In this chapter, a camera model is discussed which encompasses the standard pinhole camera model, a lens distortion model, and catadioptric cameras with parabolic mirrors. This generalized camera model is particularly well suited to being represented in conformal geometric algebra (CGA), since it is based on an inversion operation.
The pinhole camera model is the oldest perspective model which realistically represents a 3D scene on a 2D plane, where “realistically” means appearing to human perception as the original scene would. According to [25], “focused perspective was discovered around 1425 by the sculptor and architect Brunelleschi (1377–1446), developed by the painter and architect Leone Battista (1404–1472), and nally perfected by Leonardo da Vinci (1452–1519).” This concept of focused perspective also lies at the heart of projective space, which becomes apparent in Fig. 4.8, for example. In Sect. 7.1, the pinhole camera model and its representation in geometric algebra are brie y discussed. Figure 7.2 shows the mathematical construction of a pinhole camera. The point A4 in that gure is called the optical center, and P represents the image plane. If a point X is to be projected onto the image plane, the line through X and A4, the projective ray, is intersected with the image plane P. The intersection point is the resultant projection. Owing to its mathematical simplicity and its good approximation to real cameras in many cases, the pinhole camera model is widely used in computer vision.
However, the pinhole camera model is only an idealized model of a real camera. Depending on the application, it may therefore be necessary to take account of distortion in the imaging process due to the particular form of the imaging system used. This is typically the case if high accuracy is needed or if wide-angle lens systems are employed, or both. The problem is that all real lens systems different and their distortion cannot be modeled easily mathematically. The best approximations are the thin-lens and thick-lens models. However, even those are only approximations to a real lens system. Therefore, for an exact rectication of an image taken with a distorting lens system, the individual lens system has to be measured. This process is time-consuming and tedious, which is why lens distortion is typically approximated by a simple mathematical model. Various lens distortion models have been suggested for this purpose, such as the widely used polynomial model [85], the bicubic model [103], the rational model [33], and the division model [71].
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© 2009 Springer-Verlag Berlin Heidelberg
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(2009). The Inversion Camera Model. In: Geometric Algebra with Applications in Engineering. Geometry and Computing, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89068-3_7
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DOI: https://doi.org/10.1007/978-3-540-89068-3_7
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