Image Jets, Level Sets, &  Silhouettes

The boundaries of objects are a special focus of computer graphics and computer vision, and the source of some vexing unsolved problems. Object boundaries deserve special attention in image-based rendering, since both vision problems of recognition and localization, and graphics problems of interpolation and display, must be solved for the successful rendition of image-based objects.

Unlike internal edges, discontinuities in texture, and the edges of cast shadows, multi-image silhouettes do not "correspond" with one another, and cannot be used as control features in a morphing deformation. Nevertheless, it is important that silhouettes of objects change continuously and believably as the objects are transformed, and that the silhouettes of opaque objects remain opaque under general viewing conditions. Object silhouettes must exactly match the mattes of our image-based rendering when the viewpoint coincides with one of the original views.

For these purposes, and for many other useful applications in image-based rendering and its parent disciplines, level set methods recommend themselves. In particular, the level sets of phase functions of silhouettes deserve exploration. In computer graphics, such functions arise in the representation of flowfields, and in the implicitization of boundary representations of objects. In computer vision, a traditional study is the evolution of contours or level sets as a function of spatial frequency cutoff, or resolution; this is the meaning of the term, "scale," in "scale space." In image processing, pyramid and filter-bank methods have been applied to extrapolate images, or to estimate missing portions of images. Level set operations naturally extend these methods to silhouettes and binary segmentations.

One of the attractive features of implicit surfaces as modeling primitives is the ease with which shapes can be blended or interpolated, without topological restrictions. This feature is important for synthesis in image-based rendering.

The disk on the left is "inflated" on the right, by a pseudometric function of the boundary.

A signed implicitization, and a sweep of its level sets.

Automatic morphing of contours

In order to blend between or among shapes, it is possible to first convert the shapes into implicit form. Then, a blending of the density functions (for which the original shapes are level sets) produces and interpolation of shape. Booleans -- implicit functions which simply categorize space -- do not provide continuous interpolation. If the shapes are "implicitized" as signed pseudometric functions of the surface boundaries, shapes will grow or shrink continuously as the boundaries reconform to their targets.


Matte of Nefrotete, positive implicitization (inflation) of matte; inflation of matte blended with image.

This 3D model of Nefrotete is a 2-sided inflation of her matte

For symmetric objects, single silhouettes can produce reasonable approximations.

Alexander the Great

Alexander the matte, a smooth inflation of the matte, and a smooth inflation with 5% image.

The models on the left are smooth inflations; right, 5% texture blended into relief.

Shape from silhouette

"Shape from silhouette" is a vision problem of persistent interest for at least two reasons: