Title: Bootstrapping a Heteroscedastic Regression Model with Application to 3D Rigid Motion Evaluation
1Bootstrapping a Heteroscedastic Regression Model
with Application to 3D Rigid Motion Evaluation
- Bogdan Matei Peter Meer
- Electrical and Computer Engineering Department
- Rutgers University
2Bootstrap 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
3Bootstrap for Regression
4Building Confidence Regions
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
5Heteroscedasticity
- 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
6Heteroscedastic 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
7Multivariate 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
8Multivariate HEIV Algorithm
- The true values satisfy the linear constraint
- The true values are corrupted by heteroscedastic
noise
9Multivariate HEIV Algorithm
- Start with an initial solution
- Update the solution as the smallest eigenvalue of
10Error 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
11Bootstrap 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
123D Rigid Motion of a Stereo Head
- 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
13Error 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
14Evaluation 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
15Synthetic 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
16Real 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
17Real Data
- Bootstrap confidence regions with 0.95
probability of coverage
Translation estimate quaternion/SVD
Translation estimate HEIV
18Real Data
- Bootstrap confidence regions with 0.95
probability of coverage
Rotation estimate quaternion/SVD
Rotation estimate HEIV
19Conclusions
- 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