Herman J. Woltring

08-14-1991, 06:25 AM

Dear Biomch-L readers,

One of the peculiarities of electronic publication is the simplicity of

revision and modification. While this entails the risk of posterior

censorship and premature publication, it is a great advantage that useful

modifications in `objective' material can easily be implemented.

Following last week's posting on the constrained DLT, some modifiations

have been studied and implemented in the published DLTDSP routines .

Since the constraints are nonlinear, the routines are based on iterative

linearization and adjustment calculus until convergence, and finding

suitable initial estimates is rather critical if the control point

distribution is shallow (i.e., relatively small in one dimension) or

small in all three dimensions (i.e., the camera is relatively far away

/ the solid angle subtended by the control object on the camera is

rather small).

In this case, it has been observed that initial estimates can reliably

be found by the (incorrect) assumption that perspective imaging is an

equiform transformation [1] (i.e. a shape-invariant transformation

described by a position vector, an attitude matrix, and a scaling factor)

between the object coordinates {XPi,YPi,ZPi, i=1,n} and the corresponding

image co-ordinates {XIi,YIi, i=1,n}, under the assumptions of *constant*

image Z-coordinates ZIi = C and given prior estimates of the camera's

principal point (Xo,Yo) and principal distance C. If the principal

distances (Cx,Cy) per image axis are strongly different, prior scaling

per image axis is advisable. Because of the erroneous model and of

perspective compensation between various camera parameters, the estimated

values will be different from the true values, but they will act as

appropriate, inital values for the constrained DLT.

In an old study [2], a general 3-D calibration procedure was described

without the need for a voluminous 3-D calibration object as required under

the `classical' DLT; instead, a rather large *plane* of calibration points

was found to be sufficient. At ISB 1989, a modified approach was reported

emphasizing that a very small control point distribution was, in principle,

sufficient [3]; thus, landmark clusters affixed to moving body segments

could be used not only for reconstructing 3-D kinematics but also for calib-

rating the camera configuration, *without* the need to know the positions

and attitudes of those clusters (except for one defined reference) since

these parameters are calibrated *with* the unknown camera parameters.

However, initialization of the nonlinear procedure proved quite critical.

>From some simulation studies, the present approach to obtain constrained

DLT parameters (with subsequent conversion to the equivalent, conventional

internal/external camera parameters) has been found to allow rather robust

and accurate camera calibration. Now, it is merely necessary to observe a

number of planar or spatial landmark clusters (whose local coordinates must

be accurately known), and to combine measured image data on these clusters

throughout the field of view with limited-accuracy prior estimates of the

cameras' internal parameters. The latter are improved during the final,

iterative procedure of [2], together with the external camera parameters

of position and attitude.

In principle, separate calibration program runs can now be completely

avoided, by viewing the unknown and/or prior estimated camera parameters

as *nuisance* parameters to be estimated and eliminated during the

3-D kinematics reconstruction process. However, this is numerically

inefficient, and it is more advisable to select a number of time samples

from a movement record which are used for estimating the camera parameters.

In summary, the careful and expensive machining and cumbersome manipulation

of voluminous 2-D or 3-D calibration objects can be avoided by appropriate

modelling approaches in a PC environment. Both for laboratory and field

studies, this is thought a considerable advantage. However, some additional

control (e.g., at least three non-collinear points at the edges of a force

plate or of the field of view) are recommended in order to ensure a stable,

global frame of reference.

Herman J. Woltring, Eindhoven/NL

References:

[1] Journal of Biomechanics 21(1), 45-54, 1988

[2] Journal of Biomechanics 13(1), 39-48, 1980

[3] Paper #197, Proc. ISB XII, UCLA 1989

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

send the following requests to LISTSERV@HEARN.BITNET or to LISTSERV@

NIC.SURFNET.NL from your Biomch-L subscription address:

send dltdsp readme biomch-l

send dltdsp fortran biomch-l

(the last word is optional, resulting in more efficient retrieval).

One of the peculiarities of electronic publication is the simplicity of

revision and modification. While this entails the risk of posterior

censorship and premature publication, it is a great advantage that useful

modifications in `objective' material can easily be implemented.

Following last week's posting on the constrained DLT, some modifiations

have been studied and implemented in the published DLTDSP routines .

Since the constraints are nonlinear, the routines are based on iterative

linearization and adjustment calculus until convergence, and finding

suitable initial estimates is rather critical if the control point

distribution is shallow (i.e., relatively small in one dimension) or

small in all three dimensions (i.e., the camera is relatively far away

/ the solid angle subtended by the control object on the camera is

rather small).

In this case, it has been observed that initial estimates can reliably

be found by the (incorrect) assumption that perspective imaging is an

equiform transformation [1] (i.e. a shape-invariant transformation

described by a position vector, an attitude matrix, and a scaling factor)

between the object coordinates {XPi,YPi,ZPi, i=1,n} and the corresponding

image co-ordinates {XIi,YIi, i=1,n}, under the assumptions of *constant*

image Z-coordinates ZIi = C and given prior estimates of the camera's

principal point (Xo,Yo) and principal distance C. If the principal

distances (Cx,Cy) per image axis are strongly different, prior scaling

per image axis is advisable. Because of the erroneous model and of

perspective compensation between various camera parameters, the estimated

values will be different from the true values, but they will act as

appropriate, inital values for the constrained DLT.

In an old study [2], a general 3-D calibration procedure was described

without the need for a voluminous 3-D calibration object as required under

the `classical' DLT; instead, a rather large *plane* of calibration points

was found to be sufficient. At ISB 1989, a modified approach was reported

emphasizing that a very small control point distribution was, in principle,

sufficient [3]; thus, landmark clusters affixed to moving body segments

could be used not only for reconstructing 3-D kinematics but also for calib-

rating the camera configuration, *without* the need to know the positions

and attitudes of those clusters (except for one defined reference) since

these parameters are calibrated *with* the unknown camera parameters.

However, initialization of the nonlinear procedure proved quite critical.

>From some simulation studies, the present approach to obtain constrained

DLT parameters (with subsequent conversion to the equivalent, conventional

internal/external camera parameters) has been found to allow rather robust

and accurate camera calibration. Now, it is merely necessary to observe a

number of planar or spatial landmark clusters (whose local coordinates must

be accurately known), and to combine measured image data on these clusters

throughout the field of view with limited-accuracy prior estimates of the

cameras' internal parameters. The latter are improved during the final,

iterative procedure of [2], together with the external camera parameters

of position and attitude.

In principle, separate calibration program runs can now be completely

avoided, by viewing the unknown and/or prior estimated camera parameters

as *nuisance* parameters to be estimated and eliminated during the

3-D kinematics reconstruction process. However, this is numerically

inefficient, and it is more advisable to select a number of time samples

from a movement record which are used for estimating the camera parameters.

In summary, the careful and expensive machining and cumbersome manipulation

of voluminous 2-D or 3-D calibration objects can be avoided by appropriate

modelling approaches in a PC environment. Both for laboratory and field

studies, this is thought a considerable advantage. However, some additional

control (e.g., at least three non-collinear points at the edges of a force

plate or of the field of view) are recommended in order to ensure a stable,

global frame of reference.

Herman J. Woltring, Eindhoven/NL

References:

[1] Journal of Biomechanics 21(1), 45-54, 1988

[2] Journal of Biomechanics 13(1), 39-48, 1980

[3] Paper #197, Proc. ISB XII, UCLA 1989

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

send the following requests to LISTSERV@HEARN.BITNET or to LISTSERV@

NIC.SURFNET.NL from your Biomch-L subscription address:

send dltdsp readme biomch-l

send dltdsp fortran biomch-l

(the last word is optional, resulting in more efficient retrieval).