Mathematics behind Wide Angle/ Fish Eye lens Construction
Mahesh Venkitachalam
First of all, I’d like to thank the people here for maintaining a
very active newsgroup. I have just posted two messages here, and
the number and quality of responses have far exceeded my expectations.
I am trying to understand how wide angle lenses, especially
Fish-eye lenses work, and am having a tough time finding
resources on the net. Some of the questions I am trying to find
answers for are:
1) What gives a lens a wide angle of view? Is it just the small focal
length? I guess when the focal length decreases, the cone of light that
converges at the focal point is wider? But then, a 14mm lens is not
14mm long, or is it?
2) What is really happening when you have a view angle of 180 degrees,
as in the case for a fish-eye lens? It would seem like the focal point
is now inside the lens, since the ‘cone’ is now a half-plane – but then
how
is the image formed? What type of projection gives you the circular,
distorted image formed by fish-eye lenses? It looks similar to the image
formed
by the surrounding environment on a reflective sphere.
3) What is the difference between diagonal and circular fish-eye lenses?
4) I know that in 3D computer graphics, the perspective transformation
can cause objects to look distorted, depending on how you set your
projection parameters. Some of the wide angle lens manufactures
say that they have corrected for distortions. Are these distortions
really
‘distortions’ in the sense that the way you are trying to project the
image,
may be these distortions are ‘natural’? I am not talking about
engineering
defects, but the effects produced by the projection itself, the extreme
case
being a fish-eye lens.
5) Where can I find information on lens construction, not just the basic
lens equations, but some information on how these lenses with floating
elements and such are designed?
Hope that wasn’t a question too many! I’ve been think about this for a
while,
and I would be grateful if the experts here could clear up some of these
concepts
for me.
Regards,
Mahesh Venkitachalam
Phichet Chaoha
you have a 2 dimensional film plane of size 36mmx24mm.
Form a cone using the focal length as its height,
and the 36mmx24mm film plane as its base. From
this model, you can use basic mathematics to calculate
the angles of coverage using the angles at the vertex
of the cone (there are 3 of them; i.e. horizontally
along 36mm-side, vertically along 24mm-side and
diagonally along the diagonal) of each focal length.
Hope this helps.
--------------------------------
Phichet Chaoha
Math Dept., UIUC
cha...@math.uiuc.edu
http://www.math.uiuc.edu/~chaoha
Fax : (603) 372-1279
Tom
>Fish-eye lenses work, and am having a tough time finding
>resources on the net. Some of the questions I am trying to find
>answers for are:
>
>1) What gives a lens a wide angle of view? Is it just the small focal
> length? I guess when the focal length decreases, the cone of light that
> converges at the focal point is wider? But then, a 14mm lens is not
> 14mm long, or is it?
>
>2) What is really happening when you have a view angle of 180 degrees,
> as in the case for a fish-eye lens? It would seem like the focal point
> is now inside the lens, since the ‘cone’ is now a half-plane – but then
> how is the image formed? What type of projection gives you the circular,
> distorted image formed by fish-eye lenses? It looks similar to the image
> formed
> by the surrounding environment on a reflective sphere.
>
The answers to your questions are fairly complicated.
First, I think you are trying to understand focussing of a "real" lens
in terms of so-called "thin-lens" formulae as you might see in a high
school physics book. Real lenses are more complicated.
The following 3 books will get you started on understanding issues
like how focal points can be placed further from the back of a wide
angle lens than the focal length would seem to say is possible:
1) A History of the Photographic Lens -- Rudolf Kingslake; Hardcover
2) Lens Design Fundamentals -- Rudolph Kingslake; Hardcover
3) Optics in Photography (Spie, Volume 6) -- R. Kingslake(Editor);
Kingslake also has a 3 volume set on lens design that didn't show up
when I searched amazon.com. It may be out of print. These books are
quite thorough, at least as of their publication dates, but because
they are not the most recent, may not cover ultrawides as thoroughly
as you might like. Expect a quite bit of math and geometry, but
nothing too esoteric.
I think that your second fundamental interest is to have an
understanding of the coordinate transformations of different types of
lenses. Here is a short version:
For this second discussion, pretend we are only considering very small
aperature (small diameter) lenses, so we can ignore all focussing
issues, and the fact that there is a converging cone of rays meeting
at each image point. For this discussion, consider the lens as simply
a magic pinhole in a plane that transforms angles on the object side
of the plane into angles on the film side of the plane.
Perfect, distortion free rectilinear lenses map (x,y) coordinates in
an object plane into (x',y') coordinates in the image plane by a
simple, scalar multiplicative constant, the magnification, M with
x' = M*x and y'=M*y. ...easy...end of story.
A fisheye lens takes the two angular spherical coordinates of the
direction to an object point relative to the lens and its axis (ie,
theta = the angle off axis, and phi = the angle around the axis), and
maps them into an image point direction, theta-prime and phi-prime.
At minimum, the perfect fisheye will have phi-prime = phi (or phi+180
depending on how you define your coordinates). However, I don't
believe that the "correct" theta transformation of "a perfect fisheye"
is universally defined.
For example, for most photographic applications, perfect linear
angular demagnification, theta-prime = M * theta (where M<1) would
likely be just as acceptable as any general transformation,
theta-prime = f(theta) as long as it was monotonic and theta-prime
was always less than theta (eg, say it turns a 150 degree conical FOV
(ie, almost a half-space) into (say) a 45 degree diverging cone of
illumination heading towards the film. The only applications that I
can think of that require a tightly controlled theta transformation
involve measurement problems (eg, astrophysics - fisheyes looking up
at the sky to measure the angular distribution of cosmic ray showers).
To answer one more of your questions, the difference between a full
frame and circular fish eye is simply that in the full frame, the
edges of the cone of illumination on the film side of the lens is
outside the film boundaries, whereas in the normal circular fisheye,
the edge of the same cone of illumination falls totally within the
film boundary and is recorded by the film as a circle.
All in all, this is a non-trivial topic and, if you are interested in
pursuing it further, it is more appropriately done by textbook and
engineering journal article, not by newsgroup postings. You will
find that if you pursue this topic, you will also need to more
accurately / quantitatively state some of your other questions.
Hope this helps a bit,
Tom
Washington, DC