MASSACHUSETTS INST OF TECH LEXINGTON LINCOLN LAB
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In this report, a new theory analyzing the relations between 3-D convex objects and their silhouettes in orthographic projections is presented. The theory is based on three new representations of 3-D surfaces in terms of scalar, vector and tensor functions on the Gaussian sphere, and the matching representation of 2-D curves by functions on the Gaussian circle. The key advantage of these representations is that a slice through the spherical representation of a 3-D object is closely related to the circular representation of the silhouette of the object in a plane parallel to the slice. This relation is formalized in three Silhouette-Slice theorems, which underline the duality between silhouettes in object space and slices in the representation space. These theorems apply to opaque objects and have a conceptual similarity with the Projection-Slice theorem, which applies to absorbing objects. Silhouette construction with the theorems is demonstrated by examples of silhouettes of complex curved surfaces. Applications to the reconstruction of object shapes from silhouette measurements and to the recognition of objects based on their silhouettes are suggested.
- Theoretical Mathematics