Mulawa - 1
Several Aspects of Modeling and Stochastic
Analysis of a Photogrammetric Estimation of a
3D Straight Line
David Mulawa
3513 Bluff View Drive
St. Charles, MO 63303 USA
email: mulawa@aol.com
Abstract
Over the last several years there has been a considerable interest by the photogrammetric
research community in the measurement of three dimensional straight lines IL.
Inherently, the line IL is more geometrically complex than a point in space. This paper
presents some additional developments for the photogrammetric treatment of the line IL.
In particular, the stochastic properties are analyzed and a three dimensional probability
surface is constructed about an estimated line DL. A fundamental point C® related to this
stochastic surface is presented. The concept of a local reference origin C° and a method
to compute initial approximations are developed. Also, experimental models for a
cylinder and a rectangular parallelepiped are given to demonstrate the extendibility of the
modeling method.
Description of a Three Dimensional Straight Line IL:{C,p|
A straight line IL in a three dimensional space can be described by its direction p and a
fixed point C on the straight line IL:{C,p}. This description of the straight line
IL :{C,p} uses six parameters, but it known that a straight line IL in a three dimensional
object space has only four degrees of freedom. Thus, two constraint equations must be
written between the parameters {C,p}. One constraint is to select a unit length for the
direction vector iipli-i. Another constraint is to fix one degree of freedom in the
position of the point C along the straight line IL. One way to do this would be to select
the point on the straight line IL that is closest to the origin 0 of the coordinate system
C • p = 0. This last constraint will be examined further in the paper. A point P, on the
line IL can be determined by the parametric equation:
P/^C + ^p (1)
Where, s, is an arc length line parameter associated one to one with the points P t on the
line IL.
