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A panning DLT procedure for three-dimensional videography
OX-9290/93 $6.00+.00 i” 1993 Pergamon PressLtd
J. Biomechantcs Vol. 26, No 6. pp 741 751. 1993. Prmkd tn Great Britain
TECHNICAL
A PANNING
NOTE
DLT PROCEDURE FOR THREE-DIMENSIONAL VIDEOGRAPHY
BING Yu,*
TIMOTHY, J.
KoHt and JAMES G. HAY*
*Biomechanics Laboratory, Department of Exercise Science, University of Iowa, Iowa City, Iowa, U.S.A; tBiomechanics Program, Department of Biomedical Engineering and Applied Therapeutics, The Cleveland Clinic Foundation, Cleveland, Ohio, U.S.A.
Abstract-The direct linear transformation (DLT) method [Abdel-Aziz and Karara, APS Sympdsium on Photogrammetry. American Society of Photogrammetry, Falls Church, VA (1971)] is widely used in biomechanics to obtain three-dimensional space coordinates from film and video records. This method has some major shortcomings when used to analyze events which take place over large areas. To overcome these shortcomings, a three-dimensional data collection method based on the DLT method, and making use of panning cameras, was developed. Several small single control volumes were combined to construct a large total control volume. For each single control volume, a regression equation (calibration equation) is developed to express each of the 11 DLT parameters as a function of camera orientation, so that the DLT parameters can then be estimated from arbitrary camera orientations. Once the DLT parameters are known for at least two cameras, and the associated two-dimensional film or video coordinates of the event are obtained, the desired three-dimensional space coordinates can be computed. In a laboratory test, five single control volumes (in a total control volume of 24.40 x 2.44 x 2.44 m’) were used to test the effect of the position of the single control volume on the accuracy of the computed three dimensional space coordinates. Linear and quadratic calibration equations were used to test the effect of the order of the equation on the accuracy of the computed three dimensional space coordinates. For four of the five single control volumes tested, the mean resultant errors associated with the use of the linear calibration equation were significantly larger than those associated with the use of the quadratic calibration equation. The position of the single control volume had no significant effect on the mean resultant errors in computed three dimensional coordinates when the quadratic calibration equation was used. Under the same data collection conditions, the mean resultant errors in the computed three dimensional coordinates associated with the panning and stationary DLT methods were 17 and 22 mm, respectively. The major advantages of the panning DLT method lie in the large image sizes obtained and in the ease with which the data can be collected. The method also has potential for use in a wide variety of contexts. The major shortcoming of the method is the large amount of digitizing necessary to calibrate the total control volume. Adaptations of the method to reduce the amount of digitizing required are being explored.
image size and digitizing accuracy decrease as the size of the control volume increases. Finally, there is the so-called extrapolation error associated with analyzing an event which is not wholly confined to the control volume. The accuracy of the computed three-dimensional space coordinates may be poor for those parts of an event that take place outside the control volume (Wood and Marshall, 1986). To solve some of these problems for analyses of high jumping and ski jumping, respectively, Dapena (1978) developed a three-dimensional filming method using horizontally panning cameras, and Yeadon (1989) developed a method using panning and tilting cameras. Panning (and/or tilting) cameras permit a larger image size to be recorded than stationary cameras. Both of these methods required the use of a large number of accurately measured control points placed in the field of view of each camera. These control points remained in place during the event. Although reasonable accuracy was obtained with these methods, placing and measuring the necessary control points are very time-consuming, and keeping them in place during the event would not be convenient, or even possible, in many situations. The purpose of this paper is to describe a method of threedimensional videography using panning cameras for the analysis of human motion which does not require on-site
INTRODUCTION The direct linear transformation (DLT) method of AbdelAziz and Karara (1971) is widelv used in biomechanics to obtain three-dimensional space coordinates from film and video records. The use of this method in biomechanics requires, typically, at least two arbitrarily placed, stationary cameras to record the event of interest, and a control object consisting of at least six control points whose three-dimensional coordinates have been measured accurately relative to an arbitrary reference frame. The event and the position of the control object are recorded using the same camera positions and lens settings, with the position of the control object recorded before or after the event. For best results, the event should occur within the volume defined by the control object (the control volume). The DLT method has some major shortcomings when used to analyze events which take place over large areas. There are problems associated with constructing and transporting a control object large enough to match the volume in which the wide-ranging events take place. In addition, the First received 6 August 1991; accepted 8 December 1992. 741
142
measurements positions.
Technical to be taken for the determination
Note
of camera
METHODS
The use of the DLT procedure for three-dimensional analyses of human motion requires the calibration of the control volume using a two-step procedure: (1) the twodimensional film or video coordinates of each control point are obtained by digitizing the record of the control object from each camera; (2) 11 DLT parameters are then determined for each camera from these digitized coordinates and their associated known three-dimensional coordinates. The DLT (or calibration) narameters define the uosition and orientation of the camera and the characteristics of the lens. They also correct for linear components of systematic errors in the digitized film or video coordinates associated with the lens. After this calibration procedure, the record of the event from each camera is digitized and the resulting two-dimensional film or video coordinates are used with the DLT parameters to determine the desired three-dimensional space coordinates. The 11 DLT parameters for a given camera define, among other things, the position and orientation of the camera. As a camera is panned by rotating about a single fixed axis, the camera’s orientation and the 11 DLT parameters of the camera change. Assuming that the DLT parameters at several camera orientations are known, and that the DLT parameters change smoothly with changes in orientation, equations can be developed to express each of the DLT parameters as a function of camera orientation. Once these equations are defined, the DLT parameters can be estimated for any arbitrary camera orientation which lies within the bounds of the known orientations. Once the DLT parameters are estimated for at least two cameras, and the associated two-dimensional film or video coordinates of the event are obtained, the desired three-dimensional space coordinates can be computed. The orientation of a camera can be determined in many ways. In the tests described in this paper, the orientation was determined with the aid of a series of alternating black and white taped marks placed in a straight line on the ground, in the direction of the subject motion. The length of each mark was 0.619+0.005 m. At least two of these marks were within the field of view of each camera throughout the range of the panning motion used. The procedure* for determining the orientation of a given camera for a given frame of the film or video record consisted of the followmg three steps: (1) The intersection point (IP) of the vertical central line of the given frame and the line joining two of the taped marks (M,-and M2) was determined (Fig: 1). (21 The distance (in diaitizer units) between IP and one of the taped marks (the local reference) was determined, and scaled to real-life units using the known distance between the marks and simple proportion. (3) This distance was added to the known distance (in reallife units) between the local reference mark and a global reference mark. Thus, the camera orientation was measured as the distance from an arbitrarily defined global reference point, and referred to as the panning distance. The nature of the event to be analyzed determines the size of the control volume required. When a large control volume
*This computation of the panning distance contains some systematic error because the computed length of M,-IP is not the actual length of this line segment but rather its projection on a plane parallel to the plane of the videotape. This error is small when the camera is placed at a large perpendicular distance from the taped marks and (as will be shown) is compensated for in a later step of the procedure.
Fig. 1. The intersection point (IP) of (a) the vertical central line of a given frame from the video record of a given camera and(b) the line joining two of the taped marks (M, and M,).
is required, there are numerous problems with the construction, transport and use of an appropriately large control object. One alternative is to use a small control object and place it systematically in one position after another until the total control volume has been covered. The control object used in this study (Fig. 2) had 68 control points (37.5 mm diameter) distributed throughout a 2.44 m diameter sphere. The locations of these control points relative to an arbitrary fixed reference frame were measured to within 1 mm using surveying techniques. For an event that took place in a 12 m (in the direction of the panning motion of the cameras) x 2 m x 2 m volume, the control object was placed and recorded in five consecutive positions along the direction of the panning motion of the cameras. The method does not require that the control object be placed in a precise location or orientation when it is moved from place to place. In practice, it is simply picked up or rolled to the approximate location required. (The space covered by the control object in each of the five positions is called a single control uolume, and the space covered by the combination of the five single control volumes is called $e total control u&me.) The event of interest was then performed almost entirely within the total control volume. The following steps were taken to calibrate the total control volume and, thus, obtain the equations defining each DLT parameter as a function of panning distance for each single control volume and camera: (1) The control object was placed in each of the five consecutive positions and recorded while the cameras were panned. (2) The two-dimensional coordinates of some 54-68 points on the control object were obtained by digitizing a number of frames spaced through the panning motion of each camera. (The coordinates of at least six control points are required in each frame.) The first and last frames digitized for each single control volume were the first and last frames in which the entire control object was visible (Fig. 3). The panning distances for these frames are called the beginning and ending panning distance, respectively. The distance from the beginning panning distance to the ending panning distance is called the panning range for the single control volume. (3) For each single control volume and camera, the digitized coordinates of each control point were smoothed using a least-squares polynomial regression equation of digitized coordinates versus panning distance. (4) These smoothed digitized coordinates were used to determine the 11 DLT parameters for each digitized frame.
Technical
Note
Fig. 2. The three-dimensional control object supported by a photographer’s tripod. Each of the 17 rot the control object has four white table tennis balls (control points) firmly affixed to it.
743
Technical
Panning
Direction
745
Note
r>
First Digitized Frame
Beginning Panning Distance
End Panning Distance Panning Fig. 3. Panning
distances
Range
I.-
for the first and last frames digitized for a single control corresponding panning range.
(5) Each DLT parameter was expressed as a function of panning distance using least-squares polynomial regression. The polynomial regression equations of the DLT parameters versus panning distance were called calibration equations. To determine the three-dimensional space coordinates for the anatomical landmarks of a human subject, the desired frames of the videotape of the motion from each camera were digitized and the panning distance determined for each of these frames. The digitized coordinates and the panning distances from each camera were then time-synchronized using a critical-event method described below. The DLT parameters for each time-synchronized frame of each camera were then determined from the panning distances and appropriate calibration equations. Finally, the three-dimensional space coordinates of the anatomical landmarks were determined for each time-synchronized frame from the digitized event coordinates and the DLT parameters.* A critical-event method was used to time-synchronize the records from different cameras. With this method, the instant at which some critical event occurred was estimated and
*TWO steps in this procedure warrant further comment. In the first (Step 5, above) known DLT parameters for a given camera are expressed as a function of the corresponding panning distance-a measure that contains some systematic error, as previously noted. In the second (described in the preceding paragraph), regression equations (determined in Step 5) are used to predict the unknown DLT parameters using a measured panning distance as the independent variable. This measured panning distance is affected by the same source of systematic error as the panning distance mentioned above. The repeated use of panning distances affected by the same source of systematic error is selfcompensating and has no inherent effect on the determination of the required DLT parameters.
volume,
and the
designated as time zero for each camera. For example, the instant of takeoff was estimated from the last frame m which the toe of the takeoff foot was seen to be in contact with the ground and the first frame in which this contact was seen to be broken. The time for any other frame recorded was computed from the known frame rate and the number of frames recorded between the instant of the critical event and the frame. Linear interpolation was used to compute timesynchronized two-dimensional body landmark coordinates and panning distance. The DLT parameters of the two cameras for each timesynchronized frame were determined from the panning distance using the calibration equations for the current single control volume. If the panning distances of both cameras fell within the panning ranges for the same single control volume, that volume was taken to be the current single control volume. If the panning distance for one camera lay within the panning range of one single control volume and the panning distance for the second camera lay within the panning range for an adjacent single control object, a procedure designed to minimize the possible error due to extrapolation beyond the limits of the panning range was used. In this procedure, one of the two single control volumes was arbitrarily assumed to be the current single control volume. The panning distance for the camera for which this was not the appropriate single control volume was compared with the panning range of the arbitrarily chosen volume established for that camera; and the distance that the panning distance fell outside this range was noted. The roles of the two single control volumes were then reversed (that is, the single control volume not previously assumed to be the current single control volume was assigned this status) and the process repeated. The arbitrary choice of single control volume that yielded the smallest distance between the panning distance and the panning range (i.e. the least extrapolation) was used as the current single control volume.
746
Technical
Note
After the DLT parameters were determined for both cameras, the three-dimensional coordinates of the points of interest were computed from the two-dimensional digitized coordinates and the DLT parameters, using the stationary DLT procedures described by Walton (1981). The panning DLT procedure described was tested under laboratory conditions before being used to collect three-
I
-_L
L
dimensional data in the field. The experimental setup for this test is shown in Figs 4 and 5. Because the setup was symmetric about the midline of the 24.40 m dimension of a 24.40 x 2.44 x 2.44 m3 total control volume (Fig. 4), only onehalf of this volume was subjected to validation. The results of this experiment are considered characteristic of the total control volume.
24.40
m
.
2.44 m T
Fig. 4. Overhead
view of the experimental setup for the laboratory test. Numbers sequential locations of the control object.
l-5
8’m Control Object
C.
Fig. 5. Side view of the experimental
setup for the laboratory
test.
indicate
the five
Technical
Two Panasonic AG450 S-VHS camcorders were used to record the control object at a sampling frequency of 60 Hz. The control object was placed in five consecutive positions in the 12.20 x 2.44 x 2.44 m3 volume, and recorded in each position while each camcorder was panned through a range such that the control object moved into, then out of, the field of view of the camcorder. The field of view of each camcorder was a minimum of twice the diameter of the control object. This guaranteed that at least one complete single control volume was within the field of view of each camcorder throughout its panning motion. A microcomputer with a color monitor, a S-VHS videocassette recorder with a color video monitor, and PEAK2D computer software (Peak Performance Technologies, Denver, CO.) were used to digitize the videotape records of the control object. For each ofthe five single control volumes, the 68 control points were digitized at nine panning orientations. These nine panning orientations were evenly distributed through the panning motion of each camcorder. For each camera and for each single control volume. the digitized coordinates of the control points (and the DLT parameters derived from them) were expressed as leastsquares polynomial functions of panning distance. Polynomials of order one and two were used to compare the effect that the order had on the accuracy of the method. To test the accuracy of the method, 68 control points were digitized at ten camera orientations for each camcorder and each single control volume. These ten orientations were evenly distributed throughout the panning motion of each camcorder, but were different from the nine camera orientations used to produce the equations for the DLT parameters. The digitized data and the appropriate calibration equations were then used to determine three-dimensional coordinates for each control point using the procedures described above. These computed coordinates were compared to the known, measured coordinates; the resultant error (the deviation of computed position from the known, measured position) for each control point was calculated, and the mean of the resultant errors for all control points for each panning position was determined. All these error estimates were computed using a set of panning DLT (PDLT) computer programs (Yu, 1990). A two-way analysis of variance was conducted to test the effects of the order of the polynomial regression equation
141
Note
used (first or second order) and the single control volume position (numbers l-5) on the accuracy of the method--or. more specifically, on the mean resultant errors in computed three-dimensional space coordinates. A level of P < 0.05 was chosen to indicate statistical significance in these and all subsequent tests conducted. By treating each frame recorded with a panning camera as if it had been recorded by a stationary camera, the stationary (nonpanning) DLT method was used to compute the coordinates of the control points at the nine panning orientations used to develop calibration equations for each of the five single control volumes. A two-way analysis of variance was conducted to test the differences between the coordinates obtained in this manner and those obtained using the panning DLT method with the better calibration equation (first or second order), and to test the effect of control volume position on these differences. After the laboratory test, the panning DLT method was used to collect three-dimensional coordinate data on competitors in the final of the women’s triple jump at the 1990 TAC (U.S. National) Championships. Two Panasonic AG450 S-VHS camcorders were used to record the subjects’ performances during the competition and the positions of the control object after the competition. The sampling frequency for both camcorders was 60Hz. The control object was placed in nine consecutive positions in a 22 x 2.44 x 2.44 m3 volume with the 22 m dimension parallel to the runway (Fig. 6). The total calibration volume encompassed the space in which the last two strides. and the subsequent hop. step, and jump occurred. A third Panasonic S-VHS camcorder was used to record the subjects’ approach runs with a sampling frequency of 60 Hz using the procedure described by Chow (1987). In this procedure, long tapes with alternating black and white sections are lain on both sides of the runway. For each support phase, the position of the toe of the subject’s support foot and two points on each of the tapes, where a black section and a white section meet, are digitized. The equations of lines joining the leftmost and the rightmost pairs of points are then equated and solved to yield the coordinates of their point of intersection. A further equation is written for a line joining the toe of the support foot and the perspective point and, using the known locations of the points on the tapes and simple proportion, the distance of the toe from the takeoff board is computed. The videotape
Triple Jump Runway Pit
Track
Stand
Camera
3 Camera
Fig. 6. Overhead
I view of the experimental
Camera
2
setup for the field test
748
Technical Note
records obtained in this manner were used in an analysis of each subject’s approach run (Hay and Koh, 1988) and for comparison with the data collected using the panning DLT method. The equipment and methods previously described were used to digitize the control object and the body landmarks. To obtain calibration equations, the control object was digitized at nine panning orientations for each of the nine single control volumes. The number of control points digitized varied from 54 to 68 depending upon the number of control points that could be seen clearly in a given single control volume. The order of the regression equation found to provide the most accurate data in the laboratory experiment was used to develop the required calibration equations. Eighteen trials by the top four jumpers were digitized. Twenty-one body landmarks and two taped marks were digitized in each of the selected frames. The PDLT computer programs were used to compute three-dimensional space coordinates of the body landmarks. Several variables were computed from the three-dimensional coordinates of the 21 body landmarks obtained with the panning DLT method. One of these was the distance from the toe of the support foot to the pit edge of the takeoff board (the so-called toe-board distance) during the support phase of the hop. The toe-board distances computed from the data obtained with the panning DLT method were compared to those computed from the data collected with the
third camcorder using the method described by Chow (1987) and Hay and Koh (1988). RESULTS
The results of the analysis of variance to test the effects of the order of the calibration equation and the position of the single control volume on the accuracy of the computed threedimensional coordinates showed that both the main effects and the two-way interaction effect were statistically significant (Table 1). Interpretation of these results was based on ttests and one-way analyses of variance (Tables 2-4) and on a plot of mean resultant errors in computed three-dimensional coordinates, with different orders of calibration equation and different single control volumes (Fig. 7). For the single control volumes l-4, the mean resultant errors associated with the use of the linear calibration equation were significantly larger than those associated with the use of the quadratic calibration equation (Table 2, Fig. 7). For the single control volume 5, the mean resultant error associated with use of the linear calibration equation was nat significantly different from that associated with the use of the quadratic calibration equation. The position of the single control volume had a significant effect on the mean resultant errors in the computed threedimensional coordinates when the linear calibration equa-
Table 1. Two-way analysis of variance for the effect of order of calibration equation (linear or quadratic) and position of control volume (l-5) for the panning DLT me&hod in the laboratory test Source
Sum of squares
DF
Mean square
F-ratio
2886.591 2407.672 1940.984 4952.568
1 4 4 90
2886.591 601.918 485.237 55.029
52.456* 10.938* 8.818*
Order Volume Order x volume Error
* P
Order of calibration equation
1
2
3
4
5
Linear Mean (mm) SD.
44.698 17.673
27.882 8.674
25.649 8.921
24.219 6.432
16.893 3.704
Quadratic Mean (mm) SD.
19.008 1.871
16.330 2.346
14.439 1.426
17.550 1.789
18.287 3.714
4.571’
4.065’
t-value
3.923*
3.159*
0.831
* P < 0.05. Table 3. One-way analysis of variance for the effect of the position of the control volume on the accuracy of the panning DLT method (linear calibration model) Source
Sum of squares
DF
Mean square
F-ratio
Volume Error
494.068 377.273
4 45
123.517 8.384
14.733*
* P < 0.05.
Technical
749
Note
Table 4. One-way analysis of variance for the effect of the position of the control volume on the accuracy of the panning DLT method (quadratic calibration model) Source
Sum of squares
DF
Mean square
F-ratio
Volume Error
11.159 423.592
4 45
2.790 9.413
0.296
* P < 0.05.
Table 5. Two-way
Source Method Volume Method x volume Error
analysis
of variance for comparison stationary DLT methods
of the panning
and the
Sum of squares
DF
Mean square
F-ratio
713.810 34.027 17.378 374.596
1 4 4 85
713.810 8.507 4.344 4.407
161.971; 1.930 0.986
-IJNEAR BQUATION m QUADRATIC EQUATION
Fig. 7. Mean resultant errors in computed three-dimensional space coordinates associated with use of different calibration equations in different control volumes. For single control volumes 1-4, the mean resultant errors for the linear calibration equation were significantly larger than those for the quadratic calibration equation (pO.O5).
tion was used, but not when the quadratic calibration was used (Tables 3 and 4, respectively). The results suggest, therefore, that the quadratic calibration equation is better overall than the linear calibration equation, and that the resultant errors in three-dimensional coordinates computed using the quadratic calibration equation are not sensitive to the position of the single control volume. The comparison of the stationary and panning DLT methods using the quadratic calibration equation (Table 5) revealed that, under the same data collection conditions, the panning DLT method is better than the stationary DLT method. The mean resultant errors in the computed threedimensional coordinates associated with the panning and stationary DLT methods were 17 and 22 mm, respectively. The lower resultant error associated with the panning
BN 26:6-I
method is likely to be due primarily, and perhaps exclusively, to the smoothing of the digitized coordinates of the control points. This smoothing presumably reduced the random errors associated with the digitizing process and, thus, improved the subsequent estimates of the DLT parameters. These errors are not able to be removed in a like manner when the stationary procedure is used, in this case there is only a single set of coordinates available to define each control point. The quadratic calibration equation was used for the field test. The mean resultant error in the computed threedimensional coordinates of the control points used for the total control volume was 11 mm. The mean difference and the mean absolute difference between the toe-board distances for the support phase of the hop computed using the panning DLT method and using the method of Chow (1987) was 6.1
Technical Note
750
f 19.7 mm and 13.9 + 15.0 mm, respectively. Neither of these differences was significantly greater than zero. The correlation between toe-board distances computed from the data collected using the two different methods was r=0.99 with r = 28.94 (Fig. 8).
In addition to the preceeding quantitative indications of the accuracy of the panning DLT method, wire-frame diagrams of the side, top, and front views of a triple jumper during the flight phase of the step (Fig. 9) provide a graphic qualitative indication. Considered together, these results
Y = 1.05 X + 0.08 r = 0.99
/ -3ok -30
/ I
I
I
I
I
I
-20
-10
0
10
20
30
TOE-BOARDDISTANCE---
AFTROACH
40
RUN ANALYSIS (cm)
Fig. 8. Comparison of toe-board distances computed from the panning DLT data and from approach run analysis data. The high correlation between the two sets of data and the close proximity of the line of best fit to the line of identity (Y = X) indicate that the panning DLT method yields valid results in this case.
Fig. 9. Three-dimensional wire-frame diagram of step flight phase of the triple jump (from top to bottom: side, overhead, and rear views). The body positions depicted indicate qualitatively the validity of the results obtainable with the panning DLT method.
Technical
suggest that the accuracy of the three-dimensional ate data collected with the panning DLT method able for many analyses of human motion.
coordinis accept-
DISCUSSION
The panning DLT procedure described here has been developed, validated and applied successfully in practice. Its major advantage, compared to other methods of obtaining three-dimensional data for human motions that take place over a wide range, lies in the ease with which the data can be collected. Large images of the subject can be obtained over the full range of the motion with as few as two cameras. Furthermore, the time required to set up and perform the initial calibration of the total volume of interest (typically about 45 min) is rather less than in other methods requiring the use of survey techniques. Although used by the authors only for analyses of the long and triple jumps in track and field, the procedure presented has potential for use in a wide variety of other contexts. It could be used, for example, in analyses of the techniques used in running, hurdling and javelin throwing in track and field, vaulting and tumbling in gymnastics, cross-country skiing, speed skating, starting in luge and bobsled, starting, stroking and turning in swimming, and base-running in baseball. The procedure has also some limitations. Chief among these is the large amount of digitizing necessary to calibrate each single control volume and, thus, the total control volume. For the control object of Fig. 2, and the nine single control volumes of the field test reported here, as many as 5508 control points (68 control points x 9 positions/single control volume x 9 single control volumes) might have to be digitized. for each camera. There are at least two ways in which this heavy digitizing load might be reduced. First, the number of control points per position of the control object could be reduced to a number closer to six (the minimum number required in the DLT procedure). Second, the control points could be coated with a reflective material and the digitizing of these points performed automatically, rather than manually. These two options are currently being explored. As developed and used to date, the panning of the cameras requires that the motion take place within a long, narrow volume. There is nothing inherent in the method, however, that would preclude its being extended to allow analyses of events that take place in volumes of a different shape. It is conceivable, for example, that the procedure might be extended to permit analyses of techniques used in court games like badminton, tennis, volleyball and squash, or during the entire course of a free exercise in gymnastics. Such an
751
Note
extension would likely require changes in camera locations, panning and tilting of the cameras, and ground makers lain in two orthogonal directions.
Acknowledgements-The authors express their appreciation to Peak Performance Technologies, Inc., and The Athletics Congress of the U.S.A. for their financial support, and to Dr Jesus Dapena, Indiana University, for the use of his graphics subroutines.
REFERENCES
Abdel-Aziz, Y. I. and Karara, H. M. (1971) Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry. In ASP Symposium on Close Range Photogrammetry. American Society of Photogrammetry, Falls Church, VA. Alem, N. M., Melvin, J. W. and Holstein, G. L. (1978) Biomechanics applications of direct linear transformation in close-range photogrammetry. In Proc. of the 6th New England Bioengineering Conference (Edited by Jason, D.), Kingston, Rhode Island, pp. 202-206. Pergamon Press, New York. Chow, J. (1987) Maximum speed of female high school runners. Int. J. Sport Biomechanics 3, 110-127. Dapena, J. (1978) Three-dimensional cinematography with horizontally panning cameras. Sciences et Motricite 1, 3-15. Hay, J. G. and Koh, T. J. (1988) Evaluating the approach in the horizontal jumps. Int. J. Sport Biomechanics 4, 372-392. Miller, N. R., Shapiro, R. and McLaughlin, T. M. (1980) A technique for obtaining spatial kinematic parameters of segments of biomechanical systems from cinematographic data. J. Biomechanics 13, 535-541. Shapiro, R. (1978) Direct linear transformation method for three-dimensional cinematography. Res. Q. 49, 197-205. Walton, J. S. (1981) Close-range tine-photogrammetry: a generalized technique for quantifying gross human motion. Ph.D. dissertation, Pennsylvania State University, Pennsylvania. Wood, G. A. and Marshall, R. N. (1986) The accuracy of DLT extrapolation in three-dimensional film analysis. J. Biomechanics 19, 781-785. Yeadon, M. R. (1989) A method for obtaining three-dimensional data on ski jumping using pan and tilt cameras. Int. J. Sport Biomechanics 5, 238-247. Yu, B. (1990) PDLT computer software. Unpublished computer programs, University of Iowa, Iowa.
J. Biomechantcs Vol. 26, No 6. pp 741 751. 1993. Prmkd tn Great Britain
TECHNICAL
A PANNING
NOTE
DLT PROCEDURE FOR THREE-DIMENSIONAL VIDEOGRAPHY
BING Yu,*
TIMOTHY, J.
KoHt and JAMES G. HAY*
*Biomechanics Laboratory, Department of Exercise Science, University of Iowa, Iowa City, Iowa, U.S.A; tBiomechanics Program, Department of Biomedical Engineering and Applied Therapeutics, The Cleveland Clinic Foundation, Cleveland, Ohio, U.S.A.
Abstract-The direct linear transformation (DLT) method [Abdel-Aziz and Karara, APS Sympdsium on Photogrammetry. American Society of Photogrammetry, Falls Church, VA (1971)] is widely used in biomechanics to obtain three-dimensional space coordinates from film and video records. This method has some major shortcomings when used to analyze events which take place over large areas. To overcome these shortcomings, a three-dimensional data collection method based on the DLT method, and making use of panning cameras, was developed. Several small single control volumes were combined to construct a large total control volume. For each single control volume, a regression equation (calibration equation) is developed to express each of the 11 DLT parameters as a function of camera orientation, so that the DLT parameters can then be estimated from arbitrary camera orientations. Once the DLT parameters are known for at least two cameras, and the associated two-dimensional film or video coordinates of the event are obtained, the desired three-dimensional space coordinates can be computed. In a laboratory test, five single control volumes (in a total control volume of 24.40 x 2.44 x 2.44 m’) were used to test the effect of the position of the single control volume on the accuracy of the computed three dimensional space coordinates. Linear and quadratic calibration equations were used to test the effect of the order of the equation on the accuracy of the computed three dimensional space coordinates. For four of the five single control volumes tested, the mean resultant errors associated with the use of the linear calibration equation were significantly larger than those associated with the use of the quadratic calibration equation. The position of the single control volume had no significant effect on the mean resultant errors in computed three dimensional coordinates when the quadratic calibration equation was used. Under the same data collection conditions, the mean resultant errors in the computed three dimensional coordinates associated with the panning and stationary DLT methods were 17 and 22 mm, respectively. The major advantages of the panning DLT method lie in the large image sizes obtained and in the ease with which the data can be collected. The method also has potential for use in a wide variety of contexts. The major shortcoming of the method is the large amount of digitizing necessary to calibrate the total control volume. Adaptations of the method to reduce the amount of digitizing required are being explored.
image size and digitizing accuracy decrease as the size of the control volume increases. Finally, there is the so-called extrapolation error associated with analyzing an event which is not wholly confined to the control volume. The accuracy of the computed three-dimensional space coordinates may be poor for those parts of an event that take place outside the control volume (Wood and Marshall, 1986). To solve some of these problems for analyses of high jumping and ski jumping, respectively, Dapena (1978) developed a three-dimensional filming method using horizontally panning cameras, and Yeadon (1989) developed a method using panning and tilting cameras. Panning (and/or tilting) cameras permit a larger image size to be recorded than stationary cameras. Both of these methods required the use of a large number of accurately measured control points placed in the field of view of each camera. These control points remained in place during the event. Although reasonable accuracy was obtained with these methods, placing and measuring the necessary control points are very time-consuming, and keeping them in place during the event would not be convenient, or even possible, in many situations. The purpose of this paper is to describe a method of threedimensional videography using panning cameras for the analysis of human motion which does not require on-site
INTRODUCTION The direct linear transformation (DLT) method of AbdelAziz and Karara (1971) is widelv used in biomechanics to obtain three-dimensional space coordinates from film and video records. The use of this method in biomechanics requires, typically, at least two arbitrarily placed, stationary cameras to record the event of interest, and a control object consisting of at least six control points whose three-dimensional coordinates have been measured accurately relative to an arbitrary reference frame. The event and the position of the control object are recorded using the same camera positions and lens settings, with the position of the control object recorded before or after the event. For best results, the event should occur within the volume defined by the control object (the control volume). The DLT method has some major shortcomings when used to analyze events which take place over large areas. There are problems associated with constructing and transporting a control object large enough to match the volume in which the wide-ranging events take place. In addition, the First received 6 August 1991; accepted 8 December 1992. 741
142
measurements positions.
Technical to be taken for the determination
Note
of camera
METHODS
The use of the DLT procedure for three-dimensional analyses of human motion requires the calibration of the control volume using a two-step procedure: (1) the twodimensional film or video coordinates of each control point are obtained by digitizing the record of the control object from each camera; (2) 11 DLT parameters are then determined for each camera from these digitized coordinates and their associated known three-dimensional coordinates. The DLT (or calibration) narameters define the uosition and orientation of the camera and the characteristics of the lens. They also correct for linear components of systematic errors in the digitized film or video coordinates associated with the lens. After this calibration procedure, the record of the event from each camera is digitized and the resulting two-dimensional film or video coordinates are used with the DLT parameters to determine the desired three-dimensional space coordinates. The 11 DLT parameters for a given camera define, among other things, the position and orientation of the camera. As a camera is panned by rotating about a single fixed axis, the camera’s orientation and the 11 DLT parameters of the camera change. Assuming that the DLT parameters at several camera orientations are known, and that the DLT parameters change smoothly with changes in orientation, equations can be developed to express each of the DLT parameters as a function of camera orientation. Once these equations are defined, the DLT parameters can be estimated for any arbitrary camera orientation which lies within the bounds of the known orientations. Once the DLT parameters are estimated for at least two cameras, and the associated two-dimensional film or video coordinates of the event are obtained, the desired three-dimensional space coordinates can be computed. The orientation of a camera can be determined in many ways. In the tests described in this paper, the orientation was determined with the aid of a series of alternating black and white taped marks placed in a straight line on the ground, in the direction of the subject motion. The length of each mark was 0.619+0.005 m. At least two of these marks were within the field of view of each camera throughout the range of the panning motion used. The procedure* for determining the orientation of a given camera for a given frame of the film or video record consisted of the followmg three steps: (1) The intersection point (IP) of the vertical central line of the given frame and the line joining two of the taped marks (M,-and M2) was determined (Fig: 1). (21 The distance (in diaitizer units) between IP and one of the taped marks (the local reference) was determined, and scaled to real-life units using the known distance between the marks and simple proportion. (3) This distance was added to the known distance (in reallife units) between the local reference mark and a global reference mark. Thus, the camera orientation was measured as the distance from an arbitrarily defined global reference point, and referred to as the panning distance. The nature of the event to be analyzed determines the size of the control volume required. When a large control volume
*This computation of the panning distance contains some systematic error because the computed length of M,-IP is not the actual length of this line segment but rather its projection on a plane parallel to the plane of the videotape. This error is small when the camera is placed at a large perpendicular distance from the taped marks and (as will be shown) is compensated for in a later step of the procedure.
Fig. 1. The intersection point (IP) of (a) the vertical central line of a given frame from the video record of a given camera and(b) the line joining two of the taped marks (M, and M,).
is required, there are numerous problems with the construction, transport and use of an appropriately large control object. One alternative is to use a small control object and place it systematically in one position after another until the total control volume has been covered. The control object used in this study (Fig. 2) had 68 control points (37.5 mm diameter) distributed throughout a 2.44 m diameter sphere. The locations of these control points relative to an arbitrary fixed reference frame were measured to within 1 mm using surveying techniques. For an event that took place in a 12 m (in the direction of the panning motion of the cameras) x 2 m x 2 m volume, the control object was placed and recorded in five consecutive positions along the direction of the panning motion of the cameras. The method does not require that the control object be placed in a precise location or orientation when it is moved from place to place. In practice, it is simply picked up or rolled to the approximate location required. (The space covered by the control object in each of the five positions is called a single control uolume, and the space covered by the combination of the five single control volumes is called $e total control u&me.) The event of interest was then performed almost entirely within the total control volume. The following steps were taken to calibrate the total control volume and, thus, obtain the equations defining each DLT parameter as a function of panning distance for each single control volume and camera: (1) The control object was placed in each of the five consecutive positions and recorded while the cameras were panned. (2) The two-dimensional coordinates of some 54-68 points on the control object were obtained by digitizing a number of frames spaced through the panning motion of each camera. (The coordinates of at least six control points are required in each frame.) The first and last frames digitized for each single control volume were the first and last frames in which the entire control object was visible (Fig. 3). The panning distances for these frames are called the beginning and ending panning distance, respectively. The distance from the beginning panning distance to the ending panning distance is called the panning range for the single control volume. (3) For each single control volume and camera, the digitized coordinates of each control point were smoothed using a least-squares polynomial regression equation of digitized coordinates versus panning distance. (4) These smoothed digitized coordinates were used to determine the 11 DLT parameters for each digitized frame.
Technical
Note
Fig. 2. The three-dimensional control object supported by a photographer’s tripod. Each of the 17 rot the control object has four white table tennis balls (control points) firmly affixed to it.
743
Technical
Panning
Direction
745
Note
r>
First Digitized Frame
Beginning Panning Distance
End Panning Distance Panning Fig. 3. Panning
distances
Range
I.-
for the first and last frames digitized for a single control corresponding panning range.
(5) Each DLT parameter was expressed as a function of panning distance using least-squares polynomial regression. The polynomial regression equations of the DLT parameters versus panning distance were called calibration equations. To determine the three-dimensional space coordinates for the anatomical landmarks of a human subject, the desired frames of the videotape of the motion from each camera were digitized and the panning distance determined for each of these frames. The digitized coordinates and the panning distances from each camera were then time-synchronized using a critical-event method described below. The DLT parameters for each time-synchronized frame of each camera were then determined from the panning distances and appropriate calibration equations. Finally, the three-dimensional space coordinates of the anatomical landmarks were determined for each time-synchronized frame from the digitized event coordinates and the DLT parameters.* A critical-event method was used to time-synchronize the records from different cameras. With this method, the instant at which some critical event occurred was estimated and
*TWO steps in this procedure warrant further comment. In the first (Step 5, above) known DLT parameters for a given camera are expressed as a function of the corresponding panning distance-a measure that contains some systematic error, as previously noted. In the second (described in the preceding paragraph), regression equations (determined in Step 5) are used to predict the unknown DLT parameters using a measured panning distance as the independent variable. This measured panning distance is affected by the same source of systematic error as the panning distance mentioned above. The repeated use of panning distances affected by the same source of systematic error is selfcompensating and has no inherent effect on the determination of the required DLT parameters.
volume,
and the
designated as time zero for each camera. For example, the instant of takeoff was estimated from the last frame m which the toe of the takeoff foot was seen to be in contact with the ground and the first frame in which this contact was seen to be broken. The time for any other frame recorded was computed from the known frame rate and the number of frames recorded between the instant of the critical event and the frame. Linear interpolation was used to compute timesynchronized two-dimensional body landmark coordinates and panning distance. The DLT parameters of the two cameras for each timesynchronized frame were determined from the panning distance using the calibration equations for the current single control volume. If the panning distances of both cameras fell within the panning ranges for the same single control volume, that volume was taken to be the current single control volume. If the panning distance for one camera lay within the panning range of one single control volume and the panning distance for the second camera lay within the panning range for an adjacent single control object, a procedure designed to minimize the possible error due to extrapolation beyond the limits of the panning range was used. In this procedure, one of the two single control volumes was arbitrarily assumed to be the current single control volume. The panning distance for the camera for which this was not the appropriate single control volume was compared with the panning range of the arbitrarily chosen volume established for that camera; and the distance that the panning distance fell outside this range was noted. The roles of the two single control volumes were then reversed (that is, the single control volume not previously assumed to be the current single control volume was assigned this status) and the process repeated. The arbitrary choice of single control volume that yielded the smallest distance between the panning distance and the panning range (i.e. the least extrapolation) was used as the current single control volume.
746
Technical
Note
After the DLT parameters were determined for both cameras, the three-dimensional coordinates of the points of interest were computed from the two-dimensional digitized coordinates and the DLT parameters, using the stationary DLT procedures described by Walton (1981). The panning DLT procedure described was tested under laboratory conditions before being used to collect three-
I
-_L
L
dimensional data in the field. The experimental setup for this test is shown in Figs 4 and 5. Because the setup was symmetric about the midline of the 24.40 m dimension of a 24.40 x 2.44 x 2.44 m3 total control volume (Fig. 4), only onehalf of this volume was subjected to validation. The results of this experiment are considered characteristic of the total control volume.
24.40
m
.
2.44 m T
Fig. 4. Overhead
view of the experimental setup for the laboratory test. Numbers sequential locations of the control object.
l-5
8’m Control Object
C.
Fig. 5. Side view of the experimental
setup for the laboratory
test.
indicate
the five
Technical
Two Panasonic AG450 S-VHS camcorders were used to record the control object at a sampling frequency of 60 Hz. The control object was placed in five consecutive positions in the 12.20 x 2.44 x 2.44 m3 volume, and recorded in each position while each camcorder was panned through a range such that the control object moved into, then out of, the field of view of the camcorder. The field of view of each camcorder was a minimum of twice the diameter of the control object. This guaranteed that at least one complete single control volume was within the field of view of each camcorder throughout its panning motion. A microcomputer with a color monitor, a S-VHS videocassette recorder with a color video monitor, and PEAK2D computer software (Peak Performance Technologies, Denver, CO.) were used to digitize the videotape records of the control object. For each ofthe five single control volumes, the 68 control points were digitized at nine panning orientations. These nine panning orientations were evenly distributed through the panning motion of each camcorder. For each camera and for each single control volume. the digitized coordinates of the control points (and the DLT parameters derived from them) were expressed as leastsquares polynomial functions of panning distance. Polynomials of order one and two were used to compare the effect that the order had on the accuracy of the method. To test the accuracy of the method, 68 control points were digitized at ten camera orientations for each camcorder and each single control volume. These ten orientations were evenly distributed throughout the panning motion of each camcorder, but were different from the nine camera orientations used to produce the equations for the DLT parameters. The digitized data and the appropriate calibration equations were then used to determine three-dimensional coordinates for each control point using the procedures described above. These computed coordinates were compared to the known, measured coordinates; the resultant error (the deviation of computed position from the known, measured position) for each control point was calculated, and the mean of the resultant errors for all control points for each panning position was determined. All these error estimates were computed using a set of panning DLT (PDLT) computer programs (Yu, 1990). A two-way analysis of variance was conducted to test the effects of the order of the polynomial regression equation
141
Note
used (first or second order) and the single control volume position (numbers l-5) on the accuracy of the method--or. more specifically, on the mean resultant errors in computed three-dimensional space coordinates. A level of P < 0.05 was chosen to indicate statistical significance in these and all subsequent tests conducted. By treating each frame recorded with a panning camera as if it had been recorded by a stationary camera, the stationary (nonpanning) DLT method was used to compute the coordinates of the control points at the nine panning orientations used to develop calibration equations for each of the five single control volumes. A two-way analysis of variance was conducted to test the differences between the coordinates obtained in this manner and those obtained using the panning DLT method with the better calibration equation (first or second order), and to test the effect of control volume position on these differences. After the laboratory test, the panning DLT method was used to collect three-dimensional coordinate data on competitors in the final of the women’s triple jump at the 1990 TAC (U.S. National) Championships. Two Panasonic AG450 S-VHS camcorders were used to record the subjects’ performances during the competition and the positions of the control object after the competition. The sampling frequency for both camcorders was 60Hz. The control object was placed in nine consecutive positions in a 22 x 2.44 x 2.44 m3 volume with the 22 m dimension parallel to the runway (Fig. 6). The total calibration volume encompassed the space in which the last two strides. and the subsequent hop. step, and jump occurred. A third Panasonic S-VHS camcorder was used to record the subjects’ approach runs with a sampling frequency of 60 Hz using the procedure described by Chow (1987). In this procedure, long tapes with alternating black and white sections are lain on both sides of the runway. For each support phase, the position of the toe of the subject’s support foot and two points on each of the tapes, where a black section and a white section meet, are digitized. The equations of lines joining the leftmost and the rightmost pairs of points are then equated and solved to yield the coordinates of their point of intersection. A further equation is written for a line joining the toe of the support foot and the perspective point and, using the known locations of the points on the tapes and simple proportion, the distance of the toe from the takeoff board is computed. The videotape
Triple Jump Runway Pit
Track
Stand
Camera
3 Camera
Fig. 6. Overhead
I view of the experimental
Camera
2
setup for the field test
748
Technical Note
records obtained in this manner were used in an analysis of each subject’s approach run (Hay and Koh, 1988) and for comparison with the data collected using the panning DLT method. The equipment and methods previously described were used to digitize the control object and the body landmarks. To obtain calibration equations, the control object was digitized at nine panning orientations for each of the nine single control volumes. The number of control points digitized varied from 54 to 68 depending upon the number of control points that could be seen clearly in a given single control volume. The order of the regression equation found to provide the most accurate data in the laboratory experiment was used to develop the required calibration equations. Eighteen trials by the top four jumpers were digitized. Twenty-one body landmarks and two taped marks were digitized in each of the selected frames. The PDLT computer programs were used to compute three-dimensional space coordinates of the body landmarks. Several variables were computed from the three-dimensional coordinates of the 21 body landmarks obtained with the panning DLT method. One of these was the distance from the toe of the support foot to the pit edge of the takeoff board (the so-called toe-board distance) during the support phase of the hop. The toe-board distances computed from the data obtained with the panning DLT method were compared to those computed from the data collected with the
third camcorder using the method described by Chow (1987) and Hay and Koh (1988). RESULTS
The results of the analysis of variance to test the effects of the order of the calibration equation and the position of the single control volume on the accuracy of the computed threedimensional coordinates showed that both the main effects and the two-way interaction effect were statistically significant (Table 1). Interpretation of these results was based on ttests and one-way analyses of variance (Tables 2-4) and on a plot of mean resultant errors in computed three-dimensional coordinates, with different orders of calibration equation and different single control volumes (Fig. 7). For the single control volumes l-4, the mean resultant errors associated with the use of the linear calibration equation were significantly larger than those associated with the use of the quadratic calibration equation (Table 2, Fig. 7). For the single control volume 5, the mean resultant error associated with use of the linear calibration equation was nat significantly different from that associated with the use of the quadratic calibration equation. The position of the single control volume had a significant effect on the mean resultant errors in the computed threedimensional coordinates when the linear calibration equa-
Table 1. Two-way analysis of variance for the effect of order of calibration equation (linear or quadratic) and position of control volume (l-5) for the panning DLT me&hod in the laboratory test Source
Sum of squares
DF
Mean square
F-ratio
2886.591 2407.672 1940.984 4952.568
1 4 4 90
2886.591 601.918 485.237 55.029
52.456* 10.938* 8.818*
Order Volume Order x volume Error
* P
Order of calibration equation
1
2
3
4
5
Linear Mean (mm) SD.
44.698 17.673
27.882 8.674
25.649 8.921
24.219 6.432
16.893 3.704
Quadratic Mean (mm) SD.
19.008 1.871
16.330 2.346
14.439 1.426
17.550 1.789
18.287 3.714
4.571’
4.065’
t-value
3.923*
3.159*
0.831
* P < 0.05. Table 3. One-way analysis of variance for the effect of the position of the control volume on the accuracy of the panning DLT method (linear calibration model) Source
Sum of squares
DF
Mean square
F-ratio
Volume Error
494.068 377.273
4 45
123.517 8.384
14.733*
* P < 0.05.
Technical
749
Note
Table 4. One-way analysis of variance for the effect of the position of the control volume on the accuracy of the panning DLT method (quadratic calibration model) Source
Sum of squares
DF
Mean square
F-ratio
Volume Error
11.159 423.592
4 45
2.790 9.413
0.296
* P < 0.05.
Table 5. Two-way
Source Method Volume Method x volume Error
analysis
of variance for comparison stationary DLT methods
of the panning
and the
Sum of squares
DF
Mean square
F-ratio
713.810 34.027 17.378 374.596
1 4 4 85
713.810 8.507 4.344 4.407
161.971; 1.930 0.986
-IJNEAR BQUATION m QUADRATIC EQUATION
Fig. 7. Mean resultant errors in computed three-dimensional space coordinates associated with use of different calibration equations in different control volumes. For single control volumes 1-4, the mean resultant errors for the linear calibration equation were significantly larger than those for the quadratic calibration equation (p
tion was used, but not when the quadratic calibration was used (Tables 3 and 4, respectively). The results suggest, therefore, that the quadratic calibration equation is better overall than the linear calibration equation, and that the resultant errors in three-dimensional coordinates computed using the quadratic calibration equation are not sensitive to the position of the single control volume. The comparison of the stationary and panning DLT methods using the quadratic calibration equation (Table 5) revealed that, under the same data collection conditions, the panning DLT method is better than the stationary DLT method. The mean resultant errors in the computed threedimensional coordinates associated with the panning and stationary DLT methods were 17 and 22 mm, respectively. The lower resultant error associated with the panning
BN 26:6-I
method is likely to be due primarily, and perhaps exclusively, to the smoothing of the digitized coordinates of the control points. This smoothing presumably reduced the random errors associated with the digitizing process and, thus, improved the subsequent estimates of the DLT parameters. These errors are not able to be removed in a like manner when the stationary procedure is used, in this case there is only a single set of coordinates available to define each control point. The quadratic calibration equation was used for the field test. The mean resultant error in the computed threedimensional coordinates of the control points used for the total control volume was 11 mm. The mean difference and the mean absolute difference between the toe-board distances for the support phase of the hop computed using the panning DLT method and using the method of Chow (1987) was 6.1
Technical Note
750
f 19.7 mm and 13.9 + 15.0 mm, respectively. Neither of these differences was significantly greater than zero. The correlation between toe-board distances computed from the data collected using the two different methods was r=0.99 with r = 28.94 (Fig. 8).
In addition to the preceeding quantitative indications of the accuracy of the panning DLT method, wire-frame diagrams of the side, top, and front views of a triple jumper during the flight phase of the step (Fig. 9) provide a graphic qualitative indication. Considered together, these results
Y = 1.05 X + 0.08 r = 0.99
/ -3ok -30
/ I
I
I
I
I
I
-20
-10
0
10
20
30
TOE-BOARDDISTANCE---
AFTROACH
40
RUN ANALYSIS (cm)
Fig. 8. Comparison of toe-board distances computed from the panning DLT data and from approach run analysis data. The high correlation between the two sets of data and the close proximity of the line of best fit to the line of identity (Y = X) indicate that the panning DLT method yields valid results in this case.
Fig. 9. Three-dimensional wire-frame diagram of step flight phase of the triple jump (from top to bottom: side, overhead, and rear views). The body positions depicted indicate qualitatively the validity of the results obtainable with the panning DLT method.
Technical
suggest that the accuracy of the three-dimensional ate data collected with the panning DLT method able for many analyses of human motion.
coordinis accept-
DISCUSSION
The panning DLT procedure described here has been developed, validated and applied successfully in practice. Its major advantage, compared to other methods of obtaining three-dimensional data for human motions that take place over a wide range, lies in the ease with which the data can be collected. Large images of the subject can be obtained over the full range of the motion with as few as two cameras. Furthermore, the time required to set up and perform the initial calibration of the total volume of interest (typically about 45 min) is rather less than in other methods requiring the use of survey techniques. Although used by the authors only for analyses of the long and triple jumps in track and field, the procedure presented has potential for use in a wide variety of other contexts. It could be used, for example, in analyses of the techniques used in running, hurdling and javelin throwing in track and field, vaulting and tumbling in gymnastics, cross-country skiing, speed skating, starting in luge and bobsled, starting, stroking and turning in swimming, and base-running in baseball. The procedure has also some limitations. Chief among these is the large amount of digitizing necessary to calibrate each single control volume and, thus, the total control volume. For the control object of Fig. 2, and the nine single control volumes of the field test reported here, as many as 5508 control points (68 control points x 9 positions/single control volume x 9 single control volumes) might have to be digitized. for each camera. There are at least two ways in which this heavy digitizing load might be reduced. First, the number of control points per position of the control object could be reduced to a number closer to six (the minimum number required in the DLT procedure). Second, the control points could be coated with a reflective material and the digitizing of these points performed automatically, rather than manually. These two options are currently being explored. As developed and used to date, the panning of the cameras requires that the motion take place within a long, narrow volume. There is nothing inherent in the method, however, that would preclude its being extended to allow analyses of events that take place in volumes of a different shape. It is conceivable, for example, that the procedure might be extended to permit analyses of techniques used in court games like badminton, tennis, volleyball and squash, or during the entire course of a free exercise in gymnastics. Such an
751
Note
extension would likely require changes in camera locations, panning and tilting of the cameras, and ground makers lain in two orthogonal directions.
Acknowledgements-The authors express their appreciation to Peak Performance Technologies, Inc., and The Athletics Congress of the U.S.A. for their financial support, and to Dr Jesus Dapena, Indiana University, for the use of his graphics subroutines.
REFERENCES
Abdel-Aziz, Y. I. and Karara, H. M. (1971) Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry. In ASP Symposium on Close Range Photogrammetry. American Society of Photogrammetry, Falls Church, VA. Alem, N. M., Melvin, J. W. and Holstein, G. L. (1978) Biomechanics applications of direct linear transformation in close-range photogrammetry. In Proc. of the 6th New England Bioengineering Conference (Edited by Jason, D.), Kingston, Rhode Island, pp. 202-206. Pergamon Press, New York. Chow, J. (1987) Maximum speed of female high school runners. Int. J. Sport Biomechanics 3, 110-127. Dapena, J. (1978) Three-dimensional cinematography with horizontally panning cameras. Sciences et Motricite 1, 3-15. Hay, J. G. and Koh, T. J. (1988) Evaluating the approach in the horizontal jumps. Int. J. Sport Biomechanics 4, 372-392. Miller, N. R., Shapiro, R. and McLaughlin, T. M. (1980) A technique for obtaining spatial kinematic parameters of segments of biomechanical systems from cinematographic data. J. Biomechanics 13, 535-541. Shapiro, R. (1978) Direct linear transformation method for three-dimensional cinematography. Res. Q. 49, 197-205. Walton, J. S. (1981) Close-range tine-photogrammetry: a generalized technique for quantifying gross human motion. Ph.D. dissertation, Pennsylvania State University, Pennsylvania. Wood, G. A. and Marshall, R. N. (1986) The accuracy of DLT extrapolation in three-dimensional film analysis. J. Biomechanics 19, 781-785. Yeadon, M. R. (1989) A method for obtaining three-dimensional data on ski jumping using pan and tilt cameras. Int. J. Sport Biomechanics 5, 238-247. Yu, B. (1990) PDLT computer software. Unpublished computer programs, University of Iowa, Iowa.