Research Papers

Perspective Depth Extraction of Points on a Surface Under an Instantaneous Rigid Body Transformation

[+] Author and Article Information
Per Bergström

Department of Engineering Sciences and Mathematics,  Luleå University of Technology, SE-971 87 Luleå, Swedenper.bergstrom@ltu.se

J. Comput. Inf. Sci. Eng 12(2), 021003 (Feb 10, 2012) (6 pages) doi:10.1115/1.4005721 History: Received November 09, 2011; Revised November 09, 2011; Published February 10, 2012; Online February 10, 2012

We consider the computational problem of finding the point in 3D-space on a transformed surface corresponding to a coordinate pair given in a perspective mapping. The transformation is a rigid body transformation that is assumed to be small and vary. Initially, it is unknown but when it becomes known, the output must be accurate and quickly returned. Therefore, the computations are adapted for those conditions. Preprocessed shape information about the surface is computed in a perspective mapping where the surface is in an original position. We are discussing algorithms for solving the considered problem.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Idea of computational process

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Figure 2

Illustration of the perspective geometry in 3D-space. The drawn surface is in transformed position. A line goes from the perspective eye and intersects the surface.

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Figure 4

Surface of CAD-model in a perspective mapping. Perspective parameters are u and v. Each pixel represents a perspective coordinate pair. (a) Surface in original position, (b) transformed surface by, T-1, Eq. 4.

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Figure 5

Surface in original position with preprocessed shape information

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Figure 6

Illustration of the lines L and L¯ and their, respectively, surface intersection

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Figure 7

Illustration of convergence for the two algorithms. The circle marks the intersection point.

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Figure 8

Perspective view of the transformed surface. The circles mark the given perspective coordinates and the connected plus signs mark the iterative values of (μ̂,ν̂) in Eq. 10 for the two algorithms.

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Figure 9

Convergence of δ in Eq. 11 for the two algorithms. The same gray-level as in Fig. 8 distinguishes the three different given perspective coordinate pairs.




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