Article on view camera maathematics

[Leonard Evens]

I have put the first draft of an article on view camera geometry at
www.math.northwestern.edu/~len/photos/pages/vc.pdf

I should emphasize that this is a mathematics article and assumes some background in subjects like projective geometry. If you have neither the interest nor the background in such things, you should most definitely NOT look at it.

When I have finished it, I plan to write a much less technical article which describes the important facts without the detailed mathematical arguments.

As I said, it is a first draft. I haven't yet done the diagrams; one difficult section and some appendices are yet to be written. I'm not sure it is yet readable by anyone, but It is remotely possible someone will look at it and find some blooper or suggest some important issue I've missed. If you do, please let me know. I will keep updating it as I produce further drafts.

[John O'Connell]

Leonard, was this written in TeX? I saw at least one single-quote (second paragraph---'dark) that needed replacement with its mirror image.

I'd like some more articles like this on the homepage, actually. Though this one is certainly beyond my depth, it would be nice to have a local link to refer questions to rather than trying to find optics and geometry articles elsewhere on the web. Or even worse, simply saying "See Stroebel" or "See Kingslake."

[steve simmons]

In the Free Articles section of the View Camera web site is an article on camera movements and the landscape


www.viewcamera.com

Also, in the March issue there is an article on Scheimpflug with a lot of math, and in May the same article w/o the math.


steve simmons

[Greg Lockrey]

I had an instructor for a math course called "Theory of Matrices" Math 425 (Calculus of 3 dimensional matrix) by the name of Van Deventer who wore coke bottles for glasses. He liked to talk about baseball at the beginning of the class (Detroit was in the World Series that year, '68 I think) he would end his baseball discussions by saying "and now for the real world of mathmatics". Leonard, your paper brought back some memories. :)

[BarryS]

Thanks for posting this. Even though my math abilities fall far short of understanding anything beyond the basic equations, it looks like there's some useful information for the non-technical in your paper. I look forward to seeing the version with illustrations, and more importantly--the lay version.

[Jorge Gasteazoro]

I'll wait for the lay version as well, tried to read it and got lost in the first paragraph... :)

[Jim Noel]

I read the article and find it interesting and a good beginning.

However,there are some very good articles already out there, some of which are easier to understand. This was mentioned by Steve Simmons. There are also some like the one by Merklinger which are on the web, much more explanatory and get into the mathematics in a more practical manner.

[Leonard Evens]

Leonard, was this written in TeX? I saw at least one single-quote (second paragraph---'dark) that needed replacement with its mirror image.

I'd like some more articles like this on the homepage, actually. Though this one is certainly beyond my depth, it would be nice to have a local link to refer questions to rather than trying to find optics and geometry articles elsewhere on the web. Or even worse, simply saying "See Stroebel" or "See Kingslake."

Thanks for finding the typo. I hope people will look from time to time at the article. Even if they don't follow everything in it, they may still have some helpful suggestions. I've already started putting in the diagrams, and I've meade several other changes.

Yes, it was written using TeX, actually AmSLateX, which is $LaTeX$ with American Mathematical Society packages. The great bulk of mathematics articles are written using TeX, and it is a shame that so much other technical materail gets written using different versions of Word, which is not really suitable to the task. (My wife uses Word to write computer science papers because, she says, all her collaborators do, but she was much happier when she used TeX. She alwasy fighting with it and not able to predict just what it will do.)

I am having some problem doing the diagrams. I work under Linux, and there are a variety of graphics packages available, but they are all cumbersome to use and have steep learning curves. If anyone has a good suggestion about that, I would appreciate it. Right now I' trying to use xfig and inkscape.

I also wish there were a good source of links for the things you refer to, but at present I'm just barely getting by doing what I can, so I will leave that to someone else.

There are of course several other useful articles on these topics. Bob Wheeler's Notes is a classic, and for the questions he considers, it is still possibly the best source. It is where I started, and I still go back to it and find things I missed. Jeff Conrad's articles on this web site are probably the best and most comprehensive treatment of depth of field, including issues like the entrance/exit pupils. Of course there is the classic work of Merklinger, from which I learned a lot, although it took me a while to decipher the mathematics underlying it. Other articles, including those mentioned by Steve Simmons, may be useful to a lot of people. I won't try to comment on the two latest ones. I still didn't understand the first one, even after it was reprinted with additional figures, and I haven't seen the second one.

I've also been reading Q. T. Luong's book, which is full of mathematics, and which I find fascinating. Doing so encouraged me to go back and examine things from first principles, although he doesn't address many view camera issues. The primary focus is how to recreate a three dimensional image from various two dimensional projections, as needed in computer vision, an important subject with a large literature.

There are some common misconceptions which appear in some articles. The one which Doremus Scudder acknowledged is in his otherwise useful article is particularly common, and is even implied in earlier editions of Stroebel. It is that, when the lens plane is tilted/swung, the boundaries of the depth of field wedge are curved surfaces. As Wheeler indicated in his Notes, literally speaking this may be true, but it is also misleading. For all practical purposes, these surfaces may be considered to be planes. Stroebel and others confuse the depth of field boundaries which other surfaces, which are strongly curved. Wheeler made some estimates for departure from planarity along the center line of the frame, and I have, over the years, been trying to understand how things work off center. It turns out to be a rather difficult mathematical problem, probably without a nice closed answer, so one must be content with estimates. I am still not sure I completely understand all aspects of the problem, and it is one thing I hope to finally put at rest in my article. Of course, the practical photographer will note that things seem to go well in practice if you just assume they are planes, but as a mathematician, I would like to be able to prove it. One practical benefit from such an analysis would be an understanding of various extreme situations where things could go wrong. I really don't think anyone has addressed this issue and some others like it elsewhere.

[Jay Wolfe]

In the Free Articles section of the View Camera web site is an article on camera movements and the landscape


www.viewcamera.com

Also, in the March issue there is an article on Scheimpflug with a lot of math, and in May the same article w/o the math.


steve simmons

Since this is the real hub of large format photography, I think you should post all of those articles here.

[Struan Gray]

Leonard, I enjoyed reading your article, although it reminded me why I took up physics rather than mathematics :-)

Re. the curvature of the depth of field limits. This is a failure of a small angle approximation, and shows up most strongly at large tilts, wide angles and small apertures. You can see it when you use a wide angle lens and try to lay the far DOF 'plane' along a flat surface.

Set up a camera on a planar surface with a short tripod and use Scheimpflug to lay the plane of true focus along the ground. Then reduce the tilt angle, and adjust focus so that the piece of ground closest to you is in focus. If you stop down you should see that the middle distance comes into focus *after* the far distance.

I'll check tonight, but when we had all the tilt-and-DOF threads at photo.net in 2002 I seem to remember I was able to see this clearly with a 90 mm lens on a 1m tall tripod. If like me your knees hurt when you use a 1m tripod, it should be possible to see the same thing using swing and a long brick wall.

The reason is that the correction Bob Wheeler says is too small to be relevant can creep in and bite you in some cases. My take on the reasons is half way down this interminable thread on photo.net:

http://photo.net/bboard/q-and-a-fetch-msg?msg_id=003Rdn

In some of my photos I use movements to place a slice of focus along the ground. See these aspens (http://www.struangray.com/tanglings/aspens02.htm) and these rushes (http://www.struangray.com/gaels/06_raa_rushes.htm) for examples where I wanted to keep a horizontal plane crisp. For me, it's a way of making sense of visually dense environments like undergrowth. People in these sorts of threads keep telling me I'm imagining things, but when final focussing I do keep a weather eye open for a fuzzy middle distance. It works for me.

[Emmanuel BIGLER]

Great article, Leonard, although you should admit that it is written for your students in order to attract them to doing LF photography, and not for photographers in order to attract them to mathematics ;)

Regarding the question of the limits of depth of field with slanted planes I recommend to use an approximation, the "venitian blind approch". Instead of finding the real ellipse of confusion, imagine that you check for the circle of confusion on a small venitian blind located at a given point of the slanted image plane but lifted up in order to stay parallel with the exit pupil.
Look at my article (the English version is still not up to date!! shame on me ! )
see figures 5, 6, 7
http://www.galerie-photo.com/profondeur-de-champ-et-scheimpflug.html

There, you have the advantage that the circle of confusion is invariant in the image and I'm sure that you can apply the elegant formalism of projections to demonstrate in one or two powerful projective formulae that the limits of acceptable sharpness are slanted planes according to the usual rules explained in other articles.

Another issue is that at a first mathematical glance, depth of field and depth of focus issues are different since depth of field ussues use the harmonic mean to place the planes of best focus in the middle of limit planes of acceptable sharpness ; whereas depth of focus uses the regular mid-plane.
We discussed this issue with Jeff Conrad by mail, but my conclusion is that in practice with reasonable view camera settings and a reasonable choice of the circle of confusion both means are very close to each other, hence you get another simplification that allows you to demonstrate the wedge shaped depth of field in a snap.

After that, if you wish, you could go through the general shape of elliptical projection of the exit pupil on film, but I think most of future readers would be delighted to see the proof for the "DOF slanted planes" in the elegant way you demonstrate the basic Scheimpflug rule (as explained in Tuan's hand-drawn diagram on his web site)


Right now I' trying to use xfig and inkscape.

What follows is for aficionados of free software using Linux.
I've been using xfig for 8 years under unix and linux and I have no reason to switch to any other software for vectorized graphics.
The only thing painful and limited in xfig comes when you want to insert mathematics inside the diagram. Even simple subscripts are a pain.
So as soon as I have some formulae to insert into a figure, I create them with LaTeX. Then I export the formula from LaTeX into a tiny .eps file using the command dvips with the -E option.
For example assume that my LateX formula is in the file sinomega.tex

% this file is sinomega.tex
\documentclass[12pt]{article}
\usepackage{amsmath} %% of course, Leonard !!
\begin{document}%
\pagestyle{empty}
\begin{Huge}
$\sin(\omega \, t+\phi)$
\end{Huge}
\end{document}%
% end fo file

Then I create a small eps file with a tight bounding box around the formula:
latex sinomega.tex ; dvips -E sinomega.divi -o sinomega.eps
Then, I can incorporate the .eps formula inside xfig easily like I can incorporate any picture.
In the final ps or pdf output the typesetting will be perfect since xfig is clever enough to keep the .eps file as is.
You can even re-incoporate this figure inside another LaTeX document.
I use the graphicx package and from xfig I export in .eps.

See the attached pdf page where figure 2 created with xfig incorporates good-solid LaTeX mathematics and where figure 3 has (almost) no latex mathematics.

For plotting curves I've bee using gnuplot and nothing else for 15 years.
In gnuplot you can either export as .eps or export as xfig code to be re-incorporated
into another xfig figure, eventually incorporated in the final latex document.

From this latex document with minimum changes in the latex code, I convert them into html with the French & Free software HeVeA developed by Luc Maranget at INRIA.
http://pauillac.inria.fr/hevea
This is how I create both a pdf and a html for my articles on galerie-photo.

[steve simmons]

Jay Wolfe


Join Date: Jan 2005
Location: Colorado
Posts: 48 Re: Article on view camera maathematics

--------------------------------------------------------------------------------

Quote:
Originally Posted by steve simmons
In the Free Articles section of the View Camera web site is an article on camera movements and the landscape


www.viewcamera.com

Also, in the March issue there is an article on Scheimpflug with a lot of math, and in May the same article w/o the math.


steve simmons

Since this is the real hub of large format photography, I think you should post all of those articles here.


View Camera has been in publication since 1988 and has published dozens and dozens of articles on view camera technique. We have made a practice of putting user friendly articles on this topic on our web site for years.

steve simmons
publisher, view camera
www.viewcamera.com
www.foto3-2008.com

[Leonard Evens]

Many thanks Emanuel!

I have to study your comments in greater detail, but let me say a couple of things.

Your suggestions about putting mathematics in an xfig (or other) document will be very helpful. I haven't done LaTeX diagrams for a while, and I've forgotten a lot of what I once knew. I do remember that it was always a pain to put mathematics in the diagram.

About the venetian blind approach, I did try that, and an earlier version of the article used it. But in the end I decided it avoided the primary issue which, I think, is the relation between the aperture circle and the projected ellipse in what I call the reference plane, i.e., the plane parallel to the image plane through the lens. The trouble is that that this ellipse depends on everything in sight, and in order to make approximations, you have to study it in some detail. That took me considerably longer than I thought it would, mainly because I kept making mistakes, but I think I have it right now. I still have to work out the approximations in detail, but I think I understand the relevant parameters.

I haven't figured out yet just how to do things for a real lens with separate principle planes, etc., in the tilted case. I think things will work pretty well as long as the exit/entrance pupils are in the
principal planes. I will look at your articles again for enlightenment. I wish my German were up to par. I once was able to read German articles pretty easily, but I've lost the knack.

I agree with you that depth of field and depth of focus are different issues. I haven't thought about how depth of focus works in the tilted case.