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The accuracy of DLT extrapolation in three-dimensional film analysis
1. &omrhmrcs
Vol.
Prmted I” Grwt
19. No. 9. pp 7WI -7115. I986
0021-929086
Bntam
f3.W
+ .I0
Pergamon Journals Ltd
TECHNICAL
NOTE
THE ACCURACY OF DLT EXTRAPOLATION IN THREE-DIMENSIONAL FILM ANALYSIS GRAEME A. WOOD and ROBERT N. MARSHALL Department of Human Movement Studies, The University of Western Australia, Nedlands. W.A. 6009, Australia Abstract-An analysis of errors arising from the Direct Linear Transformation (DLT) approach to threedimensional reconstructions from two-dimensional images has been undertaken, the principal factor studied being the number and distribution ofcontrol points usedin thecalibration procedure. Significantly increased error was found to beassociated with extrapolation to unknown points outside thecontrol point distribution space. Differencesin accuracy between two camera position set-ups and I I vs 12 DLT parameter solutions were also examined.
INTRODlJ
A common practice in biomechanical analysis is to perform threedimensional space reconstructions from twodimensional film images using the Direct Linear Transformation (DLT) method and a sturdy three-dimensional calibration structure. Whiie more rigorous approaches are available which have less reliance on control (Woltring, 1980; Dapena et al., 1982), the computational overheads of these methods are usually quite demanding. Several researchers have reported on the acceptable accuracy alforded by the DLT method (Shapiro, 1978; Alem et al., 1978; Miller et al.. 1980), an approach that has been adopted from analytical photogrammetry (Abdel-Azh and Karara, 1971; Marzan and Karara, 197%the latter containing a complete FORTRAN listing). In brief the procedure involves an initial filming (by two or more arbitrarily placed cameras) of a reference structure containing markers of known points in space. This structure is then removed and the subject filmed in the same object space with the same camera set-up. The twodimensional images of both the reference structure and subject are then digitized and the unknown threedimensional co-ordinates of each of the subject’s landmarks are determined by solving a set of basic equations, which, for any camera, take the form
Y+bY+sY
=
LSX + L,Y + L,Z + L, L,x+L,,Y+L,,z+
1
where x and y are the image co-ordinates of a point; X, Y and Z are the three-dimensional object space co-ordinates of that point; L,-LI , are DLT parameters which define the camera position and orientation, and correct for linear components of film deformation and lens distortion; 5x and
work both Putnam (1979) and Neal (1983) have shown camera positioning and orientation not to be a critical fdctor. For a converging set-up with two cameras the optimal angle of intersection is theoretically 90” (Gosh, 1979), although in practice, where it is customary toexpress thisangle in terins of a distance: base ratio (the ratio of perpendicular distanceifrom a mid-camera point to subje-ct:distance between cambras), ratios ranging from 1:3 to 2: 1 appear to be equally reliable (Putnam, 1979; Neal, 1983). Reconstruction errors are’typitally less than 5 mm (Shapiro, 1978; Putnam, 1979; Miller et al., 1980),and are reported to improve with the addition of a greater number of control points, and also with the intrbduction of an additional term in the solution (the soqalled twelfth parameter) to account for systematic error atising from symmetrical lens distortion (Putnam, 1979). While it is frequently acknowledged that as many cantrol points as possible should be utilized and that these should be well distributed throughout the object space (Fraser, 11982), the demands imposed by such a requirement are often daunting. Large calibration structures are difficult to construct, erect and stow, and digitization is a time-consuming task. The question therefore naturally a&s-what lass of precision is to be expected by not observing these principles? Shapiro (1978) reports little loss in measurement accuracy at the extremes of a photographic field which were not covered by control points, but hiserrorsof 2-4 % (in a 1 m field) would be considered by many to be unaozeptable. A further analysis of DLT errors has therefore been undertaken which specifically addresses the question of extrapolation etrors. Comparisons of two dilTerent camera set-ups and two DLT formulations were also undertaken. METHOD
A calibration structure was built that was sufficiently large to encompass one full stride of a sprinter. It was constructed in the shape of a wedge, and of 50 mm square aluminium tubing, with sides 3.5 m Ion& a height of 2.5 m and a base of 1.5 m, and welded together with numerous diagonal struts to ensure absolute rigidity (see Fig. 1).Forty-three control point markers were painted on one side of this structure, and these were surveyed using conventional close-range photogrammetric techniques with metric cameras to determine real space co-ordinates of each marker to an accuracy better than + 1 mm (mean error over X, Y and Z). The co-ordinate
781
782
Technical Note
reference system utilized, with its origin at the front, left, bottom comer is also depicted in Fig. 1. In order to minimize the usual errors due to the small image size and relative graininess of 16 mm film, the frame was photographed on Kodachrome 64 film using an Olympus OM-1 35 mm camera. Photographs weretaken from a 12 m distant position perpendicular to the long side of the frame, and from approximately 45” to the right and left of this position with the frame approximately in the central twothirds of each photograph. Film negatives were mounted in glass slide mounts, and projected through a Leitz 35 mm slide projector onto a digitizing tablet. Two pairs of camera positions were chosen for analysis, viz right and anter (RC); and right and left (RL), providing distana:basc ratios of 1: 1and 1:2 respectively. Each position pair (RC and RL) was digitixed independently three times (trials A, B and C). For each trial 30 control points were digitized three times each, and this data file (30CP) was subsequently ‘dissected’ to produce: (1)an eleven control point file (11 CP) which sampled the total calibration structun; (2) a seven control point file (7 CP) which again sampled the total structure; and (3)an eleven control point file (11 CPX) which consisted of points clustered around the central vertical axis of the calibration structure. This last file was designed to simulate the ‘Christmas tree’type ofconfiguration used by some researchers (e.g. Shapiro, 1978; Van Gheluwe, 1978; Miller et a!., 1980; Stokes, 1984), and was included to evaluate the accuracy of extrapolation. Thus, within each trial (A, B, C), the four control point configurations utilised the same database. Every trial was then analysed twice using the DLT programme; first using the standard eleven DLT parameter solution and then with an additional term in the solution to partially correct for nonlinear symmetrical lens distortion (Manan and Karara, 1975). The unknown points digitixed for each trial consisted of 36-40 of the points marked on the calibration structure. Thus, some of the control points used were subsequently redigitized as unknown points. The same unknown point data file was used with each of the 30 CP, 11 CP, 7 CPand 11 CPX control point files in each trial, and for both 11 and 12 DLT parameter runs. In addition, the same unknown point data file from the right photograph was used in both the RC and RL analyses. The difference between the calibrated value for each unknown point and its real value was determined for each X, Y and 2 co-ordinate in each trial, and an RMS error
calculated for each camera/control point/parameter combination. A three-way analysis of variance with replication (trials A, B and C) was also performed to examine the statistical significance of differences observed between the factors studied (camera position; control point configuration; number of DLT parameters used). RESULTS
Tables 1 and 2 present a summary of the results obtained. The RMS errors displayed in Table 1 suggested a superiority of the 11 DLT parameter solution and of the right-left camera set-up. These factors were found to produce significantly better results (P c 0.01) for X and Z co-ordinates when the data were subjected to an analysis of variana (see Table 2). A significant difference was also found between the four control point configurations for X and Z co-ordinates, but not for Y. Single degree of freedom contrasts between 30 CP and 11 CP, 11 CP and 7 CP, and 11 CP and 11 CPX revealed that *the extrapolation condition (11 CPX) was consistently inferior to all other control point configurations, producing errors 50-100% greater. Transformations based on the smallest number of control points (7 CP) did not produce significantly diRerent results from more detailed configurations. The superiority of the eleven parameter solution became more apparent with diminishing control, and is most likely due to over-parameter&ion, a situation analogous to the poor modelling achieved by fitting a high-order polynomial to few data points. The non-linearity of the twelve parameter model may also be a factor particularly when extrapolation is being attempted, as is suggested by the proportionately greater errors observed under the 11 CPX compared to the 11 CP condition. However, even under the most favourable conditions examined there was no apparent benefit in using the twelve parameter model,although this may have been due to the quality of the lenses used. The average RMS error for the best situation encountered in this study was 5.7 mm (RL, 30 CP). This value is consistent with errors reported by others using the DLT approach, although the larger film size used here and the finer grain provided an advantage over the more common 16 mm format. The more consistent results in the vertical axis are likely to be due to the added information available in this dimension
Table 1. RMS errors in three-dimensional co-ordinates parentheses)
30 CP
Camera position/co-ordinate X Rightentre
(11 DLT parameter values in
Control point configurations 1lCP 7CP
10.4 (10.2)
10.6 (10.2)
13.7 (10.3)
15.5 (13.1)
(z) 4.5 (4.3) 6.9 (7.1)
(I&
(::;
$65 (6::)
(z) 10.5 (7.2)
(73; 12.7 (9.5) 12.6 (10.2)
(44;:
(z)
(44;)
17.8 (13.5)
(64:) 6.4
(z) 6.6
(2)
(6.0)
6:’
‘“e&l
5.8 (5.7)
(6.1)
(5.7)
Y Z Mean X
Right-left
Y Z Mean
All values in mm.
11 CPX
(78:; 15.5 (12.2) 13.9 (10.9)
.,
--.._c___I-_
-___-__e,_ .._.
Y
z t
X
Fig. 1. Three-dimensional calibration structure showing control markers and co-ordinate referencesystem.
183
785
Technical Note Table 2. Analysis of variance summary table
Source of variation
d.f.
X
Mean square Y
2
X
F ratio Y
Z
15.9’ IO.18 1.9
0.5 0.4 0.3
8.4’ 35.3; 7.59 1.1 1.o 1.0 0.6
Camera positions Control configurations No. of parameters
1 3 1
237.0 149.8 29.0
4.6 3.7 2.7
33.2 139.3 29.7
Camera Camera Control Camera
3
39.1
2.1
4.2
2.6
0.2
: 3
0.0 8.3 6.1
0.0 1.2 2.1
4.0 2.6
0.0 0.6 0.4
0.0 0.1 0.3
x control x parameters x parameters x control x parameters
Residual
32
14.9
8.6
3.9
Total
47
28.8
6.6
13.7
lP < 0.01.
and to the smaller amount of radial lens distortion close to the image centre. CONCLUSIONS
(1) Significant inaccuracies in three-dimensional reconstructions are likely to occur if the target points lie outside the control point distribution space. If compromises must be made then fewer control points well distributed throughout the object space will produce better results than extrapolating. (2) On the basis of the reconstruction errors observed in this study, a camera set-up with a distance:base ratio of approximately I:2 will produce better results than that with a 1: 1 ratio, as too will a solution based on the 11 DLT parameters without correction for non-liner lens distortions when few control points are used. (3) Given the inaccuracy of DLT-extrapolation and the difficulties that can arise in the construction and preservation of a calibration structure such as that used in the present study, the cost-benefits of non-DLT approaches warrant investigation. Acknowledgement-Supported by a grant from the Sir Robert Metuies Foundation of Australia. REFERENCES
Abdel-Aziz, Y. I. and Karara, H. M. (1971) Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry. ASP Symposium on Close Range Photogrammetry. American Society of Photogrammetry, Falls Church. Alem, N. M., Melvin, J. W. and Holstein, G. L. (1978) Biomechanics applications of direct linear transformation in close-range photogrammetry. Proceedings ofSixth New England Bioengineering Conference, Kingston, Rhode
Island (Edited by Jason, D.), pp. 202-206. Pergamon Press, New York. Dapena, J., Hat-man, E. A. and Miller, J. A. (1982) Threedimensional cinematography with control object of unknown shape. J. Biomechanics 15, 1I-19. Fraser, C. S. (1982) On the use of nonmetric cameras in; analytical close range photogrammetry. Can. Surwyot 36, 259-279. Gosh, S. K. (1979) Analytical Phorogrammerry, Pergamon Press, New York. Manan, G. T. and Karara, H. M. (1975) A computer program for direct linear transformation solution of the collinearity condition, and some applications of it. Symposium on Close Range Photogranvnetric Systems pp. 420-476. American Society of Photogrammetry, Falls Church. Miller, N. R., Shapiro, R. and McLaughlin, T. M. (1980) A technique for obtaining spatial kinematic parametens of segments of biomechanical systems from cinematographic data. J. Biomechanics 13, j35-S47. Neal, R. (1983) Three dimensional analysis of the golf swing. Unpublished Master’s Thesis, University of Queensland, Brisbane, Australia. Putnam, C. (1979) DLT method of three dimensional einematography: Instruction manual. Unpublished report, University of Iowa, U.S.A. Shapiro, R. (1978) Direct linear transformation method for threedimensional cinematography. Res. Q. 49, 197-205. Stokes, V. P. (1984) A method for obtaining the 3D kinematics of the pelvis and thorax during locomotion. Plum. Mvmt Sci. 3, 77-94. Van Gheluwe, B. (1978) Computerized three dimensional cinematography for any arbitrary camera s&tup. Biomechunics VI-A (Edited by Asmussen, E. and Jorgenson, K.), pp. 343-348. University Park Press, Baltimore. Woltring, H. J. (1980) Planar control in multi-camera calibration for three-dimensional gait studies. J. Biomechanics 13, 39-48.
Vol.
Prmted I” Grwt
19. No. 9. pp 7WI -7115. I986
0021-929086
Bntam
f3.W
+ .I0
Pergamon Journals Ltd
TECHNICAL
NOTE
THE ACCURACY OF DLT EXTRAPOLATION IN THREE-DIMENSIONAL FILM ANALYSIS GRAEME A. WOOD and ROBERT N. MARSHALL Department of Human Movement Studies, The University of Western Australia, Nedlands. W.A. 6009, Australia Abstract-An analysis of errors arising from the Direct Linear Transformation (DLT) approach to threedimensional reconstructions from two-dimensional images has been undertaken, the principal factor studied being the number and distribution ofcontrol points usedin thecalibration procedure. Significantly increased error was found to beassociated with extrapolation to unknown points outside thecontrol point distribution space. Differencesin accuracy between two camera position set-ups and I I vs 12 DLT parameter solutions were also examined.
INTRODlJ
A common practice in biomechanical analysis is to perform threedimensional space reconstructions from twodimensional film images using the Direct Linear Transformation (DLT) method and a sturdy three-dimensional calibration structure. Whiie more rigorous approaches are available which have less reliance on control (Woltring, 1980; Dapena et al., 1982), the computational overheads of these methods are usually quite demanding. Several researchers have reported on the acceptable accuracy alforded by the DLT method (Shapiro, 1978; Alem et al., 1978; Miller et al.. 1980), an approach that has been adopted from analytical photogrammetry (Abdel-Azh and Karara, 1971; Marzan and Karara, 197%the latter containing a complete FORTRAN listing). In brief the procedure involves an initial filming (by two or more arbitrarily placed cameras) of a reference structure containing markers of known points in space. This structure is then removed and the subject filmed in the same object space with the same camera set-up. The twodimensional images of both the reference structure and subject are then digitized and the unknown threedimensional co-ordinates of each of the subject’s landmarks are determined by solving a set of basic equations, which, for any camera, take the form
Y+bY+sY
=
LSX + L,Y + L,Z + L, L,x+L,,Y+L,,z+
1
where x and y are the image co-ordinates of a point; X, Y and Z are the three-dimensional object space co-ordinates of that point; L,-LI , are DLT parameters which define the camera position and orientation, and correct for linear components of film deformation and lens distortion; 5x and
work both Putnam (1979) and Neal (1983) have shown camera positioning and orientation not to be a critical fdctor. For a converging set-up with two cameras the optimal angle of intersection is theoretically 90” (Gosh, 1979), although in practice, where it is customary toexpress thisangle in terins of a distance: base ratio (the ratio of perpendicular distanceifrom a mid-camera point to subje-ct:distance between cambras), ratios ranging from 1:3 to 2: 1 appear to be equally reliable (Putnam, 1979; Neal, 1983). Reconstruction errors are’typitally less than 5 mm (Shapiro, 1978; Putnam, 1979; Miller et al., 1980),and are reported to improve with the addition of a greater number of control points, and also with the intrbduction of an additional term in the solution (the soqalled twelfth parameter) to account for systematic error atising from symmetrical lens distortion (Putnam, 1979). While it is frequently acknowledged that as many cantrol points as possible should be utilized and that these should be well distributed throughout the object space (Fraser, 11982), the demands imposed by such a requirement are often daunting. Large calibration structures are difficult to construct, erect and stow, and digitization is a time-consuming task. The question therefore naturally a&s-what lass of precision is to be expected by not observing these principles? Shapiro (1978) reports little loss in measurement accuracy at the extremes of a photographic field which were not covered by control points, but hiserrorsof 2-4 % (in a 1 m field) would be considered by many to be unaozeptable. A further analysis of DLT errors has therefore been undertaken which specifically addresses the question of extrapolation etrors. Comparisons of two dilTerent camera set-ups and two DLT formulations were also undertaken. METHOD
A calibration structure was built that was sufficiently large to encompass one full stride of a sprinter. It was constructed in the shape of a wedge, and of 50 mm square aluminium tubing, with sides 3.5 m Ion& a height of 2.5 m and a base of 1.5 m, and welded together with numerous diagonal struts to ensure absolute rigidity (see Fig. 1).Forty-three control point markers were painted on one side of this structure, and these were surveyed using conventional close-range photogrammetric techniques with metric cameras to determine real space co-ordinates of each marker to an accuracy better than + 1 mm (mean error over X, Y and Z). The co-ordinate
781
782
Technical Note
reference system utilized, with its origin at the front, left, bottom comer is also depicted in Fig. 1. In order to minimize the usual errors due to the small image size and relative graininess of 16 mm film, the frame was photographed on Kodachrome 64 film using an Olympus OM-1 35 mm camera. Photographs weretaken from a 12 m distant position perpendicular to the long side of the frame, and from approximately 45” to the right and left of this position with the frame approximately in the central twothirds of each photograph. Film negatives were mounted in glass slide mounts, and projected through a Leitz 35 mm slide projector onto a digitizing tablet. Two pairs of camera positions were chosen for analysis, viz right and anter (RC); and right and left (RL), providing distana:basc ratios of 1: 1and 1:2 respectively. Each position pair (RC and RL) was digitixed independently three times (trials A, B and C). For each trial 30 control points were digitized three times each, and this data file (30CP) was subsequently ‘dissected’ to produce: (1)an eleven control point file (11 CP) which sampled the total calibration structun; (2) a seven control point file (7 CP) which again sampled the total structure; and (3)an eleven control point file (11 CPX) which consisted of points clustered around the central vertical axis of the calibration structure. This last file was designed to simulate the ‘Christmas tree’type ofconfiguration used by some researchers (e.g. Shapiro, 1978; Van Gheluwe, 1978; Miller et a!., 1980; Stokes, 1984), and was included to evaluate the accuracy of extrapolation. Thus, within each trial (A, B, C), the four control point configurations utilised the same database. Every trial was then analysed twice using the DLT programme; first using the standard eleven DLT parameter solution and then with an additional term in the solution to partially correct for nonlinear symmetrical lens distortion (Manan and Karara, 1975). The unknown points digitixed for each trial consisted of 36-40 of the points marked on the calibration structure. Thus, some of the control points used were subsequently redigitized as unknown points. The same unknown point data file was used with each of the 30 CP, 11 CP, 7 CPand 11 CPX control point files in each trial, and for both 11 and 12 DLT parameter runs. In addition, the same unknown point data file from the right photograph was used in both the RC and RL analyses. The difference between the calibrated value for each unknown point and its real value was determined for each X, Y and 2 co-ordinate in each trial, and an RMS error
calculated for each camera/control point/parameter combination. A three-way analysis of variance with replication (trials A, B and C) was also performed to examine the statistical significance of differences observed between the factors studied (camera position; control point configuration; number of DLT parameters used). RESULTS
Tables 1 and 2 present a summary of the results obtained. The RMS errors displayed in Table 1 suggested a superiority of the 11 DLT parameter solution and of the right-left camera set-up. These factors were found to produce significantly better results (P c 0.01) for X and Z co-ordinates when the data were subjected to an analysis of variana (see Table 2). A significant difference was also found between the four control point configurations for X and Z co-ordinates, but not for Y. Single degree of freedom contrasts between 30 CP and 11 CP, 11 CP and 7 CP, and 11 CP and 11 CPX revealed that *the extrapolation condition (11 CPX) was consistently inferior to all other control point configurations, producing errors 50-100% greater. Transformations based on the smallest number of control points (7 CP) did not produce significantly diRerent results from more detailed configurations. The superiority of the eleven parameter solution became more apparent with diminishing control, and is most likely due to over-parameter&ion, a situation analogous to the poor modelling achieved by fitting a high-order polynomial to few data points. The non-linearity of the twelve parameter model may also be a factor particularly when extrapolation is being attempted, as is suggested by the proportionately greater errors observed under the 11 CPX compared to the 11 CP condition. However, even under the most favourable conditions examined there was no apparent benefit in using the twelve parameter model,although this may have been due to the quality of the lenses used. The average RMS error for the best situation encountered in this study was 5.7 mm (RL, 30 CP). This value is consistent with errors reported by others using the DLT approach, although the larger film size used here and the finer grain provided an advantage over the more common 16 mm format. The more consistent results in the vertical axis are likely to be due to the added information available in this dimension
Table 1. RMS errors in three-dimensional co-ordinates parentheses)
30 CP
Camera position/co-ordinate X Rightentre
(11 DLT parameter values in
Control point configurations 1lCP 7CP
10.4 (10.2)
10.6 (10.2)
13.7 (10.3)
15.5 (13.1)
(z) 4.5 (4.3) 6.9 (7.1)
(I&
(::;
$65 (6::)
(z) 10.5 (7.2)
(73; 12.7 (9.5) 12.6 (10.2)
(44;:
(z)
(44;)
17.8 (13.5)
(64:) 6.4
(z) 6.6
(2)
(6.0)
6:’
‘“e&l
5.8 (5.7)
(6.1)
(5.7)
Y Z Mean X
Right-left
Y Z Mean
All values in mm.
11 CPX
(78:; 15.5 (12.2) 13.9 (10.9)
.,
--.._c___I-_
-___-__e,_ .._.
Y
z t
X
Fig. 1. Three-dimensional calibration structure showing control markers and co-ordinate referencesystem.
183
785
Technical Note Table 2. Analysis of variance summary table
Source of variation
d.f.
X
Mean square Y
2
X
F ratio Y
Z
15.9’ IO.18 1.9
0.5 0.4 0.3
8.4’ 35.3; 7.59 1.1 1.o 1.0 0.6
Camera positions Control configurations No. of parameters
1 3 1
237.0 149.8 29.0
4.6 3.7 2.7
33.2 139.3 29.7
Camera Camera Control Camera
3
39.1
2.1
4.2
2.6
0.2
: 3
0.0 8.3 6.1
0.0 1.2 2.1
4.0 2.6
0.0 0.6 0.4
0.0 0.1 0.3
x control x parameters x parameters x control x parameters
Residual
32
14.9
8.6
3.9
Total
47
28.8
6.6
13.7
lP < 0.01.
and to the smaller amount of radial lens distortion close to the image centre. CONCLUSIONS
(1) Significant inaccuracies in three-dimensional reconstructions are likely to occur if the target points lie outside the control point distribution space. If compromises must be made then fewer control points well distributed throughout the object space will produce better results than extrapolating. (2) On the basis of the reconstruction errors observed in this study, a camera set-up with a distance:base ratio of approximately I:2 will produce better results than that with a 1: 1 ratio, as too will a solution based on the 11 DLT parameters without correction for non-liner lens distortions when few control points are used. (3) Given the inaccuracy of DLT-extrapolation and the difficulties that can arise in the construction and preservation of a calibration structure such as that used in the present study, the cost-benefits of non-DLT approaches warrant investigation. Acknowledgement-Supported by a grant from the Sir Robert Metuies Foundation of Australia. REFERENCES
Abdel-Aziz, Y. I. and Karara, H. M. (1971) Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry. ASP Symposium on Close Range Photogrammetry. American Society of Photogrammetry, Falls Church. Alem, N. M., Melvin, J. W. and Holstein, G. L. (1978) Biomechanics applications of direct linear transformation in close-range photogrammetry. Proceedings ofSixth New England Bioengineering Conference, Kingston, Rhode
Island (Edited by Jason, D.), pp. 202-206. Pergamon Press, New York. Dapena, J., Hat-man, E. A. and Miller, J. A. (1982) Threedimensional cinematography with control object of unknown shape. J. Biomechanics 15, 1I-19. Fraser, C. S. (1982) On the use of nonmetric cameras in; analytical close range photogrammetry. Can. Surwyot 36, 259-279. Gosh, S. K. (1979) Analytical Phorogrammerry, Pergamon Press, New York. Manan, G. T. and Karara, H. M. (1975) A computer program for direct linear transformation solution of the collinearity condition, and some applications of it. Symposium on Close Range Photogranvnetric Systems pp. 420-476. American Society of Photogrammetry, Falls Church. Miller, N. R., Shapiro, R. and McLaughlin, T. M. (1980) A technique for obtaining spatial kinematic parametens of segments of biomechanical systems from cinematographic data. J. Biomechanics 13, j35-S47. Neal, R. (1983) Three dimensional analysis of the golf swing. Unpublished Master’s Thesis, University of Queensland, Brisbane, Australia. Putnam, C. (1979) DLT method of three dimensional einematography: Instruction manual. Unpublished report, University of Iowa, U.S.A. Shapiro, R. (1978) Direct linear transformation method for threedimensional cinematography. Res. Q. 49, 197-205. Stokes, V. P. (1984) A method for obtaining the 3D kinematics of the pelvis and thorax during locomotion. Plum. Mvmt Sci. 3, 77-94. Van Gheluwe, B. (1978) Computerized three dimensional cinematography for any arbitrary camera s&tup. Biomechunics VI-A (Edited by Asmussen, E. and Jorgenson, K.), pp. 343-348. University Park Press, Baltimore. Woltring, H. J. (1980) Planar control in multi-camera calibration for three-dimensional gait studies. J. Biomechanics 13, 39-48.