The No-Parallax Point theory and fisheye lenses

You may read or download the complete original document (PDF): Theory of the "No-Parallax" Point in Panorama Photography

Summary:

The theory recalls that in using an ordinary camera to make panorama photographs, there is a special "no-parallax point" around which the camera must be rotated in order to keep foreground and background points lined up perfectly in overlapping frames.

Besides other findings, this theory demonstrates that:

Short history

Rik Littlefield has issued the Theory of the "No-Parallax" Point in Panorama Photography in 2006. By writing this now famous article he has then clarified a controversial subject that hampered sometimes and somehow the community of panography and especially new comers willing to learn how to make panoramas by stitching multiple source images. He has shown, besides other things, that neither of the two Nodal Points of a lens could be the point around which the camera should be pivoted so as to avoid spoiling the seams of the stitched panorama with Parallax induced stitching errors. For a decade, "Nodal Point" was been used by many on the Web -including this author- as well as by most of the developers of programs for stitching digital images. The naming of this point was plainly literally wrong and somewhat misleading scientifically and by nature. This misuse of word had led to frequent misunderstanding and lengthy discussions.

The empirical methods to find where this point is located (that were proposed in numerous Web sites) were and are still however generally correct even with the wrong use of the "nodal point" (the word).

The demonstration made by Rik with support from some other panorama photographers -cited in the article- has been very successful . The word NPP has been promptly adopted.

Shortcomings of the theory

Fisheye lenses are in fact excluded from the theory

Rik had implicitly acknowledged a void in his proposed theory when near the end of his demonstration, he admits that in some circumstances, the NPP could sometimes be elusive because its actual position depends on vignetting that happens when the entering light becomes very oblique e.g. for wide and "fast" lenses.

"Whenever significant physical vignetting occurs, the location of the limiting aperture changes across the image width, and so may the location of the no-parallax point. If it does, then in fact the lens will not have a single no-parallax point, but rather a collection of "least-parallax" points that vary with angle away from the optical axis, like a fisheye lens. This is not a good thing."
Note: Bold and underlined words were edited by this author.

In addition with the fact that the movement of the Entrance Pupil is not related to vignetting (for fisheye lenses), one could guess that Rik was eventually excepting the fisheyes from the theory when reading this convoluted sentences.

It was then already known that the fisheye lens has not a single NPP: experimental verification of this fact is easy to perform. I had issued articles back in 2001on this subject (but alas erroneously titled "Locate the Nodal Point"). The NPP seems to move (some ~20 mm) along the lens axis depending on the angle of the entering light ray. That is also to say that the NPP should be set at a different position depending on the number of rows and columns of images to be stitched or with the required number of shots on the horizontal plane.

The physical Entrance Pupil is not a point

In fact Rik had used paraxial approximation in its optical mathematics. This otherwise clever computational short-cut is IMHO inappropriate for very oblique rays of light. That makes the wide angle lenses and even more so the fisheye lenses not fully in line with the theory: paraxial approximation is probably only applicable to standard rectilinear lenses.

Furthermore, paraxial optics practice allows for ease and clarity of presentation, the confusion of the **Center** of the Entrance Pupil (i.e. the "Surface") with the Entrance Pupil (i.e. the "Principal Point"). Consequently only the latter is therefore figured on paraxial graphs and the point is labeled "Entrance Pupil" instead of the "center of the Entrance Pupil" i.e. the center of the virtual image of the aperture that in fact it is.

Generalization of the theory to fisheye lens

The theory doesn't apply to fisheye lenses. The problem is that fisheye lenses are certainly nowadays the favorite lenses of a vast majority of the (spherical) panorama (VR) photographers. The regular "long" focal rectilinear lenses are used almost only for making "Multi-Mega_or_Giga-Pixels"(e.g. cylindrical) panoramas.

Consequently this author is convinced that the (restricted) theory should be generalized to not exclude the fisheye lenses. That is the aim of this other related article:

Apropos of the Entrance Pupil and of No-Parallax Point (NPP) Introduction to the Least-Parallax Point (LPP) theory