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ISOMAP TRACKING WITH PARTICLE FILTER
Presented by Nikhil Rane
Dimensionality Reduction
Let xi be H-dimensional and yi be L-dimensional then dimensionality reduction solves the problem xi = f (yi) where H>L
Dimensionality Reduction Techniques Linear
• PCA• Transforms data into a new coordinate system so that largest
variance in on the 1st dimension, 2nd largest along 2nd dimension …
• Classical MDS• Preserves Euclidean distances between points
Nonlinear • Isomap
• Preserves geodesic distances between points
• LLE• Preserves local configurations in data
Face Database
Principal Components Analysis (PCA)
1) Make the mean of the data zero2) Compute covariance matrix C3) Compute eigenvalues and eigenvectors
of C4) Choose the principal components5) Generate low-dimensional points using
principal components
Performance of PCA on Face-data
Classical Multidimensional Scaling (MDS) Compute Distance Matrix S
Compute inner product matrix B = -0.5JSJ where J = IN – (1/N)11T
Decompose B into eigenvectors and eigenvalues
Use top d eigenvectors and eigenvalues to form the d dimensional embedding.
Performance of MDS on face-data
Locally Linear Embedding (LLE)
Find neighbors of each data point
Compute weights that best reconstruct each data point from its neighbors
Compute low-dimensional vectors best reconstructed by the weights
Performance of LLE on Face-data
Geodesic Distance
Geodesic distance – the length of the shortest curve between two points taken along the surface of a manifold
Isometric Feature Mapping (Isomap)
Construct neighborhood graph
Compute shortest paths between points
Apply classical MDS
Performance of Isomap on face-data
Tracking vs. Detection
Detection - locating an object independent of the past information
• When motion is unpredictable
• For reacquisition of a lost target Tracking - locating an object based on past information
• Saves computation time
Recursive Bayesian Framework
Estimate the pdf of state at time t given the pdf of state at time t - 1 and measurement at time t• Predict
• Predict state of the system at time t using a system-model and pdf from time t – 1
• Update
• Update the predicted state using measurement at time t by Bayes’ rule
Kalman Filtering vs. Particle Filtering
Kalman filter assumes the pdf of the state to be Gaussian at all times and requires the measurement and process noise to be Gaussian
Particle filter makes no such assumption and in fact estimates the pdf at every time-step
Resampling
Condensation algorithm Algorithm – 1) Resample 2) Predict 3) Measure
Condensation algorithm
Isomap Tracking with Particle Filtering
Create training set of a person’s face (off-line)
Use Isomap to reduce dimensionality of the training set (off-line)
Run particle filter on test sequence to track the person
Training Data
Isomap of Training Data
Isomap Discrepancy Isomap gave dimensionality of 2 when head poses moving up
were removed. Thus, the dimensionality of 3 recovered by training data can be attributed to the non-symmetry of the face about the horizontal axis.
Weighting Particles by SSD
Weighting Particles by Chamfer distance
State evolution without resampling
State evolution with resampling
Experimental Results
Videos
Videos Continued
Conclusion and Future work Isomap provides good frame-work for pose estimation
Algorithm can track and estimate a person’s pose at the same time
Use of particle filter allows parallel implementation
Goal is to be able to build an Isomap on-line so that the particle filter tracker can learn as it tracks
Thank You!