Ideal Camera
An ideal camera is an empty box with an infinitesimally small opening in one of its faces. In an ideal camera, there is no distortion in the formed image, which means that the relationship between photographic points and ground points can be expressed using a straight line. The length of the sides of the ideal camera is equal to one (one) and its photographic coordinate system corresponds to the main point. The image coordinate system in the ideal camera is called the ideal coordinate system (pinhole).
In practice, making an ideal camera without a lens has many technical difficulties, because the amount of light passing through a very small aperture makes it impossible or very difficult to form an image. In reality, we have to increase the diameter of this hole in order to have enough light to form the image. Increasing the diameter of the aperture means that we have to use a lens or a system of lenses to converge the light rays. This causes different types of distortions to appear in the image and the linear relationship between photographic points and ground points is challenged. To solve this problem, we have to use math to model the effect of distortions in a simple and effective way. Ultimately, our goal will be to move from a non-ideal real camera to an ideal camera using a mathematical model.
Image coordinate systems
The simplest possible coordinate system for an image is obtained when we consider the left and top corners of an image as the origin of the coordinates and count the pixels in these two directions.
In photogrammetric applications, this coordinate system is not so efficient, because we need to express pixels in an ideal camera coordinate system. In the coordinate system of an ideal camera, the center of the coordinates coincides with the principal point and the effects of distortions have been corrected.
Principal point is a geometric point that is obtained from the intersection of the line that passes through the optical center of the camera and is perpendicular to the photographic plane. This point has a unique feature. Amount of distortion caused by the lens is zero at this point. On the surface of the photo, the more we move away from principal point, the greater the amount of distortion caused by the lens. So, it seems wise to place the center of the ideal camera’s coordinate system on this point.
Interior orientation
With the definition stated for the principal point, we are now looking for a suitable definition of an ideal camera coordinate system. For this purpose, we define some auxiliary photo coordinate systems to move step by step from a non-ideal coordinate system with distortions that we defined at the beginning to an ideal camera coordinate system without distortion.
In the above figure, the image pyramid along with primary and auxiliary coordinate systems are defined. Distortions in an image are caused by the effect of the lens curvature on light direction. The effect of distortions is such that they cause the condition of collinearity to not be established without removing them. Therefore, to efficiently use the collinearity condition, it is necessary to eliminate lens distortions. One of the simplest models to consider the effect of distortions is by employing Brown’s distortion model. In Brown’s model, we express the effect of distortions with the following mathematical model. First, we move from a left-hand-side coordinate system to a right hand system
Then, we match the coordinate center to the principal point, then divide the coordinate values by the pixel value of the focal length
Since the amount of distortion is a function of the distance to the principal point (principal point is now located in the center of the coordinate system), so we calculate it as
In above equations, x’ and y’ are coordinates of image points in an ideal camera with unit focal length. Ten interior orientation parameters are considered in these equations. The stated model is called an interior orientation model.
By knowing good approximate interior orientation parameters values, it becomes possible to use linear relationship between pixel coordinates and object co. sys.