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Abstract
The integral photography and deconvolution techniques have been applied to identify the three-dimensional (3D) positions of particles levitating in plasma. Artifacts in the light field, i.e. ghost particles, are removed by collating between results of integral photography and direct Richardson-Lucy deconvolution (RLD). Our reconstruction system is tested with known target particles and it is found that it works well in the range of our dust experiment. By applying the integral photography and RLD techniques to the obtained experimental image, we identified the 3D positions of dust particles floating in a radio-frequency plasma. Ghost particles are eliminated from the results by deconvolution and we succeeded in obtaining the 3D structure of a dusty plasma from a single-exposure image obtained from one view port.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Generally, plasma is an ionized gas state consisting of ions and electrons that can be treated as a translucent object and continuous light source. For a wide range of plasma research fields, three-dimensional (3D) information pertaining not only for the plasma itself, but also to plasma facing material and levitating particles attracts much attention. 3D information about the positions of fine particles in a plasma is important for plasma processings as well as for studying physical processes in Coulomb crystals [1–5] and its applications [6,7]. Among the various plasma devices, several separated detectors for stereoscopic detection [8,9] and/or computed tomography [10] techniques are widely used to determine the 3D information. However, many experimental devices have a limited number of view ports and viewing areas. Thus, a method of distinguishing a 3D structure of plasma from an image obtained from one view port of limited view area is required.
Integral photography [11] is a well-known technique used in commercial plenoptic cameras. This technique has the advantage that the instantaneous 3D information of objects can be estimated from a single-exposure obtained through a single view port. It provides a 3D imaging capability by employing a lens array in order to capture scattered light rays with small parallaxes. The rays emerging from 3D objects pass through the lens array and are captured on a detector. The 3D reconstruction is then performed computationally by generating inverse propagating rays within a virtual system similar to the recorded one. As a result of the ray-tracing, it is obtained a vector function that describes the amount of light flowing in every direction through every point in virtual space. It is called light field. The technique has been applied to scientific research in diverse fields. Applications include biomedical microscopy [12,13], particle tracking for velocimetry [14], etc. Recently, the technique has been used to estimate the 3D positions of levitating particles in dusty plasmas [15–18], and number of studies is gradually increasing. In the above papers, the 3D positions of particles are detected as the extreme positions of the intensity peaks of a computationally generated light field. Applying the method to a system containing many point light sources, the number of the intersections of ray lines which located on other places of true light sauces increases. Such intersections tend to result in artifacts, i.e. ghost particles, in the reconstruction result.
To solve this problem, we propose a technique to distinguish between the real and ghost images in integral field photography with helping of 3D deconvolution. The light field is treated as the convolution result of spread lights from the true particles. The technique of 3D deconvolution would restore the spread lights to the positions of the true particles. It has been demonstrated that the Richardson-Lucy deconvolution (RLD) algorithm [19,20] can enhance the resolution of integral photography [21]. In the microscopy field, in particular several approaches and algorithms have been proposed [22–24]. In this paper, we report, for what is to our knowledge the first time, the result of applying the deconvolution technique to eliminating ghost particles in reconstruction of particles from the reconstruction of dusty plasma by integral photography. This technique could be useful to measure the 3D particle positions of small particles in many applications including plasma and aerosol sciences.
2. Method
2.1 Calculation method
We propose a reconstruction method to eliminate integral photography ghost particles of dusty plasma. A flowchart of our approach is shown as Fig. 1. The rays emerging from the dust particles pass through a multi-lens array and are captured by a detector to form a two-dimensional elemental image array. The 3D light field is then generated computationally by inverse propagating rays within a virtual system similar to the recorded one.
The luminosity $I(x,y,z)$ is estimated as the summation of the light intensity $I' (X^i,Y^i)$ at a point $(X^i,Y^i)$ on the elemental image array over all lenses, expressed as [17]
Positions of intensity peaks of the light field are stored as data set $\textbf {C}$ and are considered candidate positions of particles. The generated light field is divided into a voxel array with a relatively low spatial resolution than that of result of the ray tracing.
where, $< I(x,y,z) >$ is the volume averaged luminosity over $m$-th voxel. Subsequently, the RLD is applied to the light field in voxel space.The RLD is an iterative procedure for recovering a latent image that has been blurred by a known point spread function. Blurred data $I_m$ at the $m$-th voxel in the light field is the summation of the spread true signals:
where $\lambda _k$ is the true value of the intensity $\lambda$ of emission or reflection light sauce at the $k$-th voxel and $p(m,k)$ is the contribution coefficient from the $k$-th to the $m$-th voxel, i.e. the point spread function. The physical unit of $\lambda$ depends on what type of detector is used. Contribution $\hat {z}(m,n)$ from the $n$-th to the $m$-th voxel is expressed as The true value at the $n$-th voxel is estimated as the summation of the contribution $\hat {z}(m,n)$ from the $n$-th to the $m$-th voxel.In order to avoid that pick-upping small residual noise as a candidate of position of particle, voxels with a $\hat {\lambda }$ exceeding a certain threshold are recorded as $\textbf {V}_{hl}$. Components of $\textbf {C}$ included in $\textbf {V}_{hl}$ are adopted as the positions of particles.
2.2 Test calculation
In this subsection, we show an assessment of above technique with synthetic situation. Figure 2(a) shows a schematic of the configuration used for a test calculation with a multi pinhole camera. The multi pinhole camera has six pinholes arranged hexagonally and a seventh at its center. In the test calculation, the optical axis is along the $z$-axis and the multi pinhole is located at $z$ = 0. A detector is located on $z=0.17d$, where $d$ is the pinhole spacing. Three light sources with the value of $\lambda$ = 1000 are set at $(x,y,z)$ = $(0,0,-5d)$, $(0,0,-4d)$, and $(0.3d,0,-4d)$, respectively. These three lights are projected onto the detector and the detected intensity distribution recorded as Fig. 2(b). Figure 2(c) and (d) show the light fields on the $x-z$ plane and the $z=-4d$ calculated from the recorded data, respectively. In Fig. 2(c), three strong convergence features are evident, moreover there are some other extrema due to intersections of ray lines. In conventional integral photography diagnostics for dusty plasma, the positions of dust levitated by plasma only detect the positions of the extrema corresponding to the convergence of ray lines. Therefore, herein such intersections of ray lines are treated as ghost particles. The generated light field is divided into voxels with the volume of $(0.05d)^3$, and the RLD is applied to the voxel space. Figure 2(e) and (f) show the results of applying the RLD iteratively 200 times. Three district light sources result around the positions of the input. However, the profile of the reconstructed light sources is elongated along the optical axis as shown in Fig. 2(e). In Fig. 3, $\hat {\lambda }$ at the position $(0,0,-5d)$ is plotted as a function of the number of iterations. After 60 iterations, the value of $\hat {\lambda }$ converges to a value of approximately 150 and the integrated value of $\hat {\lambda }$ over the elongated area becomes approximately 1000. Finally, the stored positions of the light field intensity extrema are compared with the result of RLD. As a result, ghost particles are eliminated from the reconstruction. The positions of the reconstructed particles are consistent with the original positions.
Fig. 2. (a) Schematic of test calculation configuration with a multi pinhole camera that has six pinholes arranged hexagonally and a seventh at its center. (b) 2D elemental images. Calculated light fields on (c) the $x-z$ plane and (d) the $z = -4d$ plane. Distributions of $\hat {\lambda }$ in voxels (e) on the $x-z$ plane and (f) the $z = -4d$ plane after 200 iterations of RLD.
Fig. 3. The value of $\hat {\lambda }$ at the position $(0,0,-5d)$ is plotted as a function of number of iterations.
3. Results
3.1 Verification experiment
Next, we experimentally tested our reconstruction system using known target particles. The imaging system for 3D reconstruction of dusty plasmas is designed with a multi-convex lens array (9 $\times$ 6) and a typical reflex CMOS camera [18]. The difference between our method and that used in earlier commercial plenoptic camera is the absence of objective lenses located between the object and the lens array. This simpler observation system can be easily estimated a light field and the point spread functions. The number of the lenslet is designed to make uncertainty in the depth measurements as order of the ion debye length in a dusty plasma experiment. The parameters of the multi-convex lens array and those of our recording system are summarized in Table 1. The rays emerging from 3D objects pass through the lenslet array and are captured by the CMOS device. Three test particles are located at different points along the $z$-axis. In Fig. 4(a), open circles indicate the intensity of the generated light field $I_m$ along the $z$-axis, where the value of $z$ is the distance from the lens array. The positions of the target particles are indicated by filled circles. $I_m$ has a continuous distribution with three peaks. $\hat {\lambda }$ is indicated by colored solid lines. After deconvolution, the blurred 3D light field deblurred resulting in three individual narrow peaks. In Fig. 4(b), the positions of the reconstructed particles are compared with those of the real particles and can be seen to be in good agreement.
Fig. 4. (a) Open circles indicate the intensity of the generated light field $I_m$ along $z$-axis. The positions of the target particles are indicated by filled circles at the top of figure. Colored solid lines indicate $\hat {\lambda }$ obtained with the RLD. (b) Comparison of the positions of reconstructed and real particles.
Table 1. Parameters of multi-convex lens array and our recording system with the CMOS camera D810.
3.2 Real experiment result
Finally, we applied our system to a dusty plasma comprising monodisperse polymer spheres (diameter = 6.5 $\mu$m) floating in a horizontal, parallel-plate radio-frequency (RF) plasma (see [18] for details). An experimentally obtained elemental image array from horizontal port is shown in Fig. 5. Scattered light from the dust particles appears as bright dots and is recorded by the CMOS camera at 60 fps. The bright horizontal regions are reflections from the electrode. In this experiment, approximately 10 dust particles floated in the plasma. They moved chaotically and did not assume a stable state. An enlarged experimentally obtained elemental image array for this experiment is shown in Fig. 6(a). The $x$- and $y$-axes are indicated in the figure. The optical axis is along the $z$-axis, and the multi-convex lens array is located at $z$ = 0. The 3D particle positions are determined from Fig. 5 and are shown in Figs. 6(b) and (c). The positions of the peaks of convergence include true (filled green circle) and ghost (open red circle) particles and as is evident they are not obviously correlated with one another. After applying RLD, the ghost particles have been eliminated and the distribution of true dust particles in Fig. 6(b) is in good agreement with the appearance of the dust particles in Fig. 6(a). Figure 6(d) shows a bird’s-eye view of the positions of true dust particles. Green dots indicate the positions of the particles, and $\times$ symbols indicate the projections of the $x-y$ plane on $z=115$ mm. We can identify that the two particles located around $x=0.71$ mm, $y=0.42$ mm are separated approximately 0.9 mm along $z$-axis with each other, even they are almost overlapped in Fig. 6(b). From the reconstructed images, we observe that true dust particles are randomly distributed between 117 and 120 mm along the $z$-axis, the distance from the lens array. Ghost particles are eliminated from the result by RLD and we succeeded in distinguishing the 3D structure of a dusty plasma from a single-exposure image obtained from one view port.
Fig. 5. Experimentally obtained array of 54 elemental images of dusty plasma. The small bright green spots indicate the presence of dust particles. Each image element is recorded with approximately 800 $\times$ 800 square pixels.
Fig. 6. (a) Enlarged experimentally obtained elemental image after subtraction of a background image. The white dots indicate the presence of dust particles. (b) Projection in the $x-y$ plane. (c) Projection in the $x-z$ plane. Each panel shows the reconstructed positions of the particles, as obtained from Fig. 5 from which ghost particles have been eliminated by RLD. (d) Bird’s-eye view of the positions of true dust particles. $\times$ symbols indicate projections of the $x-y$ plane on $z=115$ mm.
The limitation of particle size is approximately the wave number of the irradiation laser because of the physical limit of Mie-scattering. The spatial resolutions for the $x$- and $z$-directions are approximately 6 and 108 $\mu$m, respectively. These spatial resolutions are determined by the pixel dimensions, i.e. resolution of data set C. In this experiment, the particles have approximately 3 pixels as their radii. Therefore, the limitation of number density is approximately 30 mm$^{-3}$.
4. Conclusion
We have applied integral photography and the RLD techniques to identify the 3D positions of particles levitating in a plasma. It has been demonstrated that RLD can be effectively employed to deblur the intensity extrema of the light field to peaks centered close to the actual positions of the dust particles. The artifacts in the light field, i.e. ghost particles, were removed by comparing the results of integral photography and the RLD. We tested our reconstruction system with known target particles and found that it works well in the range of our dust experiment. By applying the integral photography and the RLD techniques to the obtained experimental image, we identified the 3D positions of dust particles floating in an RF plasma. Ghost particles are eliminated from the result by the RLD and we succeeded in distinguishing the 3D structure of the fine particles in plasma from a single-exposure image obtained from one view port.
Validation of real experimental results and cross-checking the results with other methods are remained as future work. In order to further improve the accuracy of the proposed method, future work includes combining subpixel analyses, modern particle detection, and interpolation algorithms [25,26]. By combination with the Mie-Scattering ellipsometry technique [27], it would provide information about the size of particles in addition to the six-dimension of position and velocity already obtained. Further, to extend the present work to pinhole ultraviolet or soft X-ray detectors [28–31], as well as the lens array for visible light, to obtain 3D information for the emissivity distribution for high temperature plasma research.
Funding
Japan Society for the Promotion of Science (18K18750); National Institute for Fusion Science (NIFS20KLEP03).
Acknowledgments
A. Sanpei appreciates Prof. Y. Awatsuji of Kyoto Institute of Technology for fruitful suggestions from the perspective of researches for the integral photography technique. The author also thank Prof. H. Himura of Kyoto Institute of Technology, Prof. S. Masamune of Chubu University, and Prof. Y. Hayashi of Yamato University for valuable comments on this study.
Disclosures
The authors declare no conflicts of interest.
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