Journal of Mathematical Imaging and Vision

, Volume 61, Issue 7, pp 919–943

# The Highlight Ovals

Article

## Abstract

We propose a model of plane ovals based on two foci defined in the 3D space. Geometrically, some of these ovals are the isocontours of a planar specular highlight for a point light source according to Phong’s reflectance model. We, hence, call them the highlight ovals. More precisely, we define a geometric highlight oval as the locus of points p whose product of distances to the foci is proportional to the dot product of the vectors from p to the foci. When the proportionality is taken up to sign, this gives a quartic, whose roots define the algebraic highlight oval. Similarly to the Cartesian oval, the algebraic formulation hence includes a pair of geometric highlight ovals. Similarly to the Cassinian oval, the geometric definition includes non-convex closed curves. The highlight ovals have four intrinsic parameters, namely the height of the two foci relative to the oval’s plane, the distance between the two foci’s projection on the oval’s plane and a scale factor fixing the proportionality and related to the isocontour’s intensity. We thoroughly study the topology and convexity of the family of algebraic highlight ovals for any combination of its four intrinsic parameters and show how this characterizes the geometric highlight ovals, and thus the isocontours of specular highlights. We report an extensive experimental evaluation, showing that the highlight ovals form a reliable model of specular highlights.

## Keywords

Reflectance Algebraic curves Phong

## Notes

### Acknowledgements

This research has received funding from the EU’s FP7 through the ERC research Grant 307483 FLEXABLE. We thank François Chadebecq for his kind help in taking the images for the experiments.

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