Fisheye projections from spherical maps

Written by Paul Bourke
May 2003, software updated January 2016

Update, October 2019: Added support for Orthographic, Stereographic and Equisolid mappings.
Update, December 2021: Changed the -c and -u options for the more general -x,-y,-z rotations

See also: Mapping a perspective projection into a fisheye image
Converting Images from Panoramic to Fisheye
Converting an equirectangular image to a perspective projection
Converting an equirectangular image to a cylindrical panoramic projection

The source code implementing the projections below is only available on request for a small fee. It includes a demo application and an invitation to convert an image of your choice to verify the code does what you seek. For more information please contact the author.


"sphere2fish" takes a full spherical map (equirectangular projection) and extracts one of an infinite number of possible fisheye projections. The fisheye can be orientated at will by performing any number of rotations about the three coordinate axes. The fisheye field of view can be chosen up to a full 360 degrees. In addition, the user may choose the level of antialiasing (supersampling) and an option to create a circularly bounds fisheye (traditional) or a rectangularly bound fisheye (the later is usually only appropriate for smaller aperture angles). The application is in the form of a UNIX style command line interface. The usage string is shown below which also illustrates various other options.

Usage: sphere2fish [options] sphereimage
Options
   -w n       width (and height) of the fisheye image, default = 1024
   -t n       fisheye FOV (degrees), default = 180
   -x n       tilt angle (degrees), default: 0
   -y n       roll angle (degrees), default: 0
   -z n       pan angle (degrees), default: 0
   -f         full rectangular fisheye instead of circular crop, default: off
   -180       input is only 180 degrees of longitude, default: off
   -a n       antialiasing level, default = 2
   -m n       fisheye mapping, 1=stereographic, 2=equisolid, 3=orthographic, default: 0 (equidistant)
   -p n       fisheye power function for equidistant, default = 1 (disabled)
   -o s       output file name, default: name derived from input filename
   -bg r g b  background image colour (outside fisheye circle), default: black
   -d         debug mode, default: off

The conventions used with the program as drawn below. In particular, zero longitude and latitude are taken to be in the center of the spherical image.

The following spherical image will be used to illustrate how the utility may be used.

Default settings
sphere2fish thai4.jpg
  Rotated 90 degrees about vertical
sphere2fish -z 90 thai4.jpg

Rotated 45 degrees in latitude
sphere2fish -x -45 thai4.jpg
  Look directly up
sphere2fish -x -90 thai4.jpg

Roll the camera about the forward direction
sphere2fish -y 30 thai4.jpg
  Do not clip to a circle
sphere2fish -f thai4.jpg

Lookup and create a 250 degree fisheye
sphere2fish -x -90 -t 250 thai4.jpg
  90 degree fisheye
sphere2fish -t 90 thai4.jpg

The rotational command line options -x, -y, and -z can occur multiple times, the order of rotation is performed in the opposite order in which they appear. So, for example the command "sphere2fish -x -60 -z 180 thai4.jpg" will first rotate by 180 degrees about the "up" axis (pan) and then rotate by 60 degrees about the "right" axis (tilt). The result would be different if the -x and -z options were in the opposite order.

 




Mapping a perspective projection into a fisheye image

Written by Paul Bourke
August 2022

See also Converting a fisheye image into a panoramic, spherical or perspective projection.

The source code implementing the projections below is only available on request for a small fee. It includes a demo application and an invitation to convert an image of your choice to verify the code does what you seek. For more information please contact the author.


The following maps a perspective projection into a fisheye image. This utility acts as if the perspective and fisheye image represent a scene rendered with each projection. There are other ways of mapping images onto fisheye which involve placing the perspective image on a plane and orientating the plane within a 3D scene, this is discussed for realtime planetarium applications Tools for Spherical Mirror Projection.

The usage string for the command line utility is as follows.

Usage: persp2fish [options] imagefile
Options
   -w n        sets the output image size, default: 2 * perspective image width
   -a n        sets antialiasing level, default: 2
   -t n        perspective horizontal fov (degrees), default: 100
   -s n        fov of the fisheye image, default: 180
   -x n        tilt camera, default: 0
   -y n        roll camera, default: 0
   -z n        pan camera, default: 0
   -bg r g b   background image colour (outside fisheye circle), default: black
   -f          full rectangular fisheye instead of circular crop, default: off
   -d          debug/verbose mode

Notes

  • For a correct warping one needs to know the horizontal field of view (FOV) of the source perspective image. For square pixels and given aspect ratio (width/height), the vertical and diagonal field of view can be derived.

  • The destination fisheye image can be any width (height will be equal to the width) and the fisheye FOV can be anything, within reason. The utility here limits the fisheye FOV to lie between 1 and 350 degrees. For compositing, the widest fisheye lens on the market currently is 250 degrees.

  • The -f and -bg options control whether the fisheye should be cropped to the FOV circle, or not. If cropped the background colour will be used outside the fisheye circle.

  • The fisheye projection here is assumed to perfect, that is, a linear relationship between radiu from the center and field angle. For compositing not that not all physical fisheye lenses are linear, although it is straightforward to convert non-linear fisheyes into linear ones, and visa-versa, see: Fisheye lens correction.

A few example will be shown below, they are based upon the following image which has a horizontal field of view of 122 degrees.

Default settings except specifying the perspective image field of view.


persp2fish -t 122 122deg.jpg

Rotated 40 degrees about the vertical.


persp2fish -t 122 -z 40 122deg.jpg

Mapping into a 250 degree fisheye (like the Entanyia) and setting a background colour.


persp2fish -t 122 -s 250 -bg 10 10 10 122deg.jpg

 




Converting Images from Panoramic to Fisheye


Written by Paul Bourke
February 2002

Example images courtesy (and copyright) Astro Copy Service, Planetarium Augsburg, Germany

See also Angular fisheye projections.

The following will discuss the conversion of panoramic images into those suitable for display onto a dome, in particular, angular fisheye as used by the majority of planetarium domes. There are a number of panoramic image formats, they all use the same horizontal axis which ranges from 0 to 2 pi. There are however a number of vertical distortions, the two most common are considered here. One is often called a radial panoramic where the vertical axis is considered to lie on the surface of a sphere, the other is a linear panoramic where the vertical axis results from a standard perspective projection. This second is the most common, the image is obtained by projecting onto a cylinder about the camera.

As with most mappings the goal is to estimate the colour for each pixel in the destination image. The destination image in this case is the angular fisheye, the colour estimate will be determined by the corresponding pixels in the source (panoramic) image. In general there isn't a single pixel in the input image corresponding to a particular pixel in the destination image. For best results it is usual to find the closest pixel in the input image for a range of positions within each pixel in the destination image, these are averaged together to determine the final estimate of the colour. This is commonly called antialiasing, the simplest form of which is to estimate the colour by averaging over a 2x2 or 3x3 grid within each pixel.

The basic idea is to find the mapping between a coordinate system in the angular fisheye and a coordinate system in the panoramic image. The convenient coordinate system in the angular fisheye is "r" (the distance from the center of the pixel to the pixel in question) and the angle "phi" (angle of the vector to the pixel). The convenient coordinate system in the panoramic image is the same angle "phi" (the vertical axis of the panoramic), in both images this angle varies from 0 to 2 pi. The vertical axis of the panoramic is proportional to the "r" in the angular fisheye. This is illustrated in the following figure.

The conversion of the two types of panoramic image will be illustrated by transforming the following test pattern. This will be done for three values of thetamax, that is, the panoramic images will be assumed to vary vertically from 0 to 30, 60, and 90 degrees.

Radial panoramic

The hole in the center for all but the 90 degree case reflects the lack of image data from thetamax to 90 degrees which is the central portion of the fisheye image.

Linear panoramic

The most obvious feature of this case is the extreme distortion that occurs as the fisheye tends to 90 degrees. Of course, while the distortion seems extreme below that is because the test image has a equal spacing grid, something that would not normally occur in a 90 degree panoramic. A normal 90 degree panoramic would appear very distorted itself and the process of turning that into a fisheye image would correct for the distortion.

The following is a linear 45 degree angular fisheye of the panoramic shown at the top of this page.

Notes

  • It has been assumed here that the panoramic extends from 0 degrees upwards along the vertical axis. This is because the discussion here has been targeted towards turning panoramas into images for projection in planetarium domes where one only sees above the horizon in both the panoramic and fisheye.

  • Radius of the fisheye image is related to the length of the panoramic, namely, the diameter of the fisheye image is the width of the panoramic divided by pi. Other fisheye dimensions will result in a radial compression or stretching.

Further Examples

pan2sph panfilename [options]
Options:
-t n       set max theta on vertical axis of panoramic, 0...90 (default: 45)
-a n       set antialias level, 1 upwards, 2 or 3 typical (default: 2)
-w n       width of the fisheye image, height = width (default: 512)
-s         use sine function correction (default: off)
-r n       rotation angle, 0...nx (default: 0)
-f         flip insideout (default: off)
-bg r g b  set background colour, 0...255 each (default: 0 0 0)
Diagram illustrating panorama within a circular fisheye for planetarium use

Further exercise
December 2004

Usage: pano2fish [options] fisheyeimage
Options
-w n      fisheye image width and height, default = -1
-r r1 r2  inner and outer radius of the fisheye image
-h h1 h2  top and bottom panoramic edges
-a n      antialiasing level, default = 1 (no antialising)
-d        change direction of panoramic
-p n      rotate by n degrees
-c        clear region outside fisheye radius

Example

Input image

Default settings

pano2fish -w 480 pano.tga
  Set panoramic height range, measured from bottom of the image

pano2fish -a 2 -h 172 50 -w 480 pano.tga

Set the fisheye radius range, measured from the center of the fisheye

pano2fish -a 2 -h 172 50 -r 208 60 -w 480 pano.tga
  Reverse panoramic direction

pano2fish -a 2 -h 172 50 -r 208 60 -w 480 -d pano.tga

Change phase

pano2fish -a 2 -h 172 50 -r 208 60 -w 480 -d -p 180 pano.tga
  Invert mapping by swapping radius bounds

pano2fish -a 2 -h 172 50 -r 60 208 -w 480 pano.tga

Clear region outside radius range

pano2fish -a 2 -h 172 50 -r 208 60 -w 480 -c pano.tga


Rendering software solution.
Update: February 2016

There are now a number of plug-ins for compositing packages that allow one to work in polar coordinates, rather than Cartesian coordinates and also allow one to place various image projections (perspective, cylindrical, equirectangular) into the fisheye image space. But if one is used to a 3D modelling/rendering package or if one is creating realtime content then an alternative approach is to compose the non-fisheye elements in the 3D scene and then simple render the fisheye. For plane perspective images these may be placed on a billboard in the scene, at the desired position and angle. The image will appear on a flat surface in the dome oriented, positioned and sized as set in the model.

In the case relevant here, one could place a cylinder in the 3D model world with the panorama on the surface as a texture. This cylinder should have the same horizontal angular extent as the panorama, not necessarily a full 360 degrees. The height of the cylinder would normally be fixed to the right aspect ratio of the panorama but the position and angle of the panorama could be varied as desired. Indeed, rotating and tilting the cylinder is a standard type of dome transition or way of enlivening what is otherwise a static image (of course the texture could also be a video sequence). Again, rendering the fisheye of the textured cylinder will ensure the geometric effect in the dome is the same as within the 3D model.

Some examples are shown below, on the left is a view of the cylinder in relation to a polar grid representing the dome. On the right is the fisheye rendered from the middle of the dome. The examples here are created using PovRay but any 3D modelling/rendering package that supports a circular fisheye lens could be chosen.

Raising the cylinder above the spring line of the dome.

Rotating the cylinder with respect to the dome axis.