Outlier rejection

Overview

Given candidate matches or measurements, some of them wrong, estimate a model (usually a rigid transform or a bundle-adjustment solution) from the ones that agree with it. Every variant shares one rule: a match is an inlier when its residual under the current model is below an inlier threshold tau 2103.05465 eq. 6. Variants differ in how they propose a model, how they score it, and whether the score is learned. Filters that judge each match on its own, such as the ratio test on descriptor similarity, cannot use the geometry of the whole set 2103.05465 §1.

Variants

Learned spatial-consistency pruning (PointDSC)

Each correspondence is a pair of 3D keypoints. For two pairs that are both correct, the distance between their two points is about the same in both scans, so the length difference between pairs measures how consistent they are 2103.05465 §3.2 eq. 2. That term weights a nonlocal attention layer, and a neural spectral matching step scores the pairs inside each seed's feature-space neighbourhood 2103.05465 §3.4 eq. 3-5. The hypothesis with the most inliers wins 2103.05465 §3.5. Card: PointDSC: Robust Point Cloud Registration using Deep Spatial Consistency.

The paper's ablation with the same encoder shows the effect of the head: a classifier head reaches 88.17% registration recall on 3DMatch, and neural spectral matching reaches 93.28% 2103.05465 tab. 4. Removing the spatial term from the attention lowers the classifier-head result from 88.17% to 83.30% 2103.05465 tab. 4.

Sampling-based hypothesis search (RANSAC family)

RANSAC draws random minimal subsets, fits a model to each, and keeps the model with the most inliers. The paper compares RANSAC with 1k, 10k and 100k iterations and GC-RANSAC with 100k 2103.05465 tab. 1. A prioritised variant samples by predicted inlier probability instead of uniformly, which reaches 74.31% recall with 1k iterations against 40.05% for classic RANSAC-1k 2103.05465 app. Prioritized RANSAC. Card: PointDSC: Robust Point Cloud Registration using Deep Spatial Consistency.

Per-correspondence classifiers (DGR, 3DRegNet)

These learned filters label each correspondence on its own and do not use the rigid-motion constraint 2103.05465 §1. On 3DMatch with FPFH, DGR reaches 69.13% recall and 3DRegNet 26.31% 2103.05465 tab. 1.

Residual pruning of tie points in bundle adjustment

Tie points whose reprojection error is too large are removed in a loop. The steps are a prefilter on the number of images per point (at least 3), a reconstruction uncertainty limit (8) and a projection accuracy limit (8), then a pass that drops points above a reprojection error of 0.25 and re-adjusts until the largest remaining error is below 0.5 2610.01438 fig. 2. The limits are unitless 2610.01438 fig. 2. Card: The Impact of Processing Parameters on High-Accuracy Measurements in UAV Photogrammetry.

Fixed gates and robust polynomial fits on displacement fields

For a deformation survey, the same paper combines fixed gates with a robust fit. Vectors above 0.6 m horizontal or 2.0 m vertical are removed, and a first-degree polynomial is fitted with RANSAC in 150 m patches, with inlier distances of 3 cm horizontal and 5 cm vertical 2610.01438 §2.4.1. Card: The Impact of Processing Parameters on High-Accuracy Measurements in UAV Photogrammetry.

When to use which

Implementation sketch

The common core is hypothesise, score by inlier count, and refit on the inliers. The hypothesis generator is the only part that differs between variants.

# Common core: hypothesise, score by inlier count, refit  [arxiv:2103.05465 §3.5]
# corr: candidate correspondences; tau: inlier threshold, set per dataset
# make_hypotheses: random minimal subsets (RANSAC) or learned seeds (PointDSC)

def reject_outliers(corr, tau, make_hypotheses, fit, residual):
    # Fit a model to each candidate subset [arxiv:2103.05465 §3.1]
    best_model, best_count = None, -1
    for subset in make_hypotheses(corr):
        model = fit(corr[subset])
        # Count matches whose residual is below tau [arxiv:2103.05465 eq. 6]
        count = sum(residual(model, corr) < tau)
        if count > best_count:
            best_model, best_count = model, count
    # Refit on all inliers of the winning model [arxiv:2103.05465 §3.5]
    inliers = corr[residual(best_model, corr) < tau]
    return fit(inliers), inliers

# Iterative residual pruning, as in tie-point filtration  [arxiv:2610.01438 fig. 2]
# Limits 0.25 and 0.5 are the paper's values; the paper does not say which
# units they use

def prune_by_residual(points, adjust, residual, drop=0.25, stop=0.5):
    # Adjust, drop points above the per-pass limit, and repeat
    model = adjust(points)
    while True:
        points = [p for p in points if residual(model, p) <= drop]
        model = adjust(points)
        # Stop when the largest remaining residual is below the stop limit
        if max(residual(model, p) for p in points) < stop:
            return model, points

Contradictions

• Loss weight in PointDSC. The paper sets lambda = 3 2103.05465 §4. The card's Paper vs code section records that the released configurations use 1.0 for both weights, so the effective ratio differs. See PointDSC for the code details.
• Outdoor length scale in PointDSC. The paper gives sigma_d as 60 cm for outdoor scenes 2103.05465 §4. The card records that the released KITTI configuration uses 1.2 m. Reproduce with the value you cite.
• Effect of tie-point filtration in one paper. The card's pitfall says the paper finds no significant effect of filtration on the best variants. In the top 10% of variants, the table gives p = 0.231 for the northing component but p = 0.000 for the across-track component and p = 0.001 for elevation 2610.01438 tab. 4. The paper's prose and its table disagree on two of the three components.

Papers

• PointDSC: Robust Point Cloud Registration using Deep Spatial Consistency: a learned filter that uses rigid-motion consistency in attention and seeded spectral matching, with RANSAC and per-match classifiers as baselines.
• The Impact of Processing Parameters on High-Accuracy Measurements in UAV Photogrammetry: tie-point residual pruning and fixed gates in a photogrammetry pipeline, evaluated across 768 processing variants.