Goal

Robustly fit a parametric model (homography, fundamental matrix, or essential matrix) to a set $P$ of data points contaminated by outliers, without a user-tuned inlier threshold $\varepsilon$. Inputs: data set $P$; minimal sample size $s$; residual function $D(\theta, p)$; upper noise-scale bound $\sigma_\mathrm{max}$ (replaces the hard threshold $\varepsilon$); confidence target $\eta$. Outputs: weighted-least-squares model $\theta^*$ and marginalised quality score $Q^*(\theta^*, P)$. The defining property is $\sigma$-consensus: the noise scale $\sigma$ is treated as a random variable uniformly distributed on $[0, \sigma_\mathrm{max}]$, and the RANSAC quality function $Q(\theta, \sigma, P)$ is marginalised over $\sigma$ (Eq. 2); the final model is a weighted least-squares fit using the marginalised inlier likelihoods as IRLS weights (Alg. 1), with no hard inlier/outlier partition.

Algorithm

Let $P$ denote the set of data points and $N = |P|$ the point count. Let $s$ denote the minimal sample size required to instantiate the model. Let $\sigma$ denote the noise scale and $\sigma_\mathrm{max}$ its upper bound. Let $D(\theta, p)$ denote the residual of point $p$ under model $\theta$. Let $g(r \mid \sigma)$ denote the chi-squared inlier-residual density with $\rho$ degrees of freedom — a function of radius $r$ and scale $\sigma$. Let $\tau(\sigma)$ denote the effective inlier threshold at noise scale $\sigma$. Let $L(p \mid \theta)$ denote the marginalised inlier likelihood of point $p$ under model $\theta$, used as the IRLS weight. Let $Q^*(\theta, P)$ denote the marginalised quality score. Let $d$ denote the number of uniform partitions of $[0, \sigma_\mathrm{max}]$ used for discretisation (default $d = 10$). Let $\eta$ denote the confidence target for the stopping criterion. Let $\tau_\mathrm{ref}$ denote the SPRT pre-screen reference threshold (set to 1 pixel).

Procedure

Implementation

MAGSAC $\sigma$-consensus quality computation in Rust:

/// Discretised σ-consensus quality (MAGSAC).
/// Returns (Q*, per-point weights) for use in IRLS refit.
pub fn magsac_quality<F>(
    residuals: &[f64],
    sigma_max: f64,
    d: usize,
    g: F,           // chi-squared density g(r | σ)
) -> (f64, Vec<f64>)
where F: Fn(f64, f64) -> f64
{
    let dsigma = sigma_max / d as f64;
    let mut weights = vec![0.0_f64; residuals.len()];
    let mut q_star = 0.0;

    for i in 1..=d {
        let sigma = i as f64 * dsigma;
        let tau = 3.64 * sigma; // χ²(4), 0.95 quantile
        for (j, &r) in residuals.iter().enumerate() {
            let w = if r <= tau { g(r, sigma) } else { 0.0 };
            weights[j] += w * dsigma;
            q_star += w * dsigma;
        }
    }
    // Normalise by σ_max (uniform prior on σ).
    let inv = 1.0 / sigma_max;
    for w in weights.iter_mut() { *w *= inv; }
    (q_star * inv, weights)
}

Remarks

• $\sigma$-consensus eliminates the user-tuned inlier threshold $\varepsilon$ — replacing it with the upper bound $\sigma_\mathrm{max}$ (10 pixels across all experiments), which is much less sensitive to precise tuning than a hard threshold.
• Discretisation grid $d = 10$ reduces fitting cost from $O(K)$ to $O(d)$ per hypothesis. Values below $d = 5$ risk underweighting high-$\sigma$ components; the paper provides no sensitivity sweep beyond the empirical choice.
• Per-point weights $L(p \mid \theta)$ are marginalised inlier likelihoods. The final model is a weighted-least-squares fit via IRLS, not a hard inlier-set refit.
• For $\chi^2(4)$ residuals (2D point correspondences), the effective inlier threshold at each $\sigma_i$ is $\tau(\sigma_i) = 3.64\,\sigma_i$. For higher residual dimensions the constant changes; consult the appropriate $\chi^2$ quantile table.
• Float32 underflow safety: $\exp(-D^2 / 2\sigma_i^2)$ underflows in float32 when $D > {\sim}13$ px for $\sigma_i = 1$ px — safely beyond the $\tau(\sigma_\mathrm{max})$ gate, so float32 arithmetic is workable throughout the $\sigma$ loop.
• See raguram-usac for the unifying RANSAC engineering framework MAGSAC plugs into, including the When to choose USAC over MAGSAC discussion (hosted there per the older + broader-scope tiebreaker). See ransac for the four design axes that organise the modern RANSAC family.

References

1. D. Barath, J. Matas, J. Noskova. MAGSAC: Marginalizing Sample Consensus. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019. arxiv.org