1. Random Sampling Estimation Without Prior Scale
Charles V. Stewart
Department of Computer Science
Rensselaer Polytechnic Institute
Troy, NY USA

2. 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?

3. 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).

4. 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

5. 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

6. Enhancements
MSAC and MLESAC
Kernel-based methods

7. Underlying Assumptions
LMS:
Minimum fraction of inliers is known
RANSAC:
Inlier bound is known

8. 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)
45% random outliers
11% from 2nd structure
44% from 1st structure

9. 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

10. Approaches MINPRAN (IEEE T-PAMI Oct 1995)
Discussed briefly today
MUSE (IEEE CVPR 1996, Jim Miller’s PhD 1997), with subsequent, unpublished improvements
Based on order statistics of residuals
Focus of today’s 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

11. MINPRAN: Minimize 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

12. 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

13. MINPRAN: Objective Function
For a given estimate, j, monotonicity properties restrict the necessary evaluations to just the order statistics

14. MINPRAN: Discussion O(S N log N + N2) algorithm
Excellent results for single structure
Limitations:
Requires a background distribution
Tends to “bridge” discontinuities

15. 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

16. 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. Miller’s 1997 PhD thesis

17. 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:

18. 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

19. 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

20. 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.

21. 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

22. 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

23. 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 uk:N are cached.
Overall cost is O(SN logN)

24. 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.

25. Simulation: Scale Estimation
Unit-mean Gaussian plus outliers

26. Simulations (Done in 2002) Structures: Compare against
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

27. 60/40 Split, Known Scale: Step
Best scenario for MUSE

28. 60/40 Split, Known Scale: Crease
MUSE and Chen-Meer comparable

29. 60/40 Split, Known Scale: Perpendicular Planes
The poorest scenario for MUSE

30. The Effect of Incorrect Scale Estimates on MSAC
The effect of incorrect scale estimates is significant!

31. 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.

32. 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-controlled 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!

33. Exploiting Locality in the Dual-Bootstrap Algorithm

34. Exploiting Locality in the Dual-Bootstrap Algorithm

35. Exploiting Locality in the Dual-Bootstrap Algorithm

36. Locality vs. Globality Can you exploit local structure? Yes? No?
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

37. Back to MUSE Robust estimation when scale is unknown Issues:
Accuracy comparable to MSAC / MLESAC
Issues:
Stopping criteria
Efficiency improvements
Heteroscedasticity

38. 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