Experimentalists often use wind tunnels to study aerodynamic turbulence, but most wind tunnel imaging techniques are limited in their ability to take non-invasive 3D density measurements of turbulence. Wavefront tomography is a technique that uses multiple wavefront measurements from various viewing angles to non-invasively measure the 3D density field of a turbulent medium. Existing methods make strong assumptions, such as a spline basis representation, to address the ill-conditioned nature of this problem. We formulate this problem as a Bayesian, sparse-view tomographic reconstruction problem and develop a model-based iterative reconstruction algorithm for measuring the volumetric 3D density field inside a wind tunnel. We call this method WindDensity-MBIR and apply it using simulated data to difficult reconstruction scenarios with sparse data, small projection field of view, and limited angular extent. WindDensity-MBIR can recover high-order features in these scenarios within 10% to 25% error even when the tip, tilt, and piston are removed from the wavefront measurements.
Wavefront sensing is a class of imaging techniques that can be used to detect variations in a 3D density field due to high-speed airflow. 1 When a coherent light source illuminates a turbulent airflow, changes in refractive index due to variations in density can affect the optical path length (OPL) of the light. Wavefront sensing techniques such as digital holography or Shack-Hartmann wavefront sensors can precisely measure the optical path difference (OPD) over a coherent beam's aperture, allowing researchers to non-invasively detect density variations for a turbulent airflow. However, OPD is a path-integrated measurement along a line of sight and so does not yield point estimates of the underlying 3D density. Alternatively, wind tunnel flow seeding can provide 3D point information, but it is invasive and difficult to perform. 2 Computational fluid dynamics (CFD) Approved for public release; distribution is unlimited. Public Affairs release approval # 2025-5579.
is non-invasive and can model the 3D flow dynamics, but CFD simulations often fail to match experimental tests. 3,4 Wavefront tomography is an alternative non-invasive method to estimate the 3D density field of a turbulent volume. 5 While similar to parallel-beam x-ray tomography, wavefront tomography for volumetric wind tunnel reconstruction is challenging is due to sparse views, limited field of view, limitations in angular extent, and incomplete projection information. To capture the dynamic flow field of wind tunnel turbulence, all wavefront measurements must be acquired at the same time. The physical constraints of imaging inside a wind tunnel limit the number and angular extent of wavefront measurements that can be obtained simultaneously, turning this into a limitedangle, sparse view tomography problem. Furthermore, 2D OPD measurements are not complete projections of the turbulence refractive index because they lack the low order tip, tilt, and piston (TTP) information. By definition, OPD is blind to the mean OPL over the aperture (i.e., piston), and mechanical disturbances often contribute an unknown tip-tilt to the wavefront, which effectively obscures the tip-tilt contribution from the turbulence. In some cases, it is possible to recover the complete projection information if the projection field of view (FOV) is wider than the medium of turbulence. 6 However, for realistic wind tunnel experiments this is not possible because the FOV is limited by the beam diameter. Lastly, most wind tunnels have physical viewing constraints that also restrict the angular extent of the projections, making it especially difficult to resolve features along one axis of the reconstruction.
Wavefront tomography has been explored extensively in the field of microscopy for the goal of reconstructing the 3D sample refractive index. 7 While many of the methods used in this field are designed for limited angle or sparse viewing scenarios, they do not consider scenarios where the object of reconstruction is significantly wider than the projection FOV. Furthermore, since the FOV Approved for public release; distribution is unlimited. Public Affairs release approval # 2025-5579.
is wider than the sample, the additional projection information outside the sample (i.e., through the background reference medium) enables the recovery of the complete projection information (i.e., TTP included) for the sample.
Several wavefront tomography methods have been developed for wind-tunnel turbulence reconstruction, but none of them use a Bayesian model-based reconstruction framework, which is well established in the field of x-ray tomography for regularization and artifact removal. 8 Some reconstruction methods implicitly perform regularization by parameterizing the reconstruction with a low dimensional smooth orthogonal basis, 6,9 but these techniques do not use an explicit Bayesian prior model. Some sophisticated approaches have been developed to account for missing projection information, which either incorporate additional reference information 6,9 or an iterative update loop, 10,11 but in all cases they do not use an explicit Bayesian prior model to smooth out the artifacts resulting from the unknown or incorrect projection information. Furthermore, most of the methods designed for turbulence reconstruction are not designed for difficult imaging scenarios where the issues of sparse view, limited angle, small FOV, and unknown projection TTP are all present. For instance, some methods assume that the turbulent medium is radially symmetric, 9,10 which means that a single measurement can reveal all 180 degrees of projection information. Alternatively, other methods assume that the turbulence stream induces a steady flow field, 12 which means that many measurements can be taken over the course of an extended time interval. While some investigations into wind tunnel wavefront tomography did assume scenarios with a severely limited number of measurements, none of them considered c
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