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Tomislav Pribanic
03-09-1998, 01:16 AM

Last week I have asked some questions concerning DLT. I thank all of you who tried to help me. I am submiting all responses I have received, although the first
two are the most representive.
But first here is my original question:

I am new biomech postgraduate student. At the present time I am involved
in human motion analysis by means of 3-D reconstruction with DLT.
I am trying to write computer program of my own (although I am aware of contributed packages for Matlab and C on various sites e.g.http://www.kin.ucalgary.ca/isb/software/kinemat/readme1.htm). Currently I am stuck with the issue of camera calibration. According to the basic equations:

u=(L1*X + L2*Y + L3*Z + L4) / (L9*X + L10*Y+ L11*Z + 1)

v=(L5*X + L6*Y + L7*Z + L8) / (L9*X + L10*Y+ L11*Z + 1)

one needs six or more calibration points in order to calculate 11 DLT parameters.
Since we then deal with overdetermined system of equations many authors are therefore mentioning and employing linear least-square techniques to compute DLT
parameters (e.g. Hatze, H. (1988) High-precision three-dimensional photogrammetric calibration and object space reconstruction using a modified DLT-approach.
J. Biomechanics Vol 21. No. 7. pp. 533-538).
However, I have not seen anywhere the exact way how the above mentioned technique is preformed. The essence of linear least-square technique is in the form of

F(a,b)=sum( (y(i)-(a*x(i)+b))^2 ) i=1,...n
y(i),x(i) coordinates of various points
a,b parameters of line y=a*x+b.
Afterwards one takes partial derivate of F(a,b) with a and b in order to compute the parameters of line. One gets two equations with two unknowns and the rest is easy. What I do not understand is how it is applied in case of computing DLT parameters.
Do I have to write the entire thing in the form of:

F(a,b,L1,L2,...L11)=sum( (v(i)-(a*u(i)+b))^2 )

i=1,...n calibration points
u(i)=(L1*X(i) + L2*Y(i) + L3*Z(i) + L4) / (L9*X(i) + L10*Y(i)+ L11*Z(i) + 1)
v(i)=(L5*X(i) + L6*Y(i) + L7*Z(i) + L8) / (L9*X(i) + L10*Y(i)+ L11*Z(i) + 1)?
And then take the partial derivations of function F(a,b,L1,....L11) with a,b,L1,L2,....L11 in order to find parameters?
Or maybe I am supposed to go as follows:

F(L1,L2,...L11)=
sum( (u(i)-(L1*X(i) + L2*Y(i) + L3*Z(i) + L4) / (L9*X(i) + L10*Y(i)+ L11*Z(i) + 1))^2+
+ (v(i)-(L5*X(i) + L6*Y(i) + L7*Z(i) + L8) / (L9*X(i) + L10*Y(i)+ L11*Z(i) + 1))^2 )
and now also preform partial derivations of the L1,L2,..L11 unknowns?
I have tried both ways, but it did not seem to be right direction for all kinds of reasons. Also, once I acquire all parameters and try to gain the space coordinates of the unknown point out of two or more cameras picture I am again dealing with more equations than I need. Do I act in a similar fashion as above?
What is the matematical procedure when is MDLT (Modified DLT) concerned,
rather than conventional DLT?
I would be glad to post on the List all answers I recieve, once I collected them.

RESPONSES:
************************************************** *****************************
Dear Tomislav,

The best way to deal with the problem of solving linear equations with
fewer unknowns than equations
(over-determined system) is to use matrix manipulation . You can use
numerical methods to do that.
For example to deal with a system of three equations with two unknowns that
can be represented by:

[A](subscript: 3x2 ){X}(subscript:
2x1 )= {B}(subscript: 3x1)

(where {X}(subscript: 2x1) is the unknown
matrix)

You multiply both sides of the equation by the transpose of [A] to get:

[A](superscript: T)(subscript: 2x3) [A]
(subscript: 3x2 ){X}(subscript: 2x1 )= [A](superscript: T)(subscript: 2x3)
{B}(subscript: 3x1 )

you will end up with a system of two equations with two unknowns that
should be easy to solve:

[A'](subscript: 2x2 ){X}
(subscript: 2x1 )= {B'}(subscript: 2x1)

This can be applied to get the calibration coefficients for each camera and
also to get the 3D coordinates of
the targets from two or more cameras. Let me know if you have any
questions. Good luck.

Ali M. Elhagediab, PhD
Research Scientist
Aerotek Corp. (in contract with General Motors Corp.)
email: aelhaged@notes.gmr.com
************************************************** *****************************

Dear Tomislav:

I was in a similar situation some time ago when I was a Master's student in
Seoul, Korea. I've read a lot of things since then and finally could figure
out what DLT is all about. I even have written a motion analysis package
based on DLT after that.

For camera calibration with n (>6) control points, the system will be
over-determined. Then the linear system looks like this:

X B = Y

where, X = known 2n x 11 matrix, B = transpose of (L1 .... L11), and Y =
known 2n x 1 matrix made of u and v.

Multiply both sides with the transpose of X, X', to make the matrix in front
of B square. This stage is called normal equation:

(X' X) B = X' Y

Then Multiply both sides with the inverse of (X' X) or (X' X)^-1:

(X' X)^-1 (X' X) B = (X' X)^-1 (X' Y)

Then, the left side of the above equation is the same to the least-square
estimate of B:

B = (X' X)^-1 (X' Y)

With only manipulation of the matrices, you will be able to do the least
square estimation.

This procedure is in fact exactly the same to the multiple linear regression
in statistics. Consult any statistics textbook for this procedure: multiple
linear regression using matrix approach.

You may use exactly the same approach to compute the 3-D coordinates of the
unknown points with more than 2 cameras.

In reality, if you include the additional parameters to correct the optical
distortion and the de-centering distortion due to the optical system, the
situation is much more complex than this. I am in the middle of putting
these things on the web, and I'll let you know when it is done.

Hope this to be helpful.

Young-Hoo

--------------------------------------------------------------
Young-Hoo Kwon, Ph.D.
Assistant Professor
Biomechanics Lab, PL 202
Ball State University
Muncie, IN 47306
U.S.A.

Phone: +1 (765) 285-5126 (O), +1 (765) 284-4880 (H)
Internet: ykwon@bsu-cs.bsu.edu
Homepage: http://www.bsu.edu/cast/biomch/kwon/
--------------------------------------------------------------
************************************************** *******************************
(L9*X + L10*Y+ L11*Z + 1)*u = (L1*X + L2*Y + L3*Z + L4)

This is a LINEAR SYSTEM in the L1--L11 unknown.
The coefficients are Xu,Yu, Zyu X Y Z and 1.
The term on the right side is u.

N. Alberto Borghese

************************************************** *******************************
I used the methods as presented by J.S. Walton (1978)
Close range cine-photogrammetry: another approach to motion analysis.
Pages 69-97 in Terauds, J.S., editor, Sciences in Biomechanics
Cinematography, International Congress of Sports Sciences. Del Mar,
************************************************** **********************************

> F(a,b)=sum( (y(i)-(a*x(i)+b))^2 ) i=1,...n
> y(i),x(i) coordinates of various points
> a,b parameters of line y=a*x+b.
> Afterwards one takes partial derivate of F(a,b) with a and b in order to compute the parameters of line. One gets two equations with two unknowns and the rest is easy. What I do not understand is how it is applied in case of computing DLT parameters.
> Do I have to write the entire thing in the form of:
>
> F(a,b,L1,L2,...L11)=sum( (v(i)-(a*u(i)+b))^2 )
>
> i=1,...n calibration points
> u(i)=(L1*X(i) + L2*Y(i) + L3*Z(i) + L4) / (L9*X(i) + L10*Y(i)+ L11*Z(i) + 1)
> v(i)=(L5*X(i) + L6*Y(i) + L7*Z(i) + L8) / (L9*X(i) + L10*Y(i)+ L11*Z(i) + 1)?

Take the two DLT equations, and multiply left and right sides by
(L9*X(i) + L10*Y(i)+ L11*Z(i) + 1). Then you have linear equations.
Define F as:

F = sum( (leftside - rightside)^2 )

(sum over all equations, two equations (u,v) for each calibration point).

Fortunately, Matlab has a builtin linear least squares solver (say 'help slash' for
help). You simply collect the coefficients in a matrix, and use the
backslash operator. This was used in the kinemat functions.

> What is the matematical procedure when is MDLT (Modified DLT) concerned,
> rather than conventional DLT?
> I would be glad to post on the List all answers I recieve, once I collected them.

The MDLT has only 10 calibration parameters (not 11), since it assumes that the
u and v axis in image space are perpendicular. This is true for film and video.
With MDLT, you need at least 5 calibration points, but more is recommended
for reliability. Reference:

Chen, L., Armstrong, C.W., Raftopoulos, D.D., (1994) An investigation on the
accuracy of three-dimensional space reconstruction using the direct linear
transformation technique. J. Biomech. 27:493-500.
[Good paper on performance of the DLT using different sets of calibration
points. It is shown that significant improvement can be gained by
increasing the number of calibration points to 16. More than that
does not help much. Some discussion of nonlinearity and how to
correct for that.]

-- Ton van den Bogert
Department of Biomedical Engineering
Cleveland Clinic Foundation
************************************************** *****************************

I use the singular value decomposition (SVD) method to perform linear least
squares. the book Numerical Recipes in C provides C code to perform these
routines. There is also a Numerical Recipes in Pascal, Fortran, etc....

Once you solve the linear least squares procedure to determine the 11 DLT
parameters, you need to rewrite your equations to obtain XYZ in terms of
U,V, and the eleven parameters. In this way you can again use linear least
squares to find the optimum XYZ positions to fit the digitized UV coordinates.
************************************************** *****************************
Dear Tomislav:

As I told you earlier, here is the URL of my DLT page:

http://www.bsu.edu/cast/biomch/kwon/kwon3d/

It has some information on the DLT method and optical distortion-related
stuffs.

Sincerely,

Young-Hoo

--------------------------------------------------------------
Young-Hoo Kwon, Ph.D.
Assistant Professor
Biomechanics Lab, PL 202
Ball State University
Muncie, IN 47306
U.S.A.

Phone: +1 (765) 285-5126 (O), +1 (765) 284-4880 (H)
Internet: ykwon@bsu-cs.bsu.edu
Homepage: http://www.bsu.edu/cast/biomch/kwon/
--------------------------------------------------------------
************************************************** *****************************

Tomislav Pribanic
Faculty of Electrical Engineering and Computing
University of Zagreb
10 000 Zagreb
CROATIA
E-mail:Tomislav.pribanic@zesoi.fer.hr