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Paolo De Leva
04-19-2008, 04:16 AM
Dear subscribers,

Thank you for your contributions. I privately received several interesting
replies to my original posting (March 31, 2008). Two subscribers wrote that,
in their opinion, the error component which is most difficult to deal with
is that caused by total marker-occlusion (point 4b below). I wholeheartedly
agree. According to the comments I received, I adjusted my list of error
sources and reference list, and discussed the changes with some
contributors. Here’s the final result.


INTRODUCTION

The phenomenon is assumed to be known, but actually it is quite complex and,
as far as I know, nobody described it in detail and thoroughly. Examples of
sources of error affecting simpler measurement instruments are given in
engineering textbooks. There’s no space in a scientific paper for an
in-depth analysis of a phenomenon which is assumed to be known. I attached
below a list of papers each of which briefly summarizes the sources of
stereophotogrammetric error. I will give you my analysis in the next
paragraphs. The information gathered by Chiari et al. in their detailed
review on Gait & Posture (2005) was an useful starting point.

IMPORTANT NOTE: This posting is not about the “soft tissue artifact”, which
is an error in the estimated position of anatomical landmarks and bones. The
STA has been already discussed on BIOMCH-L, and the literature about it is
profuse and exhaustive.


SOURCES OF STEREOPHOTOGRAMMETRIC ERROR

The stereophotogrammetric error is defined herein as the error in the
reconstructed three-dimensional (3-D) position of a single retroreflective
or light-emitting marker. The sources contributing to this error vary with
the adopted technology. In the most widely used systems, which use
retroreflective markers and multiple video-cameras endowed with CCD or CMOS
image sensors, the stereophotogrammetric error is mainly due to these
sources (sorted by order of appearance in the data collection and processing
flow):

1. Non-uniform reflectivity of the marker surface. The information about
marker position is initially transmitted to the cameras by reflected light.
A non-uniform marker reflectivity may deform the optical image of the marker
projected on the surface of an image sensor, or cause an unexpected
distribution of the photons forming the image. Note that the circle-fitting
algorithms used to locate the marker image centroid (Frosio and Borghese,
2007) typically assume that the image is circular, and in some cases (Vicon
MX® systems) also assume that the intensity of the light decreases as the
distance from image center increases.

2. Light emitted or reflected by other objects or surfaces in the
environment. This error source produces “ghost markers” or deforms the image
of the marker (when the light reflected by the marker melts with the light
emitted or reflected by the environment). It can be minimized by turning off
the light emitters and removing or covering the reflective objects or
surfaces. When this is not possible, the error source can be removed by
using input masks, which define, for each camera, the zones of the field of
view which will be ignored by the reconstruction algorithm. Input masks,
however, are themselves a source of error because they artificially produce
partial or total occlusion of the marker (see below).

3. Reflection of marker image from nearby surfaces. If a reflective marker
is near a surface of sufficient reflectivity, then the cameras may see
reflections of the actual marker from this surface. For example, B.W. Schulz
[Brian.Schulz@va.gov] uses markers embedded on shoes “that get fairly close
to the ground (a vinyl tile floor)”, and he frequently sees ghost markers
produced by these reflections. This is a situation in which input masks
cannot be used, because the origin of the reflection changes with the
position of the marker.

4. Partial or total occlusion of the marker with respect to one or more
video-cameras. Occlusions are defined as situations in which a marker is
partially or totally hidden by parts of the subject’s body, or objects in
the environment, or other markers.
a. Partial occlusion, by deforming the marker image, causes an inaccurate
determination of the marker center position on the image plane. This error
is reduced but not eliminated by using a circle-fitting algorithm (see
above). Most systems provide an optional filter which removes “non-circular”
marker images, typically with a user-selected level of tolerance to
“non-circularity”. When activated, this filter just makes above-threshold
partial occlusion equivalent to total occlusion. Total occlusion, however,
can be regarded as an important source of error (see below).
b. Total occlusion reduces 3-D reconstruction precision and accuracy by
reducing the number of video-cameras which can provide raw 2-D coordinates.
Total occlusion is often preceded by partial occlusion. These sources may
occasionally produce very large error peaks (“wild points”) or
discontinuities in the reconstructed marker trajectory. These errors may
have an absolute value by one order of magnitude larger than the standard
deviation of the stereophotogrammetric error estimated in ideal conditions
of marker visibility. While partial occlusion is an independent error
source, total occlusion just reveals systematic errors produced by other
sources, the most important of which is the unavoidable discrepancy between
the calibration model and the actual system (point 11). Total occlusion
would not be a source of error if all the “camera rays” used for
reconstruction exactly converged at the marker centroid. In this ideal case,
two cameras would be enough to reconstruct the exact 3-D position of the
marker centroid, and the reconstructed position would not depend on the
number of cameras. Since other sources of error deviate the camera rays from
the true position of the marker, the rays actually do not converge. Thus,
the reconstruction is performed by computing a quasi-intersection, rather
than an exact intersection of camera rays. Total occlusion, by reducing the
number of camera rays used for reconstruction, may suddenly displace this
quasi-intersection. This is the reason why total occlusion often produces a
visible discontinuity in the reconstructed position of the marker. More
exactly, the discontinuity is observed if and only if in the instants
preceding the occlusion the occluded ray is systematically deviated in a
given direction, with respect to the reconstructed marker position (which,
in turn, might be systematically displaced from the true position).
Otherwise, if in the instants preceding the occlusion the occluded ray were
randomly deviated, the occlusion would just produce an increase in the
amplitude of the high frequency noise (i.e. a decrease in precision), rather
than a discontinuity in the trajectory (i.e. a change in accuracy).

5. Spatial quantization error. An image sensor contains an array of
photodetectors, and its finite spatial resolution entails a quantization of
the spatial information transmitted by the light. When a cloud of photons
strikes a photodetector, during an exposure interval, the photodetector
measures its energy, independently of the distribution of the photons on the
photodetector surface. Thus, while the optoelectronic transduction performed
by the sensor involves a natural quantization with negligible error (photons
and electrons are very small quanta), the position of the photons is
non-negligibly quantized. Typically, the light reflected by a marker hits
more than one photodetector. In this case, the spatial quantization error is
partly compensated by the more or less sophisticated circle-fitting
algorithms used to locate the marker image centroid. These algorithms
exploit a priori information about the image to achieve an apparent
sub-pixel resolution (Furneé, 1997). For instance, the shape of the image is
typically assumed to be circular. However, this produces an error when the
marker is partially occluded, or non-spherical, or non-uniformly reflective.

6. Resonance of video-camera support (wall brackets, tripods, etc.) to
possible environmental vibration produced, for instance, by vehicles. This
source of error is assumed herein to be negligible, because vibration, if it
exists, is not likely to persist and be stationary for all trials in a
motion capture session.

7. Motion blur. When the marker moves relative to a camera its recorded
image appears stretched in the direction of its relative motion. When the
velocity of the relative motion is not constant, the centroid of this
stretched image does not coincide with the mean position of the centroid of
the optical image, during the exposure interval. This error is correlated
with exposure time, which in turn can be minimized provided that the image
sensors are highly sensitive and the markers are properly illuminated
(typically by stroboscopic light projectors).

8. Electronic noise, which produces flickering of the marker image produced
by each video-camera, even when the marker is motionless relative to the
camera.

9. Quantization error in the analog-to-digital conversion of the signal
associated with each pixel.

10. Inaccuracy of the system calibration parameters, for instance due to
inaccuracy of the reference measures or limitations of the optimization
algorithm used for calibration.

11. Unavoidable limitations of the system calibration model; for instance,
discrepancies between the structural imperfections of the video-cameras
(such as lens aberrations or geometric irregularities of lenses and image
sensors, or misalignment of lenses relative to image sensors), and the
mathematical model used to represent them; in most systems, the calibration
model is designed to partly compensate for the image distortion produced by
these imperfections. A complete compensation is impossible.


REFERENCES

Cappozzo A, Della Croce U, Catani F, Leardini A, Fioretti S, Maurizi M, et
al. Stereometric system accuracy tests. In: Measurement and data processing
methodology in clinical movement analysis-preliminary. CAMARC II Internal
Report; 1993.

Cappozzo, A., Della Croce, U., Fioretti, S., Leardini, A., Leo, T., Maurizi,
M., 1994. Assessment and testing of movement analysis systems: spot checks.
Gait and Posture 3, 172.

Chiari L., Della Croce U., Leardini A. and Cappozzo A., 2005. Human
movement analysis using stereophotogrammetry. Part 2: instrumental errors.
Gait Posture, 21, 197-211.

Della Croce U, Cappozzo A. A spot check for estimating stereophotogrammetric
errors. Med Biol Eng Comp 2000;38:260–6.

DeLuzio KJ, Wyss UP, Li J, Costigan PA. A procedure to validate
three-dimensional motion assessment systems. J Biomech 1993;26:753–9.

Ehara Y, Fujimoto H, Miyazaky S, Tanaka S, Yamamoto S. Comparison of the
performance of 3-D camera systems. Gait Posture 1995;3:166–9.

Ehara Y, Fujimoto H, Miyazaky S, Mochimaru M, Tanaka S, Yamamoto S.
Comparison of the performance of 3-D camera systems II. Gait Posture
1997;5:251–5.

Frosio I., Borghese N.A., 2008. Real-time accurate circle fitting with
occlusions. Pattern Recognition, 41, 1041-1055.

Furnée H. Real-time motion capture systems. In: Allard P, Cappozzo A,
Lumberg A, Vaughan K, editors. Three-dimensional analysis of human
locomotion. New York: Wiley; 1997. p. 85–108.

Holden JP, Selbie S, Stanhope SJ. A proposed test to support the clinical
movement analysis laboratory accreditation process. Gait Posture
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Morris JRW, MacLeod A. An investigation of the sources and characteristics
of noise in a video-based kinematic measurement system. In: Models,
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functional movement analysis. CAMARC II Internal Report; 1990.

REFERENCES ABOUT ACCURACY OF SPECIFIC SYSTEMS

Cappello A, Leardini A, Benedetti MG, Liguori R, Bertani A. Application of
stereophotogrammetry to total body three-dimensional human tremor. IEEE
Trans Rehabil Eng 1997;5(4):388–93.

Everaert DG, Spaepen AJ, Wouters MJ, Stappaerts KH, Oostendorp RA. Measuring
small linear displacements with a three-dimensional video motion analysis
system: determining its accuracy and precision. Arch Phys Med Rehabil
1999;80(9):1082–9.

Haggard P, Wing AM. Assessing and reporting the accuracy of position
measurements made with the optical tracking systems. J Motor Behav
1990;22:315–21.

Klein PJ, DeHaven JJ. Accuracy of three-dimensional linear and angular
estimates obtained with the Ariel performance analysis system. Arch Phys Med
Rehabil 1995;76:183–9.

Richards JG. The measurement of human motion: a comparison of commercially
available systems. Hum Mov Sci 1999;18:589–602.

Thornton MJ, Morissey MC, Coutts FJ. Some effects of camera placement on the
accuracy of the kinemetrix three-dimensional motion analysis system. Clin
Biomech 1998;13:452–4.

Vander Linden DW, Carlson SJ, Hubbard RL. Reproducibility and accuracy of
angle measurements obtained under static conditions with the motion analysis
video system. Phys Ther 1992;72:300–5.


With kind regards,

Paolo de Leva
Department of Human Movement and Sport Sciences,
Istituto Universitario di Scienze Motorie
Rome, Italy