Formula to convert PTGUI radial distortion values to K values

[Trackside]

I need to convert the A, B and C values for radial lens distortion into K1, K2 and K3 values for use in other lens correction applications. I'm sure I have seen a formula somewhere a long while back but can't find it at all now.

[Erik Krause]

Perhaps you can get some information from Michel Thoby's article:
http://tinyurl.com/lreyhh9

If not please tell what "other lens correction applications" you mean or
what the definition for the K1, K2 and K3 model is.

--
Erik Krause
http://www.erik-krause.de

[Trackside]

Alpa lens corrector uses K values for radial distortion as does the iWitness camera calibrator ( and most other photogrammetry and optical correction applications). As they are defining the same distortion as The a, b and c values in PTGUI / panoramatools there must be a mathematical formula to convert between the 2.

[Erik Krause]

Sorry, I couldn't find anything about the used lens correction model for
those applications. However, I know that there are some other polynomial
models for lens correction. Panotools and all derived programs use a
simple third degree polynomial with a*r^3+b*r^2+c*r as core formula.

There are other models using only even exponents like
http://alumni.media.mit.edu/~sbeck/results/Distortion/distortion.html
or such using only odd exponents as mentioned on Michel Thoby's side,
both of which use k1, k2 and k3 as parameters.

There is no doubt that either model can approximate a given distortion.
However, as far as I know there is no simple formula to transfer from
one model to the other. It can be done numerically though. You'd need to
calculate the distortion curve using one model and approximate a
correction curve using the other model. But this is beyond my math, sorry...

[mone zaghi]

Searching for lens correction method I found out this page from the Panorama Tools creator...not much mathematical formula but maybe can help

http://www.panotools.org/dersch/barrel/barrel.html

[John Griffin]

Hi Mone,

Many thanks – I think this was the info I had seen long ago!

 

 

John

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[Roland Karlsson]

On Saturday, August 2, 2014 at 1:48:48 PM UTC+2, Trackside wrote:

I need to convert the A, B and C values for radial lens distortion into K1, K2 and K3 values for use in other lens correction applications. I'm sure I have seen a formula somewhere a long while back but can't find it at all now.

The formula for lens distortion in Panorama Tools is unfortunately broken. That is the only reason I do not use PTGui.

The "normal" formula for lens distortion is

r' = r * (1 + k1*r^2 + k2*r^4 + k3*r^6).

The Panorama Tools formula is

r' = r * (c + b*r + a*r^2)

In general, a working formula is

r´= r * f(r), where f is a symmetrical function

The Panorama Tool function do not fulfill that criteria. Moreover, it has by far to few terms in the polynomial.

In the normal formula, k1 and k2 is absolutely always needed and k3 is often needed to fix distortion adequately.

In the Panorama tool formula a = k1, and k2 and k3 are missing.

So - Panorama Tools is fundamentally broken. And has been so since 1998 when it was released. Can someone fix this!

[Erik Krause]

Am 11.03.2018 um 21:11 schrieb Roland Karlsson:

> The "normal" formula for lens distortion is
>
> r' = r * (1 + k1*r^2 + k2*r^4 + k3*r^6).

Why should this be normal? It's one of the possible formulas.

> The Panorama Tools formula is
>
> r' = r * (c + b*r + a*r^2)

Wrong. The panotools formula is

r' = r * (a*r^3+b*r^2+c*r)

for details see
https://wiki.panotools.org/Lens_correction_model

It works quite well, even if your theory might tell something different.
Hundreds of thousands of panoramas have been stitched successfully
with it...

[Erik Krause]

Am 11.03.2018 um 22:55 schrieb Erik Krause:
>> r' = r * (1 + k1*r^2 + k2*r^4 + k3*r^6).
> Why should this be normal? It's one of the possible formulas.

Here you have a discussion of both models:
https://wiki.panotools.org/Lens_Correction_in_PanoTools

[Roland Karlsson]

The formula I gave is the normal one for three reasons:

1. It is the one almost everyone else but PanoramTools use
2. It is the one that is normally used in scientific literature about lens distortion, for very good reasons.
   
3. The multiplicative correction function f is symmetric, which is necessary. Otherwise the function will behave very strange.
   

This is the formula I gave.

r' = r * (1 + k1*r^2 + k2*r^4 + k3*r^6) = r* f(r).

Unfortunately I did write the Panorama Tools formula wrong, here is the on from https://wiki.panotools.org/Lens_correction_model

As you can see, the "normal" model has the powers of 0, 2, 4, 6 and PanroamaTools 0, 1, 2, 3.

They are not compatible, and the powers of 1 and 3 makes it non symmetric.


/Roland

[Roland Karlsson]

BTW - that comparison web page is wrong. It claims the wrong formula that "many other" uses.

It claims that many do use:

n = N * (1 + a*N + b*N^3 + c*N^5).

But, that is not true. The formula many other do us is:

n = N * (1 + a*N^2 + b*N^4 + c*N^6).

So - any results on this page are wrong.

/Roland



On Sunday, March 11, 2018 at 11:08:58 PM UTC+1, Erik Krause wrote:

[Roland Karlsson]

BTW - take a look here

https://en.wikipedia.org/wiki/Distortion_(optics)#Software_correction

It describes the polynomial distortion correction, a.k.a. the Brown–Conrady model.

Unfortunately PanoramaTools is different.

/Roland

[PTGui Support]

Hi Roland,

If I had started PTGui from scratch it would probably have used the
Brown–Conrady model, simply because everyone uses it. But PTGui
inherited the panotools model. Over time I have considered adding the
Brown–Conrady model as a second lens model but there hasn't really been
a strong need for this. You're the first to tell this is holding you
back from using PTGui though, so now I have a good reason to reconsider
this.

It will have its own problems: in the absence of control points near the
edges, using high order terms (r^6) term will easily lead to wildly
instable results due to overfitting. It would need to be tamed with
control points very near to the edges of the image.

And I don't see any problem in using the odd order terms. In practise
they work fine, and they do reduce the control point errors. We're just
fitting a polynomial for 0 <= r <= 1. So can you please explain?

Kind regards,

Joost Nieuwenhuijse
www.ptgui.com

On 12/03/2018 00:15, Roland Karlsson wrote:
> The formula I gave is the normal one for three reasons:
>

> 1. It is the one almost everyone else but PanoramTools use
> 2. It is the one that is normally used in scientific literature about

> lens distortion, for very good reasons.

> 3. The multiplicative correction function f is symmetric, which is

> necessary. Otherwise the function will behave very strange.
>
> This is the formula I gave.
>
>
> r' = r * (1 + k1*r^2 + k2*r^4 + k3*r^6) = r* f(r).
>
>
>
> Unfortunately I did write the Panorama Tools formula wrong, here is
> the on fromhttps://wiki.panotools.org/Lens_correction_model

> <https://www.google.com/url?q=https%3A%2F%2Fwiki.panotools.org%2FLens_correction_model&sa=D&sntz=1&usg=AFQjCNGGIPEHVqY8OotdTXm1sKltdY442g>
>
> <https://wiki.panotools.org/images/math/d/4/4/d4466e5ff97cd6bbdddc514f3a28fb88.png>

>
>
> As you can see, the "normal" model has the powers of 0, 2, 4, 6 and
> PanroamaTools 0, 1, 2, 3.
>
>
> They are not compatible, and the powers of 1 and 3 makes it non symmetric.
>
>
> /Roland
>
>
>
>
>
>
>
> On Sunday, March 11, 2018 at 11:08:58 PM UTC+1, Erik Krause wrote:
>
> Am 11.03.2018 um 22:55 schrieb Erik Krause:
> >> r' = r * (1 + k1*r^2   + k2*r^4   + k3*r^6).
> > Why should this be normal? It's one of the possible formulas.
>
> Here you have a discussion of both models:
> https://wiki.panotools.org/Lens_Correction_in_PanoTools
> <https://wiki.panotools.org/Lens_Correction_in_PanoTools>
>
> --
> Erik Krause
> http://www.erik-krause.de
>

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[Erik Krause]

Am 12.03.2018 um 00:15 schrieb Roland Karlsson:

> 2. It is the one that is normally used in scientific literature about

> lens distortion, for very good reasons.

I don't know for sure, but I guess Prof. Dersch used it for practical
reasons, probably because it's suitable for rectilinear lenses as well
as for fisheyes. After all the main goal was not to scientifically
describe lens distortion, but to correct it such that images can be
stitched successfully, which it does very well.

If you have real world examples where it fails but the other one works
I'd be curious.

To answer the original question (from 4 years ago): They are partly
convertible into each other. See
http://www.panotools.org/dersch/barrel/barrel.html
(scroll to "Some additional notes")

In fact there are a couple of other models, that aim at correction as
well. Very interesting reading on Michel Thoby's page:
http://tinyurl.com/lreyhh9

And there are much more:
https://hal.inria.fr/inria-00590269/document
(Scroll down to 3.6 So Many Models...)

[Roland Karlsson]

Thanx for your responses. Seems I am partly wrong :) Nice to learn something new!

To refresh my knowledge, I did a google search and read some papers. And they did support what you are writing above. It is possible to use the PanormaTools formula. The formula should really be anti-symmetric, so that f(-x) = -f(x), but it is (of course) not necessary. And if you have both odd and even terms in the polynomial then you can have a lower order.

I am at work now and can only make brief reply. Will read your links at home.

OK - this means that I can use PTGui. But ... I am using several tools and I am not all that comfortable with using two different models. I do, when possible, convert from one tool to another.

I am the author of this tool: http://proxel.se/lens.html, which can measure lenses, and do lens corrections. I also plan to use data from Adobe lens profiles. NOTE - that the Proxel Lens Corrector tool do a special treatment of fish eyes.

It is, of course, possible to also convert (approximately) between the two different models. Could write such a program.

/Roland

PS. The reason I am looking at PTGui right now is that I have some problems with Kolor AutoPano. It is a fantastic program, but it is a bit too automatic. When it fails, it fails. And this even if you have used a panorama head, with known angles, and know the lens distortion.

[Erik Krause]

Am 12.03.2018 um 10:57 schrieb Roland Karlsson:

> It is possible to use the
> PanormaTools formula. The formula should really be anti-symmetric, so that
> f(-x) = -f(x), but it is (of course) not necessary.

Ah, now I understand your concerns. The other formulas work on image
coordinates (with 0,0 in the center) and hence have to be symmetric.
Panotools works on the radius, so the negative part of the formula isn't
used.

PTGui 11 introduced a somehow extended correction model, which better
accounts for the newer (near) stereographic fisheyes (Samyang etc.) as
well as for orthographic fisheyes like the Madoka. See
http://www.ptgui.com/beta.html for details.

I had a look at your page. Sounds interesting, although I see no real
use for me personally. If I need to correct single images I use PTGui as
well, and I guess it's ability to determine lens correction parameters
by optimizing control points from the overlap is at least as precise as
searching for straight lines, especially since it also can determine the
center offset.

I used both the Zenitar and the Peleng in the past but found the Samyang
fisheyes be optically much better, both in terms of sharpness and lens
flare.

best regards
--
Erik Krause

[Roland Karlsson]

I have made some tests now with some difficult (for Kolor Autopano) panoramas. PTGui managed them all right. They were made with a Zenitar, of some rooms.

I have also thought about getting the Samyang for my Pentax K-1. But, I see that v11 is only beta yet.

Virusfritt. www.avast.com

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[Erik Krause]

Am 12.03.2018 um 22:28 schrieb Roland Karlsson:
> I have also thought about getting the Samyang for my Pentax K-1. But, I see
> that v11 is only beta yet.

The Samyangs (I have both 12mm and 8mm) stitch very well with v10, as
many users report. In my impression probably even better than the
Zenitar. I suspect that the new improved lens model has been shaped more
towards the Madoka type.

If you get a Samyang (or renamed brand like Bower, Falcon, Polar,
Pro-Optic, Rokinon) ckeck the focus ring. It is notoriously misaligned,
and some lenses can't focus for infinity.

[Tom Sharpless]

Yes, the PT lens formula really has problems.  The worst one is that the coefficients are scaled according to image dimensions, so they represent not just the lens but a particular combination of lens and sensor -- and even image orientation!  

This fundamental mistake has caused untold misery and mystification to users of PanoTools, Hugin and PTGui.  It should have been corrected decades ago. Presumably it has survived because a lot of code depends on this way of representing the lens, including plenty of 'tweaks' that work around some of its worst consequences.

Joost could become (even more) my hero if he would install a scientifically correct lens model in PTGui, and make it inter-operable with other lens calibration systems.

[PTGui Support]

On 13/03/2018 15:38, Tom Sharpless wrote:
> Yes, the PT lens formula really has problems.  The worst one is that the
> coefficients are scaled according to image dimensions, so they represent
> not just the lens but a particular combination of lens and sensor -- and
> even image orientation!

Sorry but that's not entirely true. a/b/c are normalized to half of the
smaller side of the image. So the lens distortion correction is
orientation and size independent.

The panotools shift parameters d and e are in pixels though, so they do
suffer from this problem.

And the fact that panotools used the horizontal field of view to
characterize a lens meant that it was also orientation dependent.

The good news though: PTGui 11 fixes all those things!

> This fundamental mistake has caused untold misery and mystification to
> users of PanoTools, Hugin and PTGui.  It should have been corrected
> decades ago. Presumably it has survived because a lot of code depends on
> this way of representing the lens, including plenty of 'tweaks' that
> work around some of its worst consequences.
>
> Joost could become (even more) my hero if he would install a
> scientifically correct lens model in PTGui, and make it inter-operable
> with other lens calibration systems.

By that you mean Brown–Conrady? Perhaps some day. It will have its own
problems, and I wonder how it will work with fisheye lenses.

Joost

[Trackside]

Wow - great this topic has been resurrected after all these years!

[Tom Sharpless]

It is possible to extract pure lens parameters from PT style calibrations, but you need an additional piece of information: the factor that converts distances in the focal plane, measured in pixels, into dimensionless distances measured in focal lengths; that is, the focal length in pixels (pixels/radian), or its reciprocal, the angular size of a pixel, which is equal to (pixel width)/(focal length).  This is a fundamental parameter of all sensible lens calibration schemes.  However in PT is is estimated as the ratio of a field of view angle to an image dimension.  That is a poor estimate for many reasons, but at the time PT was written it made practical sense: it was hard to get data on the digital cameras then current; and hard for photographers to understand why such data were needed.  

Using field of view / image width as if it were pixel size works fine so long as you are going to optimize the fov value, but it has the pernicious side effect mentioned above, that the polynomial distortion coefficients become dependent on the image size, so are not suitable for export or even for re-use on a crop of the same image. 

[Roland Karlsson]

Actually - if you want to reuse the lens calibrations, no matter what units you use, you need to know if the center of the image has moved.

So, if you crop, and hope to be able to use it you either have to crop symmetrically, or you have to provide how the center has moved.

Scaling issues are not all that important, if you just give the scaling factor.

What is more important is that the A, B, C and the K1, K2, K3 values needs a rather advanced method for conversion, based on running optimisation.

/Roland

[Tom Sharpless]

I have worked with several widely used lens calibration schemes, and can honestly say that most of them are inadequate for the range of lenses used in modern photography.  The only one I would use is Gennery's model, that was developed for NASA's Mars Rover program.

This represents the basic projection curve by a function having one parameter, that can make it fit well any of  the curves that fish-eye designers have come up with, as well as all "normal" lenses, Schmidt and Newtonian reflecting telescopes, and more. 

In Gennery's scheme, as in all others, deviations from the model curve are corrected by a polynomial of even order.  But because the  model curve already fits the lens well, the polynomial corrections are small, and this avoids several serious difficulties that can happen when the polynomial has to "do a lot of the work" of matching the lens.

Both PTGui version 11 and my own PT3D contain lens models inspired by Gennery's -- not the same one, and both differ from Gennery's in detail, but the principle is the same.  So they can both handle extreme lenses better than classic PanoTools.  However both are burdened with the requirement to support the legacy PT model, and at present almost all existing PT calibrations are of that type.

[Roland Karlsson]

This seems to be very sound. You know approximately the model (distortion) of the lens. Then you optimize the deviation.

This sounds (more or less) similar to how my Proxel Lens Corrector handles fisheyes. It does not try to find a polynomial that fits the distortion. Rather it tries to find a polynomial difference to a perfect fisheye. And do not PTGui do the same? Hmmmm ... have I missed something?

/Roland

BTW - could not NASA have used lenses with totally known distortion?

[Tom Sharpless]

The PT lens model has two ideal curves, rectilinear (r = f*tan(a)) and equal-angle spherical (r = f*a) where a = angle of ray from optical axis.  Real lenses, especially ultra-wide and fish-eye, can deviate quit a lot from those curves. Then the polynomial correction needs to be big.  Since a polynomial is nothing like an actual lens curve, this destabilizes the optimization and can even generate a fitted curve that is not monotonic, making the inverse lens function indeterminate.

So it is very important that the "ideal" function comes close to the real lens.  One ideal fisheye curve is not enough, because real fish-eye lenses have lots of different curves, ranging from stereographic (with mild peripheral compression) to equal-area (with much stronger compression) and even beyond.  Gennery's function, and the one I cooked up for PT3D (and presumably Joost's in PTGui 11)  can fit all the ideal curves that lens designers have used as prototypes, and therefore come quite close to all the actual fisheye curves.

Given a good ideal curve, the exact form of the polynomial does not matter much.  Two terms, say 2nd and 4th order, is plenty for photographic purposes; metrologists might need three.

[PTGui Support]

Yet you might be surprised how well the abc polynomial works in
practise. That's why it surived for 18 years. For all fisheyes I've
tried, optimizing abc with the fisheye factor set to 0 (i.e. the old
PTGui 10 way) gives much better control point alignment than optimizing
only the fisheye factor with abc set to 0.

In other words, practical fisheye lenses deviate quite a bit from the
ideal sin() or tan() curves, and the abc polynomical can mimic the
curves quite well.

Kind regards,

Joost Nieuwenhuijse
www.ptgui.com

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[Roland Karlsson]

Thanx for this info.

Regarding metrologists. I would say stitching also needs very good correction. I am not fond of needing to have very good blending in order to hide mismatches.

/Roland

[Roland Karlsson]

Of course. You cannot assume you have an ideal fisheye. It has to be corrected.

I assume though that if you use a circular fish eye, the abc thingie will fail.

/Roland

[Erik Krause]

Am 15.03.2018 um 18:49 schrieb Roland Karlsson:

> I assume though that if you use a circular fish eye, the abc thingie will
> fail.

No, it works perfectly. As said before: hundreds of thousands of
panoramas have been successfully stitched this way. I guess Prof. Dersch
knew very well what he was doing.

I just reviewed some panoramas from 2004 (for the "once upon a time" WWP
event), shot with a Peleng 8mm then (on analog film). No problems
stitching them.

[Roland Karlsson]

Just for the record - I was talking about using the polynomial correction only. Do it still work perfectly? I would guess not.

Now, to get back to the original question regarding converting from abc to k123.

One thing that would be nice is to detect the lens and read the adobe lens database used in ACR and Lightroom. Then the values needs conversion, or PTGui supporting k123.

It would have been even cooler if cameras/lenses had the k123 parameters stored in firmware and written to EXIF. But that is SciFi :)

/Roland

[Erik Krause]

Am 18.03.2018 um 11:38 schrieb Roland Karlsson:

> Just for the record - I was talking about using the polynomial correction
> only. Do it still work perfectly? I would guess not.

panotools and PTGui until v10 don't use the polynomia alone, they use it
on a standard fisheye mapping.

[...]

> One thing that would be nice is to detect the lens and read the adobe lens
> database used in ACR and Lightroom.

RawTherapee does that, but I don't know whether all ACR built in
profiles are downloadable as .lcp files. Guess only the user supplied
ones are. However, Adobe profiles attempt to correct fisheye images to
rectilinear, which isn't desirable anyway. I found one .lcp file on my
computer (for the EF 100-400mm) but that contains only two radial
distortion parameters. That would be worse than panotools...

[Roland Karlsson]

You might be right.

I have had very good results for my Pentax camera using Pentax lens profiles. But, the Samyang 14 was crap.

/Roland

[Tom Sharpless]

Adobe's model and fitting s/w handles fish-eye lenses well, I have seen some very accurate fish-eye calibrations made with it. It is a lot of trouble though because it uses checkerboard targets, you need really big targets and lots of photos to get a good result.

It is a shame they hobble this excellent calibration scheme by assuming that the only kind of "corrected" image you might want is rectilinear.