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Laboratório de Óptica e Mecânica Experimental
A Movement Tracking Management Model A Movement Tracking
Management Model with Kalman Filtering Globalwith Kalman Filtering
Global Optimization Optimization
Techniques and Mahalanobis DistanceTechniques and Mahalanobis
Distance
Raquel Ramos Pinho, João Manuel R. S. Tavares, Miguel Velhote
Correia
Loutraki, 21–26 October 2005
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
2
Contents• Introduction;• Kalman Filter;• Incorporation of new
data:
– Simplex Method;– Mahalanobis Distance;– Marker occlusion or
appearance during movement tracking.
• Management Model;• Experimental Results;• Conclusions and
Future work.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
3
Introduction• Motion capture video systems and interactive
modelling systems can help automatic analysis and diagnosis of
objects movement;• Many tracking applications may require tracking
of several objects simultaneously, and involve problems as their
(dis)appearance of the scene;• To track objects we used:
– A Kalman filter;– Optimization Techniques;– Mahalanobis
Distance;– A Management Model.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
4
Kalman Filter• Optimal recursive Bayesian stochastic method;• In
this work:
– the system state in each time step is the set of positions,
velocities and accelerations of the tracked features (points);
– new measurements are incorporated whenever a new image frame
is evaluated.
• One of its drawbacks is the restrictive assumption of Gaussian
posterior density functions at every time step;
– Many tracking problems involve non-linear movement, human gait
is just an example.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
5
Mahalanobis Distance and Optimization• For each position
estimate there may exist at most one new measurement to correct its
predicted position.• With Kalman’s usual approach the predicted
search area for each tracked feature is given by an ellipse (whose
area will decrease as convergence is obtained and vice-versa).
– Some problems:• there may not exist any feature in the search
area or, as the
opposite, there might be several;• even if there is only one
correspondence for each feature,
there is no guarantee that the best set of correspondencesis
achieved.
-
A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
6
Mahalanobis Distance and Optimization• We propose the use of
optimization techniques to obtain the best set of correspondences
between the predictions and the measures;• To establish the best
global set of correspondences we used the Simplex method;• The cost
of each correspondence is given by the Mahalanobis distance.
Simplex Method:• Iterative algebraic procedure used to determine
at least one optimal solution for each assignment problem;•
Assignment formulation: one estimate = one measure.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
7
Mahalanobis Distance and OptimizationMahalanobis Distance:•
Distance between two features is normalized by its statistical
variations;• The Mahalanobis distance values will be inversely
proportional to the quality of the prediction/measure
correspondence; thus to optimize the correspondences, we minimize
this cost function.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
8
Mahalanobis Distance and OptimizationOcclusion/Appearance:•
Assignment restriction (1 to 1) not satisfied – problem solved with
addition of fictitious variables:
– Features matched with fictitious variables are considered
unmatched;
– Unmatched predicted position - it is assumed that the feature
has been occluded, but the tracking process is maintained by
including its predicted position in the measurement vector although
with higher uncertainty;
– Unmatched measurement - we consider it as a new feature and
initialize its tracking process.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
9
Management Model• When a feature disappear of the scene: Is it
just occluded? It was removed definitively? Should we keep its
tracking?• This decision is of greater importance if many features
are being tracked, if the image sequence is long, if the tracking
is in real-time, etc;• We use a management model in which a
confidence value is associated to each feature:
– In each frame, if the feature is visible then the confidence
value is increased, else it is decreased;
– If a minimum value of the confidence value is reached then is
considered that the feature has definitively disappeared and
itstracking will cease (if it reappears, its tracking will be
initialised);
– In this work, the confidence values are integers between 0 and
5, and initialized as 3.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
10
Experimental Results•Synthetic data:
– Points A+B translated horizontally and C+D rotated:
Prediction Uncertainty Area Measurement Correspondence
Results
A
B
C
D
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
11
Experimental Results (Cont.)
A
BC
D
Prediction Uncertainty Area Measurement Correspondence
Results
– Continuation. ...Now, Points C+D inverted the rotation
angle:
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
12
• Management of the tracking features - blobs (dis)appear
randomly:Experimental Results (Cont.)
AB
C
DE
Steps“Blobs” 0 1 2 3 4
A - 3 4 5 5B 3 4 5 5 5C 3 4 5 5 5D 3 4 3 4 5E - 3 2 3 4
Prediction Uncertainty Area MeasurementCorrespondence Result
Confidence Values:
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
13
– Tracking 6 markers in human gait:
Experimental Results (Cont.)
1
23
4
5
•Real data:
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
14
Squared difference between predictions and measures:
Experimental Results (Cont.)
StepMarker 1 2 3 4 5 6
1
2
3
4
5
6
1.2 - -0.91.2 2
2.8
2
1.1 0.7 1.3
2.2
2.8
2.2
3.3
0.8 1.0 0.2 0.9
1.3
0.9
0.1
1.3
- 1
0.1
2.2
0.8 0.8 0.8
0.0 0.0 0.0
0.3 1.0 1.5 Average=1.2
•Real data:
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
15
- Tracking 5 markers in human gait:
Experimental Results (Cont.)
Prediction Uncertainty Area Measurement Correspondence
Result
•Real data:
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
16
- Tracking persons in a shopping centre: Experimental Results
(Cont.)
(5 frames interval)
•Real data:
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
17
Conclusions and Future Work• We proposed a methodology to track
feature points along
image sequences based on:– a Kalman filter;– optimization
techniques;– Mahalanobis distance;– a Management Model;
• With our approach, the best set of correspondences is
guaranteed (with respect to the used cost function!);
• The used management model allows the tracking in continuous
image sequences in real-time, as the features simultaneously
tracked are continuously update.
-
A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
18
Conclusions and Future WorkFuture Work:• Consideration of other
stochastic methods;• Adoption of matches one to several (and
vice-versa);• Use the proposed tracking methodology in, for
example,
human gait analysis.
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A Movement Tracking Management Model with Kalman Filtering,
Global Optimization Techniques and Mahalanobis Distance
Introduction Kalman Filter Mahalanobis Distance and Optimization
Management Model Results Conclusions
Raquel R. Pinho, João Manuel R. S. Tavares, Miguel V. Correia
19
Acknowledgments• The first author would like to thank the
support of the PhD
grant SFRH / BD / 12834 / 2003 of the FCT - Fundaçãopara a
Ciência e a Tecnologia in Portugal
• This work was partially done in the scope of the project
“Segmentation, Tracking and Motion Analysis of Deformable (2D/3D)
Objects using Physical Principles”, reference
POSC/EEA-SRI/55386/2004, financially supported by FCT in
Portugal.
União Europeia FEDER
Governo da República Portuguesa
A Movement Tracking Management Model with Kalman Filtering
Global Optimization Techniques and Mahalanobis
DistanceContentsIntroductionKalman FilterMahalanobis Distance and
OptimizationMahalanobis Distance and OptimizationMahalanobis
Distance and OptimizationMahalanobis Distance and
OptimizationManagement ModelExperimental ResultsExperimental
Results (Cont.)Experimental Results (Cont.)Experimental Results
(Cont.)Experimental Results (Cont.)Experimental Results
(Cont.)Experimental Results (Cont.)Conclusions and Future
WorkConclusions and Future WorkAcknowledgments
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