ARAP

This web app demonstrates some of the principles underlying axisymmetric background-oriented schlieren (BOS) in connection with a paper by Sipkens et al. (XXXX).

The transforms

We first show a viz that plots the underlying transforms. The traditional route has involved computing the Abel transform. Sipkens et al. recently generalized this transform to not require parallel rays. Increasing the slope of the rays allows them to closer approach the center of axisymmetric objects (ASO), changing how the different annuli are weighted in computing a projected deflection field.

Changes in the slope of a single ray is represented here by progressively lighter colours. The Abel transform, corresponding to no slope (and thus parallel rays) is shown as a white dashed line.

The slider for y₀ alters where the rays passes through the center plane of the ASO, that is where z = 0. A ray where y₀ = 0 will pass directly through the origin, which is placed at the center of the ASO (where the radius is zero).

Projecting an axisymmetric object

Now consider projecting a phantom ASO, with the radial dependence of the phantom shown in the upper panel. We consider rays diverging from a single point in space o_c = [y_c, z_c]. Here, z_c control below adjusts the disantce between the camera and the ASO. Very large z_c (e.g., z_c > 20) corresponds to a camera looking directly at the ASO and yields a symmetric deflection field. The y_c = 0 case corresponds to a camera looking directly at the ASO and yields a symmetric deflection field. Larger values of y_c move the camera above the ASO and stretch the effect of the ASO in the negative direction, as more of those rays still pass through the ASO.

Progressing from darkest to lightest changes the the distance from the camera to the center of the ASO. The plot shows values for z_c = [20, 4, 2.5, 2, 1.6, 1.35, 1.2, 1.1, 1, 0.9, 0.8]. The control changes the height of the camera, i.e., y_c.