditions at an injection pressure of 300 bar. Extinction images of the spray were acquired over a 180-degree angular range in 1-degree increments. The critical issues of optical extinction tomography of sprays, related to the strong light extinction by the dense liquid core of fuel jets, were addressed. To mitigate artifacts arising from the reconstruction process, the extinction data were subjected to spatially-variant filtering steps of both the raw and post-log data, before being analytically inverted through the inverse Radon transform. This made it possible to process extinction data for very large optical depths. A nearly complete three-dimensional reconstruction of the spray was obtained, providing significant details of the spray morphology and the internal structure of the jets throughout the spray development. The different phases of the atomization process from the near-field to far-field regions of the spray were observed.


Introduction
Recent decades have seen the development and growing diffusion of gasoline direct injection (GDI) engines, by reason of improved fuel efficiency, power output, and reduced emissions (Zhao 2010;Van Basshuysen and Spicher 2009;Kalwar and Agarwal 2020;Duronio et al. 2020).Their development and advancement, as with diesel, have relied significantly on the optimization of the fuel injection process (Hoffmann et al. 2014;Pauer et al. 2017;Hassdenteufel et al. 2022;Li et al. 2022;Lee et al. 2020), driven by a strong synergy between experimental and numerical investigations (Drake and Haworth 2007;Shost et al. 2014;Postrioti et al. 2018;Shahangian et al. 2020;Arienti et al. 2021;Lien et al 2024), in which optical diagnostics played a fundamental role and still does (Drake and Haworth 2007;Hargrave et al. 2000;Zhao 2012;Fansler and Parrish 2014;Linne 2013;Berrocal et al. 2022).In this context, the usual high-speed imaging techniques for the characterization of the macroscopic parameters of sprays, based on line-of-sight optical extinction techniques, are increasingly applied in computed tomography investigations (Parrish et al. 2010; Kristensson et al. 2012;Weiss et al. 2020;Hwang et al. 2020;Lehnert et al.2022;Hwang et al. 2024;Oh et al. 2022;Karathanassis et al. 2021).In this field, optical tomography can be considered a viable complement to the use of x-rays, thanks to much more simple and accessible experimental setups.In fact, x-ray diagnostics mainly relies on large synchrotron facilities (Duke et al. 2016;Duke et al. 2017;Matusik et al. 2018;Sforzo et al. 2022), effectively limiting widespread accessibility.To overcome these limitations, more or less elaborated benchtop sources have been also developed that allow x-ray spray investigations to be performed in more accessible lab-scale facilities (Marchitto et al. 2015;Guénot et al. 2022), although they show some limitations in terms of x-ray spectrum and flux (Kastengren and Powell 2014).In any case, the high penetrating power of x-rays allows otherwise inaccessible areas of the spray to be quantitatively characterized, such as in the near-field region immediately at the injector exit and even inside the nozzle.This proves invaluable, for example, when studying the primary atomization phase of sprays (Tekawade et al. 2020;Arienti et al. 2021;Mamaikin et al. 2022).However, to the detriment of its diagnostic potential, objective experimental constraints restrict the range of operating conditions that can be explored through x-ray absorption methods.Furthermore, the signal-to-noise ratio in dilute regions decreases to unacceptable levels, effectively preventing any investigation of large areas of the spray (Linne 2013;Heindel 2018;Kastengren et al. 2017).Therefore, optical extinction tomography would supplement x-ray diagnostics, thus allowing sprays to be characterized throughout their development.On the basis of the reference literature on the subject (Natterer 2001;Kak and Slaney 2001), optical extinction tomography of sprays, in the specific case, means the cross-sectional reconstruction of the extinction coefficient of the liquid phase, starting from a set of line-of-sight extinction data (projections) measured from multiple views in a certain angular range (Kristensson et al. 2012;Weiss et al. 2020;Hwang et al. 2024).The light extinction through the spray is evaluated according the Lambert-Beer law (Bohren and Huffman 2008;Zhao and Ladommatos 1998): where I0 and I are respectively the incident and transmitted light intensity, µe(x) is the extinction coefficient of the droplet cloud at the position x and τ is the optical depth, or thickness, measured along the lineof-sight L. The extinction coefficient is a function of the size and number concentration of the spray droplets and their optical properties, and can be related to the liquid volume fraction of the spray if droplet sizes and optical properties were known (Bohren and Huffman 2008;Pickett et al. 2011, Weiss et al. 2020;Hwang et al. 2020).It is worth emphasizing that, unlike x-rays, the extinction of light through fuel sprays is mainly due to scattering, absorption being negligible.However, we did not take into account the dependence of the scattering correction on large collection angles (Wind and Szymanki 2002;Deepak andBox 1978a, 1978b), nor considered multiple scattering effects whereby the Lambert-Beer law could deviate from linearity at large optical depths (Berrocal 2007;Linne 2013;Lehnert et al. 2022), and in the following we will always refer to equation (1).
The extinction images of the spray were acquired using a focused-shadowgraph optical arrangement, taking advantage of an effective image processing method that allows distinguishing between the liquid and vapor phases of a spray in shadowgraph image (Lazzaro 2020).
The critical issues of optical extinction tomography are addressed in the paper, mainly related to the strong light extinction by the dense liquid core of fuel jets and which could give rise to serious artifacts in the reconstructed images (Fessler 1993;Schulze et al. 2011;Mori et al. 2013).Optical depths are then analytically inverted through the inverse Radon transform, using the filtered back projection (FBP) algorithm (Kak andSlaney 2001, Natterer 2001), thus obtaining an almost complete 3-D reconstruction of the spray throughout its development.However, it is worth highlighting how the stochastic and transient nature of the injection process requires averaging the optical depths over multiple injections to improve the measurement statistics.Consequently, only statistical steady characteristics of the spray should be detected.
We would like to conclude this section with some observations on the shadowgraph setup used in the present work.In our opinion, collimated lighting should be the most obvious choice in optical tomography of sprays, allowing extinction coefficients to be reconstructed using the appropriate inverse Radon algorithm for parallel illumination.Previously cited research in this field, however, is largely based on diffuse back illumination (DBI) configurations, without giving due consideration to the lighting geometry.In fact, apart from the intrinsic inaccuracy of the method (Lazzaro 2021), the DBI can be assimilated to a cone beam configuration (just swap sensor and source) and, in this case, the approximate FDK algorithm should be applied (Feldkamp et al. 1984;Hsieh et al. 2013) to analytically backproject the extinction data, acquired in the appropriate angular interval.However, the very large size of the "source", far from being point-like, would heavily influence the results and inevitably very blurred and unfaithful images of the spray would be obtained.

Experimental
An isooctane spray from a high-pressure multi-hole GDI injector was characterized by means of light extinction tomography, using a focused-shadowgraph optical setup.The experimental layout is schematically shown in Figure 1.An in-line arrangement was made by using two 500 mm focal length and 100 mm diameter planoconvex lenses spaced 2 m apart.The light source is a low-cost high-power blue cw LED (Osram LZ1-10DB00); the output wavelength is centred at 460nm and the maximum radiant flux is 800mW.The images were acquired by a high-speed camera (Photron FASTCAM SA-X2), equipped with a Nikkor 70mmf/4.5 objective.A plano-convex lens, 250 mm focal length, in front of the objective helps to adjust focus.The images, 512×440 in size, were recorded at 40000 fps with an exposure time of 0.37 µs and resolution of 0.186 mm/pixel.Optical setup parameters are summarized in Table 1.
The injector rotates integral with a small tank, of approximately 200 cm 3 in volume and designed to operate with fuel pressures up to 500 bar, whose rotation is driven by a stepper motor via a gear reducer, allowing an angular resolution of 0.1 degrees.The fuel was pressurized via a gas-driven liquid pump, which can operate up to 880 bar.The investigated injector is the six-hole counter-bore GDI injector Bosch HDEV6L used for passenger vehicles (Bosch 0 261 500 485).A picture of the injector is shown in Figure 2 on the left.This injector family has been designed for nominal injection pressures up to 350 bar.The upgrade from previous Bosch GDI injectors resulted in higher fuel injection dosing accuracy, improved fuel atomization and reduced jet penetration, reduction in nozzle wetting due to improved bore geometry.This resulted into a substantial decrease in exhaust emissions, especially in the particle number (Pauer et al. 2017;Hartmann et al. 2018;Hassdenteufel et al. 2022;Mamaikin et al. 2022).An enlarged view of the injector tip is shown in Fig. 2 on the right, where the holes have been numbered such that even and odd jets are on opposite sides of the symmetry plane of the spray plume.According to the injector specifications, approximately 73% of the total fuel flow rate is equally delivered by jets 1 through 4, while the rest is equally distributed between jets 5 and 6.Fig. 3 finally shows a schematic picture of the side and top view of the spray for the tomography projection angle of 0°.The tests were carried out in air at ambient conditions.The fuel injection pressure was kept at 300 bar and the injector energizing time was 800 µs.Looking from above, the injector was rotated clockwise and spray images were acquired over the degree range 0:179 with a 1-degree step.Images from 40 consecutive injections were acquired at each projection angle.The experimental conditions are summarized in table 2.

Macroscopic spray characteristics
The macroscopic parameters of the spray were characterized first, i.e. penetration length, projected area, cone angle and direction.Given the symmetry properties of the HDEV6 injector and the loose structure of its spray plume, the macroscopic parameters of the individual jets were characterized at the projection angle =0°, where corresponding symmetrical jets overlap.

Image processing
The spray images were segmented via an effective processing method, which allows the liquid and vapor phases to be simultaneously distinguished in a shadowgraph image.The method has been extensively described elsewhere (Lazzaro and Ianniello 2018; Lazzaro 2019; Lazzaro 2020), therefore only a brief description is provided here, providing further details later in the section dedicated to the tomographic reconstruction of the spray.
Whether delineating the boundaries of the liquid or of the vapor phase, the common steps of the processing method are an optimal filtering of the spray images, through their regularization by variational methods, and an effective thresholding procedure based on the iterative application of the Otsu's method.Furthermore, when segmenting the vapor phase, the intensity texture of the shadowgraph image is first intensified by evaluating the principal curvatures of the spray image surface.Thus, the intensity oscillations due to density gradients are greatly amplified, allowing the boundaries of the vapor phase to be easily outlined with the utmost accuracy throughout the spray development.
The raw images are regularized using the Curvature Filters (CF) (Gong and Sbalzarini 2017), which are fast discrete filters for regularizers based on "gaussian curvature", "mean curvature", and "total variation" priors.In addition to low computation times and reduced memory requirements, one of their noteworthy features is effective denoising while preserving image edges, which proves invaluable when segmenting the liquid phase (Lazzaro 2020).In fact, due to the different spatial frequencies, curvature filters effectively smooth also the light intensity pattern due to vapor density gradients, meanwhile fitting well with the intensity profile attributable to the liquid phase.It is precisely the denoised regularized images that are then processed to obtain the tomographic reconstruction of the liquid phase of the spray.

Macroscopic spray parameters
The macroscopic parameters of individual jets are defined as in Figure 4, which shows a raw image of the spray acquired at 600µs after start of injection (ASOI) and =0°.The black dash-dotted line indicates the injector rotation axis, coinciding with the injector axis.For comparison, Figure 5 shows a spray image recorded at =90°, where the individual jets can be distinguished.
The liquid and vapour phase boundaries are indicated by blue and red lines, respectively.The cone angle of the jet was measured as the angle between the linear fits of the jet outer edges between 20% and 50% of jet penetration, where the origin of the linear fits roughly coincides with the intersection of the injector and jet axes.The jet cone bisector then defines the angle of the jet axis with respect to the injector axis.It can also be seen that the spray as a whole develops approximately within a circular cone with a vertex angle of 90°.t ASOI [µs]   jets 1-2 jets 3-4 jets 5-6 t ASOI [µs] jets 1-2 jets 3-4 jets 5-6 Fig. 10.Axis angles of the liquid jets.

Computed tomography
The 3-D images of the spray extinction coefficients were analytically reconstructed through the inverse Radon transform using the Filtered Back Projection (FBP) algorithm (Kak andSlaney 2001, Natterer 2001).
The critical issues of optical extinction tomography of sprays are addressed below, where the sequential steps of the proposed solution approach are detailed.
Step 1. Image restoration One of the main drawbacks of optical tomography of sprays, in addition to their intrinsically transient nature, is the severe "photon starvation" due to the strong attenuation of light by the dense liquid core of the jets.
In fact, regardless of any considerations on noise, signal digitalisation nevertheless returns zero values when the incident light intensity is below the camera's detection limit.According the Lambert-Beer law, the optical depth,  = −(  0 ⁄ ), would not be defined in this case and extinction values could not be calculated at all.This would result in severe "streak artifacts" in the reconstructed image (Fessler 1993;Schulze et al. 2011;Prell et al. 2009;Mori et al. 2013;Hayes et al. 2018;Schofield et al. 2020), whatever method is adopted to restore the missing values.Here, however, log conversion drawbacks are partially overcome when shadowgraph images are regularized and "cleaned" of fuel vapours, thus allowing the liquid phase to be defined.This is illustrated by Figure 11 with reference to the spray images acquired at 600 µs ASOI at =0° (upper row) and =90° (bottom row).In this case, the relative extent of the photon starvation in the inner core of the jets can be appreciated from Figures 11(a  Fig. 12. Linear-scale (left) and log-scale (right) intensity profiles of the raw, regularised and "restored" images along the axial cross section 30 mm from the injector tip (dash-dotted red lines in Figure 11) for the pray images acquired at 600 µs ASOI at =0° (upper row) and =90° (bottom row).
Step 2. Optical depth averaging Following the above, the optical depth of the spray liquid phase  is calculated according the Lambert-Beer law as:  However, even though the averaging process improves the measurement statistics, rather noisy data are still returned which significantly affects the image reconstruction process.In fact, when the FBP method is applied, the tomographic image reconstruction passes through the well-known ramp or Ram-Lak filter (Kak and Slaney 1987; Ramachandran and Lakshminarayanan 1971; Zeng 2014), whose adverse effect is to amplify high-frequency measurement noise.Normally, high frequencies are attenuated by apodizing the ramp filter through a window function, whose effect is equivalent to smooth the projection measurements with a spatially invariant low-pass filter (Fessler 1993;Fessler et al. 2000;La Rivière and Billmire 2005).Additionally, when processing very noisy data, filter cutoff can be further reduced by compressing, or scaling, the frequency range of the window function.However, due to the space-invariant nature of filtering, the larger the frequency scaling, the more the oversmoothing of most of the data.The goal is therefore to optimize the resolution-noise tradeoff.Several approaches have been proposed in the literature for this purpose, which essentially consist in a spatially-variant smoothing of raw or post-log data or in the image domain too (Mori et al. 2013;Hayes et al. 2018;O'Sullivan et al. 1993;Hsieh 1998;Fu et al. 2016;Chang et al. 2016;Zeng 2022).
Step 3. Sinogram smoothing As described above, the individual raw images have already undergone a pre-log filtering phase through their regularization, before the optical depths are calculated and averaged.However, the extent of photon starvation and also the transient and stochastic nature of the injection process dictate additional measures to mitigate streak artifacts in the reconstructed image.We have followed to some extent the spline-based projection smoothing approach developed by Fessler (1993) and La Rivière and Billmire (2005), which, in our opinion, proved to be suitable for processing optical extinction data of sprays.In their method, the authors performed nonstationary smoothing of each projection data before ramp-filtering and backprojection, based on an information-weighted smoothing spline, where the weights depend on the measurements themselves.Following this approach, averaged extinction data at each projection angle were pre-filtered by weighted cubic spline interpolation, using the Matlab function csaps.The function is an implementation of the Fortran routine SMOOTH (De Boor and De Boor 1978).The smoothing spline τs,j minimizes the function: where the first term is the error measure and the second term is the roughness measure, with n being the number of data points j.The default value for the error measure weights wj is 1.The smoothing parameter p controls the tradeoff between the smoothness of the spline and its weighted agreement with the measurements and is analogous to the cutoff frequency in FBP.Interestingly, if the weights wj are identical, the spline smoothing would correspond to a low-pass Butterworth filter of order 2 times the order of the derivative in the second term and half-power frequency that is a function of p (Fessler 1993, Wahba 1990).Based on the above, the weights were varied inversely proportional to the optical depth values as follows: The smoothing parameter p was instead selected by relying on visual inspection of the reconstructed images, taken as benchmark the images reconstructed from the unsmoothed data.In this regard, with reference to the  Step 4. Filtered Back-Projection The FBP tomographic reconstruction of the spray extinction coefficients µe was then carried out by means of the Matlab function iradon, which computes the inverse Radon transform of the sinogram data.The Hann window (Chesler and Riederer 1975;Jain 1989;Fessler 2000) with a scaling factor (sf) of 0.5 was used to apodize the ramp filter, while practically the same results were obtained by selecting different apodizing windows in iradon.Figure 20 summarizes the above, by comparing the radial profiles of the extinction coefficient for jets 1 through 3 for the set of selected parameters.Although qualitatively, it can be stated that the proposed approach provides effective mitigation of streak artifacts while also preserving spray edges.Interestingly, the image reconstruction from splinesmoothed data provides satisfactory results even using the unscaled ramp filter.

3D tomographic reconstruction
The spline-smoothed sinograms of each axial plane are back-projected to reconstruct the spray slice by slice.While the main characteristics of the spray structure and individual jets can certainly be deduced from the previous figures, a more engaging and disclosing view of the internal structure of the jets is provided by Figure 24, where isosurfaces of the extinction coefficient between 1 and 5 mm -1 have been plotted with step 0.5.
The figure clearly highlights the potential of the method, detecting significant details of the spray morphology and internal structure of the jets throughout the spray development.This is supported by Figure 25, where the extinction coefficients along the jet axes were plotted as a function of the distance "s" from the jet origin.By comparing the general trend of the curves, differences are observed between the extinction values of jet1 and its symmetric jet2, probably due to small manufacturing discrepancies.Both jets 3-4 and jets 5-6, instead, develop quite similarly to each other, according to the injector specifications previously described.The extinction coefficient profile of jet1 was analysed in detail, the distinctive features of which appear more marked, as for jet2, given that they develop roughly orthogonally to the axial planes, thus fully exploiting the spatial resolution.As a support to the analysis, Figure 26 shows the front view of jet1, on whose axis some particular points have been identified and marked with letters "a" to "h" as in Figure 25.linearly with time undergoing strong atomization, thus defining the atomization or "breakup region", where "primary" and "secondary" breakup regions can also be distinguished.Afterwards the droplets formed are slowed down due to the aerodynamic interactions with the surrounding air and the subsequent jet penetration follows a fairly parabolic trend, thus defining the developed "spray region".The transition point between the two regimes defines the breakup point of the jet.As expected, the spray from the HDEV6 injector does not appear to deviate from this behaviour, as also highlighted by the penetration curves in Figure 7.However, it is worth pointing out that the penetration curves refer to the average development of the jets over time, while Figures 25 and 26 refer to a snapshot of the average jet at a certain injection time.In this case, the breakup point for jet1 at 600 µs ASOI can be reasonably located at point "f", after which the droplet crowding following their slowdown is highlighted by the increase of the extinction coefficient at point "g".The jet then evolves in the developed spray region showing the characteristic morphology.The breakup region would obviously extend from inside the injector hole up to point "f".However, optical data from the near-field region of the spray should be interpreted with caution because of time and spatial resolution limitations, narrow width of the jets, and compactness of the spray.As known, primary breakup refers to the abrupt disintegration of the liquid jet into ligaments and droplets in the near-field region at the injector exit (Beale and Reitz 1999;Bravo and Kweon 2014;Grosshans et al. 2015;Zhang et al. 2020;Li et al. 2023).It proceeds through competing fluid dynamics mechanisms whose relative weight depends on the injector design and experimental conditions, involving inertial, surface tension, viscous and drag forces (Shinjo and Umemura 2010;Brulatout et al. 2020;Yu et al. 2016;Li et al. 2023;Zeng et al. 2012;Yue et al. 2020).Regardless of the mechanisms involved, however, the net result would be the sudden increase in the number concentration of droplets.This could explain the sharp increase in the extinction coefficient when moving downstream towards point "b".In fact, since light absorption can be neglected, the extinction coefficient varies inversely to the droplets size for the same fuel concentration (Bohren and Huffman 2008).The primary droplets so formed can undergo further atomization in the secondary breakup region, as well as collision/coalescence phenomena.(Jenny et al. 2012;Banerjee and Rutland 2015;Linne 2013;Bravo and Kweon 2014;Berni et al. 2022).This region is characterized by strong droplet-gas fluid dynamic interactions, which promote air/fuel mixing and the onset and growth of turbulent structures in the gas phase (Mitroglou et al. 2007;Ghasemi et al. 2014;Lee and Park 2014;Li et al. 2022).This would translate into the observed trend of the extinction coefficient.First of all, the overall decrease of µe moving downstream towards the point "f" should reflect the jet spreading and dilution due to air entrainment, meanwhile the establishment of large-scale turbulent structures can be grasped in its oscillating trend.It should be noted how such distinctive details of the extinction coefficient profile do not appear to be blurred by the averaging process, thus suggesting a high repeatability of the spray evolution.The above analysis was then extended to the entire duration of injection, providing further significant details on the atomization characteristics of the HDEV6 injector.In this regard, Figures 27 and 28 respectively show time sequences of 3D views of the spray before and after the spray breakup at approximately 300 µs.In Figure 27 snapshots of the evolving spray are shown at full-time resolution every 25 µs, while in Figure 28, after the jet breakup, images are shown every 50 µs up to 500 µs and then every 100 µs.As previously, the development of the jet1 was examined in detail, supported in this by Figures 29 and 30, which show the temporal evolution of the extinction coefficient along the axis of the jet, respectively up to the breakup and for the entire duration of the injection.Based on the previous discussion and relying on both spray images and extinction coefficient curves, the onset and development of statistically steady large-scale flow structures in the secondary breakup region can be inferred.At the same time, it is remarkable how the transition point to the developed spray zone can be uniquely identified, thus allowing the jet breakup length to be accurately determined, as shown by the black dashed line in Figures 29 and 30.In this regard we would like to focus on the striking image of the spray at 300 µs, where the "birth" of the developed spray region for jet1 and jet2 can be observed.Furthermore, observing the time evolution of the extinction coefficient after the jet breakup, the jet dynamics seems to approach a steady flow condition right towards the end of the injection duration, which in this case does not allow definitive conclusions to be drawn.Longer injection durations should be investigated.Interestingly, despite temporal and spatial resolution limitations, details on the spray evolution during the injector opening transient are also highlighted.In this regard it can be assumed that the abrupt increase in the extinction coefficient observed at 150 µs results from the steep increase of fuel injection rate (Mohan et al. 2018;Payri et al. 2016), whereby later and faster injected fuel should build up along the jet.Furthermore, the extent of this increase should also depend on the injection rate overshoot peculiar of this type of injectors (Duke et al. 2017;Shahangian et al. 2020, Baldwin et al. 2016;Payri et al. 2022Mamaikin 2022).At the same time, the onset of turbulent flow structures, which should define the transition between the primary and secondary breakup zones, can be inferred in Figure 28 by observing the swelling and twisting of the jet heads in the images between 125 µs and 200 µs.

Summary and Concluding Remarks
This article shows the results of optical extinction tomography investigation of a high-pressure multi-hole GDI spray.The critical issue of "photon starvation" by the dense liquid core of the fuel jets have been addressed, proposing a simple pre-processing method of extinction data before their analytical inversion through the inverse Radon transform.The method, developed in a Matlab environment, is essentially based on spatially-variant and edge preserving filtering of raw and post-log extinction data.This made it possible to recover and analyse very large optical depths, allowing an almost complete threedimensional reconstruction of the spray and providing significant details on its morphology and the internal structure of the jets.The different phases of the atomization process were observed from the near-field to far-field regions of the spray.The jets breakup length was easily and uniquely detected, as well as the inception and growth of the developed spray region.The onset and development of statistical-steady large-scale flow structures in the region of secondary atomization were also inferred.
To the best of our knowledge, this is the first time such in-depth characterization of very dense sprays has been carried out through optical extinction tomography.In our opinion, the method could represent a significant step forward in this field, although it would need to be tested in more severe environments and with different spray patterns to ensure its broad applicability.

Declarations Ethical Approval
"not applicable"

Funding
"not applicable"

Fig. 3 .
Fig. 3. Schematic pictures of the side (left) and top (right) view of the spray pattern at the projection angle =0°
) and 11(e), showing the raw images I of the spray before regularization, where zero values are highlighted in yellow.The images are then regularized by setting the mean curvature prior in the CF code, which assumes that the regularized image would define a minimal surface (Gong and Sbalzarini 2017).To some extent, image regularization restores or, more properly, replaces missing values of the transmitted light intensity, while still returning high-variability data with a very low signal-to-noise ratio.The regularized images Icf are shown in Figs.11(b) and 11(f), where residual nonpositive values still persist, which are then inpainted by using the interpolating function inpaint_nans developed in Matlab by D'Errico (2024), setting the interpolation method 2. The resulting images Icf,inp are finally shown in Figs.11(c) and 11(g).The result of the regularization procedure can be further assessed in Figures 11(d) and 11(h) showing the images I-Icf,inp.The smoothing and restoring features of the regularization process can be evaluated in Figure 12, showing the intensity profiles along the axial cross section 30 mm from the injector tip, as indicated by the dash-dotted red lines in Figure 11.The linear-scale intensity profiles in Figures 12(a) and 12(c) highlight the smoothing characteristics of curvature filters, while the lower cut-off of the raw signal (red lines) due to the camera bit depth and the inpainting of the missing values can be evaluated in the log-scale plots of Figures 12(b) and 12(d).Furthermore, it can be seen how the high variability of the restored values is further amplified because of the nonlinear nature of the log conversion.
Fig.13.Optical depth profiles for the 40 injection repetitions (thin lines) and their average (thick green lines), measured at 600 µs ASOI along the axial cross section 30 mm from the injector tip at the projection angle  = 0°.

Fig. 14 .
Fig. 14.Optical depth profiles for the 40 injection repetitions (thin lines) and their average (thick green lines), measured at 600 µs ASOI along the axial cross section 30 mm from the injector tip at the projection angle  = 90°, Fig. 16.Average optical depth τ at  = 0° and spline smoothed curves τs at different smoothing parameters.
Figure 19 provides an explanatory snapshot of the proposed approach, comparing the results of unapodized and apodized FBP reconstruction of the extinction coefficient from unsmoothed and spline-smoothed sinogram data.The reconstructed images of the extinction coefficients are shown in the first column.Streak artifacts can be observed along the directions of maximum light extinction, where the jets overlap, which progressively blur as the degree of filtering increases.The second column shows instead the radial profiles of the extinction coefficients along the dashed red lines crossing the jet footprints in the selected axial plane.Finally, to substantiate the above, the last column shows the 2D fast Fourier transform (Matlab function fft2) of the reconstructed µe images, or more precisely the absolute value of fft2(µe) where the zero-frequency component has been shifted to the centre of the output (Matlab function fftshift).

Fig. 19 .
Fig. 19.Unapodized and apodized FBP reconstruction of the extinction coefficient from unsmoothed and spline-smoothed sinogram data.Left: reconstructed images of the extinction coefficients.Middle: radial profiles of the extinction coefficients along the dashed red lines.Right: 2D FFT of the reconstructed µe images.

Figure 21
Figure21shows the images of the reconstructed extinction coefficient at different axial planes (middle column) and the related sinograms (left) and radial profiles (right).It can fairly be stated that in the present case the proposed method largely mitigates the photon starvation issues by the jet core, allowing the to be nearly entirely reconstructed.A snapshot of the spray at 600 µs ASOI is then provided in Figure22, showing a 3D view of the extinction coefficients in each axial plane.For clarity a minimum threshold of 0.6 has been applied in plotting µe data.The magenta lines indicate the axes of the jets, i.e. the lines passing through the centres of the jet footprints, defined followingWeiss et  al. (2020)  as the peak value positions of the extinction coefficient.It can be observed how the spray is characterized by fairly compact and straight jets at this injection time, as can be seen in Figure23, which shows different views of the jet axes only.Figure 23 (b) refers to the top view of the spray for the projection angle of 0°, while the side views of the jet axes at  = 0° and  = 90° are shown in Figures 23 (c) and 23 (d), respectively.

τ
Fig. 21.Reconstructed images of the extinction coefficient at different axial planes (middle column) and related sinograms (left) and radial profiles along the dashed red lines (right) Fig. 22. 3D view of the extinction coefficient of the spray at 600 µs ASOI.

Fig. 30 .
Fig. 30.Time evolution of the extinction coefficient along the jet1 axis in the time range 25-800 µs.