RANSAC Robust Plane Fitting

Point Cloud

Outlier fraction 35%

Inlier scatter (noise σ) 0.08

RANSAC

Inlier distance threshold 0.18

Iterations / second 4

Live State

Iterations run

Best inliers

0 / 0

Inlier ratio w

Accuracy vs truth

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Iterations needed for 99% confidence, k = ⌈ln(1−P) / ln(1−w³)⌉

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How it works

RANSAC (RANdom SAmple Consensus) is the algorithm computer-vision pipelines use to fit a model when the data is contaminated with gross outliers — mismatched keypoints, reflections, moving objects — that a least-squares fit would be dragged off course by. Here the model is a 3D plane (the CV analogue of fitting a ground plane from stereo points, or a homography from feature matches):

repeat k times:
  1. sample s=3 points at random from all N
  2. fit the exact plane through them:
       n = (p2−p1) × (p3−p1),  d = −n·p1
  3. count inliers: points p where |n·p + d| < τ
  4. keep this model if its inlier count is the
     best seen so far
return the plane with the largest consensus set

Because only 3 points are needed to define a plane, a model built from three real surface points is exact — no averaging with outliers. Across many random trials, models built from all-inlier triplets dominate because they alone explain a large, consistent fraction of the data. The number of trials needed to be reasonably sure of drawing at least one clean triplet, with confidence P and true inlier ratio w, is:

k = ⌈ ln(1 − P) / ln(1 − w^s) ⌉,  s = 3 here

Outlier fraction — how much of the cloud is pure noise, unrelated to the surface.
Inlier scatter σ — how far real surface points wander off the true plane (sensor/measurement noise).
Distance threshold τ — how close a point must be to a candidate plane to count as an inlier; too tight rejects real points, too loose lets outliers in.
Run / Step — draws random triplets, fits, scores, and keeps the best model found so far; watch the blue plane and the green inlier set converge onto the true surface.

Real-world relevance: OpenCV's findHomography and findFundamentalMat, structure-from-motion, LiDAR ground-plane segmentation, and SLAM loop-closure all run this exact sample → fit → score → keep-best loop underneath.

Inlier (current best model) Outlier (current best model) Current random triplet Best-fit plane

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