Fisheye Lens - Mapping Function

Mapping Function

The mapping of a sideways object leads to a picture position displacement from the image center. The manner of this conversion is the mapping function. The distance of a point from the image center 'r' is dependent on the focal length of the optical system 'f', and the angle from the optical axis 'θ'.

• Original tunnel to be photographed, with camera looking from inside center to left wall.

• Gnomonical

• Gnomonical, 40° right pan

• Linear scaled (equidistant)

• Orthographic

• Equal area (equisolid angle)

• Stereographic (conform)

Normal (non-fisheye) lens:

• Gnomonical or perspective: . Works like the pinhole camera. Straight lines remain straight (distortion free). has to be smaller than 90°. The aperture angle is gaged symmetrically to the optical axis and has to be smaller than 180°. Large aperture angles are difficult to design and lead to high prices.

Fisheye lenses can have many different mapping functions:

• Linear scaled (equidistant):, where is in radians. Practical for angle measurement e.g., star maps. PanoTools uses this type.
• Orthographic: . Looks like an orb with the surroundings lying on < max. 180° aperture angle.
• Equal area (equisolid angle): . Every pixel subtends an equal solid angle, or an equal area on the unit sphere. Looks like a mirror image on a ball, best special effect (unsophisticated distances), suitable for area comparison (clouds grade determination). This type is popular but it compresses marginal objects. The prices of these lenses are high, but not extreme.
• Stereographic (conform): . This mapping would be ideal for photographers because it doesn't compress marginal objects as much. Samyang is the only manufacturer ever to be making this kind of fisheye lens. This lens is available under different brandnames. This mapping is easily implemented by software.
• Other mapping functions (for example Tailored Distortion lenses) are also possible for enhancing the off-axis resolution of fisheye lenses.

With appropriate software, the curvilinear images produced by a fisheye lens can be remapped to a conventional rectilinear projection. Although this entails some loss of detail at the edges of the frame, the technique can produce an image with a field of view greater than that of a conventional rectilinear lens. This is particularly useful for creating panoramic images.

All types of fisheye lens bend straight lines. Aperture angles of 180° or more are possible only with large amounts of barrel distortion.