Apropos of the Entrance Pupil and of No-Parallax Point (NPP)

Introduction to the Least-Parallax Point (LPP) theory


The No-Parallax Point theory and fisheye lenses

The NPP theory de facto excludes the fisheye lens for its application. This exclusion is tentatively explained in this article.

The Entrance Pupil of any lens is a surface. It is the image -most often virtual- of the aperture physical stop (i.e. the iris of the diaphragm). On a fisheye lens it's a -nearly planar- surface of a broad oval shape (an ellipse) when it is not cropped by vignetting due to internal lens obstacle(s) on one or both sides of the ellipse as the Angle gets near the edge of the coverage (say above 90° from the lens axis).

The Entrance Pupil of a fisheye lens seems to move when the observer looks at it with different angles to the optical axis.

Whatever the projection type of the fisheye, the center of the entrance pupil moves according to the direction of the light rays for example as it is shown on this image:

Quoting from a leading book for optical design:

A “fish-eye” lens covers a 180° or larger field of view by taking advantage of the heavy overcorrected spherical aberration of the powerful negative meniscus front elements, which strongly deviate the principal ray. This spherical aberration of the pupil causes the entrance pupil to move forward and off the axis and to tilt at wide angles of view. (Modern Optical Engineering: the design of optical systems; Fourth Edition; Warren J. Smith, McGraw-Hill Professional, 2007)

Apparent movement of the Entrance Pupil

(Note: Ray tracing computing requires the light rays to go from left to right, hence the unusual -reversed- position of the different groups of optical elements).

The Entrance Pupil moves away from the axis and toward the front surface of the lens, increases it size and it simultaneously rotates as the angle of incidence increases. BTW this rotation and enlarging of size is why some photons can still enter the lens at 90° or more and this is also why the loss of illluminence (a.k.a. light fall-off) towards the edge of coverage is moderate for a fisheye lens w.r.t. a standard lenses. The fisheye is "miraculously" exempt from the dreaded "Power 4" of the Cosine law!

The center of the Entrance Pupil is the point that is the virtual image of the center of the Aperture Stop (which is located on the lens axis).This point moves on a complex surface: the entrance pupil is centered on a floating point that moves about over a surface shaped rather like the flared end of a trumpet, according to the direction of the light ray. (John Houghton).

Except for 0° angle, the entrance pupil is therefore never on the lens axis. (This is not widely known). Consequently the Entrance Pupil cannot be the point around which the camera should be rotated when using a fisheye lens. The NPP theory cannot be applied for a fisheye lens if the wording and meaning are not modified adequately.

Fortunately, we shall now see that there is moreover a strong relationship between the Entrance Pupil and the...LPP.

Least-Parallax Point (LPP)

Let's go back to Rik Littlefield's paper. Near the end of the article, he called "least-parallax points" the "collection" of NPP points that would be found along the axis of a lens, like on a fisheye says he. I applaud this idea and I shall justify in the following the reason for my enthusiasm.

There is only one point that fully fits the pure NPP requirement and it's valid only when the entering ray is perpendicular to the lens axis. The semi-angle of view is then 90° and this corresponds to the case of a so-called 2-shot panorama where two full opposite hemispherical photographs are assembled.

For any other number of images to be stitched for completing a full 360° panorama, the angle of rotation between each shot shall be less than 180°: typically 120° (for 3-shot panorama), 90° (for 4-shot panorama), 60° (for 6-shot panorama), etc. Those images on the horizontal plane might be replaced or completed by other rows of images shot at different elevation angle, including 90° i.e. toward the Zenith or the Nadir to make a sphere.

With more than two images to stitch a full 360 x 180° panorama, there is no such thing as a No-Parallax Point:-(

Let us take for example a common workflow that is used to make a 360 x 180° spherical panorama with four images on the horizontal. These source photographs may be best shot 90° apart on the horizontal. We should therefore swivel the camera by 90° between two (of the four) consecutive shots around the point that could be determined by using a calibration method such as John Houghton's. Consequently, on the horizontal plane of the stitched image, pixels that were +45° from the lens axis on the first image and -45° from the lens axis on the second image are going to fit each other just fine on the seam without error if the pivot point coincides with "what was called NPP for 90 deg". By contrast, at the bottom of the panorama, the two images shall have a common seam which pixels were shot 90° from the lens axis (at the foot of the monopod for example). The ideal "NPP" there would correspond to a 180° swiveling angle.

There is no way to fit the conflicting requirements at once. To make it short:

<<While this will be fine for the horizontal parts of the scene, rays from the nadir and zenith regions will be entering the lens at angles approaching 90 degrees to the lens axis, which implies a different entrance pupil location and hence theoretically necessitating a different no-parallax rotation point.  The choice of rotation point must therefore be a compromise...>> (John Houghton)

Conclusion: the point that we are then looking for and suggested by John Houghton is... a Least-Parallax Point (LPP) which name was created by Rik Littlefield!

You may also read a sequel to this article about real shape and location of the stitching lines.

Locating the LPP

The center of perspective corresponding to a single ray that enters the lens and reaches the sensor to make one pixel, is the center of the Entrance Pupil on this ray. This ray shall go through every lens element then through the center of the Physical Stop Aperture before finally reaching the sensor. Since we know that the camera should be rotated around a point located on the lens axis, we may then extend virtually the entering ray to intersect the lens axis. We have found the Least-Parallax Point: it's beyond the Entrance Pupil when looking at it from the front of the lens:

An extended version of this movie can alternatively be viewed for a complete overview on the whole subject of this article.

Shift of the LPP due to vignetting

There is a huge difference of apparent size of the entrance pupil between a standard "fast" rectilinear lens and a fisheye lens whatever the f number of the latter. The oval bright area is shrinked to a tiny surface by the powerful negative meniscus front elements, which strongly deviate the principal ray.

When the angle approaches 85 to 90° significant vignetting effect can be observed. It crops one side of the oval then the two sides are cropped slightly. The entrance pupil location seems to be not changed by this vignetting and as a consequence the LPP is not affected, unlike what can be observed with a standard fast rectilinear lens.

Entertainment: Can we see the LPP / NPP with our own eyes?

The answer can be determined from the above paragraphs and illustrations: No, we cannot view the LPP as it is always hidden behind the Entrance Pupil!

This reminds me of these tiny cunning squirrels that -in many public parks around the world- run then climb and hid on the other side of the trunk of the tree we are looking at.

BTW there is also another related question that we frequently read: can we locate the entrance pupil by looking at it from the front of the fisheye (i.e. from the object space)? This is a movie to show how it goes:

We can see the pupil all right, no problem at all: this is this bright oval.

But it's nearly impossible to locate it in real world and with accuracy: we are in fact one-eyed here...

...because our eye are about 6 cm apart and then without a specific effort, our left eye doesn't sees the same Entrance Pupil as the right eye! This is shown on the movie above the last one. We must then place our two eye lines of sight at the same angle of incidence with the lens axis and simultaneously we must squint to have both eyes looking toward... the hidden single and common LPP (i.e. common to both eyes)! Our brain may then give you the information... if the image that is seen in the oval is without salient (or different one from the second) features.

OK, I am still training but I cannot yet do that;-)



We have demonstrated that the theory of the No-Parallax Point (NPP) must be slightly adapted and generalized to be applicable to fisheye lenses: the Entrance Pupil of a fisheye is not on the axis of the lens and it moves off the axis and forward when the oblique angle of the entering light ray increases. Therefore it is not exactly the point around which the camera must be pivoted around.

There are stricto-sensu no No-Parallax Point in a fisheye system (except for 2-shot panorama). An alternate theory about the Least-Parallax Point (LPP) is proposed: it illustrates the compromise that is needed to reduce the effect of unavoidable parallax that occurs with fisheye lenses. Parallax may induce visible stitching errors and mismatches along the seams when the subject is very close to the lens while shooting.

Looking at the Entrance Pupil shows you the direction where the LPP is located: it's on the lens axis and behind the center of the Pupil. It thus moves with it when the entering ray angle to the axis varies.

The different processes and methods that have been proposed to locate the "NPP" of a fisheye lens keep their validity, but they actually determine the location of the LPP: it is just a change of a name... and otherwise you may do as you have done before.


Michel Thoby

12 August 2009