About the various projections of the photographic objective lenses

(With an emphasis on Fisheye projections)

Foreword

This page is mainly dealing about qualitative and empirical description of the lens projections. To read more about (classical) mathematical modelling of the (classical) projection, please read this other page.

Overview

While there are other types of photographic systems (e.g. catadioptric, slit imaging, rolling shutters, etc.,), we shall deal in this article only about refractive lens projections. Amongst all possible such lens projections, the word "Fisheye" actually covers a wide variety of projection types. They all make a more or less strongly distorted images where (most of) the straight lines (in 3D real life) are not straight at all on the 2D images. The geometrical aberration resembles the infamous "barrel distortion" that may affect imaging with some Standard perspective lenses. By even more exaggerating the disconcerting bending of the straight lines and edges on the 2D image it is then possible to increase considerably the Field Angle (sometimes to more than 180°) and ultimately, to make for instance the image boundary line to ultimately become a circle and letting the four corners of the image as black areas. More than an hemisphere of the object space may thus be captured on the image plane...

.... but then straight lines of our 3D world around us are not imaged as straight lines on the 2D photograph... thus, the more frequent way of describing and analyzing the various fisheye lens projection that is found in the literature is to present this phenomenon as a "distortion". That is to say a deviation from the "normal" ideal and expected state. The latter is the central perspective projection which is itself often reduced to its simplest physical form i.e., the Pinhole camera. The fisheye lenses as a whole are consequently and casually considered as optical "alien species".

Furthermore, at first glance, any fully "circular" fisheye images looks about the same to any other one to the casual viewer. But in fact they may not really be alike and IMO they are far from being so. This common misperception of an homogenous group comprising all the sorts of fisheye lenses is deceptive. This article is a modest attempt to unveil the hidden variety and peculiarities of the "Fisheye" lenses that are nowadays very popular amongst the panorama photographers (and often these optics are their favorite).

On this page, we shall first present the four different "classical" fisheye models that are invariably listed in the literature. As a matter of fact, these conventional mathematical models have been suggested to virtually represent all fisheye lenses projection (or rather, by tradition, the opposite : fisheye lenses are, or were, constructed in order to more or less satisfy such traditional math models). Stereographic, Equidistant, Quasi-solid angle and Orthographic are the projection names that were introduced early in the 60's in Optics publications (e.g. Miyamoto, K. 1964. "Fish eye lens" in Journal of the Optical Society of America). When these four lens models are added to the standard and ideal Central Perspective (or "Rectilinear) projection, then the whole photographic refractive photographic lens domain is assumed to be fully covered.

In fact, when a "real lens" does not fit exactly the requirements for one (out of five) given "classical" model, lens manufacturers feel compelled to mention a so-called "Distortion" (with respect to this reference model) and to quantify it (% distortion) on the lens specification sheet. As a consequence and as fisheye lenses are concerned, it is presently almost impossible not to follow the tradition that forces us to label any fisheye lens with a "classical" name amongst the four discrete canonic classical types. On the positive side, this is a useful and easy (but very approximate) point of reference although it refrains the photographer from fully understand the realities of the Optics that he may use. Confused or false information about various fisheye features and other performances have subsequently spread everywhere on the web:(

Images of same Horizontal Field of View (90°) by photographic lenses with different Classical Projections

All images below on this table are simulated images. In this example, they represent the output from a 24 x 36 mm camera fit with a 54 Mega pixels sensor (9000 x 6000 px) in portrait mode. They were subsequently reduced down to 630x 420 pixels for the presentation below.

Important : The camera (sensor) is the same for all of the 5 lenses, but the projection and the focal length f (mm) of the lens is not the same. However "f" was adequately selected so as to get the same Horizontal Field of View = 90 degrees on each image.

Central Perspective = the Standard lens projection
The four "Classical" fisheye lens projections
Rectilinear projection f= 12.3 mm
Stereographic fisheye projection f= 14.5 mm
Equidistant fisheye projection f= 15.3 mm
Equisolid angle fisheye projection f= 15.7 mm
Orthographic fisheye projection f= 17 mm
  • HFov= 90°
  • VFoV; 112.6°
  • Diagonal FoV= 119.3°
  • HFov= 90°
  • VFoV= 127.3°
  • Diagonal FoV= 141°
  • HFov= 90°;
  • VFoV= 135°
  • Diagonal FoV= 153.5°
  • HFov= 90°
  • VFoV= 140°
  • Diagonal FoV= 163°
  • HFov= 90°;
  • VFoV= 180°(circle limited)
  • Diagonal FoV= 180°(circle limited)
This is an imaginary and extraordinary (sharp to the corner , undistorted and vignette-free) Ultra Wide Angle (UWGA) lens!
The image resembles the next one on its right
This is an image of the often-called "ideal" fisheye projection
The image resembles the next one on its leftt
Compression near the circular boundary of the image

If the horizontal and the vertical field of view is cropped for any reason to a moderate angle value e.g. 60° to 90°, then the apparence of the images is quite similar from one to the next on this table. The differences become obvious and important when comparing both ends though.

 

 

Photographic lenses with the same Focal Length but with various Classical Projections:

For the sake of simplifying the demonstration, the following presentation form was chosen because the images then resemble those that are usually shot by panorama photographers with a digital camera. Note that a "MouseOver mode" on the images is available:

All images are simulated images. In this example, the images would have been shot with a 24 x 36 mm camera fit with a 54 Mega pixels sensor (9000 x 6000 px) in portrait mode. They were subsequently reduced down to 630x 420 pixels for the presentation below.

Important : The focal length is set at 8 mm for all the 5 lenses although a 5 mm lens on a crop sensor camera (r= 1.6) would give exactly the same visual results.
Central Perspective: the Standard lens projection
The four "Classical" fisheye lens projections
  • Rectilinear projection (Theoretically possible -yet unrealistic- UWA. Focal length = 8 mm (or APS-C: 5 mm)
  • Horizontal Field of View= 112.6°
  • Vertical Field of View=132°
  • Diagonal Field of View= 139.4°
  • Stereographic fisheye projection (Theoretical) . Focal length = 8 mm (or APS-C: 5 mm)
  • Horizontal Field of View=147.5°
  • Vertical Field of View=194.4°
  • Diagonal Field of View= 214°
  • Equidistant fisheye projection (Theoretical). Focal length = 8 mm (APS-C: 5 mm)
  • Horizontal Field of View= 171.8°
  • Vertical Field of View=257.8°
  • Diagonal Field of View= 309.4°
  • Equisolid angle fisheye projection (Theoretical) . Focal length = 8 mm (or APS-C: 5 mm)
  • Horizontal Field of View= 194.4°
  • Vertical Field of View= 360°
  • Diagonal Field of View= 360°
  • Orthographic fisheye projection (Theoretical & practical) . Focal length = 8 mm (or APS-C: 5 mm)
  • Circular Field of View= 180° (Maximum)
  • All straight lines and edges in the object space become straight on the image.
  • The field of view is... limited
  • The image corners seems to be too stretched (e.g. look at the lamp on the left edge of the image.
  • Beware: the curved shelf below the workbench on the foreground is actually and naturally bent in real life!
  • Except for the lines passing through or near the image center, all straight lines are bent.
  • The field of view is not limited and can reach 180° and more.
  • The objects retain their general true shape even near the corner of the image.
  • The radial compression of the image is nearly imperceptible.
  • Except for the lines passing exactly through the image center, all straight lines are bent, especially when approaching the boundaries.
  • The field of view is not limited and can reach 180° and much more.
  • The objects loose their original shape when approaching the corner of the image, but less than with UWA rectilinear lenses.
  • The radial compression is moderate even in the corners.
  • Except for the lines passing exactly through the image center, all straight lines are bent, especially when approaching the boundaries.
  • The field of view is not limited (it goes to 360°!) but the image has become circularly limited. A black zone of lost pixels...
  • The compression can be marginally excessive as it renders the image nearly useless above about 190° (which is a rare occurrence).
  • Except for the lines passing exactly through the image center, all straight lines are strongly bent, especially when approaching the boundaries.
  • The field of view is limited (it cannot goes over 180°) and so is the diameter of the image on the sensor: lots of pixels are black and wasted!
  • Beyond about 165 - 170°, radial compression becomes more than hefty: the external part near the boundary of the image is therefore nearly useless:(

MouseOver (Green Crop): Maximum Practical coverage

The widest coverage of a typical real life rectilinear lens is limited to about 115-118° diagonally. That corresponds to a focal length of about 13 mm on a 24x36 mm Full Frame sensor.

The modern design for the Ultra Wide Angle (UWA) lenses often incorporates spherical lens elements. Hefty "mustache distortion" is commonly encountered with UWA but it can nowadays be easily, digitally, and perfectly corrected during post-production with a graphical tool. Some firmware of recent digital cameras allows the required correction to be performed onboard and on-the-fly...

MouseOver (Blue Crop): Practical 180° coverage

With the notable exception of the Nikkor 15 mm f/5.6, only few photographic true stereographic lenses existed or exist today, but this projection becomes widespread in video micro-cameras (e.g. automotive accessories). BTW some vendors in this field pretend to innovate by proposing this projection under fancy and IMO meaningless commercial designations such as "Taylored distortion" or "Reduced" fisheye distortion.

The widest coverage of the typical current stereographic lenses is currently limited to about 181° (e.g. 7.5 and 8 mm Samyang).

In other fields of application than photography (e.g. Infrared industrial imagery) the FoV may be larger.

New technologies (e.g. aspherical lens elements) allow to considerably reduce the size and weight of the objectives, especially for non-reflex and mirror-less cameras.

MouseOver (Black Crop): Practical 180° coverage

This projection has been selected as the "Ideal Type" by the scientific community in the 60's. It is often now the Point of Reference model for estimation and presentation of image "distortions" of fisheye lenses. For instance inventors (Patents), developers of Panorama stitching program (algorithms of), etc., are at last replacing Central Perspective by Equidistance for this important ruling role when fisheye projection is concerned.

The practical widest coverage for this type of projection (and for any FE yet constructed) is about 220° (Nikkor 6 mm f2.8).

Another rare, cumbersome and expensive equidistant lens is the famous Coastal Optical 7.45 mm f 5.6. The Peleng 8 mm f3.5 is fairly cheap in comparison!

BTW Nikon had Patented a 5.4 mm equidistant with an expected 270° FoV lens but it has never been built.

The typical FoV from commercially available equidistant lenses is about 182-185°.

MouseOver (Purple Crop): Practical 180° coverage

Equisolid projection is also known as equal-area projection, as the ratio of an incident solid angle and its resulting area in an image is constant. Very appreciated by some scientific researchers (e.g. forest canopy survey)

This is the most frequent type found on the vendors shelves. The apparent features of this lens projection resembles those of the Equidistant. The practical current widest coverage for this type of projection is about 205° on FF sensors while others including FE zoom lenses of this kind can easily provide about FoV= 195°. Unfortunately no full (uncropped) circular FE with FoV = >185° is available yet: despite manufacturers claims, on the shorter end of the focal range (i.e. ~(4.5 - 5.6) mm on APS-C and ~(7- 8) mm on FF), the actual useful FoV never have really reached the intended 185°. that's IMO a pity:(

Equisolid projection allows to design sharp and fairly compact and light lenses (e.g. Nikkor 10.5 mm, Panasonic Lumix G 8mm f/3.5 fisheye lenses).

Nikon has innovative selected this projection as the Reference for the Nikkor 10.5 mm f2.8 to in related Patent documents. BTW the inventor made a typo when writing the equidistance projection formula in its patent application!)

 

MouseOver (Red Crop): Practical 180° coverage

This projection is renown because Nikon made the famous (and now rare) "aspherical" Nikkor 10 mm f/5.6 OP. "OP" stands for "Orthographic Projection". Back then some scientific researchers wanted to get the most uniform illumination on the whole circular fisheye image and as light fall-off was difficult to physically correct on the photographic film. The orthographic projection approaches this peculiarity and thus simplifies the optical design when engineers attempted to fit the requirement.

Ironically, because of its optical specifics, this projection allows to design simple, cheap and ultra-compact lenses especially for door peephole cameras. However and for many reasons, it has rarely been seriously applied to "artistic" photography...

...until recently when Yasuhara tentatively introduced the Madoka180 to be E-mounted on the NEX cameras. It has a FoV of about 181.5° (making it presently the only circular FE for adaptor-less E-mount). It has a focal length of 7.3 mm and despite the FoV> 180°, the projection approximates quite closely the classical orthographic properties.

These images are synthetic: the standard lens projection FoV wideness, freedom from any distortion, uniform sharpness and absence of vignettting in all cases could unfortunately not be got "directly" from "real life" lenses and cameras.

 

(180°) Circular fisheye Vs (180°) Full Frame fisheye...

...with photographic lenses of adequate (different) focal lengths and of various projections. All images are limited to have a Horizontal FoV = 180° and in addition puposely:

or (alternatively):

Many fisheye lenses have a FoV of 180°. The actual FoV may even surpass this limit, except with the Orthographic projection.The same scene as for the preceding section has been photographed and simulated images with the same various lens projections are presented below in the same order as above. Again, and because the respective appearance depends proportionally on the focal length of the lens, the images have the same appearence on the monitor screen when coming from a "cropped sensor" or from a "full frame sensor". An adequate selection of the proper respective lens focal length does the trick;)

The Standard lens projection
The four "classical" Fisheye lens projections

Rectilinear projection

The specified (FoV= 180°) is impossible with this projection

Stereographic projection (Circular fisheye)

Focal length ~7 mm (APS-C: ~4.7 mm)

The image circular blue boundary simulates mechanical vignetting within the lens.

Equidistant fisheye projection (Circular fisheye)

Focal length ~7.6 mm (APS-C: ~5.1 mm)

The image circular black boundary simulates mechanical vignetting within the lens.

Equisolid angle fisheye projection (Circular fisheye)

Focal length ~9.2 mm (APS-C: ~6.2 mm)

The image circular pink boundary simulates mechanical vignetting within the lens.

Orthographic fisheye projection (Circular fisheye

Focal length ~12.2 mm (APS-C: 8.1mm)

The image circular red boundary simulates mechanical vignetting within the lens.

Real world lens focal length is limited to ~12 mm (FF camera) and ~8 mm (APS-C camera)
  • Horizontal Field of View= 180° (Specified)
  • Vertical Field of View= 180° (Cropped)
  • Diagonal Field of View=180° (Cropped)
  • Horizontal Field of View= 180° (Specified)
  • Vertical Field of View=180° (Cropped)
  • Diagonal Field of View= 180° (Cropped)
  • Horizontal Field of View= 180° (Specified)
  • Vertical Field of View= 180° (Cropped)
  • Diagonal Field of View= 180° (Cropped)
Circular Field of View= 180° (Specified and absolute possible Maximum)

MouseOver: Rectilinear projection

The specified (FoV= 180°) is impossible with this projection

MouseOver: Stereographic projection (FF fisheye)

  • Horizontal Field of View= 116°
  • Vertical Field of View= 159°
  • Diagonal Field of View= 180° (Specified)

MouseOver: Equidistant fisheye projection (FF fisheye)

  • Horizontal Field of View= 100°
  • Vertical Field of View=149.8°
  • Diagonal Field of View= 180° (Specified)

MouseOver: Equal area fisheye projection (FF fisheye)

  • Horizontal Field of View= 92.4°
  • Vertical Field of View= 144.2°
  • Diagonal Field of View= 180° (Specified)

MouseOver: Orthographic FE projection (FF fisheye)

  • Horizontal Field of View= 67.4°
  • Vertical Field of View= 112.7°
  • Diagonal Field of View= 180° (Specified)

 

 

Some test images samples

References and Links:

Perhaps the most provocative yet educated point of view about lens projections :" Perspective Projection: The Wrong Imaging Model" was written in 1995 by Margaret M. Fleck Technical report 95-01.

A brilliant paper "Nikon fisheye-Nikkor: the full story from the origins to the present, with 9 prototypes including a 5.5 mm FoV 270°!" by Marco Cavina. On another page Marco comes back to the earlier days of the fisheye lenses: a fascinating read!

The wonderful page also about Nikkor fisheye lenses by Pierre Toscani.

The incomparable fishlist by Luca Vascon.

 

Michel Thoby

6 November 2012