Video measurement of linear displacement along an oblique line using the cross-ratio.

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VIDEO MEASUREMENT OF LINEAR DISPLACEMENT ALONG AN OBLIQUE
LINE USING THE CROSS-RATIO
Mark Colpus 1 and Mark Goss-Sampson 1
Centre for Sports Science and Human Performance, University of Greenwich,
United Kingdom1
The purpose of this study was to rediscover the cross-ratio and assess its effectiveness for
measuring linear displacement when the image plane is not parallel to the object plane. In
the laboratory a fixed, 4m object length was reconstructed with a mean absolute error of
2.6mm (s.d. = 1.6mm, maximum = 4.9mm). In the field, two cameras filmed a fixed, 8m
object length with a mean absolute error of 13mm (s.d. = 5mm, maximum = 20mm). The
method is very accessible to non-specialists in projective geometry and the results are both
valid and reliable.
KEY WORDS: cross-ratio, camera, calibration, perspective.
INTRODUCTION: Ongoing technological development and increased use of video has meant
that human movement is being filmed in more environments than ever before. Whilst
qualitative feedback can be made available immediately and viewed globally, quantitative
feedback still has to be processed and confirmed to be reliable and valid. Ease of use and
affordability have secured the dominance of non-metric, digital cameras and their employment
by a wider, non-specialist user group, exacerbating the need to evidence quality of data.
Accessible techniques have not evolved accordingly.
Video data recorded on an oblique plane, such as position on a running track or football pitch,
has to be processed through the use of advanced, linear transformations within a Euclidean
space. Analysing the data in its original, projective space enables the exploitation of projective
invariant properties such as the cross-ratio. To date there appears to be no published data on
the use of cross-ratio in sports biomechanics.
This paper investigates the cross-ratio and its suitability as an accessible technique for the
digital age. Milne (1911) stated that Lazare Carnot wrote about the cross-ratio in 1803 but its
concepts can be traced back to Pappus of Alexandria (c.290 – c.350 AD). The cross-ratio
measures the projective invariance of four collinear points in Euclidean Space and directly links
the object and image space, Figure 1.
Figure 1: Projective invariance of the cross-ratio.
The cross-ratio can be defined by the relationships below:
Cross-ratio (A’B’C’D’) =
B'C'
C'A'
÷
B'D'
D'A'
=
B'C'
C'A'
×
D'A'
B'D'
=
CB
AC
×
AD
DB
= Cross-ratio (ABCD)
Object Line
Centre of
Projection, O
Image Plane
A
B
C
D
A’
B’
C’
D’

If the points A, B and C are fixed then the location of D at any other point on the line returns a
unique value for the cross-ratio (ABCD), Figure 2. Measurement of an image cross-ratio
(ABCD) and knowledge of the object points A’, B’, and C’ enables the direct calculation of the
object point D’. The points A, B, C, and D can be any order.
Figure 2: Unique variation of cross-ratio (ABCD) for fixed point A (centre, x = 50), point B (left, x
= 10), point C (right, x = 90) and variable point D (range, 0 to 100).
For planar movement the method of similar triangles means that the image:object scale is
constant, but only if the image plane is parallel to the object plane, Figure 3. For non-parallel
planes, scaling can be achieved through a 3 × 3 linear transformation but there is no accepted
methodology for the comparatively simple task of linear tracking, a task for which the cross-
ratio would appear perfectly suited.
Figure 3: Method of similar triangles ensures constant scaling only when the image and object
lines are parallel.
This study aims to introduce the cross-ratio to sports biomechanics and validate it as a simple
measurement tool suitable for modern, digital media. After the viability of the cross-ratio was
confirmed in the laboratory by reconstructing a 5m calibrated object line, its wider, practical
application was then investigated using two camera positions to film a 30m line outside.
METHODS: In the laboratory a 5m line was secured to the floor and calibrated at 0.65m, 2.65m
and 4.65m. Markers were placed at 0.1m intervals and filmed at an oblique angle using a
Panasonic HX-WA30 digital camcorder, recording in 1080-30p mode.
Outside, a 30m line was calibrated at 1.5m, 10.5m, 19.5m and 28.5m with markers placed at
1.0m intervals. The line was part filmed at either end using the same HX-WA30 camcorder
from distances of about 50m.
Using a Windows 7 computer the videos were replayed and digitised through a html5 canvas
object displayed on a modern web browser. Image resolution was 1920 × 1080 pixels. The
A = 50
B = 10
C = 90
Optical
Axis
Centre
of Projection
Image line
Parallel
object line
Oblique
object line

image co-ordinates were saved in an Excel spreadsheet and a separate cross-ratio calculated
for each marker. Combining the object calibration points with the image cross-ratios enabled
the recalculation of the original marker positions.
RESULTS: In the 5m reconstruction mean absolute error was 2.6mm (s.d. = 1.6mm,
maximum = 4.9mm). Mean error was slightly higher for the far half of the line (mean =
3.3mm, s.d. = 1.4mm) compared to the near half (mean = 1.9mm, s.d. = 1.5mm). It can be
observed from Figure 4 that the error in the far half tended to be negative whilst the error in
the near half tended to positive.
Figure 4: Variation of calculation error along the 5m calibration line (all measurements in mm).
The 30m reconstruction consisted of a near camera (2m to 19m, mean = 73mm, s.d. = 31mm)
and a far camera (11m to 28m, mean = 83mm, s.d. = 38mm). The calculation errors in Figure
5 show the same positive/negative trends about the mid-point as Figure 4. The mid-section
(11m to 19m) was recorded by both cameras and the systemic errors oppose each other,
giving a mean, absolute error of 13mm (s.d. = 5mm, maximum = 20mm).
Figure 5: Variation of calculation error along the 30m calibration line (all measurements in m).
A = 2650mm
B = 650mm
C = 4650mm
Near
camera
Far
camera

DISCUSSION: The 5m cross-ratio was accurate between the calibration points from 0.65m to
4.65m. The generally positive errors in the near half and generally negative errors in the far
half would be consistent with the displacements from the midpoint being measured in opposite
directions.
Validity was limited by the resolution of the image, causing digitised points to be quantised to
the nearest pixel, paradoxically forcing high reliability. The local maxima and minima observed
on the graph are a direct result of the quantised data. A systemic quantisation error measured
in one direction only would manifest itself as a shortened displacement in the near half and
extended displacement in the far half. This quantisation error could be reduced by using a
higher definition image and/or using a digitiser with sub-pixel resolution.
As the object points move further from the image then the image:object scale decreases and
the relative size of the quantisation errors would increase. This was observed with the slightly
higher mean error observed in the far half. Unlike many linear techniques in two-dimensional
analysis the cross-ratio clearly demonstrates its sensitivity to the changing distance between
the object line and image plane.
Though the individual results for the 30m calibration had the same characteristics as the 5m
line the errors were larger even though the curves were smoother because of better picture
resolution. Single camera results appear to be good enough for small, lab-based
displacements, e.g. single steps and jumps. For longer distances it appears the cross-ratio is
less accurate but a second, shifted cross-ratio can cancel systematic error to give results
dependent on image quality only. This would be the preferred method for displacement
calculations within large, external environments such as running tracks and football pitches.
CONCLUSION: This study shows that the cross-ratio enables a valid object line calculation
from an oblique image plane. Unsurprisingly, the quality of the results is dependent upon the
variability of the image:object scale, resolution of the image and digitising process.
Simple, laboratory displacement measures such as step length can be measured with a single,
non-metric, digital camcorder. Larger, field-based displacements are best measured using two
cameras to generate opposing cross-ratios that produce valid and reliable results.
Mathematically the calculation is simple, accessible and requires little specialist knowledge or
software.
REFERENCES:
Brannan, D. A, Esplen, M. F. & Gray, J. (2012). Geometry. (2nd ed.). Cambridge: Cambridge
University Press.
Milne, J.J. (1911). An Elementary Treatise on Cross-Ratio Geometry with Historical Notes. Cambridge:
Cambridge University Press.