Center of perspective

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Perspective

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.

Projection geometry

Figure 1. Projection geometry with a pinhole camera. The center of perspective is the tiny hole where all projection lines cross.

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.

Panoramic stitching

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).



Lens diagram

Figure 2. Diagram of a 135/2.8 lens with rotation axes through the front nodal point N and entrance pupil E.

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.

Procedure to check on parallax.

Figure 3. Configuration to reveal the presence or absence of parallax. The subject is first placed at the left side of the frame, and subsequently at the right side after rotation of the camera about a vertical axis with the help of a panoramic tripod head.

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.

Rotation about an axis through the entrance pupil.

Figure 4. Rotation about an axis through the entrance pupil.

Rotation about an axis through the front nodal point.

Figure 5. Rotation about an axis through the front nodal point.

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.

Influence of lens asymmetry on the field of view

For convenience we assume that the film/sensor covers the entire back.

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.

FOV for an asymmetrical lens

Figure 6. Field of view for an asymmetrical lens at an image magnification m and pupil factor P.

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
d
divided by
2f(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
d
divided by
2f(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.

Busy scenery

Figure 7. Scenery for inspection of the field of view of a 21-mm lens, here with zero lens extension (infinity scenario). Dash-dotted frames indicate the fields of view corresponding to angles of 55° and 75°.

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.

FOV at unit magnification

Figure 8. Field of view with the 21-mm lens mounted on a 21-mm extension tube.

If someone manages to carry out the experiment of this section with a 21-mm rangefinder lens, I will happily show the results on this page.

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



References


[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.