1. Motion of specularities on smooth random surfaces
Michael Langer Yousef Farasat
School of Computer Science, McGill University, Montreal, Canada
light source (sun)
at infinity
not
visible V
“New” observation: specular reflections and visibility windows
The set of rays in which any given specularity is visible must begin at the surface region that defines that specularity, and must pass through that specularity. Thus, at any instant of time, the local surface region defines a window in which that specularity may be visible. This window is defined by the region over which the surface is a parabola:
where fx and fy are the “focal lengths” which are roughly constant in the window. In the figure, the window is the surface region bounded by the red lines. (Previous studies of specularity motion expressed this windowing effect in terms of caustics.)
The observer moves relative to the specularity and relative to the surface window defining the specularity. The surface window and its specularity are at different depths, and thus there is motion parallax between them. This parallax gives rise to “second order motion”, namely the resulting image is the product of the image of the moving specularity and the moving window.
A surface having many concave and convex parts has many specularities and these lie in front of and behind the surface. One might expect that when an observer moves laterally, the motion of the specularities yields rich motion parallax. Unfortunately, the windowing severely restricts the visibility of the specularities.
CONCLUSION: The relative motion of specularities is primarily parallel to the observer motion. However, each specularity is visible over a limited distance of observer motion (or baseline, in case of stereo). From a computational perspective, these local windows make it difficult to track/match specularities. V
visible
Background: specularities and motion parallax
(e.g. Koenderink & van Doorn 1976, Zisserman et al. 1989, Blake & Buelthoff 1990)
Curved surfaces that have a mirror-like reflectance produce images of light sources (specularities). A convex region produces a specularity behind the surface. A concave region produces a specularity in front of the surface.
The distance from the surface to the specularity depends on surface curvature. If the curvature is high (low) then the distance is small (large).
When an observer moves laterally relative to the surface, motion parallax results. If the surface is convex (concave), then the specularity moves slower (faster) than the surface.