Dimensional Analysis through Perspective,
by James R. Williamson & Michael H. Brill
ASPRS, 1990
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| Locating the Vanishing Points |
| Locating the Principal Point |
| Determining the Perspective Orientation Matrix |
| Determining the Focal Length |
| Determining measurements/lengths |
| Definitions |
Given two lines which are known to be parallel in world space, a VP can be found at their intersection. If the lines are parallel in the image, then the VP is said to be at infinity.
Given a single line where 3 points define 2 segments and the length ratio of these 2 segments is known, the VP can be determined.
Given the x,y coordinates of one VP as well as the PP and the focal length, a second VP can be determined.
Given any two VP's and the PP, the third VP can be determined.
Given four fiducial marks (usually on the edges of the image) 1, 2, 3, 4 in clockwise order, the PP can be determined by finding the intersection of the 13 and 24.
If the image is know to be uncropped, then the four corners can be considered as fiducial marks, and the PP is the intersection of the image diagonals.
If the azimuth (a), tilt (t), and swing (s) are known, [R] is easily determined:
| [ -cos(s) | -sin(s) | 0 ] |
[ 1 | 0 | 0 ] |
[ cos(a) | -sin(a) | 0 ] | |
| [R] = | [ sin(s) | -cos(s) | 0 ] |
[ 0 | cos(t) | sin(t) ] |
[ sin(a) | cos(a) | 0 ] |
| [ 0 | 0 | 1 ] |
[ 0 | -sin(t) | cos(t) ] |
[ 0 | 0 | 1 ] |
Given known diagonals in the image of 45 degrees, two VP's can be defined to the left and right of the PP. The distance from each of these 45 degree VP's to the PP is the focal length.
Given two major VP's and the PP, the focal length can be determined.
Every object-space plane parallel to the image plane, in the depth of field or focus, has a unique scale that is constant in that plane. Then, if you know one dimension in one of those object-space planes, you can determine the scale in all of the planes parallel to the image.
The point in space occupied by the camera lens at the moment of exposure. The point acts as the perspective center of the photograph (the "lens rear nodal" or focal point) for which the bundle of visual rays from object space pass to enter image (camera) space. N.B.: Basic assumption: the object point, perspective center and image point all lie in the same straight line.
Distance from lens rear nodal point to the image plane. In close-range photogrammetry, this lenght will vary in general from image frame to image frame.
Point of the celestial sphere directly beneath to the observer or camera/eye, and opposed to the zenith (directly above the observer). In photogrammetry, this is taken as the point of intersection with the image plane of a vertical or plumb line (along the Z axis of the Object space) passing through the perspective center of the camera lens (CS). The ground/map nadir is the point on the ground/map directly beneath the CS.
The 3 x 3 orthogonal matrix, that gives the direction-cosine values of the principal ray in the object space system, relative to the vertical (down) direction. Contains nine cosines direction values.
Line of intersection of the Principal Plane with the Image Plane.
The intersection of the principal ray and the image plane. This is usually the origin of the image-space coordinate system. It contains the geometric description of the tilt angle.
Defined by the three image points of PP, nadir and the lens nodal point (CS). This plane is perpendicular to the image plane.
The ray from the CS (lens nodal point) that intersects the image plane at right angles. This is the only light ray which is perpendicular to the image plane (in a perspective projection).
The range is the distance from the camera lens (CS) to the point of focus, in the depth of field (on a given object point).
The scale is the effective focal length divided by the range: S = f' / R .
One of the 3 perspective reference lines representing the intersection of the 3 orthogonal geometry planes and the image plane. The THL (in 1 and 2 PtP) is the trace of the horizontal plane, through the camera station (CS), across the image plane. The PP lies on this THL, where the principal ray intersects the THL. Note that in this plane a circle is defined with diameter given by the 2 VPX & VPY, diameter coinciding with the THL. The CS is a point on this circle. The CS is also obtained as a vertical from the THL emerging at the position of the PP: this vertical intersects the circle at CS.
The point on the image toward which all common parallel lines in image-space converge. These parallel lines may be horizontal, vertical, diagonal, or at any angle.
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