Synthetic Images
A Brief Summary of the Synthetic Image Generator Methods
Generating each image begins with a 3-dimensional array of zero values (228 x 228 x 18) representing the $y$, $x$, and $z$ space of interest. The image outputs consider only the centered (Region size x Region size x 16) values providing particles with 228 subtract Region size pixels of $x$ and $y$, movement and one pixel of $z$ movement prior to edge interference. Particles were uniformly randomly generated (random python module) within this space until the number of particles within two standard deviations of the laser sheet width per area ($xy$) meet the desired Particle density variable. The laser sheet intensity is assumed gaussian as described by Raffel et al. [1] and follows Eq. 1.
Here $I_{p}$ indicates the maximum particle intensity, $z$ represents the depth of a particle in the direction orthogonal to the calibration plane ($x$,$y$), and $l$ represents the laser sheet width (prescribed 4 pixels). The maximum particle intensity $I_{p}$ is the Camera bit depth variable (the maximum bot depth of the camera of interest). Continuing from this understanding, the contribution of each diffraction limited particle ($i$) to the total pixel value at ($x$,$y$) is described by Eq. 2[1].
To account for particle overlap, the contribution of each particle to a given pixel value was summed over all particles, then the pixel values capped at Camera bit depth. To save computational time, Eq 2. was calculated for each particle outward from the particle center until the intensity value contribution drops below one. Using these equations, the particle centers are translated between frames using a linear approximation according to the prescribed ($u$,$v$) of the flow field. A key resulting limitation of this image generator is then the current inability of these linear approximations to accurately describe curved flows.
In addition, the potential for particles to be lost/gained from the first and second image in an image pair (i.e. out of plane motion) was accounted for through Sheet angle (the relative orientation of the laser sheet to the flow field). This approach assumes that the flow field is entirely parallel to the calibrated $x$, $y$ plane. However, the misalignment of the laser sheet to the flow field results in a $z$ component of velocity equal to the cosine of the angle between the laser sheet and the calibrated plane (θ) multiplied by the displacement along the angled direction. Equation 3 describes this $z$ displacement as a function of $x$ displacement.
An important note for Eq. 3 is that the laser sheet is assumed perfectly aligned with the calibration plane in the $y$ direction. Inspection of Eq. 1 shows that the particle peak intensity must be adjusted from image to image as the $z$ position of each particle varies across observations (i.e., across blurring and frame to frame) and this is withheld in the streaked particle calculations. Our observations also show the noise in a typical digital camera is approximately lognormally distributed. To reflect this, the synthetic image generator adds an intensity value to each particle and then these pixel intensity values are then recapped at the Camera bit depth. The intensity values are added according to a lognormal random distribution determined by the variables Noise Mean (the mean of the lognormal distribution) and Noise SD (the standard deviation of the lognormal distribution).
Finally, in addition to the standard synthetic image parameters (i.e. flow field, particle density, particle diameter, particle peak intensity, and region size), the synthetic image generator also allows the user to prescribe the Blur Count (the number of particle images per single frame image), ShutterU/ShutterV (the space between single frame particle images), and Frame Rate Ratio (the space between particle images between image 1 and image 2, determined through the Frame Rate Ratio*ShutterU/ShutterV to align with the flow physics). These variables are motivated by issues common to educational PIV, and for non-streaked particles (unchecking streaked particles), the displacement between image 1 and image 2 are prescribed by the variables U and V, with blur count assigned a value of 1. While the space between particle images and true particles from frame to frame is a function of the flow velocity, laser pulse frequency, and imaging equipment, the descriptors chosen here provide the streaking parameters in measurable terms of interest.
*Variable names are Italicized for clarification.
Resources:
[1] M. Raffel, C. Willert, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide. 1998.
Author: Jack Elliott
Date Published: June, 2022