Learning to Track Motion

Transcription

1 Learning to Track Motion Maitreyi Nanjanath Amit Bose CSci 8980 Course Project April 25, 2006

2 Background Vision sensor can provide a great deal of information in a short sequence of images Useful for determining camera position and movement Availability of local trackers: Fast Error prone Globally inconsistent

3 Motivation Global consistency as a cue for error correction Is a master algorithm possible over and above local trackers that makes corrections? Assume local tracking algorithm is a blackbox Input: sequence of images Output: image co-ordinates of successive feature positions

4 Problem Statement Given: Inaccurate estimates from a set of local trackers Objective: Devise a master algorithm for global tracking that, is based on erroneous estimates of local trackers only can make relatively accurate estimates of feature positions

5 Our Approach Hypothesis: It is possible to develop a learning algorithm that performs corrective tracking Problem naturally fits the multidimensional regression model We have seen the generalized additive (GA) algorithm of Kivinen and Warmuth We have also seen a multitude of loss functions

6 Model N local trackers Each tracker generates 6 values 2 spatial co-ordinates 2 velocity components 2 acceleration components Data-point (x, y) x: 6N+1 vector of tracker given information y: 2N vector of true spatial co-ordinates

7 Generalized Additive Algorithm 1. Initialize the parameter matrix as Θ 1 = Θ. 2. Repeat for t = 1,,l a) Get the input x t b) Compute the weight matrix Ω t = Ψ(Θ t ) c) Compute the linear activation â t = Ω t x t d) Output the prediction ŷ t = φ(â t ). 3. For j = 1,,k, update the j th row of the parameter matrix by θ t+1,j = θ t,j η(ŷ t,j y t,j )x t

8 Experiment Setup A set of 20 points is generated uniformly distributed over the unit cube The set of points is moved along a trajectory The path is projected onto 2 dimensions 6 values are generated for each point Co-ordinates, velocity and acceleration components This forms the ground truth Ground truth is perturbed by adding noise Ground truth and noisy data is input to the GA Algorithm It learns to produce corrected spatial coordinates

9 Noise added Gaussian noise, with Mean 0 Variance Variance Chi-squared noise, with Mean 0.1 Arbitrary noise In this, certain data points suddenly become zero. (which corresponds to losing a feature)

10 Transfer and Loss Functions Transfer Function Matching Loss Function Squared loss ( Φ) -1 (x) = x Φ(x) = x 2 /2 I-Divergence Logistic loss Itakura-Saito Distance Exponential loss ( Φ) -1 (x) = e x 1 ( Φ) -1 (x) = e x / (1 + e x ) ( Φ) -1 (x) = 1/x ( Φ) -1 (x) = logx Φ(x) = xlogx Φ(x) = xlogx + (1-x)log(1-x) Φ(x) = logx Φ(x) = e x

11 Training data 800 training data points: Correspond to set of image points moving randomly Data scaled and shifted To fit constraints of loss functions. Training performed for many epochs: 1, 100, 300, 500 Data shuffled randomly in every epoch Learning rate was kept fixed in a run

12 Results Test data: 400 data points Results with final weight matrix generated for both training and test data

13 Results: Gaussian Noise and Squared Loss

14 Results: Gaussian Noise and I-Divergence

15 Results: Gaussian Noise and Logistic Loss

16 Results: Gaussian Noise and Itakura-Saito Distance

17 Results: Gaussian Noise and Exponential Loss

18 Results: Other Data Sets (Chi-square and arbitrary)

19 Issues and Challenges High learning rate made weights oscillate Data preparation was a major challenge: Input had to be tuned for the loss functions for specialized domains Appropriate noise additives were needed Choice of feature dimensions Acceleration input was added to help track rotation We were unsure if the simple regression model was sufficient

20 Extensions and Future Work Extensions: Varying learning rate Use non-linear high-dimensional mapping and kernels Future work Regularization Using a structured regression model Testing on real images

21 Conclusions Over 75% reduction in error was seen in most cases I-divergence and squared loss performed comparably Itakura-Saito distance was not a good choice for this domain Better tuning of parameters may improve results

22 Q & A