Fisheye lens

This article goes into great detail about the fisheye lens. A shorter description can be found in the Fisheye Lenses section of the Wide Angle Lens article .

Modern fisheye zoom lens with a focal length of 8–15 mm

Fast fisheye lens Laowa 4 mm f / 2.8 from Venus Optics with an angle of view of 210 ° and a weight of 135 grams for Micro Four Thirds

Presented at Photokina 1970 and manufactured from 1972–1983.
The fisheye Nikkor 6mm f / 2.8 had the largest field of view of 220 ° in KB format.

In photography , the fish eye ( fisheye or fisheye lens ) describes a special lens that can depict a complete field of view with the necessary distortion . In contrast to conventional non-fisheye lenses, which proportionally depict an object plane perpendicular to the optical axis ( gnomonic projection method , see below: imaging functions ), fisheye lenses depict one hemisphere or more, with clear but not excessive distortions, on the image plane. Straight lines that do not run through the center of the image are shown curved; the image is strongly barrel-shaped (see distortion ). It usually depicts surface ratios or radial distances more faithfully than a normal, gnomonic projecting wide-angle lens and has a very large image angle (usually 180 ° in the image diagonal, in extreme cases even up to 220 °, in drafts even 270 ° and 310 °). Image angles of 180 ° or more cannot be achieved with the conventional projection method. Despite the extraordinarily large image angle, the decrease in brightness towards the edge of the image is easier to correct than with wide-angle lenses, because the image scale does not increase as much towards the edge of the image and the light does not have to illuminate large areas.

The world's first series-produced fisheye lens was introduced by Nikon in 1962 (Fisheye-Nikkor 1: 8, f = 8 mm). The lens protruded far into the camera housing, so that the mirror had to be folded up and locked, and an external viewfinder had to be attached to the hot shoe.

There are now a large number of manufacturers who produce fisheye lenses. For single-lens reflex cameras , they are usually designed as retrofocus lenses so that the mirror has enough space between the shutter and the rear lens.

Modern mirrorless camera systems with significantly shorter support dimensions allow less effort for retrofocus constructions due to the small distance between optics and sensor. This reduces weight and price or enables larger image angles or higher light intensities with the same effort.

Miniature fisheye lenses for very small sensors, such as those used in surveillance cameras or action camcorders , are even cheaper, so that even larger image angles of up to 280 ° are available and cost no more than a 180 ° retrofocus fisheye lens for the 35mm format. In the meantime, there are also enlarged versions of miniature fish eyes with image angles of up to 250 ° for mirrorless system cameras.

Types

Table 1: Types of format utilization

circular		cropped circle	Full format
circular		cropped circle	full frame
3: 2	52% sensor	78% field of view, 92% sensor	59% field of view
4: 3	59% sensor	86% field of view, 90% sensor	61% field of view
Circular fisheye for the 35mm format			Full format fisheye with a rudimentary lens hood
Shot with a circular fisheye		35mm circular fisheye on DX format camera	Interior photographed with full frame fisheye

One differentiates between fisheye lenses on the one hand according to their type of projection (see image functions below) and according to their image circle diameter in relation to the recording format.

circular

Fisheye lenses whose image circle diameter is (at most) as large as the shorter edge of the recording format of the camera are called circular fisheye (also “circular fisheye” or “round fisheye”) because they create a circular image within the rectangular recording format. The image circle of the lens is used 100%. The greatest possible sensor utilization is given in table 1; it turns out to be less due to a practically somewhat smaller image circle. In order not to crop the image circle, circular fisheyes do not have a lens hood. Circular fisheyes are the first choice when you want to capture as much of the environment (usually a hemisphere) as possible. The first fisheye lenses developed were circular fisheyes. In the absence of a full-frame fisheye, cutting out a rectangular area further reduces the sensor or film utilization to a maximum of 31% (3: 2) or 36% (4: 3) and would be extremely unfavorable.

Full format

Fisheye lenses whose image circle diameter is (at least) as large as the diagonal of the recording format of the camera are called "full-format fisheye" (also "diagonal fisheye" because of the ambiguity of the term "full format" for sensor utilization or sensor size ). They reach their largest angle of view (usually 180 °) only over the picture diagonal; their horizontal and vertical angles of view are correspondingly smaller and parts of the field of view of the lens are not used. The sensor, on the other hand, is used 100%. The lens hood turns out to be very small and limits the image field to an approximately rectangular area that only extends slightly beyond the intended recording format. As fish eyes became popular in general photography, camera manufacturers began developing full frame fish eyes. The rectangular format is most convenient for direct reproduction of the original images (without conversion).

Cropped circle

If the camera does not have the sensor format for which the fisheye lens is intended, the angle of view and format utilization change. With certain combinations, a usable image field in the form of a trimmed circle can be achieved as an intermediate format between the circular and full image. This results in good sensor utilization and usually fewer losses when rectangular cropping of the images converted into another type of projection.

Either a circular fisheye is used for the 35mm format on an APS-C or DX camera or a full format fisheye for the DX format on a full-format 35mm camera. In the second case, the lens hood will crop the image and must be removed. If it is not removable, it can be shortened with a tool (shaving the lens). Some fisheye zoom lenses can also achieve a clipped circle.

Ideally, the field of view is a circle cropped on both sides. The image circle diameter is then (at most) as large as the longer side of the recording format. The angle of view is z. B. in landscape format at the round edges of the picture both diagonally and horizontally maximum; it is only smaller vertically. In practice there is also the three-sided trimmed circle if the sensor area is not in the center of the image circle (e.g. Sigma 8 mm fisheye true to area [older model with aperture F / 4] and camera with APS-C sensor), or the Circle trimmed on four sides, if the imaging function generates a larger image circle (e.g. angularly linear Canon 8-15 mm fisheye at 8 mm and camera with APS-C sensor).

When converting to stereographic projection, after rectangular cropping, a full-format image with a 180 ° diagonal angle of view can be obtained (starting from a full-format image, the stereographic projection would be pillow-shaped. Rectangular cropping would remove the pillow tips and thus reduce the diagonal image angle). Possibly. Image angles of 180 ° are no longer possible with the original format, but only with a wider format (e.g. 16: 9) of the converted image.

Focal length

Full-format fish eyes have focal lengths of around 16 mm for the 35 mm format and are therefore in the range of powerful wide-angle lenses. For the common APS-C or DX format digital SLR cameras with crop factor 1.5 ... 1.6, the focal length is around 8 to 10 mm, depending on the type of projection.

Circular fish eyes have the shorter focal lengths. The focal length for the 35mm format is around 8 mm and for the APS-C or DX format around 4.5 mm.

Zoom fish eyes with their focal length range include the circular and full-format fish eyes if they are designed for this purpose and used for the intended recording format. Or they close the gap between a full-frame fisheye and a wide-angle or universal zoom lens and retain the barrel-shaped distortion in the entire zoom range.

Fish eyes for other formats (e.g. medium format crop factor approx. 0.5 or FourThirds crop factor 2) have different focal lengths proportional to the format size.

Image effect

Fisheye image (left) and rectified image (right)

A fisheye image should correspond to the image that a fish has when it looks from below through the water surface or from the side of an aquarium ; this is how the name of the lens came about (see section underwater view below ).

There are deviations between the visual observation of the object and the observation of the image, because by moving your head and eyes you always look centrally at every detail and perceive it without distortion, but the optics capture edge details at an angle.

curvature

The illustration of a long, straight road along its entire length must combine the following contradicting properties in the picture: The road edges are parallel in the center of the picture, and the road edges converge to vanishing points on both sides of the picture. This is only possible with barrel distortion. It is a prerequisite for fisheyes, because this is the only way to achieve image angles of 180 ° and more.

Straight lines through the center of the image remain straight lines. In the case of straight lines outside the center of the image, the curvature becomes stronger the further you pass the center. Long edges in the edge area show a strong change in direction. Above all, this alienates very large objects.

You can use the fisheye for objects that hardly have straight lines, e.g. B. for panoramic shots of landscapes. A fisheye is not recommended for objects with many straight edges (architecture). The large angle of view is practical indoors because you can depict an entire room with one picture. Either you then accept the curved lines, or you convert the image so that z. B. the vertical and the central lines of flight are straight (Pannini projection).

People at the edge of the picture appear to be crooked. They appear curved outwards. This can be accepted as a compromise for a slight deformation of the heads (see section Deformation). Alternatively, a suitable conversion (e.g. to the Pannini projection) eliminates the bowing of the body with a little more distorted heads.

Scaling

Objects of the same size lying next to one another are displayed in different sizes depending on their position in the image field. They have different reproduction scales, so that one can speak of a scale distortion.

The scaling referred to here corresponds to cartography and is defined differently than the scale distortion in conventional photography.

Scale distortion in conventional photography is the local change in size when an object plane perpendicular to the optical axis is mapped onto the image plane. This also leads to distortion. With this approach, super wide-angle lenses can also be used without scale distortion. Lateral circles that are perpendicular to the optical axis are also displayed on the same scale as if they were in the middle (and remain circles). The scale in conventional photography depends on the distance projected onto the optical axis and not on the actual (oblique) distance from the entrance pupil and describes the size perpendicular to the optical axis.

If the flat circles are replaced by three-dimensional spheres, the spherical horizons seen lie on spherical surfaces concentric to the entrance pupil and no longer on the plane of the previous circles. When projected onto the object plane, the oblique spherical horizon leads to an elliptical (egg-shaped for large objects) deformation and to an enlargement. Fish eyes do not relate to an object plane (rather to an object shell), which means that deformation and size changes are different.

Scaling describes the change in size of a small detail of a concentric spherical surface in the image compared to the central image. The scaling relates to an object with the same direct distance to the entrance pupil and to the size that appears perpendicular to the direction of the object. The deviation from the scale in the center of the image is the scaling. It apparently changes the depth of the adjacent objects. The calculation of a Tissot indicatrix makes the scaling over the areas of the distortion ellipses clear (scaling is proportional to the square root of the areas).

With the conventional super wide angle, edge details are displayed larger in relation to the direct distance. The center of the picture takes a back seat and a space that appears to be too deep is created. This makes a square room look like a corridor. With fisheye, the scaling of edge details is significantly smaller, completely canceled (true to area) or, in extreme cases, in the opposite direction (orthographic: depth compression), depending on the type of image. The apparent depth of the square space changes little. It looks bent apart (see section Curvature ).

Depth elongation

Objects lying one behind the other (at the same point in the image) seem to move apart. This effect is independent of whether the lens is a fisheye or a super wide angle. With the short focal length of all fish eyes, the depth expansion is very pronounced, so that the foreground can be separated very well from the background. Due to the fisheye image angle of 180 °, you can go very close to the object for extreme effect images and even with front lens contact you usually get the entire object on the image.

deformation

Deformation is a differently defined optical distortion than distortion .

Distortion describes the distortion of a surface perpendicular to the optical axis during imaging. Super wide-angle lenses can also be distortion-free . Lateral circles that are perpendicular to the optical axis are also displayed as circles again.

If the flat circles are replaced by three-dimensional spheres, the spherical horizons seen lie on spherical surfaces concentric to the entrance pupil and no longer on the plane of the previous circles. The oblique spherical horizon leads to an elliptical (egg-shaped for large objects) deformation when projected onto the object plane. Fish eyes do not refer to an object plane (rather to an object shell), which means that the deformation is different.

Deformation describes the distortion of a small detail of a concentric spherical surface in the image. The calculation of a Tissot indicatrix makes the deformation clear via the stretching of the distortion ellipses (ratio of major and minor axes).

While with conventional wide-angle lenses, spatial edge details are pulled apart from the center, fish eyes do the opposite and compress them. The deformation is not so strong and only noticeable in a stronger lateral position. This makes the fisheye ideal for taking pictures within a crowd or of people at a table. But you shouldn't get too close to any person in order to avoid distorting effects. For the best possible representation of people, it is recommended to convert them to conformal representation (no compression of small objects).

Mapping functions

The mapping function determines the suitability for a specific purpose. When converting to other representations or when merging to form panoramas , the correct mapping function is decisive for the quality of the end result.

Fundamental figures

Various mapping functions

Different distortion

Of the many possible mapping functions, the fundamental ones are characterized by the faithful reproduction of a parameter. The projection type of an optic is the name of the projection of the most similar fundamental mapping.

For comparison - normal (non-fisheye) lens:

• Gnomonic (free of distortion)
  looks like the pinhole camera. Straight lines are straight in the picture too. With retrofocus lenses, large image angles requireextreme effort and lead to very high prices for such a lens.

Ultra-wide-angle lenses can have the focal lengths of fisheye lenses; However, because of the way they are represented, they are not fish eyes.

Objects near the edge of the image are displayed distorted: They are stretched in the radial direction. Spatial objects on the side of the picture therefore appear too wide in the image, and people standing at the edge of the picture are then shown, for example, thicker than they are, and get "egg heads".

Fish eyes can have different imaging functions. Some special cases follow:

• True-angle ( stereographic )
  would be ideal for most photographic purposes because objects at the edge of the image are only slightly distorted. Objects whose image is only slightly extended in all directions are shown practically undistorted. People z. B. are reproduced with the correct proportions, neither too slim nor too fat. However, an object appears larger at the edge of the image than in the center, at the same distance from the camera, i.e. H. Objects on the edge appear closer than they are. For common digital SLR cameras, Samyang Optics offers optics with an approximately true-to-angle image. It is offered under various brand names, in Germany under "Walimex". For other fisheye types, the conformal imaging can be implemented using software. The stereographic representation called “Little planet” with the nadir in the center of the image is popular, but it requires a fisheye with an angle of view well over 180 ° or can be generated with panorama software.
• Equidistant (linearly divided)
  Such lenses allow angle measurements on the image in the radial direction (e.g. for star maps). This type of image is particularly suitable for extreme image angles over 180 °. Objects in the center are not as small as in the case of conformal mapping; and objects at the edge of the image are not compressed radially in contrast to the equal-area image. They nevertheless act compressed in the radial direction because they are enlarged transversely to the radial direction. People appear a little slimmer than they are. The distance between objects at the edge of the image is less underestimated than with true-to-angle imaging.

Most round-picture fish eyes, such as B. the Canon FD f / 5.6 / 7.5 mm, image this way. Software for creating panoramas, such as Hugin , are based on this type of projection.

When converting to this type of projection, image angles over 360 ° are mathematically possible. The representation is repeated in ring-shaped zones (multiple mapping).

• Equal area ( equisolid angle )
  The result is the same image as if a mirrored sphere had been placed at the camera position and viewed from a great distance compared to its diameter. The area that the image of an object occupies in the image is proportional to the solid angle at which it appears when viewed from the camera position. The distances of the objects from the camera are therefore correctly estimated. Such lenses are particularly suitable for determining the degree of coverage in the image (e.g. through vegetation or buildings, or to determine the degree of cloudiness). The radial compression of objects close to the edge is still somewhat greater than with equidistant projection. Many full-frame fish eyes depict (approximately) true to area.
• orthographically
  acts like a sphere that is viewed from a great distance and onto which the surroundings have been projected , with the center of the sphere as the projection center. Such lenses enable the analysis of lighting conditions. Lateral light areas contribute less to the brightness on a plane arranged perpendicular to the optical axis and are depicted correspondingly smaller. By default, the maximum possible image angle of 180 ° required for the lighting analysis is shown, although the distortions are only tolerable within 120 °. For pictorial photography, this type is rather disadvantageous because of the pronounced distortions; objects near the image edge are compressed even more radially than with true-to-area images.

The Nikon OP Fish-eye NIKKOR 10mm f / 5.6 follows this projection method ("OP" = orthographic projection ); it has an aspherical front lens to realize this difficult type of projection. Fish-eye attachments also usually have an approximately orthographic projection.

parameter

Spherical coordinates

Plane polar coordinates and Cartesian transformation

The surroundings are recorded in spherical coordinates , with the z-axis being the optical axis of the lens and pointing to the center of the area to be photographed. The origin of the coordinates is the entrance pupil. The position of an object is also determined. Theoretically, the spherical radius has no influence on the imaging function (real lenses: see section Notes ). In this way, all objects can be scaled to exactly one distance, and a surrounding sphere is created. The focal length of the lens ( = ) is chosen for the radius of the surrounding sphere . ${\displaystyle (r,\varphi ,\theta )}$ ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle f}$

In Cartesian coordinates, the image plane is at = and touches the surrounding sphere at the pole ( ). Plane polar coordinates are applied to the image . The location of an image detail is described by. In this geometric arrangement, the mapping functions can be well illustrated and usually also constructed. The object coordinates (azimuth) and (polar angle) are mapped in the image coordinates (azimuth) and (radius). ${\displaystyle z}$ ${\displaystyle f}$ ${\displaystyle \theta =0}$ ${\displaystyle (r,\phi )}$ ${\displaystyle \varphi }$ ${\displaystyle \theta }$ ${\displaystyle \phi }$ ${\displaystyle r}$

The principle is the same as in the azimuthal map network design of the earth. There applies . However, in contrast to the globe, the surrounding sphere is viewed from the inside. The front of the image plane then shows the surroundings as a mirror image. For the rear view required here, the image coordinates (polar and 2D Cartesian) are oriented differently, so that the azimuth 0 ° direction and the azimuth direction of rotation can differ between the surroundings and the image. Azimuth angular distances are mapped without being falsified, and mapping function and azimuth do not influence one another. This gives an azimuthal image. ${\displaystyle \phi =\varphi }$

The mapping function describes how an object appears shifted by a radius from the center in the polar angle . The lens has the (central) focal length . With = 1 the normalized mapping function becomes . ${\displaystyle \mathbf {g} }$ ${\displaystyle \theta }$ ${\displaystyle r}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle \mathbf {g} }$ ${\displaystyle \mathbf {h} }$

${\displaystyle r=\mathbf {g} (\theta ,f)=f\cdot \mathbf {h} (\theta )}$

Lateral objects appear in a different size compared to the central position, which usually also differs meridional and sagittal .

meridional scaling:    ${\displaystyle S_{m}={\frac {\mathrm {d} \mathbf {h} (\theta )}{\mathrm {d} \theta }}}$

sagittal scaling:          ${\displaystyle S_{s}={\frac {\mathbf {h} (\theta )}{\sin \theta }}}$

Solid angle scaling  , linear (effective) scaling and deformation can be derived from this: ${\displaystyle S_{\Omega }}$ ${\displaystyle S}$ ${\displaystyle D}$

${\displaystyle S_{\Omega }=S_{m}\cdot S_{s}}$

${\displaystyle S={\sqrt {S_{\Omega }}}}$

${\displaystyle D={\frac {S_{m}}{S_{s}}}}$

The fundamental mapping functions is noticeable that there is a power there between deformation and scaling that whatever   is  . Because and are derived from and , there is also a -independent power with between and . This becomes the key figure of the respective fundamental function. ${\displaystyle B}$ ${\displaystyle \theta }$ ${\displaystyle D=S^{B}}$ ${\displaystyle D}$ ${\displaystyle S}$ ${\displaystyle S_{m}}$ ${\displaystyle S_{s}}$ ${\displaystyle S_{m}}$ ${\displaystyle S_{s}}$ ${\displaystyle \theta }$ ${\displaystyle N}$ ${\displaystyle S_{m}=S_{s}^{N}}$ ${\displaystyle N}$

${\displaystyle B={\frac {2\,(N-1)}{N+1}}}$     (Balance between deformation and scaling)

The curvature   factor is the increase in curvature in the paraxillary region. ${\displaystyle C}$

${\displaystyle C={\frac {N-2}{3-N}}}$     ( Negative curvature with barrel distortion)

The special cases C = 0 (curvature-free, distortion-free), D = 1, S _m = 1, S = 1, S _s = 1 (constant scale of the respective size) lead to the five fundamental functions of the previous section and the following table. None of these basic functions can cover more than one special case. Gnomonic (C = 0) normal and telephoto lenses apparently fulfill all special cases because the scales D, S _m , S and S _s only differ slightly from the value one due to the small angle of view , which means that the deviations are barely perceived or compensated for by the perspective ( Image viewing angle similar to recording angle).

Table 1: Fundamental functions

Projection type (fundamental)	gnomonic	Fisheye
		conformal	equidistant	true to area	orthographic
sketch
${\displaystyle N}$	${\displaystyle 2}$	${\displaystyle 1}$	${\displaystyle 0}$	${\displaystyle -1}$	${\displaystyle \infty }$
${\displaystyle B}$	${\displaystyle {\tfrac {2}{3}}}$	${\displaystyle 0}$	${\displaystyle -2}$	${\displaystyle \infty }$	${\displaystyle 2}$
${\displaystyle C}$	${\displaystyle {\boldsymbol {C}}=0}$	${\displaystyle -{\tfrac {1}{2}}}$	${\displaystyle -{\tfrac {2}{3}}}$	${\displaystyle -{\tfrac {3}{4}}}$	${\displaystyle -1}$
Mapping function	${\displaystyle {\boldsymbol {r=f\tan \theta }}}$	${\displaystyle {\boldsymbol {r=2f\tan {\frac {\theta }{2}}}}}$	${\displaystyle {\boldsymbol {r=f\,\theta }}}$	${\displaystyle {\boldsymbol {r=2f\sin {\frac {\theta }{2}}}}}$	${\displaystyle {\boldsymbol {r=f\sin \theta }}}$
meridional scaling	${\displaystyle S_{m}={\frac {1}{\cos ^{2}\theta }}}$	${\displaystyle S_{m}={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}}}$	${\displaystyle {\boldsymbol {S_{m}=1}}}$	${\displaystyle S_{m}=\cos {\frac {\theta }{2}}}$	${\displaystyle S_{m}=\cos \theta }$
sagittal scaling	${\displaystyle S_{s}={\frac {1}{\cos \theta }}}$	${\displaystyle S_{s}={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}}}$	${\displaystyle S_{s}={\frac {\theta }{\sin \theta }}}$	${\displaystyle S_{s}={\frac {1}{\cos {\frac {\theta }{2}}}}}$	${\displaystyle {\boldsymbol {S_{s}=1}}}$
Scaling (effective)	${\displaystyle S={\frac {1}{\sqrt {\cos ^{3}\theta }}}}$	${\displaystyle S={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}}}$	${\displaystyle S={\sqrt {\frac {\theta }{\sin \theta }}}}$	${\displaystyle {\boldsymbol {S=1}}}$	${\displaystyle S={\sqrt {\cos \theta }}}$
deformation	${\displaystyle D={\frac {1}{\cos \theta }}}$	${\displaystyle {\boldsymbol {D=1}}}$	${\displaystyle D={\frac {\sin \theta }{\theta }}}$	${\displaystyle D=\cos ^{2}{\frac {\theta }{2}}}$	${\displaystyle D=\cos \theta }$
Angle of view  ${\displaystyle {\boldsymbol {\alpha _{\text{max}}}}}$	<180 °	<360 °	360 ° (and more)	360 °	180 °
${\displaystyle \alpha _{\text{stark}}}$	102 °   1)	180 °   1)	217 °   2)	180 °   2)	120 °   2)
${\displaystyle \alpha _{\text{mittel}}}$	75 °   1)	131 °   1)	159 °   2)	131 °   2)	90 °   2)
${\displaystyle \alpha _{\text{schwach}}}$	54 °   1)	94 °   1)	115 °   2)	94 °   2)	66 °   2)
Image circle  = 180 ° ${\displaystyle {\boldsymbol {\alpha }}}$	${\displaystyle \varnothing \infty }$	${\displaystyle \varnothing 4f}$	${\displaystyle \varnothing \pi f}$	${\displaystyle \varnothing {\sqrt {8}}f}$	${\displaystyle \varnothing 2f}$
${\displaystyle \alpha }$ = 360 °		${\displaystyle \varnothing \infty }$	${\displaystyle \varnothing 2\pi f}$	${\displaystyle \varnothing 4f}$
Front area  : back		${\displaystyle 1:\infty }$	${\displaystyle 1:3}$	${\displaystyle 1:1}$

¹⁾   limited by scaling S
²⁾   limited by deformation D

The sketch in the table is to be understood in such a way that the scene is first projected onto a spherical surface, with the center of the sphere as the projection center, in order to then be mapped from the sphere onto the image plane as shown.

${\displaystyle \alpha _{\text{stark}}}$, and are sample angles of view. With strong , medium and weak the amount of distortion is by scaling and deformation meant. It lies within the following arbitrarily selected areas: ${\displaystyle \alpha _{\text{mittel}}}$ ${\displaystyle \alpha _{\text{schwach}}}$ ${\displaystyle S}$ ${\displaystyle D}$

strong		${\displaystyle 0{,}5\dotsc 2}$
medium		${\displaystyle {\sqrt {0{,}5}}\dotsc {\sqrt {2}}}$
weak		${\displaystyle {\sqrt[{4}]{0{,}5}}\dotsc {\sqrt[{4}]{2}}}$

In the middle of the image, the relevant value is 1 (distortion-free) and becomes steadily smaller or larger towards the edge, depending on the type of projection and parameter. Even larger angles of view are possible with a fundamental intermediate function (section Mathematical Models below). With gnomonic projection, these areas fit well with the division into super wide-angle , moderate wide-angle and normal lens .

Parametric mappings

Commercially available fish eyes are not developed for measuring purposes, but for photographers. In the upper range of the characteristic curve, a few percent deviation from the fundamental basic functions is possible in order to limit the effort and thus weight and price. A better approximation of the real characteristic curve can be achieved with models that also enable intermediate and special functions via one or more parameters.

The mostly non-fundamental figures is   (see section parameter ) is not solely dependent on corresponding parameters, but also on  . The parameters  are then converted for comparison purposes for and results . The non-fundamental function corresponds to the corresponding fundamental intermediate function for smaller  values ​​and takes a different course for larger  values. ${\displaystyle N}$ ${\displaystyle \theta }$ ${\displaystyle N}$ ${\displaystyle \theta =0}$ ${\displaystyle N_{0}=N(\theta =0)}$ ${\displaystyle \theta }$ ${\displaystyle \theta }$

Table 2: Parametric functions

Mapping type (parametric)	general fundamental function	Examples of non-fundamental functions
		polynomial	angle scaled	shifted projection center	Fish view (chapter underwater view )
sketch
parameter	${\displaystyle N}$	${\displaystyle a_{3}...a_{2i+1}}$	${\displaystyle L}$	${\displaystyle Z}$	Refractive index   ${\displaystyle n}$
${\displaystyle N}$	${\displaystyle {\color {Gray}N}}$	${\displaystyle {\color {Gray}{\frac {\ln {S_{m}}}{\ln {S_{s}}}}}}$
${\displaystyle N_{0}=N(\theta {\to }0)}$		${\displaystyle {\frac {18a_{3}}{6a_{3}+1}}}$	${\displaystyle {\frac {L\max(6L,-3L)}{L\max(2L,-L)+1}}}$	${\displaystyle Z+2}$	${\displaystyle 3-n^{2}}$
${\displaystyle B}$	${\displaystyle {\color {Gray}{\frac {2N-2}{N+1}}}}$	${\displaystyle {\color {Gray}{\frac {\ln D}{\ln S}}}}$
${\displaystyle B_{0}=B(\theta {\to }0)}$		${\displaystyle {\frac {24a_{3}-2}{24a_{3}+1}}}$	${\displaystyle {\frac {L\max(8L,-4L)-2}{L\max(8L,-4L)+1}}}$	${\displaystyle {\frac {2(Z+1)}{Z+3}}}$	${\displaystyle {\frac {4-2n^{2}}{4-n^{2}}}}$
${\displaystyle C}$	${\displaystyle {\color {Gray}{\frac {N-2}{3-N}}}}$	${\displaystyle {\frac {6a_{3}-2}{3}}}$	${\displaystyle {\frac {L\max(2L,-L)-2}{3}}}$	${\displaystyle {\frac {Z}{1-Z}}}$	${\displaystyle {\frac {1-n^{2}}{n^{2}}}}$
Mapping function	... developed from with  = f  at θ = 0 and  r = 0 ${\displaystyle {\boldsymbol {{\frac {\mathrm {d} r}{\mathrm {d} \theta }}=\left({\frac {r}{\sin \theta }}\right)^{N}}}}$ ${\displaystyle {\tfrac {\mathrm {d} r}{\mathrm {d} \theta }}}$	${\displaystyle {\boldsymbol {r=f\cdot \sum _{i=0}^{n-1}a_{2i+1}\,\theta ^{2i+1}}}}$	${\displaystyle {\boldsymbol {r=f\,{\frac {\sin(L\,\theta )}{L\,\cos(\max(L,0)\cdot \theta )}}}}}$	${\displaystyle {\boldsymbol {r=f\,{\frac {(1-Z)\sin \,\theta }{\cos \,\theta -Z}}}}}$	${\displaystyle {\boldsymbol {r=f{\frac {\sin \theta }{\sqrt {1-{\frac {\sin ^{2}\theta }{n^{2}}}}}}}}}$
meridional scaling	${\displaystyle {\color {Gray}S_{m}={\frac {\mathrm {d} r}{f\mathrm {d} \theta }}}}$	${\displaystyle S_{m}=\sum _{i=0}^{n-1}(2i+1)\,a_{2i+1}\,\theta ^{2i}}$	${\displaystyle S_{m}={\frac {\cos(\min(0,L)\cdot \theta )}{\cos ^{2}(\max(L,0)\cdot \theta )}}}$	${\displaystyle S_{m}={\frac {(1-Z)(1-Z\cos \theta )}{(\cos \theta -Z)^{2}}}}$	${\displaystyle S_{m}={\frac {\cos \theta }{(1-{\frac {\sin ^{2}\theta }{n^{2}}})^{1{,}5}}}}$
sagittal scaling	${\displaystyle {\color {Gray}S_{s}={\frac {r}{f\sin \theta }}}}$		${\displaystyle S_{s}={\frac {\sin(L\theta )}{L\sin \theta \cos(\max(L,0)\cdot \theta )}}}$	${\displaystyle S_{s}={\frac {1-Z}{\cos \theta -Z}}}$	${\displaystyle S_{s}={\frac {1}{\sqrt {1-{\frac {\sin ^{2}\theta }{n^{2}}}}}}}$
Scaling (effective)	${\displaystyle {\color {Gray}S={\sqrt {S_{m}\cdot S_{s}}}}}$		${\displaystyle S={\sqrt {\frac {\sin(L\theta )\cos(\min(0,L)\cdot \theta )}{L\sin \theta \cos ^{3}(\max(L,0)\cdot \theta )}}}}$	${\displaystyle S=(1-Z){\sqrt {\frac {1-Z\cos \theta }{(\cos \theta -Z)^{3}}}}}$	${\displaystyle S={\frac {\sqrt {\cos \theta }}{1-{\frac {\sin ^{2}\theta }{n^{2}}}}}}$
deformation	${\displaystyle {\color {Gray}D={\frac {S_{m}}{S_{s}}}}}$		${\displaystyle D={\frac {L\sin \theta \cos(\min(0,L)\cdot \theta )}{\sin(L\theta )\cos ^{2}(\max(L,0)\cdot \theta )}}}$	${\displaystyle D={\frac {1-Z\cos \theta }{\cos \theta -Z}}}$	${\displaystyle D={\frac {\cos \theta }{1-{\frac {\sin ^{2}\theta }{n^{2}}}}}}$
Angle of view   ${\displaystyle {\boldsymbol {\alpha _{\text{max}}}}}$			${\displaystyle {\frac {180^{\circ }}{|L|}}}$	${\displaystyle 2\arccos {\frac {Z}{\max(1,Z^{2})}}}$	${\displaystyle 180^{\circ }}$ at  ${\displaystyle n>1}$
${\displaystyle \alpha _{\text{stark}}}$	245 ° at N = ${\displaystyle {\tfrac {1}{3}}}$		244 ° at L = 0.2684	236 ° at Z = −1.2734
${\displaystyle \alpha _{\text{mittel}}}$	182 ° at N = ${\displaystyle {\tfrac {1}{3}}}$		181 ° at L = 0.2587	178 ° at Z = −1.43
${\displaystyle \alpha _{\text{schwach}}}$	132 ° at N = ${\displaystyle {\tfrac {1}{3}}}$		131 ° at L = 0.2542	130 ° at Z = −1.536

Gray text: general formula, if there is no special formula.Some
formulas can contain special cases, e.g. B. L = 0, do not calculate. The special case formulas have not been included in the interests of clarity. With the max and min function, several parameter range-dependent formulas are combined into one.

${\displaystyle \alpha _{\text{stark}}}$, and are the image angles for    , for which scaling and deformation have the following arbitrarily selected values: ${\displaystyle \alpha _{\text{mittel}}}$ ${\displaystyle \alpha _{\text{schwach}}}$ ${\displaystyle B(\theta ={\tfrac {\alpha }{2}})=-1}$ ${\displaystyle S}$ ${\displaystyle D}$

strong		${\displaystyle D\approx 0{,}5}$		${\displaystyle S\approx 2}$
medium		${\displaystyle D\approx {\sqrt {0{,}5}}}$		${\displaystyle S\approx {\sqrt {2}}}$
weak		${\displaystyle D\approx {\sqrt[{4}]{0{,}5}}}$		${\displaystyle S\approx {\sqrt[{4}]{2}}}$

Mathematical models

• fundamental intermediate function

For fundamental functions, the balance between deformation and scaling remains independent of . The same applies to , so that becomes a parameter and can form any intermediate functions. Based on   the direction field results (ordinary differential equation) ${\displaystyle \theta }$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle S_{m}=S_{s}^{N}}$

${\displaystyle {\frac {\mathrm {d} r}{\mathrm {d} \theta }}=\left({\frac {r}{\sin \theta }}\right)^{N}}$ .

For   becomes the initial condition ${\displaystyle \theta \to 0}$

${\displaystyle {\frac {\mathrm {d} r}{\mathrm {d} \theta }}=f}$ 

which selects the field line of the desired function. The function curve of the mapping function can be developed using a numerical method.  Every fundamental intermediate function can be formed with the parameter  , except with N = 3 and N =  . This also includes all fundamental functions with the exception of orthographic functions. The calculation is complex. ${\displaystyle N}$ ${\displaystyle \infty }$

At the same time limiting value of deformation and scaling (  ,   ,  ) to get the largest possible angle of view , and (values see Table 2). The other case simultaneous limit valency   is unusable (  ,  0 ° ). ${\displaystyle B=-1}$ ${\displaystyle N={\tfrac {1}{3}}}$ ${\displaystyle C=-{\tfrac {5}{8}}}$ ${\displaystyle \alpha _{\text{stark}}}$ ${\displaystyle \alpha _{\text{mittel}}}$ ${\displaystyle \alpha _{\text{schwach}}}$ ${\displaystyle B=1}$ ${\displaystyle C=\infty }$ ${\displaystyle \alpha _{\text{max}}=}$

• scaled angle function

For some fundamental functions, the angle functions have a factor of 0.5 or 1 in front of the angle . With this factor as a parameter (linearity factor  , actually non-linearity) the tangent and sine functions can be scaled. The sign chooses between tangent and sine. ${\displaystyle \theta }$ ${\displaystyle L}$

positive L:     with L = +  for 0 <N ₀ <3 ${\displaystyle r=f\,{\frac {\tan(L\,\theta )}{L}}}$ ${\displaystyle {\sqrt {\tfrac {N_{0}}{6-2\,N_{0}}}}}$

negative L:     with L = -  for either N ₀ <0 or N ₀ > 3 ${\displaystyle r=f\,{\frac {\sin(L\,\theta )}{L}}}$ ${\displaystyle {\sqrt {\tfrac {N_{0}}{N_{0}-3}}}}$

The two cases can be combined into a formula with the max function:

${\displaystyle L\neq 0}$:     with the N ₀ -dependent separate L calculation as above. ${\displaystyle r=f\,{\frac {\sin(L\,\theta )}{L\,\cos(\max(L,0)\cdot \theta )}}}$

L = 0 requires special treatment with  . Thus, all fundamental basic functions are included. ${\displaystyle r=f\,\theta }$

The Thoby-Fisheye is also a scaled angle function and is calculated as follows:

${\displaystyle r=k_{1}\,f_{\text{Nenn}}\sin(k_{2}\,\theta )}$.

The values and were found empirically for the AF DX Fisheye Nikkor 10.5mm f / 2.8G ED ( )  . should actually be if the most suitable focal length is used instead of the nominal focal length printed on the lens . Then the general formula given above with  11.28 mm and  = - 0.731 can be used. The slight deviation from to or to indicates a scaled function (see below). ${\displaystyle f_{\text{Nenn}}=10{,}5\,\mathrm {mm} }$ ${\displaystyle k_{1}=1{,}47}$ ${\displaystyle k_{2}=0{,}731}$ ${\displaystyle k_{1}}$ ${\displaystyle {\tfrac {1}{k}}_{2}}$ ${\displaystyle k_{2}}$ ${\displaystyle f=k_{1}k_{2}f_{\text{Nenn}}\approx }$ ${\displaystyle L}$ ${\displaystyle k_{1}}$ ${\displaystyle {\tfrac {1}{k}}_{2}}$ ${\displaystyle f}$ ${\displaystyle f_{\text{Nenn}}}$

• polynomial

Mapping functions are centrally symmetric and therefore odd:

${\displaystyle r=f\cdot \sum _{i=0}^{n-1}a_{2i+1}\,\theta ^{2i+1}}$  with  = 1,  =  and other odd terms. ${\displaystyle a_{1}}$ ${\displaystyle a_{3}}$ ${\displaystyle {\tfrac {N_{0}}{18-6N_{0}}}}$

${\displaystyle n\,}$ is the number of polynomial terms. Parameters are , ...  ( constant). For a good reproduction of an optic several links and thus parameters are necessary. ${\displaystyle a_{3}}$ ${\displaystyle a_{5}}$ ${\displaystyle a_{2n-1}}$ ${\displaystyle a_{1}}$

A polynomial can be calculated from points of incidence angle and image radius which has as many coefficients as there are points. An infinite polynomial series can be derived from an analytic function , but the points at hand always deviate slightly from an analytic function. As a result, the polynomial becomes more and more wavy as the number of points increases. There is a risk of an oscillating curve with huge wave peaks and valleys between the precisely formed point values.

A spline interpolation based on low-order polynomial pieces leads to a better but more complicated solution. With support points placed on the spline that are derived from Chebyshev points , an approximately error-optimized polynomial can be generated. The support points are (because of the odd polynomial in -scaling) very close together at the range limits and further apart in the middle, so that the flat waves in the middle and the steep waves at the edge are the same height. It may then be necessary to readjust the interpolation points in order to achieve this optimum. The degree of the polynomial should only be as high as the accuracy (e.g. image pixel size) requires. ${\displaystyle \theta ^{2}}$

Geometric models

• shifted projection center

This function is based on the projection of a surrounding sphere onto an image plane from a projection center which is shifted from the center of the surrounding sphere along the optical axis. With the z-axis as the optical axis, the z-value of the projection center becomes the parameter Z.

${\displaystyle r=f\,{\frac {(1-Z)\sin \,\theta }{\cos \,\theta -Z}}}$ with Z = N ₀ - 2

This function includes the following fundamental functions: gnomonic, triangular, orthographic. Even if the functions “equidistant” and “true to area” can only be achieved approximately, the model fits well. In this model, the edge areas are compressed more. This is usually the same with real lenses.

The general panini projection uses this method in the horizontal, but the formula contains the parameter d (distance, Z = - d). The panorama viewer "krpano" enables a fisheye effect by fading from straight (gnomonic) to fisheye distorted with the help of the parameter view.distortion (distortion, Z = - view.distortion). And the room distortion with almost light-fast movement can also be described with this formula (   ). ${\displaystyle Z=-{\tfrac {v}{c}}}$

Constructive models

• Optics calculation

If the structure of the optical system is known, the path of light rays can be calculated. The parameters here are the sequence of lens surfaces (vertex position, radius of curvature and refractive index of the subsequent medium), as well as the position of the aperture diaphragm and image plane. Make sure that the beam goes through the center of the aperture diaphragm. The beam path through the optical system must be calculated for each angle of incidence or, in the case of backward calculation, for each pixel. The mapping function results from the angle of incidence and the point of incidence of the image plane. In comparison to the other mapping functions, instead of one calculation step, several calculation steps are required according to the number of interfaces. Nevertheless, the calculation is still relatively simple because, in contrast to the development of optics, no calculation with broad bundles of rays or wavefronts, meridional and sagittal image shells or aberrations is necessary.

• Interpolation curve

Based on the optics calculation, a table is created with a sufficient number of angle of incidence-image radius pairs. These pairs form a series of points that are connected by spline interpolation using polynomial pieces. This reduces the computing effort compared to the pure optics calculation . Some manufacturers publish such tables.

Corrective models

• Deviation polynomial

Instead of simulating a fisheye lens very precisely with a high-grade polynomial, it is cheaper to approximate the optics with a simple model. The remaining deviation between the real optics and the model is described by a polynomial:

${\displaystyle r=f\,\sum _{i=0}^{n-1}a_{2i+1}\left({\frac {r_{\rm {Modell}}}{f}}\right)^{2i+1}}$  (  odd links) ${\displaystyle n\,}$

Parameters are , and further coefficients up  . The model function must be one-to-one (a bijective function ) within the entire image field . The most similar fundamental mapping or a parametric mapping with only one but appropriately chosen parameter is suitable as a model function. The deviation polynomial is then less complex. If the model function in the paraxillary region agrees well with the real optics, then . However, the uniqueness of the model function has absolute priority over the paraxillary agreement. ${\displaystyle a_{1}=1}$ ${\displaystyle a_{3}={\frac {N_{0}}{18-6\,N_{0}}}-{\frac {N_{0,{\rm {Modell}}}}{18-6\,N_{0,{\rm {Modell}}}}}}$ ${\displaystyle a_{2n-1}}$ ${\displaystyle a_{3}=0}$

Special case: If the model function is angularly linear (equidistant), the polynomial and deviation polynomial have the same coefficients.

Separate polynomials of deviation for red, green and blue can capture the lateral color errors of the optics. For this, mostly and / or the individual color channels are varied slightly from one another. Sometimes you limit yourself to the detection of color defects and ignore the general deviation, e.g. B. no panoramas should be created. ${\displaystyle a_{1}}$ ${\displaystyle a_{3}}$

The correction polynomial is inverse to the deviation polynomial

${\displaystyle r_{\rm {Modell}}=f\,\sum _{i=0}^{n-1}b_{2i+1}\left({\frac {r}{f}}\right)^{2i+1}}$  (  odd terms). ${\displaystyle n\,}$

For and for the following terms ( ), the correction polynomial can be generated with sufficient accuracy using and ( ). With the correction polynomial, the image can be converted into the selected model function and the lateral chromatic aberration can be reduced - usually as the first step for conversion into another projection type. ${\displaystyle a_{1}\approx 1}$ ${\displaystyle |a_{2i+1}|\ll 1}$ ${\displaystyle i\geqq 1}$ ${\displaystyle b_{1}={\frac {1}{a_{1}}}}$ ${\displaystyle b_{2i+1}=-a_{2i+1}}$ ${\displaystyle i\geqq 1}$

• Scaled function

Sometimes there are only a few not quite suitable mapping functions available. Usually they deviate on one side. With a different focal length , an imaging function can be optimized in such a way that the actual imaging function is crossed and the deviations on both sides are the same and thus as small as possible. The focal length becomes a parameter. The prerequisite is that the actual function is known at least with a few points. Due to the changed focal length, center details are misinterpreted in terms of size (mostly not disturbing). ${\displaystyle f_{\text{adapted}}}$ ${\displaystyle f_{\text{adapted}}}$

• Dummy function

Sometimes there are only a few not quite suitable nonparametric mapping functions available. For these, the focal length can be optimized in such a way that certain transformed image properties are improved. ${\displaystyle f_{\text{pseudo}}}$

For example, you will vary the focal length for equal area imaging until z. B. a gnomonic conversion results in straight edges - hereinafter referred to as straight line calculation. If the converted line becomes wavy, you can try optimizing the orthographic, angular linear and conformal mapping and opting for the mapping function that produces the least waviness when calculating straight lines. Instead of straightness you can z. B. also optimize for freedom from deformation by conformal conversion. then has a different value. The straight line calculation, however, is the simpler and very sensitive method, with which it can be determined exactly to a fraction of a percentage (by comparison with a drawn line). ${\displaystyle f_{\text{pseudo}}}$ ${\displaystyle f_{\text{pseudo}}}$

The straight line calculation enables an almost perfect gnomonic conversion. Converting with the same focal length into the conformal image leads to a minor but noticeable error. There are bigger problems when trying to create panoramas. The reason for this is that, in contrast to a scaled function, there is no adaptation to the true mapping function curve and the angles of the objects are also misinterpreted due to the wrong focal length.

The true focal length and the true imaging function can be determined by taking a photo of a test arrangement with a known object-side angle . To do this, however, the wrong angle must also be calculated from the image using the inverse sham mapping function. ${\displaystyle f}$ ${\displaystyle \theta _{\text{pseudo}}}$

In the gnomonic conversion, the following applies:

${\displaystyle r=f\tan \theta =f_{\text{pseudo}}\tan \theta _{\text{pseudo}}}$

This can be used to determine the true focal length:

${\displaystyle f=f_{\text{pseudo}}{\frac {\tan \theta _{\text{pseudo}}}{\tan \theta }}}$

We define a correction factor

${\displaystyle k={\frac {f}{f_{\text{pseudo}}}}={\frac {\tan \theta _{\text{pseudo}}}{\tan \theta }}}$,

which becomes a parameter to use the true focal length.

With the standardized mapping functions and the apparent and the true mapping function can be compared. ${\displaystyle \mathbf {h_{\text{pseudo}}} }$ ${\displaystyle \mathbf {h} }$

${\displaystyle r=f\,\mathbf {h} (\theta )=f_{\text{pseudo}}\mathbf {h_{\text{pseudo}}} (\theta _{\text{pseudo}})={\frac {f}{k}}\mathbf {h_{\text{pseudo}}} (\arctan(k\tan \theta ))}$   (Straight line method)

The core of the correction is . This bends the angular characteristic in a wave shape, whereby the values ​​for 0 °, 90 ° and 180 ° remain. This is a very special correction that not every parametric mapping function can provide. ${\displaystyle \theta _{\text{pseudo}}=\arctan(k\tan \theta )}$

A dummy function makes a hidden correction by entering an incorrect focal length. Only the conversion for which the dummy function has been optimized works. The extraction of the true correction enables a new parametric mapping function that works for all conversions.

Examples of even calculated apparent functions (left, apparent function, right resolved to the true function):

${\displaystyle r=2\,f_{\text{pseudo}}\sin \left({\frac {\theta _{\text{pseudo}}}{2}}\right)={\frac {2\,f}{k}}\sin \left({\frac {\arctan(k\tan \theta )}{2}}\right)}$   apparently true to area

${\displaystyle r=f_{\text{pseudo}}\,\theta _{\text{pseudo}}={\frac {f}{k}}\arctan(k\tan \theta )}$       apparently equidistant

The effects of incorrect focal length show that for all other functions it is important to determine the true angles and focal lengths. A photo of a test arrangement is essential for this. With the true focal length there are fewer problems with conversion. The resolved apparent functions (with the substituted expressions for true angles and focal lengths) can be used without any problems and are parametric by k. Already parametric functions can also be upgraded with the parameter k in order to adapt the function curve even better to the real optics.

Hints

Mapping functions of real fish eyes differ more or less from the fundamental basic functions. If the accuracy requirements are high, parametric functions should be used. The "shifted projection center" function is quite suitable. Otherwise one can make do with dummy functions.

The mapping functions must have their zero point in the deformation-free center of the mapping (the pixel in which neither horizontal nor vertical lines are bent). This point is offset to the center of the image and, in the case of real lenses, also not exactly on the optical axis of the fisheye or exactly on the center of the image circle of a circular fish eye. If this offset is not taken into account when calculating straight lines, all lines are bent in the same direction. Test recordings in landscape and portrait format with lines at the top and bottom can help to determine the vertical and lateral offset between the deformation-free center and the center of the image with calculations. The center of an image circle or the position of the optical axis can be used for normal requirements. The center of the image is shifted by the sensor installation and the position of the used part of the sensor area compared to the bayonet and should only be used for low demands.

The mapping functions only apply to objects that are sufficiently far away. The entrance pupil of fisheyes is not stationary, but moves with an increasing angle of incidence on an arc forwards and to the side of the incident ray. A parallax error occurs for close objects; the displaced entrance pupil sees them at a larger angle of incidence and a smaller distance than would be expected from the frontal entrance pupil. Objects that are close by are also stretched radially. This more or less compensates for the edge compression of most fish eyes. The mapping function changes depending on the distance. Nearly straight edges are therefore unsuitable as a reference for straight calculations. A suitable reference is e.g. B. the ocean horizon.

The conversions are often based on the image, i.e. backwards. Accordingly, inverse mapping functions are to be used. In the case of backward calculation, instead of the angle of incidence, either a direct conversion is made into a target function or into a 3D point in space (back projection).

conversion

If the distortions typical of fisheye lenses are not desired as a design element, digital fisheye recordings can be corrected with the help of image processing programs. However, this is usually associated with decreasing quality towards the edge of the image, or with a restriction of the field of view shown.

There are different projections into which a fisheye image can be converted:

• azimuthal is a group of the following projections: gnomonic, true-angle, linear-angle, true-area and orthographic (chapter Fundamental mappings ). Conventional lenses and fisheye lenses ideally depict this. Parametric azimuthal functions such as scaled angle function and shifted projection center can be the source of a conversion, but not a desirable goal.

Lines through the center of the image (e.g. vanishing lines) remain straight and in the original direction. Center circles remain circular. A tilted recording can be corrected very easily by rotating the image even after the conversion.

• Cube network: The special case of the cubic cross is the development of a surrounding cube and contains contiguous gnomonic projections.

The picture lines are shown kinked on the folded edges of the cube. Other edges must be cut open for unwinding. Points on either side of such edges swing apart and lose their neighborhood. The special case of the cross shape requires an image area of ​​4 × 3 partial image widths to display the 6 partial images. This has the disadvantage that only 50% of the image area is used.

The special case series has all partial images next to each other and is not a development. The resulting image has an aspect ratio of 6: 1 and consists e.g. B. from the arrangement front-right-back-left-top-bottom. The image area is used 100%. Some panorama viewers recognize the "cube projection" because of this aspect ratio and can use it as a source in addition to the spherical panorama (2: 1).

• Panorama is a group of cylindrical projections that unfold onto a plane. With a normally vertical cylinder axis, there is a horizontal angular linear division. The vertical direction (axis direction) is scaled according to a function, but independently of the rotation angle. According to this scaling, there are the following panoramas: gnomonic (cylinder development), true to angle (Mercator), angle linear (spherical panorama), true to area (Lambert).

Vertical lines in the axial direction remain straight. All directions around the axis are equal. A tilted recording leads to a wavy horizon and S-shaped verticals. The maximum possible angle of view is 360 ° horizontally and 180 ° vertically.

Transverse panoramas work the same way, but the cylinder axis is horizontal so horizontal lines stay straight. The maximum possible angle of view is 180 ° horizontally and 360 ° vertically.

• Multiple gnomonic panorama ( Multiple rectilinear panorama ) according to Dersch is a projection onto a prismatic surface that is developed.

As with the panorama, the verticals are straight. Horizontal lines are no longer curved, but kinked at the edges of the lateral surface (possibly with rounding) and straight in between. The outer surface must be set up to match each image. At a house front z. B. the creases are to be placed on the boundaries between the individual houses.

In contrast to the cube network, nothing has to be cut open.

• Pannini projection (alternative notation: Panini projection ) is a cylindrical projection that is not developed, but projected onto the image plane. In the horizontal direction of rotation, the function is analogous to an azimuthal projection (angular, angular linear, orthographic, or shifted projection center). In the vertical axis direction, the projection is gnomonic and scaled to the side angle in such a way that the central alignment lines are straight and run at the original angle.

Vertical lines in the axial direction and central alignment lines are straight. A tilted receptacle bends off-center verticals in an S-shape.

In the case of a very large vertical angle of view, the gnomonic projection can be replaced by the Mercator projection in the vertical axis direction in order to reduce distortions. This is then no longer a real pannini projection and is e.g. B. called veduta mercator . Inclined alignment lines are then also slightly S-shaped.

• Round rectangle projection according to Chang, Hu, Cheng, Chuang is the projection of a square ellipsoid onto the image plane. The azimuthal projection and the pannini projection are also possible as special cases.

A circular 180 ° image field is transformed into a rectangle, with the corners rounded to avoid sharp-edged kinks. The rectangular diagonals divide the image field into four triangles. In the upper and lower triangles, horizontal lines become straight. In the left and right triangles, vertical lines become straight. Image angles of over 180 ° are possible.

• Wieden's rectangle is the superposition of a normal and a transversal panorama.

By adopting the x-values ​​from one panorama and the y-values ​​from the other panorama, both straight horizontal and straight vertical lines can be achieved regardless of their position in the image. Inclined lines run in an S-shape and are bent towards the edge of the picture in the 45 ° or 135 ° direction. The opening angle is limited to 180 °. Depending on the type of panorama, the rectangular projection can be angular, angular linear, or area-true.

• RectFish is a projection that fits a circular image field trimmed at the top and bottom into the rectangle of the image format.

As much as possible of the original image is retained. As with panorama, vertical lines become straight. The horizontal division, however, remains non-linear, and the radial edge compression also remains.

There are other projections that can be converted into, e.g. B. the Quincunx map projection (sample photo 360 ° in the English edition - accessed on December 20, 2015). Depending on the subject, one or the other projection is more suitable.

In difficult cases, the direction of the camera can also be changed using software, e.g. B. to correct converging lines or to compensate for tilting. This correction is made by projecting the image back into the 3D space onto the surrounding sphere or an image shell using the source mapping function, rotating the space using a transformation matrix, and then opening the straight or suitably rotated space again using the target mapping function the image plane is calculated back.

Applications

• Scientists and resource managers (e.g. biologists, foresters, geographers and meteorologists) use fisheye lenses to capture a hemispherical area of ​​plant vegetation in ecophysiology or the potential short-wave radiation from the horizon elevation ( sky view factor ) and from it to predict the long-wave radiation to be derived. From the analysis of these images, relevant structural parameters of tree tops such as leaf area index (LAI), leaf angle distribution and light availability close to the ground can be derived . Fisheye lenses also help with the evaluation of the forest condition , the recording of the wintering places of monarch butterflies and the management of vineyards . In topoclimatology, the formation of deep frosts in inversion weather conditions can be derived from the elevation of the horizon of ground depressions, as well as statements about the causes of cold-air lake phenomena ( Funtensee ).
• From data of the parameters of the sky view factor obtained from geo-referenced fisheye recordings , meteorological relationships of the radiation balance of urban heat islands are examined in urban climatology.
• Meteorologists determine the cloud cover (degree of sky cover).
• Astronomers record a large part of the sky and use it to record constellations , the Milky Way , meteors , polar lights and light pollution .
• Many planetariums use fisheye projection lenses to project the night sky or other digital content onto the interior of a dome.
• Surveillance cameras with fisheye lenses can capture an entire room at once. In contrast to pivoting cameras, there is no temporary dead area and no susceptible drive.
• Video door intercom systems with a particularly large angle of view ( peephole function).
• With the IMAX dome system (previously 'OMNIMAX'), the film is recorded through a circular fisheye lens, and the cinema film is projected through a similar lens onto a hemispherical screen.
• Photographers and videographers use fisheye lenses to bring the camera as close as possible to the crucial point for action shots, while also capturing the overall context. For example, when skateboarding , the focus is on the board and the skater can still be seen in the picture.
• The first music video recorded entirely with a fisheye lens was the 1989 Beastie Boys song " Shake Your Rump " .
• Flight simulators and visual combat simulators use fisheye projection lenses to create a seamless environment for pilots, air traffic controllers or military personnel to train
• In computer graphics , circular fish eyes can be used to create an environment mapping of the physical world. A complete 180 degree fisheye image can fill half of a cubic environment mapping with a corresponding algorithm. Environment maps can be used to reproduce 3D objects embedded in virtual panoramas.

gallery

• Example shots with fisheye lenses
• 

  Space view of a tropical cyclone over the Bahamas .

• 

  The nave of the Saint-Julien de Brioude basilica in Auvergne .

• 

  Aerial view of Diósgyőr Castle in Hungary .

• 

  The Moul waterfall in Wells Gray Provincial Park ( British Columbia , Canada ) seen from the neighboring rock cave.

• 

  Hemispheric image of the night sky with the Milky Way in the Star Park Westhavelland .

• 

  Giant Arborvitae on the Giant Cedar Trail in Mount Revelstoke National Park ( British Columbia , Canada ).

Alternatives

Lens systems

Fisheye attachment lens for smartphone

Interior photographed with smartphone and fisheye lens

In order to achieve the fisheye effect, special attachments are also available that are screwed, snapped on, clamped (clips or clamps) or (e.g. on smartphones ) attached to a normal lens at the front . The combination of normal lens and attachment then behaves like a fisheye. The resolutions are available as supplementary lens and a conversion lens.

Auxiliary lens

A fisheye attachment (fisheye converter) is a greatly reducing wide-angle converter with barrel-shaped distortion. It consists of several lenses and works like a Galilean telescope used in reverse . The refractive power required for imaging comes exclusively from the following lens.

In order to keep the loss of image quality low, the entry pupil of the objective must be at a certain distance from the attachment and must not be too large ( stop down ). Only at a certain distance between the attachment and the objective is an image position seen through the appropriately corrected position of the rear lens of the attachment. Seen from the entrance pupil of the objective, the edge of the field of view should fit exactly on the edge of the rear lens (applies to fisheye attachments). Depending on the lens or smartphone, the distance is often not optimal, and the image quality then deteriorates considerably. Newer smartphone cameras have a larger opening due to their more powerful lenses, which in conjunction with additional lenses also leads to poorer image quality.

Zoom lenses with fisheye attachment can be used over the entire focal length range, whereby the fisheye effect is visible in the wide-angle position. With poorly corrected attachments, the image can be blurred in the greater telephoto range.

A peephole works similarly, but the eye takes over the function of the following lens.

Auxiliary lens

A specially shaped diverging lens, often called semi-fisheye , creates a reduced virtual image with barrel-shaped distortion. Due to the fairly flat front, the entire area in front of the lens (180 ° image angle) in the glass is refracted to twice the angle of total optical reflection (approx. 80 °). A hemispherical recess on the back leaves or reduces this angle of view when it emerges from the glass. This lens has a negative focal length and the following lens must focus at an extreme close-up distance (a few centimeters). Not every lens can do that. At best, zoom lenses can focus that close in the wide-angle range. Additional lenses are not zoomable. The quality is even worse than that of an auxiliary lens.

mirror

Mirror images on convex mirrors are barrel-shaped like fish-eye lenses. When viewed from a great distance, a mirror ball creates an equi-solid image. The back of a parabolic mirror creates an image that is conformal. Other types of imaging are also possible with the appropriate mirror shapes. Due to the self-imaging of the photo equipment, only a ring-shaped zone can be used.

advantages

Image angles over 180 ° are possible without any problems. This allows 360 ° cylinder panoramas to be generated from an image or video (moving panorama). In the case of videos, it is not possible to take several pictures between the picture phases for later assembly (so-called stitching).

disadvantage

The camera (possibly also tripod and photographer) can be seen in the middle of the mirror image. With a long focal length macro or telephoto lens, this area can be kept small, but it cannot be avoided if you only aim for a single image instead of a panorama.

The area behind the mirror is lost. This is not a real disadvantage, because lens fish eyes cannot see all the way back.

There are no mirror fish eyes on the free market. You have to create suitable solutions by building your own.

But there are also ready-made omnidirectional cameras with a catadioptric system. A lens optic is supplemented with a mirror. This creates a coordinated, compact system that usually also includes the camera. For panoramas, the lens optics should be directed upwards, with the mirror above it. A thin-walled glass cylinder carries the mirror (individual mirror supports would interrupt the panorama). The ring panorama can be converted into a normal panorama. Nadir (covered by the lens optics) and zenith (covered by the mirror) cannot be displayed. Several mirrors are also possible, or the mirroring takes place within a glass body by total reflection or mirror coating.

Underwater view

Water ideal n = 1.33349

Real water n is color dependent.

When a fish looks out of the water, it sees the outside world reduced in size and distorted into a barrel. The outside world (180 °) appears in a cone with an angle of view of 96 ° (double the angle of total optical reflection). This area is also called the Snell window . Outside this window, the underwater world is reflected on the surface of the water through total reflection. The same effect occurs with a side view from an aquarium with flat walls.

The outside world appears reduced to 75% in the middle and compressed at the edge. A diving mask or an underwater camera housing with a flat lens cancels this effect, since the direction of view is broken at the water level created by the lens. Only with the eyes located directly in the water or from the center of the sphere of a transparent hood can one see into the water without distorting the direction and thus perceive the fish-eye effect. The water is formed into an auxiliary lens through the eyeball or the hood (see above under "Lens systems"). In the case of eye-water contact, one cannot see clearly. In addition, waves on the water surface prevent a clear view.

Robert W. Wood, Professor of Experimental Physics, described this effect and built a water bucket camera for looking up and a water-filled pinhole camera for any viewing direction, and thus took the first fisheye photos. This gave the later developed lenses with the same point of view the name fisheye lens.

The formula describes the image of a pinhole camera that is filled with an optically refractive medium:

${\displaystyle r=f{\frac {\sin \theta }{\sqrt {1-{\frac {\sin ^{2}\theta }{n^{2}}}}}}}$

${\displaystyle r}$ Image position as the distance to the center of the image

${\displaystyle f}$ (central) focal length, also takes the reduction into account 

${\displaystyle f={\frac {s}{n}}}$

${\displaystyle s}$ Back focus, distance between aperture diaphragm (hole) and image plane

${\displaystyle \theta }$ Polar angle of the external object to be depicted

${\displaystyle n}$ Refractive index of the medium

Air has an index of refraction close to . In this case the figure is gnomonic. If the refractive index is very high, the image becomes orthographic. The other functions from the “Mapping functions” chapter can be approximated in the center of the image ( ), but cannot be reproduced for large angles. For a refractive index greater than one, the viewing angle into the outside world is always 180 ° and the edge is compressed. ${\displaystyle n=1}$ ${\displaystyle n={\sqrt {3-N_{0}}}}$

Stitching

Fish-eye effect using the app

Panorama from camera with two fish eyes

Several recordings from the same point of view can be combined with stitching or panorama software to form an image or a 360-degree video . There are also panorama cameras that are equipped with multiple lenses and image sensors that are triggered simultaneously.

The software can usually output several types of projection and thus also simulate a fish-eye image. If the software can only output a panorama, this can be converted into a fisheye image with other software.

For example, there are also camera apps for smartphones that create the fisheye effect as a variant of the panorama image.

Web links

Commons : Fisheye Recordings  - collection of images, videos, and audio files

Wiktionary: Fisheye lens  - explanations of meanings, word origins, synonyms, translations

• Illustrated history of the Nikon fisheye lenses (German)
• Additional Information on Fisheye-Nikkor lenses (English)
• About projection and distortion (PDF; 949 kB) and about design aspects (PDF; 2.4 MB), two basic articles about fish-eye lenses
• Fisheye in olypedia.de
• Beam paths, animations (English), original in French
• Entaniya Fisheyes 200 ° / 250 ° HAL - imaging function. fisheyelens.de, accessed on February 15, 2018 .  - Example of a polynomial calculation

Individual evidence

1. ^ Chadwick B. Martin: Design issues of a hyper-field fisheye lens. (PDF; 207 kB) University of Arizona , archived from the original on June 6, 2014 ; accessed on June 5, 2014 . 
2. ↑ Additional Information on 8mm f8 Fisheye-Nikkor Lens (English)
3. ↑ M.ZUIKO DIGITAL ED 8mm 1: 1.8 Fisheye PRO - OM-D & PEN lenses - Olympus. Retrieved March 1, 2017 . 
4. ↑ Entaniya Super Wide Fisheye Lens for P0.5 / M12. Entaniya Co., Ltd., Accessed February 15, 2018 . 
5. ↑ ^{a b} HAL 250/200 - Entaniya. Entaniya Co., Ltd., Accessed February 15, 2018 .  , (Tables under Technical Specification → MFT / Image Height: More Info , accessed on 2018-02-15)
6. ↑ Canon EF8-15mm f / 4L Fisheye USM on camera with sensor in 35mm format.
7. ↑ smc Pentax-DA FISH-EYE 10-17mm 1: 3.5-4.5 ED for camera with crop sensor ( Downloads tab for sample images ).
8. ↑ ^{a b} Panini projection (English).
9. ^ ^{A b c} Thomas K. Sharpless, Bruno Postle, Daniel M. German: Pannini: A New Projection for Rendering Wide Angle Perspective Images. (PDF 16.35 MB) Computational Aesthetics in Graphics, Visualization, and Imaging, 2010, accessed on August 12, 2014 . 
10. ↑ ^{a b c} Spatial motifs , in: Wikibook digital imaging methods - image recording , accessed on December 31, 2016
11. ↑ Samyang Fish-eye at lenstip.com (English)
12. ↑ Samyang 8 mm f3.5 fisheye CS lens, shave and test report (English).
13. ↑ Little planet in the English Wikipedia
14. ↑ DB Gennery: Generalized camera calibration including fish-eye lenses. (PDF 5.06 MB, Chapter 2.3. "Basic Lens Model", p. 12) Jet Propulsion Laboratory , 2003, accessed on August 10, 2014 (English). 
15. ↑ Fisheye Projection. Retrieved August 20, 2014 . 
16. ↑ krpano XML Reference at krpano.com (English)
17. ↑ DB Gennery: Generalized camera calibration including fish-eye lenses. (PDF 5.06 MB, Chapter 2.2. "Moving Entrance Pupil", p. 7) Jet Propulsion Laboratory , 2003, accessed on August 10, 2014 (English). 
18. ↑ ^{a b c} Peter Wieden: Zimmer , accessed on December 12, 2015.
19. ↑ Bredenfeld, Thomas: Digitale Fotopraxis Panoramafotografie . 2nd, updated and expanded edition. Galileo Press, Bonn 2012, ISBN 978-3-8362-1861-0 , 5.12 Basic recording examples , p. 66 .
20. ↑ H. Dersch: Multiple Rectilinear Panoramas. 2010, accessed December 23, 2015 . 
21. ↑ Che-Han Chang, Min-Chun Hu, Wen-Huang Cheng, Yung-Yu Chuang: Rectangling Stereographic Projection for Wide-Angle Image Visualization. (PDF; 11.5 MB) 2013, accessed on December 11, 2015 (English). 
22. ↑ RectFish projection comparisons , accessed December 20, 2015.
23. ^ John P. Snyder, Philip M. Voxland: An Album of Map Projections (Professional Paper 1453). (PDF; 12.6 MB, 249 pages) US Geological Survey , 1989, accessed on December 26, 2015 . 
24. ↑ Geiger, R. Aron, RH, Todhunter, P. 2009: The Climate Near the Grond. 7th edition, Rowman & Littlefield, London. ISBN 978-0-7425-5560-0 . (Rudolf Geiger: Das Klima der Bodenennahen Luftschicht . Verlag F. Vieweg & Sohn Braunschweig 1927 = Die Wissenschaft Vol. 78; 2nd edition ibid. 1942; from 3rd edition title with the addition Handbuch der Mikroklimatologie ibid. 1950; 4 . Ed. Ibid. 1961. 5th edition under the title The Climate near the Ground, edited by Robert H. Aron and Paul Todhunter. Verlag F. Vieweg & Sohn Braunschweig 1995).
25. ↑ Snell's window in the English Wikipedia
26. ↑ RW Wood: Fish-Eye Views, and Vision under Water. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Volume 12, Number 6. In: Taylor & Francis Online. Johns Hopkins University, August 1906, pp. 159-161 , accessed June 3, 2019 .  
27. ↑ Prof. RW Wood: Fish-eye view and seeing under water (translation). 1906, Retrieved August 25, 2019 .