Shading from shape, the eikonal equation solved by grey-weighted distance transform

Abstract For an optical or acoustical wavefront running through a medium of space variant refraction index the eikonal equation connects local front arrival time with local refraction index. So-called difference approximation methods are known for solving the spatial wavefront development with time and thus, indirectly, the eikonal equation. Here a novel fast method for the calculation of an approximative solution of the eikonal equation is proposed. From literature it is known that by solving an eikonal equation one can construct a line pattern rendition of a given image. We have generalized this method and made it fit for line engravings. We have found yet another kind of image display based on solving an eikonal equation: shading from shape. We propose to construct a matte 3-D surface (shape) that, when illuminated perpendicularly and imaged in eye or camera, yields a grey value (shading, luminance) field that renders the image. Both methods have been applied in a recent design for a Dutch coin.