Abstract
Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal projections in a number of directions. We discuss relations between (1) a space curve and the projected curves from several distinct directions, and (2) a surface and the apparent contours of projections from several distinct directions, in terms of differential geometry and singularity theory. In particular, formulae for recovering certain information on the original curves or surfaces from their projected images are given.
Highlights
As is well known via triangulation, when we look at a point from two known viewpoints, we can calculate where the point is
When we look at a surface, we observe an apparent contour, which gives us some information about the surface
To obtain the Gaussian curvature of a surface, information about the second order derivatives of the surface is required, and in [13, 14], Koenderink showed that one can obtain the Gaussian curvature of a surface as the product of the curvature of the contour and the normal curvature along a single direction
Summary
As is well known via triangulation, when we look at a point from two known viewpoints, we can calculate where the point is. Note that we give explicit formulae for reconstructing basic information of curves and surfaces from a finite number of projected images (see Remark 1.1). Explicit formulae for reconstructing information about surfaces and curves from their projected images are useful tools in practical settings. In addition to Koenderink’s famous result [13, 14] explained above, [10] provided a formula for recovering a surface from continuous data of the apparent contours. Their works have received attention in the context of visual perception and computer vision (cf [1,2,3, 8, 22]). We remark that the k-th order information of the given function ψ at 0 represents the k-jet of ψ at 0 in the terminology of singularity theory (cf. [12])
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