A photograph is a point projection of a three-dimensional scene,
commonly referred to as a perspective view, with the center of
perspective at the camera lens itself [1]. However,
"at the lens" is an imprecise indication for a compound lens, and
for certain applications a more accurate description is required. The
purpose of the present article is to indicate the precise location of
the center of perspective, and discuss its influence on the field of
view.

A simple but adequate formulation of the center of perspective is furnished
by considering the pinhole camera in Fig. 1. The tiny hole ensures a
reasonably sharp image at the rear face of the box,
where it may be recorded by a light-sensitive medium. Any image-forming light
has to pass through the pinhole, and the position of this hole determines how
different elements of the outside world are imaged relative to one another.
Another page discusses apparent perspective effects such as converging
verticals.
For instance, a distant object is rendered smaller than an identical object close to
the pinhole, and by altering the position of the pinhole the knapsack in
Fig. 1 may be hidden behind the head—or be revealed if it is already
hidden. The viewpoint associated with the position of the hole is called
perspective, and the hole itself is the center of perspective.

When the pinhole in Fig. 1 is replaced by a thin lens with an adjoining
aperture stop, the perspective does not change. In that case there is no
ambiguity when the center of perspective is said to be "at the lens." However,
when a camera is equipped with a compound lens the situation is less intuitive
since such a lens has a certain length. Where is the center of perspective of
a compound lens? The answer appears to surprise many people, but it should not
as a compound lens also has
a hole through which all light must pass. This hole is known as the entrance
pupil, which is the lens aperture that is seen when you look into a lens
from the front. The analogy with the pinhole camera is particularly strong
if we close down the leaf diaphragm of the lens to yield a small aperture. Any
light from the outside world must be directed toward this hole in
order to contribute to image formation, and the center of
perspective is the center of the entrance pupil
[2,3]. There is simply no choice.

For a photographic technique like panoramic stitching, whereby camera and lens are rotated as a unit between successive images before these are stiched in post-processing, the rotation axis must be carefully chosen for the various elements in the 3D world not to change position relative to one another between images (parallax). It is frequently asserted that "the nodal point" is the proper pivot point, but this is incorrect unless it happens to coincide with the entrance pupil. Note that a lens has two nodal points, which are the points that the principal planes H and H' have in common with the optical axis. H corresponds to the front nodal point and H' to the rear nodal point. These principal planes are of paramount importance for the size and position of the image, for the depth of field, but not for the viewpoint. As a case in point we consider the lens shown in Fig. 2. This lens is chosen for the demonstration because a) the manufacturer provides the necessary information and b) the entrance pupil E is conveniently separated from both nodal points. The figure shows the lens elements, the principal planes H and H', the front nodal point N, the position of the variable leaf diaphragm, and the image of its opening formed by the elements in front of it: the entrance pupil E (here depicted at an arbitrary f-number).

A studio setup is used with the dog of Fig. 1 placed in front of a vertical bar. Fig. 3 shows two pictures. One with the bar and knapsack at the left side of the frame, and one with these elements at the right side after rotation of camera plus lens about a vertical axis. Two rotation axes are investigated as indicated in Fig. 2: one through the entrance pupil E and one through the front nodal point N. Their positions are simply adopted from the lens sheet, and the realization of the pivot point is accurate within an uncertainty of 2 mm or smaller.

Figure 4 shows crops containing the knapsack and background bar when the
rotation axis passes through the entrance pupil of the lens. There is no
parallax, i.e., no displacement of the knapsack relative to the bar, and this
choice of pivot point seems admirably suited for panoramic stitching. By
contrast, a clear parallax is observed if the rotatation axis passes through the
front nodal point (Fig. 5). These observations confirm that the entrance
pupil is the center of perspective, and that the front nodal point is a poor
choice for rotation if perspective is to be preserved. Since the two principal
planes are close to each other for the employed lens, it is left to the
imagination of the reader that a rotation axis through the rear nodal point will
be a poor choice as well for panoramic stitching. Moreover, in the present case
study light from marginal image elements such as the knapsack, directed toward
the nodal point(s), does not even reach the image at reduced apertures.

Whereas the position of the entrance pupil is fixed for rectilinear lenses such as
the 135/2.8 used in the example above, the situation is more complicated for fisheye
lenses. In a fisheye lens the position of the entrance pupil is not fixed, but
varies varies with the field angle. See
[4, 5] for more detail.

In Fig. 1, the field of view (FOV) is determined by the size of the
rear face of the box and its distance from the pinhole. The back may be
displaced closer to the pinhole to increase the FOV, or moved further away to
decrease the FOV. So long as the location of the pinhole remains fixed the
perspective does not alter. A similar relation holds for cameras equipped with
lenses. Traditionally, the field of view is governed by the lens focal length,
the format (sensor dimensions), and the object distance, whereas perspective is
determined solely by the position of the lens entrance pupil. However, since the
FOV originates from the entrance pupil and since object distance is measured
from the front principal plane, traditional FOV equations require a correction
when asymmetrical lenses are used at close focus
[2, 6].

In order to quantify the field of view, it is convenient to introduce the pupil
magnification *P*, defined as the ratio of the exit pupil diameter to the
X is in front of H' for *P* > 1, and behind H' for
*P* < 1.
Likewise, E is located in front of H for *P* > 1, and behind H for
*P* < 1.
entrance pupil diameter. On another page
it is shown that the exit pupil X is separated from the rear principal plane H'
by a distance (*P*-1)*f*. Using the familiar lens equation

1/f |
= | 1/v |
+ | 1/b |
(1) |

where *f* is the focal length and *v* and *b* the object and image distances
The lens conjugate equation also applies to the pupils.
measured from the respective principal planes, one finds that the entrance pupil is separated from the
front principal plane H by a distance (1 - 1/*P*)*f*. This leads to the diagram
in Fig. 6, which sketches the FOV geometry for an asymmetrical lens with *P*>1.
The sketch is for a close-up photograph at some image magnification *m*, because it
is in this regime that lens (a)symmetry influences the field of view. At a magnification
*m* ≡ *b*/*v*, the object distance is derived from Eq. 1 as
*v* = *f* + *f*/*m*.

Often image-space geometry is considered, with the same outcome.
As Fig. 6 indicates, the object distance *f* + *f*/*m* is
measured from the front principal plane. It is then routinely assumed that the FOV
can be calculated by adopting an object size of *d*/*m*, where
*d* is the sensor format diagonal. This yields a full diagonal FOV

α |
= 2×atan |
ddivided by
2
f(1+m) |
(2) |

However, since we have already seen that the center of perspective is not some nodal point but the center of the entrance pupil, the commonly used Eq. 2 is amenable to refinement. From the geometry in Fig. 6 it follows that the FOV measured from the center of the entrance pupil equals

α |
= 2×atan |
ddivided by
2
f(1+m/P) |
(3) |

At large object distances the contribution of the term *m*/*P* is negligible,
and there will be no noticeable influence of lens symmetry (i.e. the *P* value)
on the field of view. For symmetrical lenses with *P* = 1 the entrance pupil
coincides with the front principal plane, and Eq. 3 degenerates into Eq. 2. For such
lenses there is no influence at any object distance. However, the effect is quite
noticeable for asymmetrical lenses (*P*≠1) employed at close focus.
To illustrate this, we consider a 21-mm retrofocus wideangle lens with a pupil
magnification *P* = 3. Full lens details are given in the manufacturer's
lens sheet.
The scenery is of course not at infinity but at a
distance of several meters. However, at a focal length of 21 mm this is
close enough.
The lens was used on a camera with a 24×36 mm sensor,
and set to infinity for a studio experiment conducted with the busy scenery
shown in Fig. 7.

After the picture of Fig. 7 was taken, a 21-mm extension ring was inserted
between the lens and the camera. Nothing else changed. This scenario
corresponds to an image magnification
*m* = 1 and an object distance *v* = 42 mm.
A suitable calculator also gives the fields of view
pertinent to the present experiment:

VWDOF 2.1---------input---------------------------------- Format 24x36 mm 24x36 mm 24x36 mm Focal length 21 mm 21 mm 21 mm Pupil factor 3 1 3 Extension 0 mm 21 mm 21 mm ------------------output--------------------------------- Object dist Inf 42.0 mm 42.0 mm Magnification 0 1.00 1.00 Field of view 91.7 deg 54.5 deg 75.4 deg

The first scenario gives the FOV for the infinity setting of Fig. 7.
The second scenario considers the presence of the extension tube, but ignores
the issue of lens (a)symmetry and uses *P* = 1. This is just
application of Eq. 2. Finally, the third scenario gives the FOV predicted
by Eq. 3 with due allowance for the true pupil magnification
*P* = 3. The three angles are indicated in Fig. 7 and sizable
differences are observed between, first, the infinity shot and both
close-focus scenarios, and second, between adopting the front nodal
point (55°) or entrance pupil (75°) as the center of perspective.
The factual result of the experiment is finally shown in Fig. 8.
There is the unavoidable (background) blur, but at an aperture of f/22 various
picture elements are still individually recognizable and there is
no doubt that the computation yielding a FOV of 75° is the correct one.

Note that 21-mm rangefinder lenses are (almost) symmetrical
and *will* reveal a 55° FOV on the 24×36 mm format
when extended by a distance of 21 mm.
Two lenses of the same focal length with the same infinity FOV,
when given the same extension, may thus yield substantially different
close-up FOVs.

It was demonstrated that a lens should be rotated about an axis through the
entrance pupil in order to avoid parallax with panoramic stitching. Although this
was already stated in David Jacobsen's excellent 1996 lens FAQ [7],
it has taken the photographic community many years to come around. Even today
many sources still refer to the nodal point, but the balance is decidedly changing in
favor of the entrance pupil. Compare for example
this lens sheet (January 2004 download)
with today's
version. The present article also revealed that the position of the entrance pupil is
important for the field of view. Authors who consider the issue of lens symmetry
important enough to take it into account in depth-of-field or exposure calculations,
should also acknowledge its importance for the field of view. This is perhaps the only
omission in Jacobsen's lens FAQ (or tutorial [8]).

© Paul van Walree 2009–2014

[1] | Rudolf Kingslake, Optics in Photography, SPIE Optical Engineering Press (1992). |

[2] | Rik Littlefield, Theory of the "no-parallax" point in panorama photography. |

[3] | John Houghton, Finding the no-parallax point. |

[4] | Michel Thoby, Laser results. |

[5] | Douglas A. Kerr, The proper pivot point for panoramic photography. |

[6] | Douglas A. Kerr, Field of view in photography. |

[7] | David Jacobsen, http://photo.net/photo/optics/lensFAQ |

[8] | David Jacobsen, http://photo.net/photo/optics/lensTutorial |

Frank van der Pol is acknowledged for assistance with the studio-setup photographs.

spherical aberration | astigmatism and field curvature | distortion | chromatic aberrations | vignetting | lens hoods | flare | filter flare | depth of field | dof equations | vwdof | bokeh | spurious resolution | center of perspective | MTF measurements | corrections | misconceptions