#
Quantification of Sub-Pixel Dynamics in High-Speed Neutron Imaging^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{1}H in hydrocarbon fuels, and high-speed detectors that offer a field of view of several centimeters with a spatial resolution in the order of 50–100 μm and temporal resolution in the order of 1 ns to 1 µs [17,18,19]. This spatial resolution can capture the geometric detail of all but the smallest features of an injector (nozzle holes and needle seat region), making neutron and X-ray imaging highly complementary tools for obtaining geometric and compositional information via tomography. However, mechanical dynamics, such as needle lift and wobble, have been observed with X-ray imaging to occur below 5 and 50 μm, respectively [8], meaning that such dynamics are below the pixel size of existing high-speed neutron imaging detectors.

## 2. Materials and Methods

#### 2.1. Neutron Imaging Configurations

_{2}O

_{3}diffuser just after the aperture was used to spatially homogenize the beam. The beam profile was further controlled by a He-filled flight tube between the aperture and the detector that was equipped with silicon windows and motorized boron–nitride exit slits that defined the final beam size [17]. Typical open-beam neutron flux at the detector was ~10

^{7}n/cm

^{2}/s at maximum aperture.

^{10}B-doped microchannel plate (MCP) was used to convert neutrons to an electron cascade, which was further amplified by a standard glass MCP. The resultant electron pulse was detected by a 2 × 2 Timepix readout positioned behind the MCP stack. This configuration is referred to here as the “MCP detector,” and has 512 × 512 pixels with 2.8 × 2.8 cm field of view, a physical pixel size of 55 µm, and 1 µs timing capability [17,18]. The fuel injector was mounted in an Al spray chamber at the sample position as shown in Figure 1C and was fired synchronously with the detector. The chamber was continuously purged with gaseous Ar at controlled temperature and pressure to provide the ambient condition for the injected spray and to evacuate the sprayed fuel from the chamber.

^{6}LiF/ZnS scintillator that converts the incoming neutrons into visible light, along with a camera and optics in a light-tight box. The CCD detector had a ~7 × 7 cm field of view, a pixel size of 37 µm, 80–100 µm spatial resolution, and ~1 s timing resolution [17,22]. The injector was mounted in a custom Al holder on a rotation stage at the sample position. Further details of the neutron CT configuration and comparison to X-ray CT are described by Duke et al. [16].

#### 2.2. Injector and Operating Conditions

^{6}injection events were recorded with a time bin length of 5.12 µs.

#### 2.3. Neutron Attenuation Model

_{0}(λ), transmitted intensity I(λ), attenuation coefficient μ(λ), path length d, and neutron wavelength λ. In general, the transmission will be wavelength-dependent; however, for the present experiment, the full polychromatic beam available at HFIR CG-1D was used with no wavelength selection. For a dynamic, nested multiphase system as depicted in Figure 3, the time-dependent transmission through the entire system as measured at a given detector pixel is a function of the path lengths and macroscopic attenuation coefficients Σ for each phase (A, B, C, …):

_{ref}, the transmission through the non-moving phases (e.g., C and external) will be the same in either condition, and the expression therefore reduces to one dependent only on the attenuation coefficients and dynamic path length differences through the phases A and B, which share a moving interface:

_{OB}, as well as the intensity of the “dark frame” I

_{DF}, with the neutron shutter closed. The measured intensity I

_{meas}can then be normalized to transmission by

_{DF}is generally non-zero and “structured” as a characteristic of a CCD or complementary metal–oxide–semiconductor (CMOS) sensor, whereas the dark frame of the Timepix-based MCP detector used here can be considered zero for all practical purposes [18], with any counts being caused by random gamma or cosmic rays. The open-beam image I

_{OB}is also structured because of imperfections in the detector and spatial inhomogeneity of the incident neutron beam caused by the guides. The total intensity may also evolve over time due to variation in reactor output. However, despite the high-speed imaging experiments described here being performed continuously over a period exceeding 24 h, intermittent open-beam measurements were not necessary because of the dynamic normalization approach. As described previously and illustrated in Figure A1, the MCP detector output for a single pixel at time bin t

_{i}is the sum of all counts that were recorded as occurring within that pixel and that time bin over all cycles c

_{k}, or an ensemble time bin:

_{i}of cycle c

_{k}can be written as

_{a}to t

_{b}directly preceding the injection event, during which the injector is in a static condition, one obtains a reference transmission T

_{ref}:

_{i}) to T

_{ref}, which can be reduced to an expression dependent only on I

_{meas}(t

_{i}) if it is assumed that I

_{OB}does not change significantly over the course of a single cycle. For the present experiment, which was conducted at 25 Hz, this assumption is quite reasonable. The dynamic normalization is expressed as

#### 2.4. Path Length Model

_{dyn},b

_{dyn}) and the static, or reference, circle at (a

_{ref},b

_{ref}) is then given by

_{X}(x), the expected value E of a function g(X), which is dependent on X, is given by

_{a}(x) will be given by

_{a}(x) are replaced by the dummy variable α within the integral. An example calculation for a single value of x in a unitless system is shown in Figure 5A, where f

_{a}(α) follows a normal distribution with mean µ

_{a}and standard deviation σ

_{a}. The interval was discretized and E[d(x,a)] was calculated by performing numerical integration of the shaded region.

_{a}(x) and d(x) were centered at zero and the result of the convolution was translated to µ

_{a}. To guarantee that the tails of the distribution were adequately captured, the domain for convolution when applied to the high-speed images was defined as

_{a}in a unitless system is shown in Figure 5B, which illustrates that treating the value of a probabilistically blurs the path length profile, adding tails to the ends while decreasing the expected value in the center. Both effects become more pronounced as µ

_{a}approaches and exceeds r.

_{a}

_{,dyn}− µ

_{a}

_{,ref}is set as equal to 1% of the radius $r$, and the standard deviations σ

_{a}

_{,dyn}and σ

_{a}

_{,ref}are set as equal over a range of 0 to 50% of $r$. The effect of increasing standard deviation is to add tails to the expected value of path length difference and decrease the peak values.

_{ref},a

_{dyn})] approaches zero everywhere for large values of σ

_{a}

_{,dyn}and σ

_{a}

_{,ref}, which would also occur in the case of near-zero displacement when µ

_{a}

_{,dyn}≈ µ

_{a}

_{,ref}. The possibility of non-unique solutions requires that some informed constraints be placed on the fitting procedure to avoid non-physical results. One example is that the PDF describing the dynamic displacement will be bounded by the static container that surrounds the moving phase (in this case, the injector needle is contained within the body of the injector).

_{a}

_{,dyn}and σ

_{a}

_{,ref}, but, in principle, could be separated if the ESF were known. These parameters will be referred to as “total blur” to emphasize the fact that they are the combination of multiple effects.

#### 2.5. Image Processing

#### 2.6. Extraction of Sample Parameters from Neutron Radiographs and CT

_{body,phys}= 7.5 ± 0.05 mm and the pixel dimensions of the outer body radius r

_{body,px}= 97.85 ± 0.69 px and needle radius r

_{needle,px}= 21.28 ± 1.03 px, the physical dimension of the needle radius was calculated as r

_{needle,phys}= 815.4 ± 40.4 µm.

_{fuel}− Σ

_{steel}) that should be seen in the 2D high-speed images. Due to the small displacement of the needle relative to its size, a fixed value of ∆Σ = 2.82 cm

^{−1}was determined to be appropriate. A relative error (or uncertainty) of ±15% in the fitted displacement was attributed to the uncertainty in the attenuation coefficients.

#### 2.7. Model Fitting Procedure

_{i}= 1/σ

_{i}

^{2}such that data points with higher variance had a less significant effect on the fit.

## 3. Results

^{6}injections. If a high variability existed in the displacement direction and/or magnitude, the normalized images would become blurrier during deflection periods rather than displaying a consistent structure.

## 4. Discussion

## 5. Conclusions

^{6}injection events and employing an image normalization procedure to generate maps of dynamic neutron path length variation. The internal geometry of the injector was extracted from a neutron computed tomography reconstruction to develop a model of how the neutron path lengths through the different phases in the device would change with a given displacement of the injector needle. This model was then fit to each frame in the normalized high-speed image sequence to generate a time-resolved measurement of needle displacement, resulting in the measurement of motions on the scale of ~6% of the pixel size. This approach opens a path to the in situ, noninvasive measurement of cyclic dynamics in real devices and systems at temporospatial scales that were not previously achievable.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Experimental Conditions

Parameter | Value |
---|---|

Fuel | iso-octane (C_{8}H_{18}, 2,2,4-trimethylpentane) |

Fuel temperature (°C) | 90 |

Injector temperature (°C) | 90 |

Chamber temperature (°C) | 60 |

Fuel pressure (MPa) | 20 |

Chamber pressure (kPa) | 100 |

Argon flow rate (slpm) | 41 |

Injection rate (Hz) | 25 |

Injections per cycle | 1 |

Injection trigger delay (ms) | 1 |

Injection command duration (µs) | 680 |

Shutter | Dead Time (ms) | Start Time (ms) | End Time (ms) | Time Bins | Time Bin Length (μs) | Total Recorded |
---|---|---|---|---|---|---|

0 | 0.1 | 0.1 | 7.0 | 1348 | 5.12 | 1,339,654 |

1 | 0.4 | 7.4 | 9.0 | 78 | 20.48 | 1,291,400 |

2 | 0.4 | 9.4 | 15.0 | 273 | 20.48 | 1,292,873 |

3 | 0.4 | 15.4 | 25.0 | 449 | 20.48 | 1,309,046 |

4 | 0.4 | 25.0 | 28.0 | 146 | 20.48 | 1,288,513 |

5 | 0.4 | 28.4 | 35.0 | 322 | 20.48 | 1,307,914 |

## Appendix B. Attenuation Coefficients and Beam Hardening

^{−1}) is overlaid on Figure 7D and compares well with the range of values of the attenuation coefficient within the injector components obtained from the neutron CT.

_{8}H

_{18}), the density and atomic composition are known exactly. However, the cross section data for

^{1}H in the ENDF are for free atoms, whereas the binding of protons in molecules is known to increase their neutron cross section considerably. Therefore, we should expect that NEUIT will underpredict the macroscopic attenuation coefficient for C

_{8}H

_{18}. Molecule-specific bound cross section data for several hydrocarbons have been manually added to NEUIT based on empirical values from literature [37], but unfortunately those do not include C

_{8}H

_{18}. Two hydrocarbons with bound cross section entries in NEUIT were chosen to bracket the prediction for the fuel based upon their similar H/C ratios to C

_{8}H

_{18}(H/C = 2.25): C

_{16}H

_{34}(H/C = 2.125) and C

_{4}H

_{10}(H/C = 2.5). All three compounds were set to the density of C

_{8}H

_{18}(0.692 g/cm

^{3}). As expected, the measured value of Σ for the fuel falls between the NEUIT predictions for C

_{16}H

_{34}and C

_{4}H

_{10}at zero thickness, whereas the NEUIT value for C

_{8}H

_{18}, which uses the free cross section of

^{1}H, is considerably lower than measured. It is also observed that, as the thickness increases, the measured value of Σ for the fuel decreases more rapidly than predicted by NEUIT due to incoherent scattering from

^{1}H and the short sample-to-detector distance (~5 cm), which is consistent with the literature [24].

**Figure A2.**Measurement of macroscopic attenuation coefficients for fuel and injector. (

**A**) Radiograph of fuel-filled injector in Al sample holder. (

**B**) Radiograph of empty injector in Al sample holder. (

**C**) Normalization of same axisymmetric region in fuel-filled and empty injector radiographs results in transmission image of the fuel. (

**D**) Normalization of the empty injector by an empty region of the axisymmetric sample holder results in transmission image of the injector. (

**E**) Plot of −ln(T) vs. material thickness at each pixel in C and D shows a linear response for the steel injector and a non-linear response for the iso-octane fuel, indicating that beam hardening and scattering effects from the fuel are significant in this geometry. (

**F**) Same data from E plotted as macroscopic attenuation coefficient Σ are compared to predictions from NEUIT for three hydrocarbons and Fe.

_{fuel}− Σ

_{steel}) that should be seen in the 2D high-speed images. The average inner and outer diameters of the injector body were extracted from the CT over the region seen in the high-speed imaging in a similar manner as shown in Figure 7, and the expected value of path length through both the fuel and steel for a filled injector were calculated by Equations (12) and (16) assuming a Gaussian blur of 175 µm. The modeled system and resulting path length as a function of radial distance are shown in Figure A3A, where the peak in the expected value of path length through the fuel is 4.55 mm and through the steel is 3.6 mm. The position-dependent path lengths were used to estimate position-dependent Σ using the fitted data from Figure A2F, with results shown in Figure A3B. The attenuation coefficient difference ∆Σ was also calculated and was compared against a fixed value of ∆Σ = 2.82 cm

^{−1}, which is representative within the region where the needle motion is found. The shaded regions in Figure A3B correspond to the 95% confidence intervals for Σ obtained from the fits shown in Figure A2F.

**Figure A3.**Investigation of beam hardening effects. (

**A**) Cross section of fuel and injector system with position-dependent expected path length. (

**B**) Position-dependent expected value of attenuation coefficient in each component. Shading indicates 95% confidence interval from attenuation coefficient fits. (

**C**) Model of expected value in log-ratio normalized images due to displacement of injector needle show that position-dependent and constant ∆Σ are virtually indistinguishable. (

**D**) Error from use of constant ∆Σ in displacement fitting model is negligible over the range of displacements expected in this system. Confidence interval for relative error due to uncertainty in attenuation coefficient fits is roughly constant at ±15%.

## Appendix C. Parametric Investigation of Displacement Model

_{fuel}− Σ

_{steel}acts as a gain term in the attenuation model and has a significant effect on the predicted displacement as shown in Figure A4A. The R

^{2}fit statistic shows essentially no dependence on ∆Σ at any filtering level, and the value of ∆Σ = 2.82 cm

^{−1}chosen for the time-series fitting was based on beam hardening calculations shown in Figure A3, which is, in turn, based on the injector geometry extracted from the CT and the values of Σ extracted from images of the fuel-filled and empty injector.

^{2}, and the displacement prediction is relatively insensitive near this value. The needle radius of 815.4 µm extracted from the neutron CT agrees well with the peak in R

^{2}seen in Figure A4C, though R

^{2}is nearly flat for values from 810–840 µm.

^{2}and that the displacement prediction is relatively insensitive within ±1 px of this value.

^{2}optimum at the high deflection condition. Figure A4E shows displacement and R

^{2}results for a sweep of blur magnitude in which the reference and dynamic blur were set to be equal. The R

^{2}peak at ~3.1 px agrees with intuition, and displacement is also relatively insensitive at this value. Figure A4F shows a sweep of the ratio of dynamic to reference blur with an obvious peak near a value of unity. Lacking the detailed ESF required to separate the components contributing to blur, both the reference and dynamic blur parameters were set to a value of 3.1 px for the time-series fitting. Fits of the entire time series were performed with the dynamic total blur allowed to vary as an optimization parameter in the fitting process, and the results remained close to the same value of 3.1 pixels used to obtain the fits shown, with no apparent time-dependent structure that would indicate higher blur during injection.

^{2}is improved considerably by filtering, but there is a comparably minor effect on the predicted displacement. The fits of the unfiltered data generally have a negative R

^{2}, which means that the model offers a poorer fit to the data than a horizontal line at the mean value. The reason for this can be seen in the fitting results shown in Figure 8, for which, the unfiltered data have a bias toward negative values due to the low signal-to-noise ratio, while the model is constrained to a value of zero in the tails. The lowpass filter in the time domain significantly reduces the noise and nearly eliminates the negative bias, yielding R

^{2}> 0.73 at the selected set of fit parameters. The addition of the Poisson filter in the spatial domain offers a further improvement to noise reduction, yielding R

^{2}> 0.86 with the same fit parameters.

**Figure A4.**Effect of fit parameters on displacement prediction and fit quality. Fit parameters were independently varied at a high-displacement condition (t = 2.27 ms). Symbols indicate fit parameters chosen for the time-series displacement fitting. (

**A**) Attenuation coefficient difference has a large effect on predicted displacement but no impact on R

^{2}. Image pixel size (

**B**) and injector needle radius (

**C**) together set the scaling used in the model, and the peaks in R

^{2}for both parameters are near the values obtained from scale measurements with low sensitivity in predicted displacement at those values. (

**D**) Reference position (nominal needle center) has clear R

^{2}peak and low sensitivity in predicted displacement at same value. (

**E**) With reference and dynamic total blur set as equal, the filtered cases exhibit an R

^{2}peak and mild sensitivity in predicted displacement. (

**F**) Varying the ratio of dynamic to reference total blur does not provide compelling evidence for a value greater than unity.

## References

- Bilheux, H.Z.; McGreevy, R.; Anderson, I.S. Neutron Imaging and Applications: A Reference for the Imaging Community; Springer: New York, NY, USA, 2009. [Google Scholar] [CrossRef]
- The 2018 EPA Automotive Trends Report: Greenhouse Gas Emissions, Fuel Economy, and Technology since 1975. Available online: https://www.epa.gov/automotive-trends (accessed on 1 October 2020).
- Myung, C.L.; Park, S. Exhaust nanoparticle emissions from internal combustion engines: A review. Int. J. Automot. Technol.
**2011**, 13, 9. [Google Scholar] [CrossRef] - Jatana, G.S.; Splitter, D.A.; Kaul, B.; Szybist, J.P. Fuel property effects on low-speed pre-ignition. Fuel
**2018**, 230, 474–482. [Google Scholar] [CrossRef] - Strek, P.; Duke, D.; Swantek, A.; Kastengren, A.; Powell, C.F.; Schmidt, D.P. X-ray Radiography and CFD Studies of the Spray G Injector; SAE International: Warrendale, PA, USA, 2016. [Google Scholar]
- Neroorkar, K.; Schmidt, D. A Computational Investigation of Flash-Boiling Multi-Hole Injectors with Gasoline-Ethanol Blends; SAE International: Warrendale, PA, USA, 2011. [Google Scholar]
- Saha, K.; Som, S.; Battistoni, M.; Li, Y.; Pomraning, E.; Senecal, P. Numerical Investigation of Two-Phase Flow Evolution of In-and Near-Nozzle Regions of a Gasoline Direct Injection Engine During Needle Transients. SAE Int. J. Engines
**2016**, 9, 1230–1240. [Google Scholar] [CrossRef] - Baldwin, E.T.; Grover, R.O.; Parrish, S.E.; Duke, D.J.; Matusik, K.E.; Powell, C.F.; Kastengren, A.L.; Schmidt, D.P. String flash-boiling in gasoline direct injection simulations with transient needle motion. Int. J. Multiph. Flow
**2016**, 87, 90–101. [Google Scholar] [CrossRef] [Green Version] - Battistoni, M.; Xue, Q.; Som, S.; Pomraning, E. Effect of off-axis needle motion on internal nozzle and near exit flow in a multi-hole diesel injector. SAE Int. J. Fuels Lubr.
**2014**, 7, 167–182. [Google Scholar] [CrossRef] - Powell, C.F.; Kastengren, A.L.; Liu, Z.; Fezzaa, K. The Effects of Diesel Injector Needle Motion on Spray Structure. J. Eng. Gas Turbines Power
**2010**, 133, 012802. [Google Scholar] [CrossRef] - Kastengren, A.L.; Powell, C.F.; Liu, Z.; Fezzaa, K.; Wang, J. High-Speed X-ray Imaging of Diesel Injector Needle Motion. In Proceedings of the ASME 2009 Internal Combustion Engine Division Spring Technical Conference, Milwaukee, WI, USA, 3–6 May 2009. [Google Scholar]
- Kastengren, A.L.; Tilocco, F.Z.; Powell, C.F.; Manin, J.; Pickett, L.M.; Payri, R.; Bazyn, T. Engine Combustion Network (ECN): Measurements of nozzle geometry and hydraulic behavior. At. Sprays
**2012**, 22, 1011–1052. [Google Scholar] [CrossRef] - Fansler, T.D.; Parrish, S.E. Spray measurement technology: A review. Meas. Sci. Technol.
**2014**, 26, 012002. [Google Scholar] [CrossRef] - Kolakaluri, R.; Subramaniam, S.; Panchagnula, M.V. Trends in Multiphase Modeling and Simulation of Sprays. Int. J. Spray Combust. Dyn.
**2014**, 6, 317–356. [Google Scholar] [CrossRef] - Tekawade, A.; Sforzo, B.A.; Matusik, K.E.; Fezzaa, K.; Kastengren, A.L.; Powell, C.F. Time-resolved 3D imaging of two-phase fluid flow inside a steel fuel injector using synchrotron X-ray tomography. Sci. Rep.
**2020**, 10, 8674. [Google Scholar] [CrossRef] - Duke, D.J.; Finney, C.E.; Kastengren, A.; Matusik, K.; Sovis, N.; Santodonato, L.; Bilheux, H.; Schmidt, D.; Powell, C.; Toops, T. High-resolution x-ray and neutron computed tomography of an engine combustion network spray G gasoline injector. SAE Int. J. Fuels Lubr.
**2017**, 10, 328–343. [Google Scholar] [CrossRef] - Santodonato, L.; Bilheux, H.; Bailey, B.; Bilheux, J.; Nguyen, P.; Tremsin, A.; Selby, D.; Walker, L. The CG-1D Neutron Imaging Beamline at the Oak Ridge National Laboratory High Flux Isotope Reactor. Phys. Procedia
**2015**, 69, 104–108. [Google Scholar] [CrossRef] [Green Version] - Tremsin, A.S.; McPhate, J.B.; Vallerga, J.V.; Siegmund, O.H.W.; Hull, J.S.; Feller, W.B.; Lehmann, E. Detection efficiency, spatial and timing resolution of thermal and cold neutron counting MCP detectors. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2009**, 604, 140–143. [Google Scholar] [CrossRef] - Tremsin, A.S.; Vallerga, J.V.; Siegmund, O.H.W. Overview of spatial and timing resolution of event counting detectors with Microchannel Plates. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2020**, 949, 162768. [Google Scholar] [CrossRef] - Liu, D.; Hussey, D.; Gubarev, M.V.; Ramsey, B.D.; Jacobson, D.; Arif, M.; Moncton, D.E.; Khaykovich, B. Demonstration of achromatic cold-neutron microscope utilizing axisymmetric focusing mirrors. Appl. Phys. Lett.
**2013**, 102, 183508. [Google Scholar] [CrossRef] - Trtik, P.; Hovind, J.; Grünzweig, C.; Bollhalder, A.; Thominet, V.; David, C.; Kaestner, A.; Lehmann, E.H. Improving the Spatial Resolution of Neutron Imaging at Paul Scherrer Institut—The Neutron Microscope Project. Phys. Procedia
**2015**, 69, 169–176. [Google Scholar] [CrossRef] [Green Version] - Koerner, S.; Lehmann, E.; Vontobel, P. Design and optimization of a CCD-neutron radiography detector. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2000**, 454, 158–164. [Google Scholar] [CrossRef] - Tomviz 1.9.0. Available online: https://tomviz.org/ (accessed on 1 October 2020).
- Kang, M.; Bilheux, H.Z.; Voisin, S.; Cheng, C.L.; Perfect, E.; Horita, J.; Warren, J.M. Water calibration measurements for neutron radiography: Application to water content quantification in porous media. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2013**, 708, 24–31. [Google Scholar] [CrossRef] - Boillat, P.; Carminati, C.; Schmid, F.; Grünzweig, C.; Hovind, J.; Kaestner, A.; Mannes, D.; Morgano, M.; Siegwart, M.; Trtik, P.; et al. Chasing quantitative biases in neutron imaging with scintillator-camera detectors: A practical method with black body grids. Opt. Express
**2018**, 26, 15769–15784. [Google Scholar] [CrossRef] - Carminati, C.; Boillat, P.; Schmid, F.; Vontobel, P.; Hovind, J.; Morgano, M.; Raventos, M.; Siegwart, M.; Mannes, D.; Gruenzweig, C.; et al. Implementation and assessment of the black body bias correction in quantitative neutron imaging. PLoS ONE
**2019**, 14, e0210300. [Google Scholar] [CrossRef] [Green Version] - MATLAB R2018b; The MathWorks, Inc.: Natick, MA, USA, 2018.
- Kaestner, A.P.; Kis, Z.; Radebe, M.J.; Mannes, D.; Hovind, J.; Grünzweig, C.; Kardjilov, N.; Lehmann, E.H. Samples to Determine the Resolution of Neutron Radiography and Tomography. Phys. Procedia
**2017**, 88, 258–265. [Google Scholar] [CrossRef] - Tremsin, A.S.; Vallerga, J.V.; McPhate, J.B.; Siegmund, O.H.W. Optimization of Timepix count rate capabilities for the applications with a periodic input signal. J. Instrum.
**2014**, 9, C05026. [Google Scholar] [CrossRef] - Azzari, L.; Foi, A. Variance Stabilization for Noisy+Estimate Combination in Iterative Poisson Denoising. IEEE Signal Processing Lett.
**2016**, 23, 1086–1090. [Google Scholar] [CrossRef] - Ballabriga, R.; Campbell, M.; Llopart, X. Asic developments for radiation imaging applications: The medipix and timepix family. Nucl. Instrum. Methods Phys. Res. Sect. A Accel. Spectrometers Detect. Assoc. Equip.
**2018**, 878, 10–23. [Google Scholar] [CrossRef] [Green Version] - Llopart, X.; Alozy, J.; Ballabriga, R.; Campbell, M.; Egidos, N.; Fernandez, J.M.; Heijne, E.; Kremastiotis, I.; Santin, E.; Tlustos, L.; et al. Study of low power front-ends for hybrid pixel detectors with sub-ns time tagging. J. Instrum.
**2019**, 14, C01024. [Google Scholar] [CrossRef] - Losko, A.S.; Han, Y.; Schillinger, B.; Tartaglione, A.; Morgano, M.; Strobl, M.; Long, J.; Tremsin, A.S.; Schulz, M. New perspectives for neutron imaging through advanced event-mode data acquisition. Sci. Rep.
**2021**, 11, 21360. [Google Scholar] [CrossRef] - Zhang, Y.; Bilheux, J.C.; Bilheux, H.Z.; Lin, J.Y.Y. An interactive web-based tool to guide the preparation of neutron imaging experiments at oak ridge national laboratory. J. Phys. Commun.
**2019**, 3, 103003. [Google Scholar] [CrossRef] - Zhang, Y.; Bilheux, J.C. ImagingReso: A tool for neutron resonance imaging. J. Open Source Softw.
**2017**, 2, 407. [Google Scholar] [CrossRef] [Green Version] - Brown, D.A.; Chadwick, M.B.; Capote, R.; Kahler, A.C.; Trkov, A.; Herman, M.W.; Sonzogni, A.A.; Danon, Y.; Carlson, A.D.; Dunn, M.; et al. ENDF/B-VIII.0: The 8th Major Release of the Nuclear Reaction Data Library with CIELO-project Cross Sections, New Standards and Thermal Scattering Data. Nucl. Data Sheets
**2018**, 148, 1–142. [Google Scholar] [CrossRef] - Melkonian, E. Slow Neutron Velocity Spectrometer Studies of O
_{2}, N_{2}, A, H_{2}, H_{2}O, and Seven Hydrocarbons. Phys. Rev.**1949**, 76, 1750–1759. [Google Scholar] [CrossRef]

**Figure 1.**Experimental setups for high-speed neutron imaging and neutron CT. Beam travels from right to left in all panels. (

**A**) High-speed setup uses the MCP detector with a custom sample environment that includes the injector and spray chamber. (

**B**) Tomography setup uses the CCD detector with injector mounted on a rotation stage. (

**C**) Photo of aluminum spray chamber mounted in front of MCP detector.

**Figure 2.**Neutron CT reconstruction of single-hole gasoline direct injector. (

**A**) Slice from CT reconstruction with an illustration of regions targeted in the high-speed neutron image sequence and time-series displacement fits. Fuel flows from top to bottom. (

**B**) Sectioned volumetric rendering of the injector illustrating the internal geometry and features of the device.

**Figure 3.**Illustration of neutron path length variation in moving phases. Neutron arriving at a given detector pixel after passing through a nested multiphase system (

**A**–

**C**) with one moving phase (

**A**) will encounter a shorter path length through (

**A**) when that phase moves as shown.

**Figure 4.**Illustration of path length variation through moving circles with discrete or probabilistic location. (

**A**) Two circles at different discrete positions (reference and dynamic) in the xy plane with neutron path in the y direction denoted by n. (

**B**) Path lengths and path length difference for paths in the y direction as a function of x. (

**C**) Probability distributions for locations of two circles at different positions (reference and dynamic) in the xy plane. (

**D**) Expected values for path lengths and path length difference for paths in the y direction as a function of x are smoothed out in comparison to the discrete path lengths in (

**B**).

**Figure 5.**Development of the probabilistic path length model. (

**A**) Example calculation of expected value for the path length function for a given value of x. E[d(x,a)] was obtained by integrating the product of the path length function d(x,a) and the PDF f

_{a}(x) over all possible values of a (dummy variable α) as indicated by the shaded region. (

**B**) Expected value of path length through a circle of given radius and mean x location with varying standard deviation of x location. Example calculation from A is indicated by the open circle. (

**C**) Effect of varying the standard deviation of circle location on the expected value of path length difference for circles displaced by 1% of the radius. (

**D**) Effect of varying the ratio of standard deviation of the dynamic and reference circle locations on the expected value of path length difference for circles displaced by 1% of the radius.

**Figure 6.**Filtering and normalization methods for high-speed imaging. Image is from a time bin with a relatively large needle deflection (t = 2.27 ms). Rows correspond to filtering, and columns correspond to normalization. The injector body outline has been overlaid on the normalized images for clarity. Variations in neutron count rate due to displacement of the injector needle are apparent in the normalized images due to the difference in attenuation coefficient between the steel needle and the surrounding hydrogenous fuel, with movement of the needle toward light and away from dark in the subtraction images, and toward blue and away from red in the log-ratio images.

**Figure 7.**Process for extracting needle radius from neutron CT. Radial regions extending from the needle center were defined for both the valve needle (red) and outer body (blue) fits. (

**A**) Axial and radial extent of each region overlaid on frontal slice. (

**B**,

**C**) Transverse view within each region with illustration of radial extent. (

**D**) Attenuation coefficient of each voxel within each region plotted by radius with edge fits overlaid. Horizontal dashed line shows attenuation coefficient calculated from projections. Fit of outer body is used to set the image scaling based on the known outer body diameter. This scaling is used with the fit of valve needle radius to measure its size in microns.

**Figure 8.**Examples of attenuation model fitting for each filtering level. Log-ratio normalized images are from a time bin with a relatively large needle displacement (t = 2.27 ms), and the injector body outline has been overlaid on the normalized images for clarity. Negative (red) in images corresponds to a decrease in neutron count rate relative to the reference, whereas positive (blue) indicates an increase. The blue and red vertical bands indicate the cylindrical needle moving to the left, because the steel needle has a lower attenuation coefficient than the surrounding hydrogenous fuel. Boxed region in each log-ratio image was averaged in the z direction to create 1D data for fitting. Although signal-to-noise ratio and fit metrics improved dramatically with filtering, the resulting displacement prediction from the fitting procedure is similar in each case.

**Figure 9.**Results of high-speed imaging measurements and displacement model fitting. Selected frames from the 95% subtraction (

**A**) and log-ratio (

**B**) normalizations with lowpass + Poisson filtering highlighting motion of the injector needle. The injector body outline has been overlaid on the normalized images for clarity. (

**C**) Time-series fits of needle displacement indicate sub-pixel resolution relative to the 55 μm pixel size with significant noise reduction for the filtered cases but similar shape and magnitude to the unfiltered case.

Parameter | Unit | Value |
---|---|---|

$\mathrm{Macroscopic}\text{}\mathrm{attenuation}\text{}\mathrm{coefficient}\text{}\mathrm{difference}\text{}\u2206\mathsf{\Sigma}={\mathsf{\Sigma}}_{\mathrm{fuel}}-{\mathsf{\Sigma}}_{\mathrm{steel}}$ | cm^{−1} | 2.82 |

Image pixel size | µm | 55 |

Needle radius | µm | 815.4 |

$\mathrm{Reference}\text{}\mathrm{position}\text{}{\mu}_{a,\mathrm{ref}}$ | px | 84 |

$\mathrm{Reference}\text{}\mathrm{total}\text{}\mathrm{blur}\text{}{\sigma}_{a,\mathrm{ref}}$ | px | 3.1 |

$\mathrm{Dynamic}\text{}\mathrm{total}\text{}\mathrm{blur}\text{}{\sigma}_{a,\mathrm{dyn}}$ | px | 3.1 |

Displacement range | px | ±5 |

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## Share and Cite

**MDPI and ACS Style**

Wissink, M.L.; Toops, T.J.; Splitter, D.A.; Nafziger, E.J.; Finney, C.E.A.; Bilheux, H.Z.; Santodonato, L.J.; Zhang, Y.
Quantification of Sub-Pixel Dynamics in High-Speed Neutron Imaging. *J. Imaging* **2022**, *8*, 201.
https://doi.org/10.3390/jimaging8070201

**AMA Style**

Wissink ML, Toops TJ, Splitter DA, Nafziger EJ, Finney CEA, Bilheux HZ, Santodonato LJ, Zhang Y.
Quantification of Sub-Pixel Dynamics in High-Speed Neutron Imaging. *Journal of Imaging*. 2022; 8(7):201.
https://doi.org/10.3390/jimaging8070201

**Chicago/Turabian Style**

Wissink, Martin L., Todd J. Toops, Derek A. Splitter, Eric J. Nafziger, Charles E. A. Finney, Hassina Z. Bilheux, Louis J. Santodonato, and Yuxuan Zhang.
2022. "Quantification of Sub-Pixel Dynamics in High-Speed Neutron Imaging" *Journal of Imaging* 8, no. 7: 201.
https://doi.org/10.3390/jimaging8070201