Collinearity equation

The Kollinearitätsgleichung on the mathematical and geometric foundations of collinear image. A typical example of a collinear diagram shows a central projection. Here are precisely mapped back to straight, preserved division ratios.

Applications of Kollinearitätsgleichung can be found in all fields of optics and optical imaging, especially in the optical surveying, photogrammetry and other indirect measurement techniques (eg, flow rate of a water body, bending strength of materials). Most will be recalculated on the coordinates of the observed object points of the recorded pixels. Pixel, the center of projection and the observed object point lie on a straight line.

In known 3D coordinates of the object points to their image coordinates can be calculated. This corresponds to the photographic image of the object points with known camera position. The calculation is based on the model of a pinhole camera which is the technical implementation of the central projection in the ideal case. As a mathematical formulation of the central projection serving collinearity equations for transformation of the coordinates of the individual points. This is multiplied substantially with a 3x3 rotation matrix:

It xyz coordinate system with the x and y axes in the image plane. The point P is mapped by means of a central projection on the image surface. The imaged point P in this system, the coordinates of the image P, the coordinates x and y, and the coordinates of the projection center. In a central projection, there is a fixed relationship between

And

C, the distance of the projection center of the image surface. So:

Resolution from the last equation and substituting in the other two equations yields the result:

The point P is usually determined by the coordinates X, Y, and Z in any coordinate system " outside " of the camera. In this system, the center of projection, the coordinates. This system can be transformed in the system of the camera by means of a rotation and a translation. Translation influences the differences of the coordinates do not, and the rotation, often indicated as the camera transformation is determined by a 3 x 3 matrix R, and leads into:

And

Substitution of these expressions leads to two equations which are called ' collinearity equations "

With the indexing for the camera position and recording, and a correction term for the aberration of the lens, look at the equations as follows:

The symbols mean:

  • I - index for numbering the different cameras
  • J - index for numbering the different object and image points
  • C - constant chamber, roughly equivalent to the focal length of the lens
  • R - 3 × 3 rotation matrix to define the viewing direction of the camera
  • - Vector to describe the asymmetry of the pixels of array sensors
  • - Vector defining the projection center
  • - Vector to define the 3D coordinates of the object points
  • - Vector defining the position of the image principal point on the film or sensor
  • And - functions to specify the distortion corrections
Mathematics Line (geometry) Translation (disambiguation)
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