Unique solvability for the problem of recovering an optical surface from its three images

Vladimir S. Ladyzhets

doi:10.1117/12.251817

30 September 1996 Unique solvability for the problem of recovering an optical surface from its three images

Vladimir S. Ladyzhets

Proceedings Volume 2826, Vision Geometry V; (1996) https://doi.org/10.1117/12.251817
Event: SPIE's 1996 International Symposium on Optical Science, Engineering, and Instrumentation, 1996, Denver, CO, United States

Abstract

The following problem is considered: given three image intensities (photographs) of an opaque object surface, reconstruct the shape of the surface and the surface local reflecting properties. This problem is formalized on the basis of rigorous description of object photoimage formation (in the framework of geometrical optics and photometry) and resulted in three concepts: an optical surface (the mathematical description of an opaque object surface), a projection schema (the mathematical description of an image- formatting optical apparatus as a (photo)camera), a projection-schema image (the mathematical description of an image intensity). The usage of these concepts allows one to formulate the opaque object surface reconstruction problem as the mathematical problem of reconstructing an optical surface from its three projection-schema images. It is shown that the optical surface reconstruction (OSR) problem takes the form of a Cauchy problem for the system of three first- order, non-linear, partial differential equations. A general uniqueness theorem for the OSR problem is proven. Also, the OSR problem for a specific class (`diffuse-speculum' surfaces) of optical surfaces is investigated. The conditions providing the unique reconstruction of a diffuse- speculum optical surface have been found in the form of `pure' geometric conditions (the restrictions on the relative positions in the space of the source of illumination and cameras) as well as in the form of some restrictions on image intensity distributions which to be used if the geometric conditions are not fulfilled.

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Vladimir S. Ladyzhets "Unique solvability for the problem of recovering an optical surface from its three images", Proc. SPIE 2826, Vision Geometry V, (30 September 1996); https://doi.org/10.1117/12.251817

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KEYWORDS

Geometrical optics

Opacity

Photography

Cameras

Projection systems

Reflectivity

Reflection

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