Fig. 1. (a) A road segment with outlying pixels; (b) its OLS line fit; (c) its LMS line fit.

Fig. 2. Dual transformation and properties.

Fig. 3. The strip bounded by lines ℓ₁ and ℓ₂ in the primal plane dualizes to the bracelet in the dual plane. A point such as p₁ that lies inside the strip dualizes to a line intersecting the bracelet, and a point such as p₂ that lies above the strip dualizes to a line that passes below the bracelet.

Fig. 4. Line arrangement (a), and levels and and the 4-trapezoids they define (b).

Fig. 5. Counting arrangement vertices through inversion counting.

Fig. 6. Quantile approximation for LMS.

Fig. 7. Proof of Lemma 2.

Fig. 8. Sorted intersection points along slab sides (for k^-=3) (a), and pseudo-levels and the lower bound (b).

Fig. 9. The hybrid approximation algorithm.

Fig. 10. Counting the lines intersecting L_j and L_j^′.

Fig. 11. Examples of point distributions. Each example contains 500 points in a square of side length 2. There are roughly 30% inliers and 70% outliers. The inliers are perturbed by Gaussian distributed residual noise with a standard deviation of 0.01. In (c) and (d) there are 10 clusters of line segments and circles, respectively.

Fig. 12. Running time versus data size and quantile error factor.

Fig. 13. Number of stages versus: (a) data size and (b) quantile error factor. In the second case we plot the function 10(1+log₁₀(1/ε_q)) for comparison.

Fig. 14. Running time versus quantile and residual error factors.

Fig. 15. Running time versus residual standard deviation for the topological plane-sweep algorithm (exact) and (ε_q=ε_r=0) for the: (a) line+unif; and (b) line+segments distributions.

Fig. 16. CPU time speed-up of ALMS over topological plane sweep versus residual error ε_r for various distributions.

Running time and actual error of the topological plane-sweep and ALMS algorithms for n=5000 points, various distributions and various values of ε_r

The speed-up is the ratio of the two running times. In all cases ε_q=0.