Fisheye lens correctionWritten by Paul Bourke
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.
A normal lens has pincushion or barrel distortion which can be corrected for to give a perfect perspective projection, a pin-hole camera. This is called "rectification" and is often applied before any warp/stitch/blending is applied, for example, in panoramic photography.
In the same way there is a perfect circular fisheye projection, that is, one in which the distance r from the center of the fisheye circle is linearly proportional to the latitude of the corresponding 3D vector. Such a fisheye lens is often referred to as a "tru-theta" lens and while such lenses can and have been manufactured, in real life and for lower cost lenses the relationship is non-linear. The non-linearity normally occurs towards the periphery of the fisheye and results in a compression artefact.
In many cases the lens manufacturer can supply data for the curve relating "r", the distance on the sensor or fisheye circular image, to latitude of the 3D vector corresponding to that radius. In some cases for high grade lenses the manufacturer will supply curves for the particular lens in question, an acknowledgement that lenses can vary typically by 5% in the manufacturing process. In the situation where no such data is available one needs to construct a rig which when photographed will allow the angles to be measured.
The approach to correcting for the non-linear relationship is to fit a suitable polynomial to the data points relating "r" to latitude. A general function for the latitude φ might be
Since the fisheye is assumed to be radially symmetric and r=0 is the center of the fisheye corresponding to a latitude of 0, a0 is zero. In practice the highest order polynomial needed for the fitting is n=3, in some extreme examples a 4th order (n=4) is required. So the polynomial for a least squares fit is:
The null case, where the fisheye already has a linear relationship would be a1 = φmax / rmax, a2 = 0, and a3 = 0. Where φmax is the half the circular fisheye angle and rmax the radius on the image or sensor corresponding to φmax. Noting that the r is often normalised to 1, in which case rmax = 1.
As an example, the following data relating distance on the sensor to latitude is illustrated below, black curve, for a 196 degree fisheye. The blue line shows the tru-theta relationship for a true fisheye, and the relationship after the correction is applied. The horizontal axis is field of view (field angle) in degrees and the vertical axis is normalised fisheye image coordinates.
Usage: fishcorrect [options] tgafile Options -w n sets the output image size, default: same as input fisheye -a n sets antialiasing level, default: 2 -s n fisheye aperture (degrees), default: 180 -c x y fisheye center, default: center of image -r n fisheye radius, default: half the fisheye image width -z n roll angle, default: 0 -p n n n third order correction terms -d debug mode
The following image is captured with a 210 degree fisheye which has quite significant compression towards the rim.
The following is the undistorted version, that is, radius on the fisheye images is now proportional to latitude.
This lens curve seems to need a 4'th order term in order to get a good fit at the extreme field of view angle of 125 degrees. Graph is radians horizontally, and normalised fisheye image (sensor) vertically.
Sigma seems to be one of those companies who refuse to provide linearity data on their lenses. Totally unreasonable, this is a fundamental characteristic of a lens that an owner has every right to insist upon. Fortunately one can determine it with only a modest effort. The equipment required is shown below, the procedure is as follows
Canon is another company who behave badly and don't supply this fundamental information of their products. It is more complicated with this lens since the field of view and the curves change with different zoom values. So there is actually a whole family of curves required in order to deal with any particular zoom value, alternatively one could create a 3D surface fit. This could be automated given that the focal length is supplied in the exif data on the photograph. Two curves are provided, based upon the usual use of the lens by the author: fully zoomed out (8mm) and 12mm is the maximum zoom to fill full frame Canon5D Mk III sensor horizontally.