Jungfraujoch/image_analysis/scale_merge/StillsPartialityRefine.cpp

// SPDX-FileCopyrightText: 2025 Filip Leonarski, Paul Scherrer Institute <filip.leonarski@psi.ch>
// SPDX-License-Identifier: GPL-3.0-only

#include "StillsPartialityRefine.h"

#include <algorithm>
#include <array>
#include <atomic>
#include <cmath>
#include <future>
#include <thread>
#include <vector>

#include <Eigen/Dense>
#include <ceres/rotation.h>

#include "Merge.h"
#include "../../common/Logger.h"
#include "../../common/ThreadAffinity.h"

namespace {
    constexpr size_t MIN_FIT_REFLECTIONS = 20;
    constexpr double kRadToDeg = 180.0 / 3.14159265358979323846;

    double SafeInv(double x, double fallback) {
        if (!std::isfinite(x) || x == 0.0)
            return fallback;
        return 1.0 / x;
    }

    // One accepted reflection reduced to the physical partiality fit: the base reciprocal vector q (crystal
    // frame at the stored per-image orientation), the reference full intensity, the measured intensity, the
    // reciprocal of the prescaling correction (the factor that turns a true intensity back into counts)
    // and the weight. dist_ewald / partiality are recomputed from q under the refined tilt.
    struct FitObs {
        double qx, qy, qz;
        double Iref;
        double Iobs;
        double inv_corr;     // 1 / (prescaling_corr * qe_corr * flight_corr)
        double weight;       // 1 / sigma
    };

    double VecLen(double x, double y, double z) { return std::sqrt(x * x + y * y + z * z); }

    // Analytic partiality for a reflection whose base reciprocal vector is q, tilted by (psi_x, psi_y).
    // Mirrors BraggPrediction: dist_ewald = |S| - 1/lambda with S = q_rot + S0, and
    // p = exp(-dist_ewald^2 / 2 sigma^2), sigma^2 = gamma0^2 + (gamma_e*d*)^2 + (bw*|q_z|)^2 - three
    // independent broadenings added in quadrature: the reciprocal-lattice point's own radius (gamma0,
    // ~1/domain size, resolution-INdependent), the mosaic/divergence spread (gamma_e*d*, proportional to
    // d*) and the bandwidth smear along the beam. In practice the fit below returns gamma_e ~ 0 and the
    // width is essentially gamma0 - see there.
    double ComputeP(double qx, double qy, double qz,
                    double psi_x, double psi_y,
                    double s0x, double s0y, double s0z, double inv_lambda,
                    double gamma0, double gamma_e, double bw) {
        double aa[3] = {psi_x, psi_y, 0.0};
        double q[3] = {qx, qy, qz};
        double qr[3];
        ceres::AngleAxisRotatePoint(aa, q, qr);
        const double Sx = qr[0] + s0x, Sy = qr[1] + s0y, Sz = qr[2] + s0z;
        const double de = std::sqrt(Sx * Sx + Sy * Sy + Sz * Sz) - inv_lambda;
        const double dstar = std::sqrt(qr[0] * qr[0] + qr[1] * qr[1] + qr[2] * qr[2]);
        const double sig_ang = gamma_e * dstar;
        const double sbw = bw * std::fabs(qr[2]);
        const double sig2 = gamma0 * gamma0 + sig_ang * sig_ang + sbw * sbw;
        if (!(sig2 > 0.0))
            return 1.0;
        return std::exp(-0.5 * de * de / sig2);
    }

    // The prediction window cuts the rocking curve. Reflections fill a window of half-width W evenly, so
    // their plain moment of de is u = W^2 / 3, and the intensity-weighted moment m of a Gaussian of width
    // s cut at +-W is u * g(a), a = W / s, with g(a) = 3/a^2 (1 - 2 a phi(a) / erf(a / sqrt 2)): 1 for a
    // curve much wider than the window, s^2 / u for a much narrower one. Solves g(a) = m / u for s^2 by
    // bisection (g falls with a). A curve wider than the window cannot be measured through it, so it is
    // taken as wide as the window (a = 1).
    double UntruncatedVariance(double m, double u) {
        auto g = [](double a) {
            const double phi = std::exp(-0.5 * a * a) / std::sqrt(2.0 * 3.14159265358979323846);
            return 3.0 / (a * a) * (1.0 - 2.0 * a * phi / std::erf(a / std::sqrt(2.0)));
        };
        const double t = m / u;
        double lo = 1.0, hi = 50.0;
        if (!(t < g(lo)))
            return 3.0 * u;
        for (int i = 0; i < 60; ++i) {
            const double mid = 0.5 * (lo + hi);
            (g(mid) > t ? lo : hi) = mid;
        }
        const double a = 0.5 * (lo + hi);
        return 3.0 * u / (a * a);
    }

    // Robust per-crystal scale G (linear in G given the model coefficients), identical objective to
    // ScaleOnTheFly::SolveScaleIRLS: minimise sum Cauchy_k( w (G*coeff - Iobs) ) over G >= 0.
    double SolveScaleIRLS(const std::vector<double> &coeff, const std::vector<double> &Iobs,
                          const std::vector<double> &weight, double robust_k) {
        auto weighted_scale = [&](auto robust_weight) {
            double num = 0.0, den = 0.0;
            for (size_t i = 0; i < coeff.size(); ++i) {
                const double rw = robust_weight(i);
                const double w2 = weight[i] * weight[i];
                num += rw * w2 * coeff[i] * Iobs[i];
                den += rw * w2 * coeff[i] * coeff[i];
            }
            return den > 0.0 ? num / den : NAN;
        };

        double G = weighted_scale([](size_t) { return 1.0; });
        if (!std::isfinite(G))
            return 1.0;
        G = std::max(0.0, G);

        const double k2 = robust_k * robust_k;
        for (int iter = 0; iter < 30; ++iter) {
            const double G_prev = G;
            const double G_next = weighted_scale([&](size_t i) {
                const double res = weight[i] * (G * coeff[i] - Iobs[i]);
                return 1.0 / (1.0 + res * res / k2);
            });
            if (!std::isfinite(G_next))
                break;
            G = std::max(0.0, G_next);
            if (std::abs(G - G_prev) <= 1e-7 * std::max(G, 1.0))
                break;
        }
        return G;
    }

    // The rotation by the angle-axis vector (psi_x, psi_y, 0) as a matrix (Rodrigues), so that one tilt
    // is applied to all the reflections of a crystal by a matrix product each.
    std::array<double, 9> TiltMatrix(double psi_x, double psi_y) {
        const double theta = std::sqrt(psi_x * psi_x + psi_y * psi_y);
        if (theta < 1e-15)
            return {1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 1.0};
        const double kx = psi_x / theta, ky = psi_y / theta;
        const double c = std::cos(theta), s = std::sin(theta), t = 1.0 - c;
        return {c + t * kx * kx, t * kx * ky, s * ky,
                t * kx * ky, c + t * ky * ky, -s * kx,
                -s * ky, s * kx, c};
    }

    // ComputeP with the tilt given as its rotation matrix.
    struct TiltModel {
        double s0x, s0y, s0z, inv_lambda;
        double gamma0, gamma_e, bw;

        double P(const std::array<double, 9> &R, const FitObs &o) const {
            const double qx = R[0] * o.qx + R[1] * o.qy + R[2] * o.qz;
            const double qy = R[3] * o.qx + R[4] * o.qy + R[5] * o.qz;
            const double qz = R[6] * o.qx + R[7] * o.qy + R[8] * o.qz;
            const double de = VecLen(qx + s0x, qy + s0y, qz + s0z) - inv_lambda;
            const double sig_ang = gamma_e * VecLen(qx, qy, qz);
            const double sbw = bw * std::fabs(qz);
            const double sig2 = gamma0 * gamma0 + sig_ang * sig_ang + sbw * sbw;
            if (!(sig2 > 0.0))
                return 1.0;
            return std::exp(-0.5 * de * de / sig2);
        }
    };

    // Refine the orientation tilt (psi_x, psi_y) holding the scale G fixed: minimise
    //   sum_i rho(r_i^2) / 2 + |psi / sigma_prior|^2 / 2,   r_i = weight_i (G p_i(psi) Iref_i / corr_i - Iobs_i),
    // with the Cauchy rho(s) = k^2 log(1 + s / k^2) - the intensity-space objective of ScaleOnTheFly, and
    // the Gaussian prior a proper MAP term since r_i is already in sigma units. Levenberg-Marquardt on the
    // normal equations reweighted by rho', the Jacobian by central differences, every trial tilt held inside
    // the bound. Returns false when the cost cannot be evaluated, leaving psi as it came in.
    bool RefineTilt(const std::vector<FitObs> &obs, const TiltModel &model, double G, double robust_k,
                    double inv_sigma_prior, double max_tilt, double psi[2]) {
        constexpr int MAX_ITERATIONS = 25;
        constexpr double STEP = 1e-7;   // rad, for the central differences
        const double k2 = robust_k * robust_k;
        const double prior2 = inv_sigma_prior * inv_sigma_prior;

        auto cost = [&](const double p[2]) {
            const auto R = TiltMatrix(p[0], p[1]);
            double c = 0.5 * prior2 * (p[0] * p[0] + p[1] * p[1]);
            for (const auto &o: obs) {
                const double r = o.weight * (G * model.P(R, o) * o.inv_corr * o.Iref - o.Iobs);
                c += 0.5 * k2 * std::log1p(r * r / k2);
            }
            return c;
        };

        double current = cost(psi);
        if (!std::isfinite(current))
            return false;
        double lambda = 1e-4;
        for (int iter = 0; iter < MAX_ITERATIONS; ++iter) {
            const auto R = TiltMatrix(psi[0], psi[1]);
            const auto Rxp = TiltMatrix(psi[0] + STEP, psi[1]), Rxm = TiltMatrix(psi[0] - STEP, psi[1]);
            const auto Ryp = TiltMatrix(psi[0], psi[1] + STEP), Rym = TiltMatrix(psi[0], psi[1] - STEP);
            double A00 = prior2, A01 = 0.0, A11 = prior2;
            double b0 = prior2 * psi[0], b1 = prior2 * psi[1];
            for (const auto &o: obs) {
                const double a = o.weight * G * o.inv_corr * o.Iref;
                const double r = a * model.P(R, o) - o.weight * o.Iobs;
                const double jx = a * (model.P(Rxp, o) - model.P(Rxm, o)) / (2.0 * STEP);
                const double jy = a * (model.P(Ryp, o) - model.P(Rym, o)) / (2.0 * STEP);
                const double w = 1.0 / (1.0 + r * r / k2);   // rho'
                A00 += w * jx * jx;
                A01 += w * jx * jy;
                A11 += w * jy * jy;
                b0 += w * jx * r;
                b1 += w * jy * r;
            }

            bool accepted = false;
            double trial[2] = {psi[0], psi[1]};
            double trial_cost = current;
            while (lambda < 1e10) {
                const double M00 = A00 * (1.0 + lambda), M11 = A11 * (1.0 + lambda);
                const double det = M00 * M11 - A01 * A01;
                if (!(det > 0.0))
                    break;
                trial[0] = std::clamp(psi[0] - (M11 * b0 - A01 * b1) / det, -max_tilt, max_tilt);
                trial[1] = std::clamp(psi[1] - (M00 * b1 - A01 * b0) / det, -max_tilt, max_tilt);
                trial_cost = cost(trial);
                if (std::isfinite(trial_cost) && trial_cost < current) {
                    accepted = true;
                    lambda = std::max(lambda / 10.0, 1e-12);
                    break;
                }
                lambda *= 10.0;
            }
            if (!accepted)
                break;
            const double step = std::hypot(trial[0] - psi[0], trial[1] - psi[1]);
            const double decrease = current - trial_cost;
            psi[0] = trial[0];
            psi[1] = trial[1];
            current = trial_cost;
            if (decrease <= 1e-6 * current || step <= 1e-10)
                break;
        }
        return true;
    }
}

StillsPartialityRefine::StillsPartialityRefine(const DiffractionExperiment &x)
    : experiment_(x),
      hkl_key_generator_(x.GetScalingSettings().GetMergeFriedel(), x.GetSpaceGroupOrP1()),
      d_min_limit_(x.GetScalingSettings().GetHighResolutionLimit_A()),
      d_max_limit_(x.GetScalingSettings().GetLowResolutionLimit_A()),
      min_partiality_(x.GetScalingSettings().GetMinPartiality()),
      bandwidth_sigma_(x.GetBandwidthFWHM().value_or(0.0f) / 2.3548f) {}

double StillsPartialityRefine::RefineOne(IntegrationOutcome &outcome,
                                         const std::map<HKLKey, double> &reference,
                                         const std::optional<RunWidth> &run_width) const {
    if (outcome.reflections.empty())
        return 0.0;

    const Coord Astar = outcome.latt.Astar();
    const Coord Bstar = outcome.latt.Bstar();
    const Coord Cstar = outcome.latt.Cstar();
    const Coord S0 = outcome.geom.GetScatteringVector();
    const double inv_lambda = 1.0 / outcome.geom.GetWavelength_A();

    auto base_q = [&](const Reflection &r) {
        return Astar * static_cast<float>(r.h) + Bstar * static_cast<float>(r.k)
               + Cstar * static_cast<float>(r.l);
    };

    // Collect the reflections that constrain the fit (accepted, non-ice, finite, present in the reference).
    std::vector<FitObs> obs;
    obs.reserve(outcome.reflections.size());
    // Moments of the excitation error against resolution: de^2 ~ gamma0^2 + gamma_e^2 * d*^2, fitted per
    // crystal by ordinary least squares on (d*^2, de^2). Both components come out of the data.
    double m_n = 0.0, m_x = 0.0, m_xx = 0.0, m_y = 0.0, m_xy = 0.0;
    size_t n_de = 0;
    for (const Reflection &r: outcome.reflections) {
        if (r.on_ice_ring || !AcceptReflection(r, d_min_limit_, d_max_limit_))
            continue;
        if (!std::isfinite(r.I) || !std::isfinite(r.sigma) || r.sigma <= 0.0f)
            continue;
        const auto it = reference.find(hkl_key_generator_(r));
        if (it == reference.end() || !std::isfinite(it->second))
            continue;

        const Coord q = base_q(r);
        obs.push_back(FitObs{
            .qx = q.x, .qy = q.y, .qz = q.z,
            .Iref = it->second,
            .Iobs = static_cast<double>(r.I),
            .inv_corr = SafeInv(r.prescaling_corr * r.qe_corr * r.flight_corr, 1.0),
            .weight = SafeInv(r.sigma, 1.0),
        });

        // Excitation error at the stored orientation (psi = 0), collected as the moments of de^2 against
        // d*^2, so BOTH width components are fitted rather than one being forced to zero. Forcing the
        // width to be purely angular (gamma0 = 0) pins it to the high-resolution edge - it is fitted over
        // a d*^2-dense population - and it then collapses at low d*, giving p ~ 0 for reflections that
        // were plainly recorded, which inflated the merged low-resolution intensity scale ~3.6x.
        const double dstar = VecLen(q.x, q.y, q.z);
        const double de0 = VecLen(q.x + S0.x, q.y + S0.y, q.z + S0.z) - inv_lambda;
        if (dstar > 1e-9) {
            const double x = dstar * dstar, y = de0 * de0;
            m_n += 1.0;
            m_x += x;
            m_xx += x * x;
            m_y += y;
            m_xy += x * y;
            ++n_de;
        }
    }

    if (obs.size() < MIN_FIT_REFLECTIONS || n_de == 0)
        return 0.0;

    // Without the run's measured width (too few reflections in the run to measure it), the width comes
    // from this crystal's own reflections. Solve the 2x2 normal equations for de^2 = A + B d*^2. A degenerate spread in d* (all reflections in
    // one shell) leaves B undetermined, so fall back to the pure angular width there; a negative fitted
    // component is unphysical and is clamped to zero, which reduces to the previous model.
    //
    // Measured outcome, worth knowing before touching this: the fit does NOT split the width between the
    // two terms - it returns gamma0 ~ 4e-4 1/A and gamma_e ~ 0 (their cross-over sits at d = 0.3 A, far
    // outside any measured range), i.e. a width constant in the LINEAR Ewald distance. That is
    // structural, not a fluke: prediction accepts reflections on a fixed linear |dist_ewald| cutoff, so
    // the accepted population's de^2 is flat in d*^2 by construction and the slope is genuinely ~0. The
    // truncated population cannot constrain an angular term, and its width is the window's, not the
    // crystal's - which is why the run's measured width is used wherever there is one.
    const double det = m_n * m_xx - m_x * m_x;
    double A = 0.0, B = 0.0;
    if (std::fabs(det) > 1e-30) {
        A = (m_xx * m_y - m_x * m_xy) / det;
        B = (m_n * m_xy - m_x * m_y) / det;
    } else {
        B = m_x > 0.0 ? m_y / m_x : 0.0;
    }
    double gamma0 = std::sqrt(std::max(0.0, A));
    const double gamma_e_fit = std::max(std::sqrt(std::max(0.0, B)), 1e-9);
    double gamma_e = settings_.gamma_e > 0.0 ? settings_.gamma_e : gamma_e_fit;
    double bw = bandwidth_sigma_;
    if (run_width) {
        gamma0 = run_width->gamma0;
        gamma_e = std::max(run_width->gamma_e, 1e-9);
        bw = run_width->bandwidth_sigma;
    }

    double psi[2] = {0.0, 0.0};
    double G = 1.0;

    const TiltModel model{.s0x = S0.x, .s0y = S0.y, .s0z = S0.z, .inv_lambda = inv_lambda,
                          .gamma0 = gamma0, .gamma_e = gamma_e, .bw = bw};
    const bool refine_tilt = obs.size() >= settings_.min_reflections;
    const int inner = refine_tilt ? settings_.inner_iterations : 1;

    for (int it = 0; it < inner; ++it) {
        // (1) Solve G given the current partialities.
        std::vector<double> coeff(obs.size()), Iobs(obs.size()), weight(obs.size());
        for (size_t j = 0; j < obs.size(); ++j) {
            const double p = ComputeP(obs[j].qx, obs[j].qy, obs[j].qz, psi[0], psi[1],
                                      S0.x, S0.y, S0.z, inv_lambda, gamma0, gamma_e, bw);
            coeff[j] = p * obs[j].inv_corr * obs[j].Iref;
            Iobs[j] = obs[j].Iobs;
            weight[j] = obs[j].weight;
        }
        G = SolveScaleIRLS(coeff, Iobs, weight, settings_.robust_k);
        if (!(G > 0.0) || !std::isfinite(G))
            return 0.0;

        if (!refine_tilt)
            break;

        // (2) Refine the tilt holding G fixed. A cost that cannot be evaluated keeps the tilt this crystal
        // came in with and stops refining it - G alone is still a usable model.
        const double inv_sigma_prior = settings_.prior_sigma_deg > 0.0 ? kRadToDeg / settings_.prior_sigma_deg : 0.0;
        if (!RefineTilt(obs, model, G, settings_.robust_k, inv_sigma_prior, settings_.max_tilt_rad, psi))
            break;
    }

    // Keep what the crystal came in with. This refinement is on by default, so a crystal the tilt model
    // happens to suit WORSE than the fixed partiality it replaces must not be made worse by it - and
    // whether it suits is only known once the corrections are written and the CC re-measured.
    const auto cc_before = outcome.image_scale_cc;
    const auto g_before = outcome.image_scale_g;
    std::vector<std::pair<float, float>> before;   // partiality, image_scale_corr
    before.reserve(outcome.reflections.size());
    for (const auto &r: outcome.reflections)
        before.emplace_back(r.partiality, r.image_scale_corr);

    // Write the refined partiality + scale correction onto every reflection of the crystal (not only the
    // fit subset), so the merge sees a consistent model. image_scale_corr = prescaling_corr * qe_corr * flight_corr /
    // (partiality * G), the same composition ScaleOnTheFly writes.
    for (auto &r: outcome.reflections) {
        const Coord q = base_q(r);
        const double p = ComputeP(q.x, q.y, q.z, psi[0], psi[1], S0.x, S0.y, S0.z, inv_lambda,
                                  gamma0, gamma_e, bw);
        r.partiality = static_cast<float>(p);
        const float corr = r.prescaling_corr * r.qe_corr * r.flight_corr;
        const double denom = p * G;
        r.image_scale_corr = (std::isfinite(corr) && std::isfinite(denom) && denom > 0.0)
                                 ? static_cast<float>(corr / denom)
                                 : NAN;
    }
    outcome.image_scale_g = static_cast<float>(G);

    // The corrections just changed, so the CC that ScaleOnTheFly measured no longer describes them.
    // Refresh it here: it is reported per image and --min-image-cc drops images by it, so it has to be
    // the CC of the data that is actually merged.
    const auto [cc, cc_n] = ImageReferenceCC(outcome.reflections, reference, hkl_key_generator_,
                                             d_min_limit_, d_max_limit_, min_partiality_);

    // Adopt the refined model only if it correlates with the reference at least as well as the model it
    // replaces. Rejecting puts the crystal back exactly as it arrived, which is the same state a
    // crystal with too few reflections to fit ends in. A CC that cannot be measured at all counts as
    // worse, not as no opinion: ImageReferenceCC returns NaN when too few reflections clear the
    // partiality cut, which is precisely what a refinement that collapsed the partialities produces -
    // and adopting it would leave image_scale_cc NaN, on which --min-image-cc drops the image outright.
    if (cc_before.has_value() && std::isfinite(*cc_before) && !(std::isfinite(cc) && cc >= *cc_before)) {
        for (size_t i = 0; i < outcome.reflections.size(); ++i) {
            outcome.reflections[i].partiality = before[i].first;
            outcome.reflections[i].image_scale_corr = before[i].second;
        }
        outcome.image_scale_g = g_before;
        return 0.0;   // nothing adopted, so no tilt to report
    }
    outcome.image_scale_cc = cc;
    outcome.image_scale_cc_n = cc_n;

    const double tilt_deg = std::sqrt(psi[0] * psi[0] + psi[1] * psi[1]) * kRadToDeg;
    return tilt_deg;
}

std::optional<StillsPartialityRefine::RunWidth> StillsPartialityRefine::MeasureRunWidth(
        const std::vector<IntegrationOutcome> &outcomes, const std::optional<UnitCell> &reference_cell) const {
    // The rocking width measured on the crystals rather than on the prediction window. A lattice point
    // sits anywhere across the window with equal likelihood, while what it records is I_full * p(de), so
    // the intensity-weighted second moment of the excitation error, sum I de^2 / sum I, is the width s^2
    // of a Gaussian p - corrected for the window cutting the curve's tails (UntruncatedVariance). The
    // plain moment of de over the predicted reflections, which the per-crystal fit in RefineOne uses, is
    // instead the window's own (W^2 / 3). Measured in shells of d*^2 over the whole run, where the
    // intensities of thousands of crystals average out, and fitted as s^2 = gamma0^2 + gamma_e^2 d*^2 +
    // (bandwidth_sigma lambda d*^2 / 2)^2.
    const auto &indexing = experiment_.GetIndexingSettings();
    struct Point { double x, de2, w, var_w; };
    std::vector<Point> pts;
    for (const auto &o: outcomes) {
        if (!o.image_scale_g || !(*o.image_scale_g > 0.0))
            continue;
        if (reference_cell && !o.latt.GetUnitCell().is_close(*reference_cell, indexing.GetUnitCellDistTolerance(),
                                                               indexing.GetUnitCellAngleTolerance_deg()))
            continue;
        for (const auto &r: o.reflections) {
            if (r.on_ice_ring || !AcceptReflection(r, d_min_limit_, d_max_limit_))
                continue;
            const double corr = r.prescaling_corr * r.qe_corr * r.flight_corr / *o.image_scale_g;
            const double w = r.I * corr, sigma_w = r.sigma * corr;
            if (!std::isfinite(w) || !std::isfinite(sigma_w) || !std::isfinite(r.dist_ewald) || !(r.d > 0.0f))
                continue;
            pts.push_back({1.0 / (static_cast<double>(r.d) * r.d),
                           static_cast<double>(r.dist_ewald) * r.dist_ewald, w, sigma_w * sigma_w});
        }
    }
    constexpr size_t N_SHELLS = 8;
    if (pts.size() < 100 * N_SHELLS)
        return std::nullopt;
    std::sort(pts.begin(), pts.end(), [](const Point &a, const Point &b) { return a.x < b.x; });

    Logger logger("StillsPartialityRefine");
    // A shell's moment is no better than the summed intensity it is weighted by, so a shell is used only
    // where that sum is measured to 10 % (sum I / sigma(sum I) >= 10): the shells beyond the diffraction
    // limit, where the sum is noise and can even be negative, say nothing about the width.
    constexpr double MIN_SHELL_SIGNAL = 10.0;
    std::vector<double> sx, sy;
    for (size_t k = 0; k < N_SHELLS; ++k) {
        double sw = 0.0, svar = 0.0, swde2 = 0.0, sde2 = 0.0, sumx = 0.0;
        const size_t lo = k * pts.size() / N_SHELLS, hi = (k + 1) * pts.size() / N_SHELLS;
        for (size_t i = lo; i < hi; ++i) {
            sw += pts[i].w;
            svar += pts[i].var_w;
            swde2 += pts[i].w * pts[i].de2;
            sde2 += pts[i].de2;
            sumx += pts[i].x;
        }
        const double n = static_cast<double>(hi - lo);
        if (!(sw >= MIN_SHELL_SIGNAL * std::sqrt(svar)) || !(swde2 > 0.0))
            continue;
        const double x = sumx / n, y = UntruncatedVariance(swde2 / sw, sde2 / n);
        sx.push_back(x);
        sy.push_back(y);
        logger.Debug("Stills rocking width at d = {:.2f} A: {:.2e} 1/A ({:.3f} deg), I/sigma {:.0f}", 1.0 / std::sqrt(x),
                     std::sqrt(y), std::sqrt(y / x) * kRadToDeg, sw / std::sqrt(svar));
    }
    if (sx.size() < 3)
        return std::nullopt;

    // Least squares for s^2 = A + B d*^2 + C d*^4 over the shells. Every term is a variance, so a negative
    // one is unphysical: of the fits on each subset of the three terms, the best with none negative is kept.
    std::array<double, 3> best{};
    double best_residual = INFINITY;
    for (int mask = 1; mask < 8; ++mask) {
        std::vector<int> terms;
        for (int t = 0; t < 3; ++t)
            if (mask & (1 << t))
                terms.push_back(t);
        const int m = static_cast<int>(terms.size());
        Eigen::MatrixXd M = Eigen::MatrixXd::Zero(m, m);
        Eigen::VectorXd v = Eigen::VectorXd::Zero(m);
        for (size_t k = 0; k < sx.size(); ++k)
            for (int i = 0; i < m; ++i) {
                const double fi = std::pow(sx[k], terms[i]);
                v(i) += fi * sy[k];
                for (int j = 0; j < m; ++j)
                    M(i, j) += fi * std::pow(sx[k], terms[j]);
            }
        const Eigen::VectorXd c = M.ldlt().solve(v);
        if (!c.allFinite() || (c.array() < 0.0).any())
            continue;
        std::array<double, 3> coef{};
        for (int i = 0; i < m; ++i)
            coef[terms[i]] = c(i);
        double residual = 0.0;
        for (size_t k = 0; k < sx.size(); ++k) {
            const double r = sy[k] - coef[0] - coef[1] * sx[k] - coef[2] * sx[k] * sx[k];
            residual += r * r;
        }
        if (residual < best_residual) {
            best_residual = residual;
            best = coef;
        }
    }
    if (!std::isfinite(best_residual) || !(best[0] + best[1] + best[2] > 0.0))
        return std::nullopt;
    // The d*^4 term is a bandwidth: its radial smear bw |q_z|, with |q_z| = lambda d*^2 / 2 on the sphere.
    const RunWidth width{.gamma0 = std::sqrt(best[0]), .gamma_e = std::sqrt(best[1]),
                         .bandwidth_sigma = 2.0 * std::sqrt(best[2]) / experiment_.GetWavelength_A()};
    logger.Info("Stills rocking width measured: {:.2e} 1/A + {:.3f} deg angular + {:.4f} bandwidth (FWHM)",
                width.gamma0, width.gamma_e * kRadToDeg, width.bandwidth_sigma * 2.3548);
    return width;
}

double StillsPartialityRefine::Run(std::vector<IntegrationOutcome> &outcomes, size_t nthreads,
                                   const std::optional<UnitCell> &reference_cell) const {
    const auto &indexing = experiment_.GetIndexingSettings();
    auto merged_crystal = [&](const IntegrationOutcome &o) {
        return !reference_cell || o.latt.GetUnitCell().is_close(*reference_cell,
                                                                indexing.GetUnitCellDistTolerance(),
                                                                indexing.GetUnitCellAngleTolerance_deg());
    };
    if (nthreads == 0)
        nthreads = AvailableCpus();
    nthreads = std::max<size_t>(1, nthreads);

    double last_mean_tilt = 0.0;
    const auto run_width = MeasureRunWidth(outcomes, reference_cell);

    for (int outer = 0; outer < settings_.outer_iterations; ++outer) {
        // Reference full intensities from the current corrections.
        const std::vector<MergedReflection> merged = MergeAll(experiment_, outcomes, nthreads, reference_cell);
        std::map<HKLKey, double> reference;
        for (const auto &m: merged)
            reference[hkl_key_generator_(m)] = m.I;

        std::atomic<double> tilt_sum{0.0};
        std::atomic<size_t> tilt_n{0};
        std::atomic<size_t> next{0};

        auto worker = [&]() {
            size_t i = next.fetch_add(1);
            while (i < outcomes.size()) {
                const double t = merged_crystal(outcomes[i]) ? RefineOne(outcomes[i], reference, run_width) : 0.0;
                if (t > 0.0) {
                    double prev = tilt_sum.load();
                    while (!tilt_sum.compare_exchange_weak(prev, prev + t)) {}
                    tilt_n.fetch_add(1);
                }
                i = next.fetch_add(1);
            }
        };

        const size_t nt = std::min(nthreads, std::max<size_t>(1, outcomes.size()));
        if (nt <= 1) {
            worker();
        } else {
            std::vector<std::future<void>> futures;
            futures.reserve(nt);
            for (size_t t = 0; t < nt; ++t)
                futures.emplace_back(std::async(std::launch::async, worker));
            for (auto &f: futures)
                f.get();
        }

        last_mean_tilt = tilt_n > 0 ? tilt_sum.load() / static_cast<double>(tilt_n.load()) : 0.0;
    }

    return last_mean_tilt;
}