Bootstrapping a Heteroscedastic Regression Model with Application to 3D Rigid Motion Evaluation

1
Bootstrapping a Heteroscedastic Regression Model
with Application to 3D Rigid Motion Evaluation

• Bogdan Matei Peter Meer
• Electrical and Computer Engineering Department
• Rutgers University

2
Bootstrap Principle Efron, 1979

• Rigorous method based on resampling the data
• Data must be independent and identically
  distributed (i.i.d.)
• Statistical measures computed from one data set

Data
Sampling with replacement
Bootstrap samples
Bootstrap replicates

• Example covariance of the estimate

3
Bootstrap for Regression

• Model
• Measurements

4
Building Confidence Regions

• The ellipsoid

contains the true estimate with probability

• pseudoinverse of the bootstrapped covariance
  matrix
• the percentile of the distribution

• Relation to error propagation
• does not imply linearization
• provides more accurate coverage
• trades computation time for analytical derivations

5
Heteroscedasticity

• Point dependent errors
• Appears in many 3D vision problems
• due to linearization
• multi-stage tasks

e.g. estimating the 3D rigid motion of a stereo
head
6
Heteroscedastic Regression

• Total least squares (TLS) algorithm assumes
  i.i.d. data. Under heteroscedasticity yields
  biased solutions.
• Non-linear methods, like Levenberg-Marquard
• may converge to local minima
• are computationally intensive
• Proposed methods
• renormalization Kanatani, 1996
• HEIV algorithm Leedan Meer, ICCV 98 Matei
  Meer, CVPR 99

7
Multivariate HEIV Algorithm

• Iterative method
• Can start from random initial solution
• Central module solves the generalized eigenvalue
  problem
• Provides consistent estimate
• Converges in less than 5 iterations
• It is the Maximum Likelihood solution for normal
  noise

semi-positive definite matrices
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Multivariate HEIV Algorithm

• The true values satisfy the linear constraint

• The true values are corrupted by heteroscedastic
  noise

9
Multivariate HEIV Algorithm

• Start with an initial solution

• Compute

• Find the scatter

• Update the solution as the smallest eigenvalue of
  

10
Error Analysis for Heteroscedastic Problems

• First order approximation of the HEIV estimate
  covariance

• To analyze any algorithm applied to
  heteroscedastic data the bootstrap samples must
  be based on the HEIV residuals

11
Bootstrap for Heteroscedastic Regression

• The measurements are not i.i.d.
• Need a consistent estimator for the residuals
• Use a whiten-color cycle to generate bootstrap
  samples
• Outliers must be eliminated with robust
  preprocessing

Data
Data correction
Coloring
Residuals
B. samples
Whitening
B. replicates
12
3D Rigid Motion of a Stereo Head

• True values are related

• 3D points recovered from stereo have
  heteroscedastic noise Blostein et al., 1987
• In quaternion representation the rigid motion
  constraint is

• Rigid motion estimation of a stereo head is a
  multivariate heteroscedastic regression problem

13
Error Analysis of 3D Rigid Motion

• The corrected measurements are

• The covariance of the residuals

• The covariance matrices of the 3D points
  , are obtained through
  bootstrap

14
Evaluation of 3D Rigid Motion Methods

• Methods
• quaternion Horn et al., 1988 and SVD Arun et
  al., 1987 algorithms give identical results.
  Both are TLS type (biased).
• HEIV algorithm
• B 200 bootstrap replicates were used for the
  covariances (confidence regions) of the motion
  parameters
• Angle-axis representation for the rotation matrix
• Using error propagation is very difficult Pennec
  Thirion, 1997

15
Synthetic Data

• Bootstrap compared with Monte Carlo analysis
• Monte Carlo uses the true data and the true noise
  distribution
• bootstrap uses only the available measurements

bootstrap o HEIV x
quaternion/SVD bootstrap HEIV
quaternion/SVD
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Real Data

• Four images, planar texture sequence (CIL-CMU)
• ground truth about the relative position of the
  frames available

Frame 1
Frame 4

• Points were matched using Z. Zhangs program
• 3D data recovered by triangulation Hartley,
  1997

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Real Data

• Bootstrap confidence regions with 0.95
  probability of coverage

Translation estimate quaternion/SVD
Translation estimate HEIV
18
Real Data

• Bootstrap confidence regions with 0.95
  probability of coverage

Rotation estimate quaternion/SVD
Rotation estimate HEIV
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Conclusions

• The HEIV algorithm is a general tool for 3D
  vision
• Bootstrap can supplement the execution of a
  vision task with statistical information which
• captures the actual operating conditions
• reduces the dependence on simplifying assumptions
• Confidence regions in the input domain can
  provide uncertainty information about the true
  locations of features