Random Sampling Estimation Without Prior Scale

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Random Sampling Estimation Without Prior Scale

• Charles V. Stewart
• Department of Computer Science
• Rensselaer Polytechnic Institute
• Troy, NY 12180-3590 USA

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Motivating Example
Is it possible to automatically estimate the
structures in the data without knowing the noise
properties and without any one structure
containing at least a minimum fraction of the
data?
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Notation

• Set of data points, e.g.
• (x,y,z) measures
• Corresponding pixel coordinates
• Parameter vector - the goal of the estimation
• Function giving the fitting error or residual
• The form of the model, e.g. a line, a plane, a
  homography, is implicit here
• Objective function to be minimized
• Order statistics of the residuals, usually
  unsigned (non-negative).

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Outline of Generic Random Sampling Algorithm

• For j 1,,S
• Select a minimal subset xj,1,,xj,k of the
  x1,,xN points
• Generate an estimate ?j from the minimal subset
• Evaluate h(?j) on the N-k points not in the
  minimal subset
• Retain ?j as the best if h(?j) is the minimum
  thus far. Denote the best as ?j.
• Gather the inliers to the best estimate, ?j, and
  refine using (weighted) least-squares

Variations on this algorithm have focused on
efficiency improvements
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Objective Functions of Classical Estimators

• Least-Median (Rousseeuw 1984) objective function
  uses the median of the (square) order statistics
• RANSAC (Fischler and Bolles 1981) objective
  function
• This reverses the original form of RANSAC.
  As written here, RANSAC is designed to
  minimize the number of points outside an inlier
  bound, T

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Enhancements

• MSAC and MLESAC
• Kernel-based methods

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Underlying Assumptions

• LMS
• Minimum fraction of inliers is known
• RANSAC
• Inlier bound is known

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Why Might This Be Unsatisfying?

• Structures may be seen in data despite unknown
  scale and large outlier fractions
• Potential unknown properties
• Sensor characteristics
• Scene complexity
• Performance of low-level operations
• Problems
• Handling unknown scale
• Handling varying scale (heteroscedasticity)

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Goal

• A robust objective function, suitable for use in
  random-sampling algorithm, that is
• Invariant to scale,
• Does not require a prior lower bound on the
  fraction of inliers

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Approaches

• MINPRAN (IEEE T-PAMI Oct 1995)
• Discussed briefly today
• MUSE (IEEE CVPR 1996, Jim Millers PhD 1997),
  with subsequent, unpublished improvements
• Based on order statistics of residuals
• Focus of todays presentation
• Code available in VXL and on the web
• Other order-statistics based methods
• Lee, Meer and Park, PAMI 1998
• Bab-Hadiashar and Suter, Robotica 1999
• Kernel-density techniques
• Chen-Meer ECCV 2002
• Wang and Suter, PAMI 2004
• Subbarao and Meer, RANSAC-25 2006

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MINPRANMinimize Probability of Randomness
26 inliers within /- 8 units of
random-sample-generated line
72 inliers within /- 7 units of
random-sample-generated line
55 inliers within /- 2 units of
random-sample-generated line
65 outliers
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MINPRAN Probability Measure

• Probability of having at least k points within
  error distance /- r if all errors follow a
  uniform distribution within distance /- Z0
• Lower values imply it is less likely that the
  residuals are uniform
• Good estimates, with appropriately chosen values
  of r (inlier bound) and k (number of inliers),
  have extremely low probability values

r
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MINPRAN Objective Function

• For a given estimate, ?j, monotonicity properties
  restrict the necessary evaluations to just the
  order statistics

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MINPRAN Discussion

• O(S N log N N2) algorithm
• Excellent results for single structure
• Limitations
• Requires a background distribution
• Tends to bridge discontinuities

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Toward MUSE Ordered Residuals of Good and Bad
Estimates

• Objective function should capture
• Ordered residuals are lower for inliers to good
  estimate than for bad estimate
• Transition from inliers to outliers in good
  estimate

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Statistics of Order Statistics

• Density and distribution of absolute residuals
• Scale normalized residuals and order statistics
• To a good approximation, the expected value of
  kth normalized order statistic is simply
• The variance of the kth normalized order
  statistic is obtained using a Taylor expansion of
  F-1
• Details omitted! See James V. Millers 1997 PhD
  thesis

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Scale Estimates from Order Statistics

• Repeating, the kth normalized order statistic,
  assuming all residuals follow distribution F with
  unknown ?, is
• Taking expected values on both sides
• Rearranging isolates the unknown ??
• Finally, we can obtain an unbiased estimate of ?,
  one for each order statistic

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Scale Estimates from Order Statistics Good
and Bad Estimates

• We have only assumed that the residuals follow a
  Gaussian distribution
• Scale estimates are much lower for the inliers to
  the good estimate
• Q How do we determine which scale estimate to
  choose?

Before addressing this we consider a scale
estimate based on trimmed statistics
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Trimmed Statistics Scale Estimates

• Expected value of sum of first k order statistics
  (remember, these are unsigned)
• Generating a scale estimate
• Computing the denominator based on a Gaussian
  distribution yields
• This estimate is more stable than the quantile
  version and is used in practice

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Which of the O(N) Possible Scale Estimates?

• Original idea choose smallest scale estimate.
• Problem variance in scale estimates leads to
  instability and reintroduces bias toward small
  estimates

This shows the ratio of the expected residual of
the order statistic to its standard deviation.
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Minimum (Variance) Unbiased Scale Estimate

• Choose the scale estimate from a set of order
  statistics that has the smallest standard
  deviation
• There is a strong tendency for this choice to
  occur just before the transition from inliers to
  outliers.

std?k vs. k for good estimate
std?k vs. k for bad estimate
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MUSE Objective Function

• Given parameter vector ?? calculate the
  (unsigned) residuals and their order statistics,
• Calculate the scale estimates (for the quantile
  scale estimate)
• Choose k that minimizes
• call this k
• The objective function is

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More About MUSE

• Remainder of the basic random sampling method is
  unchanged
• In practice
• Use trimmed scale estimate with quantile-based
  standard deviation calculation
• Evaluate the standard deviation at a small set of
  k values, e.g. 0.2N, 0.25N, , 0.85N
• Expected values and standard deviations of the
  ukN are cached.
• Overall cost is O(SN logN)

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S-K Refinement

• For given k, find the number of residuals within
• Call this number N
• Re-evaluate
• This produces less bias in good scale estimates
  while leaving bad scale estimates essentially
  unchanged.

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Simulation Scale Estimation
Unit-mean Gaussian plus outliers
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Simulations (Done in 2002)

• Structures
• Single or multiple planar surfaces in R3
• Single or multiple plane homography estimation
• Compare against
• LMS
• MSAC
• Chen-Meer, ECCV 2002, kernel density algorithm
• Weak dependence on prior scale estimate
• Bias is measured as the integral of the square
  estimation error, normalized by area

Experiments done by Ying-Lin (Bess) Lee
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60/40 Split, Known Scale Step
Best scenario for MUSE
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60/40 Split, Known Scale Crease
MUSE and Chen-Meer comparable
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60/40 Split, Known Scale Perpendicular Planes
The poorest scenario for MUSE
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The Effect of Incorrect Scale Estimates on MSAC
The effect of incorrect scale estimates is
significant!
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Results Discussion

• Similar results with plane homography estimation
• In most cases
• Performance of MUSE is comparable to MSAC (RANSAC
  or MLESAC) when scale is known
• Performance of MUSE is better than MSAC (RANSAC /
  MLESAC) when scale is unknown
• Performance of kernel-density based algorithms is
  comparable to MUSE
• These have a very weak dependence on prior scale
  or use Median Absolute Deviation (MAD) to provide
  rough scale estimates.

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Our Current Work at Rensselaer

• Dual-Bootstrap estimation of 2d registration
  with
• Keypoint indexing
• Start from single keypoint match
• Grow and refine estimate using statistically-contr
  olled region growing and re-estimation
• MUSE as initial scale estimator
• M-estimator as parameter estimator
• Reliable decision criteria
• Algorithm substantially outperforms keypoint
  matching with RANSAC!

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Exploiting Locality in the Dual-Bootstrap
Algorithm
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Exploiting Locality in the Dual-Bootstrap
Algorithm
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Exploiting Locality in the Dual-Bootstrap
Algorithm
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Locality vs. Globality

• Can you exploit local structure?
• Yes?
• Careful scale estimation
• M-estimator (really gradient-based)
• No?
• RANSAC with appropriately chosen cost function
• MSAC or MLESAC when scale is known
• MUSE or kernel-based method when scale is unknown
• Efficient algorithms
• Lack of local exploration is both the blessed
  and the curse of RANSAC

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Back to MUSE

• Robust estimation when scale is unknown
• Accuracy comparable to MSAC / MLESAC
• Issues
• Stopping criteria
• Efficiency improvements
• Heteroscedasticity

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C Code Available

• http//www.vision.cs.rpi.edu
• Stand-alone MUSE objective functions, including
  statistics of order statistics
• Dual-Bootstrap executable and test suite
• vxl.sourceforge.net
• rrel library contributed by Rensselaer