Optical-flow-based background-oriented schlieren technique for measuring a laser-induced underwater shock wave

The basic equations in the BOS technique are briefly recapitulated for convenience of reading. The working principle of the BOS technique is illustrated in Fig. 1 (Venkatakrishnan and Meier 2004; Yamamoto et al. 2015). A typical BOS system consists of a camera, a background pattern plate (such as a random-dot pattern plate), a light source, and a computer for image/data processing. For simplicity, it is assumed that the background reference plane (x, z) is parallel to the image plane. Thus, the image coordinates are equal to \(\lambda (x,z),\) where \(\lambda\) is a proportional constant in the orthographical projection. The light ray radiated from the background pattern plate is deflected from the original straight path through a fluid domain with the density gradient due to optical refraction, and therefore, the image of the background pattern has an apparent displacement (shift) field.

The displacement on the image plane is proportional to the path integral of the small density gradient through a fluid domain. For BOS setting, Hinsberg and Rösgen (2014) gave the following relation for the in-plane displacement vector \({{\varvec{w}}}^{\prime }\) in the background platewhere c (100 μm in this case) is the thickness of the density gradient domain, \(l_{0}\) (8 mm in this case) is the distance from the density gradient domain to the background plate, \(n_{0}\) is the refractive index of water at the condition without a shock wave, n is the refractive index of water in the test condition, and \(\nabla\) is the gradient operator on the coordinate plane (x, z). The relation between the refractive index and the fluid density is given by the Gladstone–Dale equation (Merzkirch 2012; Raffel 2015)where K is the Gladstone–Dale constant (3.34 × 10⁻⁴/kg for water) and \(\rho\) is the density of the fluid. Substitution of Eqs. (2) into (1) yieldswhere \(\rho _{0}\) is the fluid density under hydrostatic pressure. It is emphasized that the displacement vector \({{\varvec{w}}}^{\prime }\) obtained by the BOS technique is the path-integrated (or projected) quantity across the measurement domain with the density gradient. In general, to extract the 3D field from the path-integrated quantity requires tomographic reconstruction from data at multiple viewing directions.

Further, applying the dot product of \(\nabla\) to Eq. (3), we have the Poisson’s equation for the densitywhere the source term S is proportional to the divergence of the displacement vector \({{\varvec{w}}}^{\prime }\) byIn principle, when the displacement vector \({{\varvec{w}}}^{\prime }\) is measured, the density field can be determined by solving Eq. (4). The Gauss–Seidel method is used to solve Eq. (4) numerically here. Then, the pressure field can be further determined by applying the Tait equationwhere \(p_{0}\) is the hydrostatic pressure, B is a constant of 314 MPa, and the exponent \(\alpha\) is 7 (Brujan 2010; Yamamoto et al. 2015). In this study, we measure a spherical shock wave in water with the peak overpressure of about 1 MPa. In this case, the density change due to the overpressure is >1 kg/m³. This means that the change in the refractive index is very small [O(10⁻⁴)].

The BOS technique acquires an undisturbed background pattern image and the corresponding disturbed image. Figure 2 shows the typical images obtained in the experimental setup in this work. Figure 2i shows a shadowgraph snapshot of a shock wave propagating spherically around a laser-induced bubble, where the high-pressure region is expected at a shock front. Figure 2ii, iii shows an undisturbed background reference image and the corresponding disturbed image, respectively. The zoomed-in image of the particle pattern is shown as well. Without image processing, it is difficult to see directly the shock wave in Fig. 2iii since the change of the refractive index of water is so small.

The key element in the BOS technique is to determine the displacement vector field in the undisturbed background reference image and the corresponding disturbed image. This computation is usually carried out by using the cross-correlation method in PIV. An effort has been made to adopt the classical optical flow methods in computer vision science in BOS (Atcheson et al. 2009). The optical flow method can be further developed based on rational physical foundations for various flow visualizations (Liu and Shen 2008). The optical flow is described by the generic equation in the image plane, i.e.,where \({\mathbf {u}}\) is the velocity in the image plane referred to as the optical flow, g is the normalized image intensity, \(\nabla\) is the spatial gradient in the image plane, and f is a term related to the diffusion and boundary fluxes which are negligibly small in most cases. In velocimetry, the optical flow in Eq. (7) has a clear physical meaning, that is, the optical flow is proportional to the light-ray-path-averaged velocity of fluid or particles in flows. We call this method as the physics-based optical flow technique. In a special case where \(\nabla \cdot {\mathbf {u}} = 0\) and \(f = 0\), Eq. (7) is reduced to the Horn–Schunck brightness constraint equation which is the foundation of the classical optical flow method developed by Horn and Schunck (1981). It is noted that in a general case the optical flow is not divergence-free and \(\nabla \cdot {\mathbf {u}} = 0\) is not physically true. For BOS applications, the difference \(\partial g\)/\(\partial t\) \(\approx\) \(\Delta g\)/\(\Delta t\) is used in Eq. (7), where \(\Delta g = g - g_{ref}\) is the difference between the disturbed image and the undisturbed reference image, and the nominal time interval is unitary (\(\Delta t = 1\)). Therefore, the optical flow in Eq. (7) is interpreted as the displacement vector field in the image plane (i.e., \({\mathbf {u}} = {{\varvec{w}}}\)) that is generated by the deflected light ray through the density gradient domain. The displacement vector in the background plane is related to that in the image plane by \({{\varvec{w}}} = \lambda {{\varvec{w}}}^{\prime }.\)

To determine the displacement vector field in the image plane, a variational formulation with a smoothness constraint is typically used (Liu and Shen 2008). By minimizing the functional, the Euler–Lagrange equation is given for the optical flow. The standard finite difference method is used to solve the Euler–Lagrange equation with the Neumann condition on the image domain boundary for the optical flow. The optical flow algorithm used in this work has the routines: the Horn–Schunck estimator for an initial solution (Horn and Schunck 1981) and the Liu–Shen estimator for a refined solution of Eq. (7) (Liu and Shen 2008). The relevant parameters in preprocessing and optical flow computation should be suitably selected. The main parameters are the Lagrange multipliers for the Horn–Schunck and Liu–Shen estimators. Other parameters are the number of iterations in successive improvement of optical flow computation by using a coarse-to-fine iterative scheme, and the sizes of the Gaussian filters for correcting the effect of a local illumination intensity change and removing small random noise in images. A mathematical analysis of the physics-based optical flow and an iterative numerical algorithm are given by Wang et al. (2015). Quantitative comparison between the optical flow and cross-correlation methods for PIV images has been evaluated by Liu et al. (2015). The variational optical flow method based on Eq. (7) is a differential approach that can achieve the theoretical spatial resolution of one vector per pixel. Figure 3 illustrates the difference between the optical flow method as a differential approach and the cross-correlation method as a region-based integral approach. The optical flow method is particularly suitable to detect small displacements in narrow regions (such as shock wave).

The tomographic reconstruction technique is used to reconstruct the relevant physical quantity (such as the fluid density or the divergence of the displacement vector) on a spherical surface of a propagating shock wave from the path-integrated or projected quantity obtained from the BOS measurements. In volumetric flow visualizations, an image can be modeled as a set of 1D projections of a field of a quantity with the given projection angles through a 2D domain as illustrated in Fig. 4. The basic tomographic problem is how to reconstruct the 2D field from these projections. The projection of the quantity q(x, y) can be expressed as an integral along a ray, i.e.,where \(l=x\cos \theta +y\sin \theta\) is the coordinate along a projection line, \(\theta\) is the angle defining the projection lines, R(q) denotes the Radon transform, and \(\delta\) denotes the Dirac delta function. An inversion of the Radon transform is sought for q(x, y). This tomographic problem has been studied in 3D flow measurements (Feng et al. 2002; Venkatakrishnan and Meier 2004). We introduce the Fourier transformThen, the solution of the integral equation Eq. (8) can be formally given byClearly, the tomographic reconstruction of q(x, y) requires many projections in \(\theta\) \(\in\) \([0,\pi ].\) In actual computation, the inverse Radon transform in MATLAB is used in this work. In this process, we generate the sinogram digitally. Although the Abel transform is simpler for the reconstruction of an axial symmetrical field, we do not utilize it because of its high sensitivity to noise (Venkatakrishnan and Meier 2004).

In the BOS measurement of a spherical shock wave, the following procedures are proposed as illustrated in Fig. 5.It is necessary to comment the selection of the projected quantity for tomographic reconstruction. In the BOS measurements proposed by Venkatakrishnan and Meier (2004), the projected fluid density was obtained first by solving the Poisson’s equation, and then, the local density was reconstructed by using the tomographic technique. In contrast, the present procedures conduct the tomographic reconstruction before solving the Poisson’s equation. The projected quantity is the divergence of the projected displacement vector, i.e., \(Q(l,\theta ) = \nabla \cdot {{\varvec{w}}}^{\prime },\) and thus, the divergence of the local displacement vector is first reconstructed by using the tomographic technique. Next, the local density is calculated by solving the Poisson’s equation. In principle, since the Poisson’s equation is valid for both the local and path-integrated (projected) quantities, the approach used by Venkatakrishnan and Meier (2004) should be equivalent to the present approach. Nevertheless, the rationale behind the present arrangement is to avoid the propagation of the error in solving the Poisson’s equation into tomographic reconstruction that is more sensitive to the errors.

A typical case with the laser power of 1.2 mJ is considered to compare the results obtained by using the OF-BOS and the PIV-BOS. Figure 8 shows the displacement magnitude fields obtained by using the OF-BOS and the PIV-BOS. The OF-BOS gives \(2008\times 1085\) vectors, while the PIV-BOS gives \(250\times 138\) vectors in the same domain in Fig. 8. In optical flow computations, the Lagrange multipliers for the Horn–Schunck estimator and the Liu–Shen estimator are 20 and 2000, respectively. In PIV computations, the window size is iteratively reduced from the initial size of 32 × 32 pixels to 16 × 16 pixels and then to 8 × 8 pixels. Both the OF-BOS and the PIV-BOS capture the sharp change induced by the shock wave in the displacement fields. As expected, the cross-correlation computations in windows in the PIV-BOS tend to smooth out sharp features like the shock waves. Figure 9 shows the profiles of the measured displacement along a ray aligned with the hydrophone, where the averaging operation over 50 lines for the OF-BOS and 20 lines for the PIV-BOS in the z-direction is applied to reduce the noise. The displacements were estimated from pressure data by using Eqs. (3)–(6) with an assumption of constant speed of propagation. It is indicated that both the OF-BOS and the PIV-BOS give results consistent with that given by the hydrophone at that location. In particular, the displacement induced by the shock wave is well captured by both the techniques. Nevertheless, the OF-BOS achieves a much higher spatial resolution.

Figure 10 shows the reconstructed pressure fields obtained by using the OF-BOS and the PIV-BOS. The OF-BOS capture correctly the sharp pressure change across the shock wave, while the PIV-BOS has a larger error in the reconstructed pressure field particularly near the ‘north pole’ and ‘south pole’ due to its low spatial resolution. For quantitative comparison, as shown in Fig. 11, the pressure profiles reconstructed based on the displacement vector fields given by the OF-BOS (2008 × 1085 vectors) and PIV-BOS (250 × 138 vectors) are plotted against the data obtained by the hydrophone. The pressure profile given by the OF-BOS along a ray aligned with the hydrophone is consistent with that given by the hydrophone. In contrast, the pressure profile obtained by the PIV-BOS exhibits a considerably broader distribution extending to the inner region although its peak value at that location agrees with the data given by the hydrophone. Clearly, the OF-BOS provides the more accurate reconstruction of the pressure field of the shock wave. In contrast, the PIV-BOS has a much larger deviation from the data given by the hydrophone, which is particularly contributed by the large errors near the ‘north pole’ and ‘south pole’ (as shown in Fig. 10). In this case, the major issue of the PIV-BOS is its much lower spatial resolution that tends to corrupt the tomographic reconstruction of the shock wave.

Measurements at different levels of the laser power were conducted to further compare the data obtained by using the OF-BOS, the PIV-BOS, and the hydrophone. Figure 12 shows the typical displacement distributions induced by the laser-induced shock wave at three levels of the laser energy, where the OF-BOS and PIV-BOS data are obtained along the ray aligned with the hydrophone. Overall, the OF-BOS and PIV-BOS data are in good agreement with the hydrophone data in the low and medium levels of the laser energy (0.6 and 1.4 mJ). The maximum pixel displacements obtained by OF-BOS are 0.8, 1.7 and 2.3 pixel, respectively. In the cases of the higher levels of the laser energy (2.1 and 2.9 mJ) with larger displacements, a coarse-to-fine iterative scheme is adopted to improve the accuracy of optical flow computation (Liu et al. 2012; Liu et al. 2015). In this scheme, images are initially downsampled by 2 for a coarse-grained velocity field and then a refined velocity field with the full image resolution is obtained in iterations for correction of the large displacements. Two and three iterations are applied to the cases of 2.1 and 2.9 mJ, respectively. Figure 13 shows the corresponding pressure distributions of the laser-induced shock wave at three levels of the laser energy. The OF-BOS is able to detect the weak shock wave and the pressure peaks of the shock wave are consistent with those given by the hydrophone. Figure 14 shows the peak pressure value of the shock wave as a function of the laser energy, indicating the favorable comparison between the data obtained by using the OF-BOS and the hydrophone. The PIV-BOS also detects the peak pressure in the lower energy cases, but the pressure distribution inside the shock front does not match that of hydrophone.

It is conjectured that the spatial resolution of the extracted displacement field would have a significant impact on the tomographic reconstruction of the pressure field of the shock wave. This is because the spatial derivatives of the displacement vector field should be calculated accurately for the tomographic reconstruction of the divergence of \(\nabla \cdot {{\varvec{w}}}^{\prime },\) which will have the direct influences on the extracted density and pressure fields. To examine this problem, the displacement field of 2008 × 1085 vectors in the half-circle domain obtained by the OF-BOS is downsampled into several fields with the lower resolutions. Then, the tomographic reconstruction procedures are applied to these fields to observe the effect of the spatial resolution. Figure 15 shows the reconstructed pressure fields based on the selectively downsampled displacement fields, where the field obtained by the PIV-BOS is also shown for reference. It can be observed that the quality of the reconstructed pressure field is degraded as the spatial resolution decreases. The degraded result given by the PIV-BOS is particularly obvious.

The quality of the pressure field particularly near the shock wave directly depends on the spatial resolution of the reconstructed field of the divergence of the displacement field, which is illustrated in Fig. 16. Furthermore, as shown in Fig. 17, the effect of the spatial resolution on the reconstructed pressure field can be quantitatively observed in the pressure profiles along the ray aligned with the hydrophone. For the OF-BOS, the displacement field of 2008 × 1085 vectors is downsampled to that of 268 × 148 vectors. As shown in Fig. 17a, the reconstructed pressure distribution exhibits that the pressure peak is broadened increasingly with the decrease in the spatial resolution and the peak value deviates from the data given by the hydrophone. In addition, the displacement field obtained by the PIV-BOS can be interpolated to generate a pseudo-high-resolution field of 2091 × 1000 vectors. As shown in Fig. 17b, such interpolation could indeed improve the quality of the reconstructed pressure field even though the peak value becomes lower.