Errors Induced by Model Deformation
Model deformation generated by aerodynamic loads causes a displacement Ax = x’ – x of the wind-on image relative to the wind-off image. This
displacement leads to the deviations of the quantities Dx(Ax), h/href, c/cref,
and q0/q0ref in Eq. (7.2) from unity because the distributions of the luminescent
intensity, paint thickness, dye concentration and illumination level are not spatially homogeneous on a surface. After the image registration technique is applied to re-align the wind-on and wind-off images, the estimated variances of these quantities are var[Dx(Ax)] ~ W(V )/V2, var(h/href ) ~ W(h)/(href )2,
and var(c/cref ) ~ W(c)/(cf )2. The operator W(•) is defined as
W( •) = (д /д x)2 a 2x + (д/д y)2 a2, where ax and ay are the standard deviations of least-squares estimation for image registration.
The uncertainty in q0(X)/q0ref(X’) is caused by a change in the illumination
intensity on a model surface after the model moves with respect to the light sources. When a point on the model surface travels along the displacement vector AX = X’ – X in the object space, the variance of q0/q0ref is estimated by
var[q0(X )/q0ref(X’)] »(q0ref )~21(Xq0)• (AX )|2. Consider a point light source with unit strength that has a light flux distribution q0(X-Xs) = | X-Xs, where n is an exponent (normally n = 2) and | X – Xs | is the distance between the point X on the model surface and the light source location Xs. Thus, the variance of q0/q0ref for a single point light source is var[q0(X )/q0ref(X’)]
= n2 X – Xs I 41( X – Xs) • (AX )|2. The variance for multiple point light
sources can be obtained based on the principle of superposition. In addition, model deformation leads to a small change in the distance between the model surface and the camera lens. The uncertainty in the camera performance parameters due to this change is var(nc/ncref ) ~ [R2/(Rj + R2)]2(AR1/R1)2,
where R1 is the distance between the lens and model surface and R2 is the distance between the lens and sensor. For R1 >> R2, this error is very small.