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Abstract
The key to uncovering underlying fluid mechanisms lies in high-resolution and large-scale three-dimensional (3D) measurements of flow fields. Currently, the mainstream methods that are capable of volumetric measurements, such as tomographic background oriented schlieren and conventional plenoptic background oriented schlieren (plenoptic BOS), suffer system complexity and low axial resolution, respectively, prohibiting their application in high fidelity 3D flow measurement. This paper proposed an isotropic resolution plenoptic BOS (ISO plenoptic BOS) system that employed a mirror to create a second image view for the region of interest, thereby can achieve isotropic spatial resolution with only one camera. We comprehensively assessed the feasibility of the system by imaging the density field induced by candle flames, heat gun, and the Mach disk produced by the underexpanded jet through the high-pressure nozzle exit. All results proved that the dual-view plenoptic BOS system has higher axial resolution and can provide a more accurate 3D density field than the conventional system. As a BOS system that can achieve high-resolution volumetric imaging without the additional cost of cameras, data acquisition, hardware synchronization, and scanning, our ISO plenoptic BOS can expand the road to large-scale and high-resolution aerodynamic imaging.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
By taking advantage of the relationship between refractive index and density, schlieren-based flow visualization techniques are able to sensitively reveal inhomogeneous density variation of the measured region in a qualitatively and non-invasive way. Background oriented schlieren (BOS) is one of the main schlieren systems [1] that features simple system implementation and has been widely used for density field measurement. Different from the conventional schlieren system that requires multiple lenses to produce a parallel beam to illuminate the measured region, BOS only needs a background pattern as a reference and then computationally resolves the density change by quantifying the displacement of two patterns imaged with and without density disturbance. The easy implementation of BOS has enabled it to work over broad domains, ranging from candle flow imaging in stationary conditions [2], to shock/boundary interaction imaging for supersonic flow [3,4].
One big challenge for conventional BOS is that it can only provide a 2D image of the measured region, as the acquired image is a 2D line-of-sight integrated quantity of the inhomogeneous density volume. Even though volumetric density variation can also be measured, this requires multiple cameras to image one region over different angles [5], making the whole system bulky and impractical for optical access limited applications. Fortunately, recently developed plenoptic BOS is able to image a 3D density field with only one light field camera that captures volumetric information by placing a microlens array (MLA) at the focus of the main lens [6]. Because the light field camera captures 2D spatial and 2D angular information simultaneously, compared to conventional BOS, plenoptic BOS is able to achieve an extended depth of field while still maintaining a large numerical aperture for high efficiency light collection [7]. So far, plenoptic BOS has demonstrated imaging of swept shock/boundary layer interaction [8], Mach 3.3 free air jet [9] and flame [6].
However, the limited angular information collected by a single plenoptic camera is not enough for high axial resolution density field reconstruction, prohibiting plenoptic BOS for accurate 3D flow field measurement. Experiment has demonstrated that increasing the number of plenoptic cameras to form a tomographic configuration and then capturing the density field through different views can help improve the axial resolution, but this modality inevitably faced additional costs in the budget, system alignment and calibration [10], and extra limitations in system allocation for space and optical access constrained condition. Therefore, it is necessary to develop a single camera based plenoptic BOS system with high spatial resolution.
To improve the resolution of plenoptic BOS without increasing additional cost and workload, we propose a mirror-enhanced resolution plenoptic BOS by adding a mirror on the right of the field of view (FOV) of a conventional single-camera plenoptic BOS system (Fig. 1). Because the angel between the mirror and the camera’s sensor plane is 45°, the camera can image the density field along the lateral direction through the mirror, which is equivalent to adding a lateral imaging view to the FOV. We acquired both the conventional view and the lateral view images with the left half and the right half of the sensor array of the same camera, respectively. By fusing images from two views, our approach has significantly improved the axial resolution of conventional light field systems without increasing the number of cameras and the workload in synchronization costs. We demonstrated the superior axial resolution of our system through the imaging of flames, hot air gun streams, and high-pressure nozzle jets.
Fig. 1. Working principle of dual-view plenoptic BOS. (a) Experimental setup diagram of dual-view plenoptic BOS. (b) Schematic showing the imaging principle of dual-view plenoptic BOS. (c) The high-resolution reconstruction procedure of dual-view plenoptic BOS. The schematics of two light-field volumes are shown in light red (volume in normal view) and blue (volume in mirror view), and the combined isotropic resolution volume.
2. Methods
Isotropic resolution plenoptic BOS (ISO plenoptic BOS) system consists of one mirror, two background plates and one light field camera (Fig. 1(a)). Specifically, the system uses a light field camera to obtain a front view (view 1 in Fig. 1 (b)) and a lateral view (view 2 in Fig. 1 (b)) of the image region in the left and right halves of the sensor array, respectively. In view 1, the camera directly images the background plate 1. This view is the same as the conventional 3D-3C BOS, which is of low axial resolution. To improve axial resolution, we added view 2 by simply placing the mirror in the imaging area at 45 ° to the front view direction. In view 2, the camera images the region of interest through the mirror, which corresponds to the lateral view of the imaging area. In this way, view 2 can convert the axial direction of view 1 into its lateral direction. Since the lateral resolution of light field imaging is much better than the axial resolution, the axial resolution of view 1 can be improved by fusing the data of view 2, resulting in isotropic spatial resolution [11].
2.1 Plenoptic image reconstruction
The plenoptic camera contains a microlens array located in front of the image sensor to record light field information. Generally, the light field is denoted as L(u,v,s,t), where the light ray intersects the aperture plane at (u, v) for angular information, and intersects the microlens plane at (s, t) for spatial information. The raw plenoptic image can be calculated to shift the focal plane, and the refocus equation [12] is written as:
As shown in Fig. 1(c), the raw plenoptic images acquired in the presence and absence of density disturbance were refocused to generate the same number of perspective views. Perspective image pairs are segmented according to the two views for subsequent BOS calculations.
2.2 Volumetric calibration pipeline
The accuracy of system calibration determines the quality of volume reconstruction. For our dual-view imaging system, the imaging depth range is -200 mm to 200 mm (Fig. 2(a)), which is the range that requires calibration. When the FOV is scaled, the system needs to be re-calibrated to achieve accurate data fusion. Taking the experimental FOV in Section 3.1 as an example, we used the direct light field calibration method [13] to perform calibration in five steps. The first step is to image the calibration board. The plane of the board was placed on the translation stage parallel to the plane of the camera sensor, and the black dots of the board (1 mm in diameter) were evenly distributed over a 2 mm space. We successively acquired light field images of the board at intervals of 5 mm within the imaging depth range of -200 mm to 200 mm. The second step is to extract light field information. We extract the spatial and angular information of each image (Fig. 2(b)), thereby generating a matrix that stores the relationship between any point in the imaging area and its light field information. The third step is to generate the mapping function. We established a mapping function P through direct light field calibration so that the known object space coordinates (X, Y, Z) of each point correspond to the position on the microlens plane, i.e.:
where (u,v) and (s,t) represent the angle and position coordinates of the light field, respectively.Fig. 2. Calibration of the light field camera. (a) Diagram of the calibration area. (b) Original light field image and calibration flow block. A total of 81 light field images were for calibration. (c) Validation results of the calibration. The dot card at depths from -198 mm to -8 mm and 8 mm to 198 mm with 20 mm interval was imaged for validation. Blue triangles represent the reconstruction depths of the dot cards and red circles represent the mean errors from the true value (mean ± std.).
The fourth step is to combine the mapping function and the refocusing algorithm. At this point, the refocusing equation is rewritten as:
The fifth step is to verify the mapping function. To verify the feasibility and accuracy of the mapping function, we imaged the dot board at another depth range (-198 mm to -8 mm and 8 mm to 198 mm) with an interval of 20 mm, then reconstructed these images and used the mapping function to correct the reconstructed image. Within the main measurement range, the error between the true depth and the resolved depth is in the range of -3 mm to 3 mm (Fig. 2(c)), proving that the calibration algorithm was reliable for subsequent experiments. Finally, the above calibration function and superposition algorithm are used as templates for following image reconstruction. In addition, a fitting curve could be generated with the calibrating data for multiple FOVs at different scales, so that the calibration data for any intermediate FOV can be attained through data fitting, which can greatly save the calibration time.
2.3 Density reconstruction
Density reconstruction involves computing the displacement field, solving the Poisson equation and quantifying the density field. First, for the three-dimensional displacement field computation, we use the PIVlab [14] toolbox to calculate the background pattern movement caused by the density disturbance through the cross-correlation algorithm performed on the perspective image pairs of each view. Secondly, the Poisson Eq. (4) is solved to obtain the refractive index n. Then the density field of a single-view is obtained by the Gladstone-Dale Eq. (5).
As shown in Fig. 1(c), we rotated the lateral view by 90° to make it in the same direction as the front view, and then superimposed it with the front view. Therefore, regions where density changes intersect each other produced local high (or low) intensities, and the location of the perturbation can be easily extracted. In order to reduce the cross-shape PSF due to direct superposition, we first removed all non-overlapping parts, and then we applied a Gaussian filter to smooth the image and reduce noise.To demonstrate the resolution improvement of the dual-view system, we imaged tracer particles (2 mm) in the same manner as our previously published paper [11] and quantified the full widths at half-maximum (FWHM) of the particle imaged by single-view and dual-view system. The lateral FWNMs of the particle in the single-view and double-view systems are both 2.17 mm, while the axial FWHMs are 16.89 mm and 2.25 mm respectively. The results showed that the axial spot size of the single-view system was severely poorer than the lateral size, but the dual-view system can approximately achieve isotropic resolution.
For image acquisition, we used a Lytro Illum light field camera that contains 7728 × 5368 pixels and a 625 × 434 microlens array. All images were acquired with a fixed aperture of F/2.0. Regarding the design of background patterns, it is important to consider the pixel size of the background pattern on the camera sensor rather than its actual size. A speckle generator was used to generate background images for different views such that the point image size is about 3 pixels, and the point-to-point spacing is 2 to 4 pixels, which has the best performance [16]. For all experiments, the background was printed on paper and mounted in a fixed position during image acquisition. Background lighting was provided by two LED table lamps placed on either side of the camera.
3. Result
3.1 Dual candle flames imaging demonstrated the superior resolution of dual-view plenoptic BOS
To verify the feasibility of dual-view plenoptic BOS, we first imaged two candle flames within an imaging field of 100 × 100 × 100 mm3 (Fig. 3(a)), aiming to qualitatively and simultaneously observe the density gradients produced by two heat sources at different depths. We fixed the two candle flames in known positions and adjusted their height so that one flame was behind the other when viewed from the front. Specifically, the nominal focal plane of the system was set at the background plate 1, the front candle (A) was placed 140 mm in front of the nominal focal plane, and the two candles were 70 mm apart and 130 mm away from the background plate 2. The front and lateral views were obtained from the left and right half of the plenoptic camera sensor array, respectively.
Fig. 3. Validation experiment with two candles. (a) Schematic of the validation experiment and the flame profile in front view, lateral view and dual-view images (b) Comparison of densities reconstructed with front-view, lateral-view, and dual-view images (minimum intensity projection). (c) 3D density field reconstructed in dual-view.
Due to the low axial resolution, the reconstructed candle flame density field obtained by single-view imaging is obviously elongated along the axial direction. Consequently, the two flames (flames A and B) positioned axially along the front view cannot be distinguished in the front view image (Fig. 3(b) top left), therefore, only one flame was shown in this view. This result clearly demonstrates the resolution limitation of single-view plenoptic imaging. In contrast, by converting the axial resolution of the front view to the lateral resolution of the lateral view, the lateral view can distinguish the two flames (Fig. 3(b) top right). The information of the front view and lateral view images complemented each other in the xy, yz, and xz directions of the actual object space, respectively. After careful fusion of the dual-view data, the two candle flames were clearly resolved with high resolution in dual-view imaging (Fig. 3(c)), proving that the poor resolution of single-view imaging is successfully enhanced through dual-view imaging. For clear comparison, we plotted the profile of the flame in different views, it is obvious that the reconstructed result of the dual-view system identified two peaks that correspond to the two flames, but the profile for the conventional system failed to identify two peaks (Fig. 3(a)). This experiment showed that the axial resolution of the density field reconstruction can be effectively improved by adding one more view to conventional plenoptic imaging.
Adding a mirror view will inevitably affect the FOV of the BOS system. In the BOS experimental setup, both the measured object and the background pattern should be within the depth of field of the plenoptic camera, which requires that the optical path difference between the two views is as small as possible. Therefore, the pattern in the added view needs to be close to the mirror, resulting in a sacrifice in FOV.
3.2 Dual-view plenoptic BOS measurement of heat gun flow field
We further verified the feasibility of the dual-view system by heat gun flow field imaging. We reconstructed the 3D position of the thermal density field within an imaging field of 150 × 150 × 150 mm3. Due to the higher resolution and accuracy of the dual-view system, the reconstructed cone-shaped density field is significantly better than the strip-shaped density field reconstructed by the single-view system.
Using the refocusing property of plenoptic imaging, we obtained the magnitude of the BOS background displacement at the center depth of the two views (Fig. 4(a)). The flow field expanded outward from the nozzle and had a relatively symmetrical structure. It is important to note that both the x and y displacement component values can be used separately to generate a focused BOS image. This allows independent observation of displacements occurring in either direction, and is equivalent to changing the direction of the knife edge in the traditional schlieren experiment to observe the displacement in a certain direction. Hence, we generated BOS images (Fig. 4 (a)) using the x and y components of displacement, respectively. The result revealed that the x-component displacement played the most important role in generating the displacement magnitude because the strongest density gradient occurred in that direction.
Fig. 4. Measurement results of heat gun flow field. (a) Displacements in each direction of image reconstruction from the front view (left) and lateral view (right). (b) Comparison of densities reconstructed with front-view, lateral-view, and dual-view images (minimum intensity projection). (c) 3D density field reconstructed in dual-view.
We then reconstructed the BOS image using both the x and y displacements (Fig. 4(b)). Similar as the flame image, the single-view system reconstructed the density field with low resolution, and failed to reveal the true shape of the heat gun. In contrast, the dual-view data accurately depicts the 3D density field of the real heat gun airflow. This verification successfully proved that our dual-view system can reduce the axial elongation effect in conventional plenoptic system reconstructions.
3.3 Dual-view plenoptic BOS enabled complex flame flow field measurement
In order to verify the feasibility of the dual-view plenoptic system for imaging complex flow fields, we conducted experiments on the flame flow field of a mineral oil lamp. Due to the inhomogeneous combustion of the mineral oil lamp and the artificial additional disturbance, the flame flow field became unstable. We reconstructed the density field of the flame in a 150 × 150 × 150 mm3 area by dual-view fusion.
There was a clear difference in the flame displacement trends in the front view and the lateral view, unlike the axisymmetric shape presented by the conventional candle flame (Fig. 5(a)). The information of a single-view was less than enough to faithfully reflect the flame information. The x-component of displacement is also the most critical factor determining the overall displacement in this experiment, reflecting that the large density gradient occurs in this direction.
Fig. 5. Measurement results of complex flame flow field. (a) Displacement in each direction of image reconstruction from the front view (left) and lateral view (right). (b) Comparison of densities reconstructed with front-view, lateral-view, and dual-view images (minimum intensity projection). (c) 3D density field reconstructed in dual-view.
For comparison, Fig. 5(b) shows the density diagram for the single-view plenoptic system and the dual-view plenoptic system. The erratic combustion of mineral oil lamps resulted in a flame that actually appeared to be a combination of two disturbed flames. Because of the poor axial resolution of the single-view system, the density field cannot accurately reflect the flow field in the xy, yz and xz planes (Fig. 5(b) top). However, the minimum density projection of dual-view can clearly show the flame flow field (Fig. 5(b) bottom). In summary, by combining the superposition of the lateral view in the dual-view plenoptic BOS, the measurement accuracy and resolution of the single-camera plenoptic BOS in the depth direction can be significantly improved.
3.4 Dual-view plenoptic BOS enabled high-pressure nozzle jet measurement
To verify the feasibility of a dual-view plenoptic BOS system for supersonic flow imaging, we conducted experiments on the jet of a high-pressure nozzle, aiming to observe the Mach disk, which is invisible to the naked eye at low temperatures. We used a MISUMI ALVA1 Laval nozzle with a hole diameter of 2.7 mm, so that the exit Mach number can reach about 1.2. In this system, the nominal focal plane was set at the background plate 1, and the nozzle was fixed vertically at 60 mm in front of the nominal focal plane. The imaging volume of each view is 8 × 8 × 14 mm3.
In order to verify the correctness of the reconstruction results, we used traditional schlieren to image the same jet (Fig. 6(a)). Results obtained with both schlieren and plenoptic BOS showed similar structural features, which revealed the Mach disk [17] generated due to the inequality between the pressure in the supersonic fluid and the ambient pressure. Mach disk is a standing wave formed by the mutual interference of compression waves and expansion waves in supersonic airflow. The nozzle used in our experiment is a convergence-expansion type, and the ejected gas accelerates from subsonic speed to sonic speed in the convergence section. When the jet emerges from the nozzle at a higher pressure than the ambient air, it is called “underexpan sion”. According to the previous study [18,19], the flow field structure of the under-expansion supersonic impact jet is shown in Fig. 6(b). After the jet is ejected, a shock wave is formed near the center line, which is usually called a Mach disk. There are oblique shock waves around the Mach disk. A shear layer forms between the subsonic flow behind the Mach disk and the supersonic flow behind the oblique shock. The total pressure lost behind the Mach disk is greater than the total pressure lost by the flow through the oblique shock, which causes the pressure at the center of the plate to be lower than outside, creating a recirculation zone. A contact surface is formed between the recirculation zone and the airflow behind the Mach disk. Fluid passing through the Mach disk enters the shear layer, away from the axis of the jet. A fan-shaped expansion wave appears at the intersection of the oblique shock wave and the jet boundary, which is reflected by the shear layer to form an annular compression wave.
Fig. 6. Measurement results of the high-pressure nozzle jet flow field. (a) Comparison of traditional schlieren and plenoptic BOS results. (b) Schematic diagram of Mach disk in underexpanded jet. (c) Displacement in each direction of image reconstruction from the front view (top) and lateral view (bottom). (d) Comparison of densities reconstructed with front view, lateral view, and dual-view images (maximum intensity projection, left), and 3D density field reconstructed in dual-view (right).
The displacement maps of the two views (Fig. 6(c)) show the high-pressure gas at the exit expanded and compressed multiple times to form a series of Mach disks. Similarly, due to the low axial resolution of the single-view system, the front view, while reflecting the high-density Mach disk structure typical of an underexpanded jet, has a density field that is elongated along the axial direction (Fig. 6(d) left). In contrast, the high resolution of dual-view plenoptic BOS enables it to perfectly and accurately reconstruct the density field in three-dimensional space (Fig. 6(d) right). The pattern of disks would repeat indefinitely if the gases were ideal and frictionless. However, real gases are not ideal, and friction along the free-jet boundary between air and exhaust results in a turbulent shear layer. This layer creates viscous damping that causes the wave structure to gradually dissipate with distance. This experiment has proven that dual-view plenoptic BOS can be used to measure supersonic flow fields, and significantly improve the measurement accuracy and resolution of single-view BOS in the depth direction.
4. Conclusion
This study introduces the dual-view plenoptic BOS system for isotropic resolution volumetric density field measurement. Leveraging the volumetric imaging capabilities of light field cameras, this system can achieve high-resolution 3D reconstruction without additional scanning or camera. We evaluated the system performance by imaging stable and unstable heat sources, as well as the density of a high-pressure nozzle jet. When imaging candle flames that overlap front and back, the dual-view plenoptic BOS system can clearly distinguish the positions of the two candles, but the single-view mode cannot distinguish the two flames in the axial direction. Imaging of the heat gun airflow further demonstrated the superior axial resolution of the dual-view system. In complex combustion imaging, the dual-view system reconstructed a more comprehensive and accurate density field than the conventional system. Furthermore, when imaging high-pressure nozzle jets, the dual-view system captured the underexpanded jet structure and reconstructed the high-resolution Mach disk density field. This verified the feasibility of our system for supersonic flow imaging, and with appropriate accommodation it is possible to translate our technique into high-speed flow imaging of the wind tunnel. All results demonstrated that our technique was suitable for high-resolution, large-field aerodynamic measurements.
We can further optimize the dual-view plenoptic BOS system from five aspects: FOV, MLA, BOS experimental settings, imaging speed, and data processing efficiency. Since we divide the camera sensor into dual-view imaging, half of the imaging area is lost in a single-view. Using a camera that can accommodate more and larger sensors helps increase our FOV. The MLA in a plenoptic camera system is key to providing multi-field-angle information and determining the resolution of the refocused image. A new nanoimprinted MLA [20] consisting of one larger primary microlens and surrounding satellite microlenses can help improve the spatial resolution of our images. In terms of the experimental setup, adjusting BOS parameters (such as sensitivity S) helps to obtain the maximum signal from a given schlieren object [21]. In terms of imaging speed, our current limitation is the frame rate of the camera. A newly developed ultrafast camera improves imaging frame rates for transient flow imaging [22]. Optically multiplexed schlieren videography combines schlieren technique with videography based on image multiplexing to capture the time-resolved sequences in a single photograph by illuminating the sample with a rapid burst of uniquely spatially modulated light pulses, thus enabling, in principle, infinite frame rates to be realized [23]. In terms of data processing efficiency, we can apply deep learning based methods to accelerate light field reconstruction [24] and displacement field reconstruction [25,26], making it possible to achieve real-time high-resolution volumetric imaging.
Instantaneous high-resolution volumetric imaging will generate more possibilities for flow field measurement studies. We hope that our study can inspire more aerodynamic studies and further extend to supersonic flow field measurements to help us better understand flow mechanisms.
Funding
Postgraduate Research & Practice Innovation Program of NUAA (xcxjh20220201); National Natural Science Foundation of China (U20A2070, 12025202); Natural Science Foundation of Jiangsu Province (BK20230876); Fundamental Research Funds for the Central Universities (NS2023007); Startup of Nanjing University of Aeronautics and Astronautics (90YQR23004); Young Elite Scientist Sponsorship Program (YESS20210238).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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