More accurately and in summary: the center of the Entrance Pupil of all real lenses is never at a constant point when the light angle varies
Definition of Diacaustic:
This article is based on the following principle:
One of the important properties of any lens is the position of the entrance pupil point. Ideally this is the point towards which incoming chief rays (each of which is the center ray of a bundle of rays that make it through the lens and aperture stop to create a pixel of the image) are headed (extended as a straight lines). Therefore, this is the position from which the camera seems to view the world. This position often is referred to as the perspective center or Viewpoint of the optical imaging system. This specific point is generally thought as being
This is absolutely true when there is no distortion i.e. when the lens is perfectly rectilinear where:
In short: this belongs exclusively to the central projection model.
However in the real life, the Entrance Pupil of real lens is not necessarily on the lens axis: although for an ideal lens the entrance pupil is fixed, for actual lenses the position of the entrance pupil can change as the off-axis angle of an incoming ray increases. BTW we shall see that the shift is the necessary and basic feature of some lenses (e.g.Fisheye).
Depending on the type of application that needs using an imaging system based on refracting lens, the displacement of the center of the Entrance Pupil may be considered as neglegible or not. That is the case whan the distance from this point to the subject and the whole scene (in the object space) is very high with respect to the distance of the shift of the Pupil. In panorama photography this might not be always the case as severe stitching errors may be caused by this parallax shift that depends upon the angle of the entering light ray in the lens if the subject is distant by less than a meter or so.
For most lenses the effect of Pupil movement can be "relatively" small but for fish-eye lenses it is always quite large: or some specific photographic fisheye lenses it can be of the order of the size of the lens, while it is usually less than about 20 mm in current DSLR fisheye lenses. This shift can be very important especially for objects that are close to the lens.
In fact, unlike standard lenses, real fish-eye lenses may have ﬁelds of view greater than 120°. Such large ﬁeld angles result in very large distortion, so large that one really should not even quote distortion as such. The lens, of course, has focal length, but it has little meaning considering the large distortion. According to the "cos4 law" of illumination, the illumination of optical systems is proportional to the fourth power of cos(theta). However, this assumes no distortion and a constant entrance pupil diameter with ﬁeld angle. Therefore, lenses obeying this “law” would have no illumination past semi-ﬁeld angles of 90°. Whereas in the case of a fisheye lens and with increasing ﬁeld angle, the entrance pupil moves from inside the lens toward the front of the lens and substantially increases in size. Pupil aberration is of course considered during the design of the lens.
The incoming rays that pass through the center of the entrance pupil for different incident angles progressively move with a peculiar pattern: the (dia)caustic formed by these rays is the locus of the actual entrance pupil points.(please read the definition of diacaustic at the top as a foreword to this article). This is illustrated on a specific page by this author when introducing the Least Parallax Point (LPP) theory and principle. Least Parallax Points are the successive projection of the center of the pupil onto the optical axis, i.e. the intersection of the rays with the optical axis. This intersection, that is the projection on the axis of the Entrance Pupil Center, exists if we assume that the pupil shift is radial symmetrical, even though this assumption is not exactly true in the presence of lens decentering, but this would be another topic...
There are an infinite number of rays i coming from the space side toward the lens and among these some millions of them eventually hit the sensor elements to make an image after going through the whole lens. You may imagine that all this successful rays i were instead captured by the same number of tiny 1-pixel cameras and each of them would be centered on the diacaustic and aligned on one of the ray i directed toward space.
The movement of the Entrance Pupil (of all fisheye lenses whatever their projection type), is easily observed by looking at it from the front while the lens is pivoted. It happens to be not so easy -as it wrongly seems at first glance- to measure its actual location in 3D space though, but this also would be another topic...
BTW Pierre Toscani (in French) has drawn some very nice illustrations of the movement of the Entrance Pupil for most of the historic Nikon fish-eye lenses. These illustrations are truly amazing!
A standard photographic lens is a lens that yields images where straight features and edges in the 3D space side appear with straight lines, as opposed to being curved on the recorded image. In other words, it is a lens with no barrel or pincushion distortion.
The ideal standard photographic lens is based on the principle of the rectilinear projection. As stated above, the rectilinear projection has the fundamental property that straight lines in real 3D space are mapped to straight lines in the projected image. The radial distortion on the image follows the rectilinear mapping rule R = f x tan (Theta) so as it's impossible to make a rectilinear lens with 180 degree (hemispheric) coverage. In fact it's already very difficult to make a fine rectilinear lens with more than about 110 degrees of horizontal coverage!
It is easy to demonstrate that the center point of the Entrance Pupil of a perfect orthoscopic lens (i.e. without aberration) is a fixed point: two parallel lines in the object space defines a plane. If this plane would also contain the center point of the Entrance Pupil (that is also the perspective center) then the two parallels should be represented as one single straight line on the image plane (sensor). If the perspective center moved when incident angle varies, then and consequently some distortion would appear and the lines would be bent on the image plane. As any level of distortion is absolutely banned (by definition of the word recrilinear), one can conclude that in this ideal case the center point of the Entrance Pupil must stay fixed. Some simple experiments can be built on this principle (e.g. in the section "about distortion" below).
Unfortunately there is no real perfect rectilinear lens in the real world. The ideal rectilinear lens exists only on the wish list to the lens designer. It is a specification term and tolerances are necessarily to be simultaneously taken in account early in the preliminary design. Some distortion of the straight lines that are projected on the image plane (sensor) have to be tolerated to a certain degree. In fact the constraints that drive the design of a lens are conflicting (as it is always the case in a complex system). The straightness of the lines on the photographic image to faithfully represent a straight line in the 3D space is a parameter that shall be balanced in a very complex trade-off study against all and every other parameters, including cost. The R = f x tan (Theta) rule shall then never faithfully and exactly obeyed and the entrance pupil may shift on a similar way to the fisheye pupil:
Distortion in general may be impacting the workflow and the quality of the work output of the panorama photographer as well as standard photography (e.g. architectural).
UPDATE: Pierre Toscani has published an animation that spectacularily illustrates this topic in stunning detail: view Fig 12a (in French and English) on his page (in French) about Pupils, Ports and Apertures
Then what general guidelines does a lens designer follow in the course of the preliminary design of a lens (any type)?
Some major guidelines are stated by lens design knowledgeable teachers. One of the more renown authors in the field is Milton Laikin who wrote the Lens Design book. Mr Laikin, as an industrial entrepreneur has actually produced many special lenses, including fisheye lenses. He has published the detailed construction description of some of these lenses and beside the classical (ideal) examples from the literature, these (real world) models are still used as test subjects for learning how to use sophisticated optical design tools. Many articles on optics that are presented all over the world have based their demonstrations on case-studies of one or more of these lenses originally designed by Milton Laikin.
From Lens Design (e.g. latest 4th edition -2006- by Taylor & Francis Group, LLC), in the first section of the first chapter that is titled "OPTIMIZATION METHODS" one can read the following about photographic lenses:
Below are some design considerations for photographic lenses. Keep in mind that this represents a generalization and is indicative of photographic lenses for single-lens reflex (SLR) type cameras.
(End of quote\)
This text which was writen before the digital age. It could be updated to be in line with the state of arts digital technologies. For example I use an excellent new lens (Samyang 14 mm f2.8) that have been measured to have 7.4% of distortion but this can be easily corrected by software. Anyhow the above list of 7 items still nowadays is an engineering reminder that a geek photographer should know, even if he doesn't completely adhere to it.
The word "vignetting" here means "optical and mechanical" vignetting and it excludes natural light radial fall-off or loss of luminance. Many lens reviews or lens test reports found on the Web do not separate the two groups of effects one from the other as the cheap optical test facilities that are used are not designed for that purpose. Mr Laikin perfectly knows better than anyone that some light fall-off in unavoidable when the light comes obliquely on the lens even "at half the aperture".
The #1item on the above Laikin's list (i.e. 1. Distortion) is in fact concerning only the photographic "standard" lenses. Fisheye lenses were not at all for a long time considered very usefull for strict photographic usage and was reserved for scientific purposes or for exotic projection schemes e.g. in some 70 mm movie theaters). BTW some of Mr Laikin's fanous industrial products are of the latter kind.
Distortion -meant here as deviation from the ideal R = f x tan (Theta) rule- and Entrance Pupil movement are the two faces of the same coin. We are not going to give a mathematical lecture about this subject here. From experiments, we shall first simply look at photographs that have spectacular distortion. Then from the analysis of our observation, we shall derive some rather simple principles.
Using a wide angle lens (a standard very-wide or a standard extremely-wide angle if possible) mounted on your favorite camera, take the picture of a straight edge in the 3D space. Let's assume that this is a plumb line hanging from the ceiling. You should not shoot directly at the string line: in the viewfinder (and in landscape mode) let the vertical line be away from the center. Get as close as you possibly can to the object. The focus had been set to the minimum distance possible and you may have to accept the subject to be a bit too close and consequently a bit unfocused (blur). Look now at the photo: is the straight line (in 3D space) represented as a straight line on the photo? If it is straight, then you are a lucky photographer who possesses a perfect lens (as distortion is concerned). However the line in general is bent on the image. More or less bent, but definitely bowed.
Of course you would not be surprised by this result if you had been using a fisheye. But we assumed from the start that you were shooting here with a "rectilinear" lens. Well, you are observing what is distortion. That's simple.
At about 150 mm (six inches) from the first plumb line it would be nice if you could hang a second one. Two 2-meter long plumb lines are now hanging from the ceiling and are parallel one with the other.
I did it! To get less blur due to object that is out of focus, I closed the iris of the diaphragm to the minimum (f/22 or more). For stability, the camera (Canon EOS 5D Mk2) was installed in landscape mode on a standard photographic 3-axis-head on top of a tripod. The tripod was placed at a position that allowed the front of the lens to be only a few centimeters away from the nearest plumb line and the head did let me to vary the alignment of the two plumb lines with respect to the axis of the lens. The tripod was placed in such a way that the two strings were first roughly "aligned" when seen through the viewfinder (or on the LCD with Live View activated). The distance from the front of the lens to the nearest plumb line was small enough (about 40 mm) to get a blurred (out-of-focus) but yet recognizable image. As a matter of fact I wanted to have the two strings of the plumb lines to be strictly coincident at the top and bottom of the image for a "moderately" oblique incident angle. The tripod position was moved on the floor surface until satisfaction (Thanks to the live-view x10 mode).
Following are three photographs: with a Tokina 10-17 mm (shaved full frame fisheye) @ 17 mm , a Nikkor 10.5 mm (shaved circular fisheye) and with a Samyang 14 mm (standard lens):
Tokina 10-17 mm @ 17 mm
(shaved full frame fisheye)
Nikkor 10.5 mm
(shaved circular fisheye)
Samyang 14 mm
Canon 24-105 mm
@ 24 mmm
From left to right: The most convex line between the two strings is the farthest plumb line in the 2 fisheye lenses cases (it is then the leftmost of the two strings). Whereas this is reversed in the two cases of standard lenses where the nearest string looks here more convex!
Analysis of the observations:
In addition, it is clear that the direction of the movement of the projection point of the entrance pupil (aka LPP) on the lens axis is opposite in the two cases. That is why the bowing lines relative position is reversed on the respective images.
The distance of the shift of the projection on the LPP of the Samyang lens and its direction have been measured with a LASER pointer test bench. The results have been reported on this page. The other lenses have also been measured and published a while ago by the author here and here.
Large distortion creates a unique problem: lens distortion (among other aberrations) are sensitive to object distance changes and consequently it is a source of parallax. Because of the pupil movement, there is distortion but conversely I postulate that when distortion can be observed and measured on an uncorrected image (e.g. directly downloaded from the camera), there must have been a movement of the projection of the entrance pupil on the axis (whatever is the lens projection type). As distortion is always present for real standard lenses to a certain extent, the entrance pupil is therefore never completely fixed and the LPP principle is applicable for any type of projection.
Fortunately the effect of the movement in the case of standard lens is generally rather benign compared to fisheye lenses. It may be not noticed at all. It is a problem that affects mainly wide and very wide-angle lenses because long focal lenses can (generally) only be focused for greater distances i.e. on more distant subjects. One should note that:
23 September 2010
Rev 14 oct 2010