Bootstrapping a Heteroscedastic Regression Model with Application to 3D

Rigid Motion Evaluation

Bogdan Matei Peter Meer

Electrical and Computer Engineering Department

Rutgers University

• Rigorous method based on resampling the data

• Data must be independent and identically distributed (i.i.d.)

• Statistical measures computed from one data set

Bootstrap Principle [Efron, 1979]

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Sampling with replacement

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• Example: covariance of the estimate

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Bootstrap for Regression

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• Obtain bootstrap samples as

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• Model

• Measurements

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– pseudoinverse of the bootstrapped covariance matrix

– the percentile of the distribution

Building Confidence Regions

• Relation to error propagation– does not imply linearization

– provides more accurate coverage

– trades computation time for analytical derivations

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Heteroscedasticity

• Point dependent errors

• Appears in many 3D vision problems

– due to linearization

– multi-stage tasks

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

• 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]

Heteroscedastic Regression

• 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

Multivariate HEIV Algorithm

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Multivariate HEIV Algorithm

• The true values satisfy the linear constraint

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• The true values are corrupted by heteroscedastic noise

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• Find the scatter

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• Update the solution as the smallest eigenvalue of

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Error Analysis for Heteroscedastic Problems

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

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• First order approximation of the HEIV estimate covariance

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

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Coloring

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3D 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

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• True values are related

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

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• The corrected measurements are

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• The covariance of the residuals

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

Error Analysis of 3D Rigid Motion

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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]

• Bootstrap compared with Monte Carlo analysis– Monte Carlo uses the true data and the true noise distribution

– bootstrap uses only the available measurements

Synthetic Data

bootstrap: ‘o’ HEIV ‘x’ quaternion/SVD

bootstrap ‘ ’ HEIV ‘+’ quaternion/SVD

Translation error tCt ˆtr RCr ˆtrRotation error

Real Data

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

Frame 1

• Points were matched using Z. Zhang’s program

• 3D data recovered by triangulation [Hartley, 1997]

Frame 4

Real Data

Translation estimate quaternion/SVD

Translation estimate HEIV

• Bootstrap confidence regions with 0.95 probability of coverage

Real Data

Rotation estimate quaternion/SVD

Rotation estimate HEIV

• Bootstrap confidence regions with 0.95 probability of coverage

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